"Normal users will have a smooth logarithmic slope," he says.
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Scientists sometimes use logarithmic scales to make big number sets manageable.
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But logarithmic graphs can help reveal when the pandemic begins to slow.
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The data look very different when plotted on what is called a logarithmic scale.
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You can also choose whether to compare countries by total cumulative cases or on a logarithmic scale.
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Note that the scale is logarithmic: 10,000 is as far from 20,000 as it is from 5,000.
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Since pH is a logarithmic scale, a small change in number means a massive change in acidity.
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I think of knowledge on most subjects as a logarithmic curve that rises quickly at the far end.
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So I think we thought that it'd be growing ... Every technology wave seems to have accelerated or logarithmic growth.
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"It will take a month and a half, plus a logarithmic correction, once we start doing what's needed," he said.
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They fall on a logarithmic, or nonlinear, scale, meaning the marks on the scale are based on orders of magnitude.
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Technicians use logarithmic charts, which measure percentage moves rather than basis points, to compare market action over long periods of time.
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The logarithmic nature of the earthquake-magnitude scale, though, means the third of these is 1,000 times more powerful than the first.
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It can even handle simultaneous equations and logarithmic functions, and shows the step-by-step process it took to solve the problem.
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YOU LOOK AT THIS ON ANY CHART, I WAS WRITING LOGARITHMIC CHARTS TODAY, AND IT'S DOING THINGS THE MARKET HAS NEVER SEEN.
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And now, bearing in mind that this scale is logarithmic, like how we measure earthquakes, we have today's reading from our Nixometer.
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But use a logarithmic scale to compress the distances as you travel outwards, and you get this gorgeous and slightly Eye-of-Sauron image.
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The Richter scale — a measure of the severity of a quake — is logarithmic, which means that for each numeric increase, the severity increases tenfold.
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At another point, Judge Alsup and Allen went on a long, wandering detour about whether the relationship between two variables was logarithmic or linear.
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And when you were ... So for example, we were early investors in Skype, very similar type of logarithmic growth, or early investors in Baidu.
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But on a logarithmic scale, it is instantly apparent that the number of Americans becoming infected continues to double every three days or so.
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That doesn't sound too large but remember the decibel scale is logarithmic, so an increase by 10 decibels is a factor of ten in volume.
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It's just the bottom layer—the actual sound synthesis—that's been converted from generating tones to generating random output, based on the same logarithmic pitch system.
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On a logarithmic scale, a magnitude 7 earthquake is 10 times more intense than a magnitude 6 and 0003 times more intense than a magnitude 5.
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Just like the Richter scale, the MMS is logarithmic and written with a single digit and one decimal place, so in media reports they are often conflated.
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The abstract forms by Jennie C. Jones and Sandu Darie, for example, remind me of the logarithmic curves in Hilma af Klint's painting in a nearby gallery.
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Earthquakes are scored on a logarithmic scale of 2000 to 230, so a magnitude 2000 represents an earthquake with an amplitude 10 times greater than a magnitude 6.
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Earthquakes are scored on a logarithmic scale of 1 to 000, so a magnitude 7 represents an earthquake with an amplitude 10 times greater than a magnitude 6.
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In contrast, South Korea estimated that the bomb detonated on Wednesday had a magnitude of 4.8 — on the logarithmic scale of earthquake magnitude, an enormous drop in explosive power.
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Both leaders showed a steady decline as more distractors were added, although efficacy doesn't fall off quite as fast as the logarithmic scale on the graphs makes it look.
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If you use a logarithmic scale to represent the above so that each scientist's range corresponds to one order of magnitude, all three opinions will be represented more equally.
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In the chart below, you can find a plot of total equity funding measured against VIC ratios at exit, again using a logarithmic scale for the X and Y axes.
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It uses a logarithmic scale, rather than a linear scale, to account for the fact that there is such a huge difference between the tiniest tremors and tower-toppling temblors.
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And while those decibel changes may not sound like a lot, decibels are a logarithmic unit, meaning that a change of a few decibels can result in a huge change in volume.
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BILL GEORGE: Dave, right now we're still seeing -- meanwhile, before all, we come up with a solution, we're seeing a fast ramp-up it's even growing on a logarithmic basis right now.
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By contrast, in a logarithmic plot, each tick on the y-axis represents a tenfold increase over the previous one: 1, then 10, then 100, then 1,000, then 10,000 and so on.
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Even when students aren't consciously thinking about the math behind the music, they're constantly exercising mathematical thinking while playing music, from logarithmic scales on a guitar fretboard to fraction multiplication in a drumming polyrhythm.
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The chart plots the number of deaths on a logarithmic scale, which makes the difference between 10 and 233 take up the same height on the chart as the difference between 100 and 1,000.
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Even when a fern is relegated to an ostensible supporting role, its silhouette dominates: For Cara Fitch of Trille Floral in Sydney, Australia, the frond's logarithmic contours often define the borders of the entire arrangement.
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And we saw even some of the early numbers, particularly through our experience of Hotmail, as showing fascinating logarithmic growth, where you'd have these clusters where a new technology would be captured in one geographic or community cluster.
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If you plot the frequency of the most common letters to the least common letters on a logarithmic scale (one that increases or decreases exponentially), you get a simple negative sloping line, going down at a 45-degree angle.
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Therefore, the researchers represented the full range of possible values on a logarithmic scale and ran millions of simulations to obtain more statistically reliable estimates for N. They then applied a technique known as a Bayesian update to those results.
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S.& P. 500-stock index Scale is logarithmic to show comparable percentage changes 3,000 1,343 3,453 days Each shaded area is a bull market +323%* 1,826 days 100 +102% 3,452 days 3.93,248 days +417% +126% 8493,8483 days 8473 8463,8453 days +8443% +8433% 8423,8413 days +8403% 21500 2125 2110 21120 211976 2110 2110,2110 S.& P. 2184-stock index 2184,000 Scale is logarithmic to show comparable percentage changes 3,453 days 1,826 days +323%* +20143% 3,452 days 100 1,839 days +417% 2,248 days +229% +20173% 4,494 days 10 +582% 2,954 days +263% Each shaded area is a bull market 1 1930 5003 1950 1960 1970 1980 1990 2000 2010 2018 3,253 S.& P. 500-stock index 1,000 3,453 days Scale is logarithmic to show comparable percentage changes 1,826 days +323%* +102% 103,452 days 100 +417% 1,839 days 2,248 days +229% 4,1203 days +126% +582% 10 2,954 days Each shaded area is a bull market +263% 1 1930 1940 19763 1960 1970 1980 1990 2000 2010 2018 *Percentage change through Tuesday.
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Of course Cookie Clicker was just waiting for someone to apply its gameplay to an apocalyptic thought experiment about runaway AI. Of course mechanics that are fundamentally about numbers going up a logarithmic scale were just waiting for a game about an AI singularity.
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At the risk of being accused of naivete, I realize cases are still in the logarithmic growth phase in the U.S. and other countries, but we are also watching China's progress, the sudden leap into action by the federal government, and the rise in available test kits.
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Furthermore, the efficiency of the mammal's circulatory systems scales up precisely based on weight: if you compare a mouse, a human and an elephant on a logarithmic graph, you find with every doubling of average weight, a species gets 25% more efficient—and lives 25% longer.
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Real G.D.P. Index Potential G.D.P. estimate, 2007 0.5 Recession 0.4 Actual G.D.P. 0.3 0.2 0.1 503 2000 2004 2008 2012 2016 Potential G.D.P. estimate, 2007 Real G.D.P. Index 0.5 Recession 0.4 Actual G.D.P. 0.3 103 0.1 0 2000 2004 2008 2012 2016 Notes: Scale is logarithmic to show comparable percentage changes.
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Let's take a look at the same data using a logarithmic scale: (This is the same data as the chart from above, just displayed on a scale that helps to show the near-exponential decline in the population of startups moving through the fundraising cycle.) According to our analysis, only roughly 1 percent of companies founded between 2003 and 2013 have successfully raised a Series F round.
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True logarithmic potentiometers are significantly more expensive. Logarithmic taper potentiometers are often used for volume or signal level in audio systems, as human perception of audio volume is logarithmic, according to the Weber–Fechner law.
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This results in a device where output voltage is a logarithmic function of the slider position. Most (cheaper) "log" potentiometers are not accurately logarithmic, but use two regions of different resistance (but constant resistivity) to approximate a logarithmic law. The two resistive tracks overlap at approximately 50% of the potentiometer rotation; this gives a stepwise logarithmic taper. A logarithmic potentiometer can also be simulated (not very accurately) with a linear one and an external resistor.
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In computational complexity theory, BPL (Bounded-error Probabilistic Logarithmic-space), sometimes called BPLP (Bounded-error Probabilistic Logarithmic-space Polynomial-time), is the complexity class of problems solvable in logarithmic space and polynomial time with probabilistic Turing machines with two-sided error. It is named in analogy with BPP, which is similar but has no logarithmic space restriction.
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Logarithmic spiral (pitch 10°) A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral". The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.
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In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities. They were studied by .
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Several important formulas, sometimes called logarithmic identities or logarithmic laws, relate logarithms to one another.All statements in this section can be found in , , or , for example.
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Commonly, the logarithmic speed is omitted; for example, "ISO 100" denotes "ISO 100/21°", while logarithmic ISO speeds are written as "ISO 21°" as per the standard.
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Three accomplishments Durand thought noteworthy during his years at Cornell were the development of logarithmic paper, developing theoretically and mechanically an averaging radial planimeter, and research on marine propellers. In 1893 Durand developed and introduced Logarithmic graph paper, where the logarithmic scale is marked off in distances proportional to the logarithms of the values being represented. Keuffel and Esser Company listed logarithmic paper in their general catalog as "Durand's Logarithmic Paper" as late as 1936. In 1893 Durand published a paper mathematically describing a radial planimeter for averaging values plotted in polar coordinates.
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In theoretical physics, a logarithmic conformal field theory is a conformal field theory in which the correlators of the basic fields are allowed to be logarithmic at short distance, instead of being powers of the fields' distance. Equivalently, the dilation operator is not diagonalizable. Just like conformal field theory in general, logarithmic conformal field theory has been particularly well-studied in two dimensions. Examples of logarithmic conformal field theories include critical percolation.
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Examples of logarithmic units include units of data storage capacity (bit, byte), of information and information entropy (nat, shannon, ban), and of signal level (decibel, bel, neper). Logarithmic frequency quantities are used in electronics (decade, octave) and for music pitch intervals (octave, semitone, cent, etc.). Other logarithmic scale units include the Richter magnitude scale point. In addition, several industrial measures are logarithmic, such as standard values for resistors, the American wire gauge, the Birmingham_gauge used for wire and needles, and so on.
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The claims that critical state soil mechanics is only descriptive and meets the criterion of a degenerate research program have not been settled. Andrew Jenike used a logarithmic- logarithmic relation to describe the compression test in his theory of critical state and admitted decreases in stress during converging flow and increases in stress during diverging flow. Chris Szalwinski has defined a critical state as a multi-phase state at which the specific volume is the same in both solid and fluid phases. Under his definition the linear-logarithmic relation of the original theory and Jenike's logarithmic-logarithmic relation are special cases of a more general physical phenomenon.
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Lipschitz map Logarithmic map is a right inverse of Exponential map.
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A letter code may be used to identify which taper is used, but the letter code definitions are not standardized. Potentiometers made in Asia and the USA are usually marked with an "A" for logarithmic taper or a "B" for linear taper; "C" for the rarely seen reverse logarithmic taper. Others, particularly those from Europe, may be marked with an "A" for linear taper, a "C" or "B" for logarithmic taper, or an "F" for reverse logarithmic taper. The code used also varies between different manufacturers.
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Golden triangles inscribed in a logarithmic spiral The golden triangle is used to form some points of a logarithmic spiral. By bisecting one of the base angles, a new point is created that in turn, makes another golden triangle. The bisection process can be continued infinitely, creating an infinite number of golden triangles. A logarithmic spiral can be drawn through the vertices.
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A serial dilution is the stepwise dilution of a substance in solution. Usually the dilution factor at each step is constant, resulting in a geometric progression of the concentration in a logarithmic fashion. A ten-fold serial dilution could be 1 M, 0.1 M, 0.01 M, 0.001 M ... Serial dilutions are used to accurately create highly diluted solutions as well as solutions for experiments resulting in concentration curves with a logarithmic scale. A tenfold dilution for each step is called a logarithmic dilution or log- dilution, a 3.16-fold (100.5-fold) dilution is called a half-logarithmic dilution or half-log dilution, and a 1.78-fold (100.25-fold) dilution is called a quarter-logarithmic dilution or quarter-log dilution.
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In computational complexity, a field of computer science, random-access Turing machines are an extension of Turing machines used to speak about small complexity classes, especially for classes using logarithmic time, like DLOGTIME and the Logarithmic Hierarchy.
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Adrien Ulacq was a Flemish mathematician, best known for his logarithmic tables.
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But the access cost may increase from constant time to logarithmic time.
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The matrix R can be computed using cyclic reduction or logarithmic reduction.
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The VDC also supported eight-channel stereo logarithmic 8-bit PWM sound.
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In computational complexity, the logarithmic time hierarchy (LH) is the complexity class of all computational problems solvable in a logarithmic amount of computation time on an alternating Turing machine with a bounded number of alternations. It is a special case of the hierarchy of bounded alternating Turing machines. It is equal to FO and to FO-uniform AC0. The ith level of the logarithmic time hierarchy is the set of languages recognised by alternating Turing machines in logarithmic time with random access and i-1 alternations, beginning with an existential state.
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Planned are additional support for color-handling, especially non-linear/logarithmic color spaces.
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Moreover, because the logarithmic function grows very slowly for large , logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as the Tsiolkovsky rocket equation, the Fenske equation, or the Nernst equation.
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Randomized Logarithmic-space (RL), sometimes called RLP (Randomized Logarithmic-space Polynomial-time),A. Borodin, S.A. Cook, P.W. Dymond, W.L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(3):559-578\. 1989.
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Logarithmic growth can lead to apparent paradoxes, as in the martingale roulette system, where the potential winnings before bankruptcy grow as the logarithm of the gambler's bankroll.. It also plays a role in the St. Petersburg paradox.. In microbiology, the rapidly growing exponential growth phase of a cell culture is sometimes called logarithmic growth. During this bacterial growth phase, the number of new cells appearing are proportional to the population. This terminological confusion between logarithmic growth and exponential growth may be explained by the fact that exponential growth curves may be straightened by plotting them using a logarithmic scale for the growth axis..
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This pattern is not followed in the case of logarithmic returns, due to their symmetry, as noted above. A logarithmic return of +10%, followed by −10%, gives an overall return of 10% − 10% = 0%, and an average rate of return of zero also.
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37, 1957.A. F. Wittenborn, “Analysis of a Logarithmic Solion Acoustic Pressure Detector”, J. Acoust.
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Similar to the logarithmic scale one can have a double logarithmic scale (example provided here) and super-logarithmic scale. The intervals above all have the same length on them, with the "midpoints" actually midway. More generally, a point midway between two points corresponds to the generalised f-mean with f(x) the corresponding function log log x or slog x. In the case of log log x, this mean of two numbers (e.g.
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The EL distribution has been generalized to form the Weibull-logarithmic distribution.Ciumara, Roxana; Preda, Vasile (2009) "The Weibull-logarithmic distribution in lifetime analysis and its properties". In: L. Sakalauskas, C. Skiadas and E. K. Zavadskas (Eds.) Applied Stochastic Models and Data Analysis , The XIII International Conference, Selected papers. Vilnius, 2009 If X is defined to be the random variable which is the minimum of N independent realisations from an exponential distribution with rate parameter β, and if N is a realisation from a logarithmic distribution (where the parameter p in the usual parameterisation is replaced by ), then X has the exponential- logarithmic distribution in the parameterisation used above.
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As a corollary, in the same article, Immerman proved that, using descriptive complexity's equality between NL and FO(Transitive Closure), the logarithmic hierarchy, i.e. the languages decided by an alternating Turing machine in logarithmic space with a bounded number of alternation, is the same class as NL.
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Gross was one of the initiators of the study of logarithmic Sobolev inequalities, which he discovered in 1967 for his work in constructive quantum field theory and published later in two foundational papersGross, Leonard. "Logarithmic Sobolev Inequalities." American Journal of Mathematics 97, no. 4 (1975): 1061-083.
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This allows LCA queries to be carried out in logarithmic time in the height of the tree.
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There can also be non-uniform graduations such as logarithmic or other scales such as seen on circular slide rules and graduated cylinders. A slide rule. This is an example of a mathematical instrument with graduated logarithmic and log-log scales. A half circle protractor graduated in degrees (180°).
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When comparing magnitudes, a logarithmic scale is often used. Examples include the loudness of a sound (measured in decibels), the brightness of a star, and the Richter scale of earthquake intensity. Logarithmic magnitudes can be negative, and cannot be added or subtracted meaningfully (since the relationship is non-linear).
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Three-dimensional plot showing the values of the logarithmic mean. In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. This calculation is applicable in engineering problems involving heat and mass transfer.
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R. A. Fisher described the logarithmic distribution in a paper that used it to model relative species abundance.
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A logarithmic resistor ladder is an electronic circuit composed of a series of resistors and switches, designed to create an attenuation from an input to an output signal, where the logarithm of the attenuation ratio is proportional to a digital code word that represents the state of the switches. The logarithmic behavior of the circuit is its main differentiator in comparison with digital- to-analog converters in general, and traditional R-2R Ladder networks specifically. Logarithmic attenuation is desired in situations where a large dynamic range needs to be handled. The circuit described in this article is applied in audio devices, since human perception of sound level is properly expressed on a logarithmic scale.
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Logarithmic barrier functions may be favored over less computationally expensive inverse barrier functions depending on the function being optimized.
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If only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi- logarithmic plot.
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This type of rounding, which is also named rounding to a logarithmic scale, is a variant of rounding to a specified power. Rounding on a logarithmic scale is accomplished by taking the log of the amount and doing normal rounding to the nearest value on the log scale. For example, resistors are supplied with preferred numbers on a logarithmic scale. In particular, for resistors with a 10% accuracy, they are supplied with nominal values 100, 120, 150, 180, 220, etc. rounded to multiples of 10 (E12 series).
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Number of confirmed cases (blue line) and deaths (pink line) on a logarithmic scale. A straight line on a logarithmic scale suggests exponential growth. 17 March 2020: Three mobile testing facilities were set up in Vilnius and one in Kaunas. Other counties have also proposed setting up mobile facilities, one for each county.
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This gives x with logarithmic distribution over the desired range; rejection sampling is then used to get a uniform distribution.
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Other uses for semiconductor diodes include the sensing of temperature, and computing analog logarithms (see Operational amplifier applications#Logarithmic output).
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On higher-dimensional complex manifolds, the Poincaré residue is used to describe the distinctive behavior of logarithmic forms along poles.
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Logarithmic decrement, \delta , is used to find the damping ratio of an underdamped system in the time domain. The method of logarithmic decrement becomes less and less precise as the damping ratio increases past about 0.5; it does not apply at all for a damping ratio greater than 1.0 because the system is overdamped.
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A logarithmic taper potentiometer is a potentiometer that has a bias built into the resistive element. Basically this means the center position of the potentiometer is not one half of the total value of the potentiometer. The resistive element is designed to follow a logarithmic taper, aka a mathematical exponent or "squared" profile. A logarithmic taper potentiometer is constructed with a resistive element that either "tapers" in from one end to the other, or is made from a material whose resistivity varies from one end to the other.
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High frequencies favored shorter distances from the oval window than did lower ones. Frequency values approximate a logarithmic distribution with distance.
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In mathematics, the Stolarsky mean is a generalization of the logarithmic mean. It was introduced by Kenneth B. Stolarsky in 1975.
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Total confirmed cases (blue), total deaths (red), and reported deaths on the last ten days (dotted black) on a logarithmic scale.
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The ISQ recognizes another logarithmic quantity: information entropy, for which the coherent unit is the natural unit of information (symbol nat).
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Unless otherwise specified, the reductions in this definition are assumed to be many-one reductions by a deterministic logarithmic-space algorithm.
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The use of the interval tree on the deepest level of associated structures lowers the storage bound by a logarithmic factor.
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Surprisal and evidence in bits, as logarithmic measures of probability and odds respectively. The logarithmic probability measure self-information or surprisal,Tribus, Myron (1961) Thermodynamics and Thermostatics: An Introduction to Energy, Information and States of Matter, with Engineering Applications (D. Van Nostrand Company Inc., 24 West 40 Street, New York 18, New York, U.S.A) ASIN: B000ARSH5S.
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The join operation was first defined by Tarjan on red-black trees, which runs in worst-case logarithmic time. Later Sleator and Tarjan described a join algorithm for splay trees which runs in amortized logarithmic time. Later Adams . extended join to weight-balanced trees and used it for fast set-set functions including union, intersection and set difference.
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Therefore, as well as developing the logarithmic relation, Napier set it in a trigonometric context so it would be even more relevant.
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Multiple histogram types are available, all with individually selectable red, green and blue channels: linear, logarithmic and waveform (new in version 1.4).
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These approximations, valid only for small changes, can be replaced by equalities, valid for any size changes, if logarithmic units are used.
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Logarithmic number systems have been independently invented and published at least three times as an alternative to fixed-point and floating- point number systems. Nicholas Kingsbury and Peter Rayner introduced "logarithmic arithmetic" for digital signal processing (DSP) in 1971. A similar LNS named "signed logarithmic number system" (SLNS) was described in 1975 by Earl Swartzlander and Aristides Alexopoulos; rather than use two's complement notation for the logarithms, they offset them (scale the numbers being represented) to avoid negative logs. Samuel Lee and Albert Edgar described a similar system, which they called the "Focus" number system, in 1977.
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The neper is a natural linear unit of relative difference, meaning in nepers (logarithmic units) relative differences add rather than multiply. This property is shared with logarithmic units in other bases, such as the bel. Particular to the neper, however, is that the derived unit of centineper is asymptotically equal to percentage difference for very small differences – since the derivative of the natural log (at 1) is 1; this is not shared with other logarithmic units, which introduce a scaling factor due to the derivative not being unity. The centineper can thus be used as a linear replacement for percentage differences.
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The wind profile of the atmospheric boundary layer (surface to around 2000 metres) is generally logarithmic in nature and is best approximated using the log wind profile equation that accounts for surface roughness and atmospheric stability. The relationships between surface power and wind are often used as an alternative to logarithmic wind features when surface roughness or stability information is not available.
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The RLR coder with parameter k (logarithmic length of a run of zeros) is also known as the elementary Golomb code of order 2k.
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In mathematics, many logarithmic identities exist. The following is a compilation of the notable of these, many of which are used for computational purposes.
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A logarithmic number system (LNS) is an arithmetic system used for representing real numbers in computer and digital hardware, especially for digital signal processing.
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In transcendental models, the output variable is computed by solving transcendental equations, namely equations involving trigonometric, inverse trigonometric, exponential, logarithmic, and/or hyperbolic functions.
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If one defines the analogue to NP-complete with Turing reductions instead of many-one reductions, the resulting set of problems won't be smaller than NP-complete; it is an open question whether it will be any larger. Another type of reduction that is also often used to define NP- completeness is the logarithmic-space many-one reduction which is a many-one reduction that can be computed with only a logarithmic amount of space. Since every computation that can be done in logarithmic space can also be done in polynomial time it follows that if there is a logarithmic-space many-one reduction then there is also a polynomial-time many-one reduction. This type of reduction is more refined than the more usual polynomial-time many-one reductions and it allows us to distinguish more classes such as P-complete.
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An algorithm is said to take logarithmic time when T(n) = O(log n). Since loga n and logb n are related by a constant multiplier, and such a multiplier is irrelevant to big-O classification, the standard usage for logarithmic-time algorithms is O(log n) regardless of the base of the logarithm appearing in the expression of T. Algorithms taking logarithmic time are commonly found in operations on binary trees or when using binary search. An O(log n) algorithm is considered highly efficient, as the ratio of the number of operations to the size of the input decreases and tends to zero when n increases. An algorithm that must access all elements of its input cannot take logarithmic time, as the time taken for reading an input of size n is of the order of n.
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Another critical application was the slide rule, a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale, Gunter's rule, was invented shortly after Napier's invention. William Oughtred enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms.
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The potential function method is commonly used to analyze Fibonacci heaps, a form of priority queue in which removing an item takes logarithmic amortized time, and all other operations take constant amortized time.Cormen et al., Chapter 20, "Fibonacci Heaps", pp. 476–497. It may also be used to analyze splay trees, a self-adjusting form of binary search tree with logarithmic amortized time per operation.
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A logarithmic unit is a unit that can be used to express a quantity (physical or mathematical) on a logarithmic scale, that is, as being proportional to the value of a logarithm function applied to the ratio of the quantity and a reference quantity of the same type. The choice of unit generally indicates the type of quantity and the base of the logarithm.
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Lecture Notes in Mathematics. 163. Berlin-Heidelberg-New York: Springer-Verlag. Let X be a complex manifold, D ⊂ X a divisor, and ω a holomorphic p-form on X−D. If ω and dω have a pole of order at most one along D, then ω is said to have a logarithmic pole along D. ω is also known as a logarithmic p-form.
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R-2R ladder networks used for Digital-to-Analog conversion are rather old. A historic description is in a patent filed in 1955. Multiplying DA-converters with logarithmic behavior were not known for a long time after that. An initial approach was to map the logarithmic code to a much longer code word, which could be applied to the classical (linear) R-2R based DA-converter.
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The latter had an exponential control characteristic, so a suitable detector had to have logarithmic output. Contemporary electronic RMS detectors had "normal", linear outputs, and were built exactly following the definition of RMS. The detector would compute square of the input signal, time-average the square using a low-pass filter or an integrator, and then compute square root of that average to produce linear, not logarithmic, output. Analog computation of squares and square roots was performed using either expensive variable-transconductance analog multipliers (which remain expensive in the 21st century) or simpler and cheaper logarithmic converters employing exponential current-voltage characteristic of a bipolar transistor.
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The log semiring also arises when working with numbers that are logarithms (measured on a logarithmic scale), such as decibels (see ), log probability, or log-likelihoods.
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Number of cases (blue) and number of deaths (green) on a logarithmic scale. This the following is a timeline of the COVID-19 pandemic in Canada.
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It is also possible to perform non-linear regression directly on the data, without involving the logarithmic data transformation; for more options, see probability distribution fitting.
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This article attempts to document the timeline of the COVID-19 pandemic in Indonesia. Number of cases (blue) and number of deaths (red) on a logarithmic scale.
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Apparent magnitude is a logarithmic measure of apparent brightness. The distance determined by luminosity measures can be somewhat ambiguous, and is thus sometimes called the luminosity distance.
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For all regular polygons, each mouse traces out a pursuit curve in the shape of a logarithmic spiral. These curves meet in the center of the polygon.
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For faster-than-logarithmic growth, the model does not produce small worlds. The special case of O(\ln\ln N) is known as ultra-small world effect.
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Further, the most recent periods of evolution hold the most interest for us. We need therefore increasingly more space for our outline the nearer we approach modern times, and the logarithmic scale fulfills just this condition without any break in the continuity. Two examples of such timelines are shown below, while a more comprehensive version (similar to that of Sparks' "Histomap") can be found at Detailed logarithmic timeline.
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Audio amplifier power, normally specified in watts, is not always as significant as it may seem from the specification. Due to the logarithmic nature of human hearing, audio power or sound pressure level (SPL), must be increased by ten times to sound twice as loud. This is why SPL is measured on a logarithmic scale in decibels (dBs). An increase of 10dBs results in a perceived doubling of loudness.
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In some situations, especially in RF domain, monolithic logarithmic amplifiers are also used to reduce number of components and space used, as well improve bandwidth and noise performance.
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The distance modulus is a way of expressing distances that is often used in astronomy. It describes distances on a logarithmic scale based on the astronomical magnitude system.
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"The Book of Origins: The first of everything – from art to zoos". Hachette UK which he developed to calculate logarithmic tangents.Eli Maor (2013). "Trigonometric Delights", Princeton University Press.
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It was powered by a National Semiconductor MM57134ENW/M integrated circuit. The President Scientific added logarithmic and trigonometric functions. It was powered by a General Instrument CF-599.
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In 2005 Omer Reingold introduced an algorithm that solves the undirected st-connectivity problem, the problem of testing whether there is a path between two given vertices in an undirected graph, using only logarithmic space. The algorithm relies heavily on the zigzag product. Roughly speaking, in order to solve the undirected s-t connectivity problem in logarithmic space, the input graph is transformed, using a combination of powering and the zigzag product, into a constant-degree regular graph with a logarithmic diameter. The power product increases the expansion (hence reduces the diameter) at the price of increasing the degree, and the zigzag product is used to reduce the degree while preserving the expansion.
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The formula as given can be applied more widely; for example if f(z) is a meromorphic function, it makes sense at all complex values of z at which f has neither a zero nor a pole. Further, at a zero or a pole the logarithmic derivative behaves in a way that is easily analysed in terms of the particular case :zn with n an integer, n ≠ 0\. The logarithmic derivative is then :n/z; and one can draw the general conclusion that for f meromorphic, the singularities of the logarithmic derivative of f are all simple poles, with residue n from a zero of order n, residue −n from a pole of order n. See argument principle.
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These mathematical tables from 1925 were distributed by the College Entrance Examination Board to students taking the mathematics portions of the tests Tables of common logarithms were used until the invention of computers and electronic calculators to do rapid multiplications, divisions, and exponentiations, including the extraction of nth roots. Mechanical special-purpose computers known as difference engines were proposed in the 19th century to tabulate polynomial approximations of logarithmic functions - that is, to compute large logarithmic tables. This was motivated mainly by errors in logarithmic tables made by the human computers of the time. Early digital computers were developed during World War II in part to produce specialized mathematical tables for aiming artillery.
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A boundary Q-divisor on a variety is a Q-divisor D of the form ΣdiDi where the Di are the distinct irreducible components of D and all coefficients are rational numbers with 0≤di≤1. A logarithmic pair, or log pair for short, is a pair (X,D) consisting of a normal variety X and a boundary Q-divisor D. The log canonical divisor of a log pair (X,D) is K+D where K is the canonical divisor of X. A logarithmic 1-form on a log pair (X,D) is allowed to have logarithmic singularities of the form d log(z) = dz/z along components of the divisor given locally by z=0.
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An example In mathematics, a conchospiral a specific type of three-dimensional spiral on the surface of a cone (a conical spiral), whose floor projection is a logarithmic spiral.
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But in 1963 Donald Mackay showed and in 1978 John Staddon demonstrated with Stevens' own data, that the power law is the result of logarithmic input and output processes.
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Microautophagy is also connected with organellar size maintenance, composition of biological membranes, cell survival under nitrogen restriction, and the transition pathway from starvation-induced growth arrest to logarithmic growth.
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This is different from usual code concatenation where the inner codes are the same for each position. The Justesen code can be constructed very efficiently using only logarithmic space.
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As search points follow logarithmic spiral trajectories towards the common center, defined as the current best point, better solutions can be found and the common center can be updated.
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The theory underpinning it was laid out in the BCG perspective The Product Portfolio in 1970. One unusual characteristic of this visualization is that the x-axis is generally represented on an inverted scale (greater values are on the left), and on a logarithmic scale. The logarithmic scale may have been carried over from the work done on the Experience Curve (which uses a log-log scale), and is not always consistently used as such.
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G.711 defines two main companding algorithms, the μ-law algorithm and A-law algorithm. Both are logarithmic, but A-law was specifically designed to be simpler for a computer to process. The standard also defines a sequence of repeating code values which defines the power level of 0 dB. The μ-law and A-law algorithms encode 14-bit and 13-bit signed linear PCM samples (respectively) to logarithmic 8-bit samples.
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Harbisson's Sonochromatic Music Scale (2003) is a microtonal and logarithmic scale with 360 notes in an octave. Each note corresponds to a specific degree of the color wheel. The scale was introduced to the first eyeborg in 2004. Modern Painters, The International Contemporary Art Magazine pp 70-73 (New York, June 2008) Harbisson's Pure Sonochromatic Scale (2005) is a non-logarithmic scale based on the transposition of light frequencies to sound frequencies.
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The power may be either in linear units, or logarithmic units (dBm). Usually the logarithmic display is more useful, because it presents a larger dynamic range with better detail at each value. An RF sweep relates to a receiver which changes its frequency of operation continuously from a minimum frequency to a maximum (or from maximum to minimum). Usually the sweep is performed at a fixed, controllable rate, for example 5 MHz/sec.
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While a helix is produced by projecting a straight line onto the surface of a cylinder, Sang's method requires that a series of logarithmic curves be projected onto a cylindrical surface, hence its name. French; Ives, 1902, op. cit., p. 100. In terms of strength and stability, a skew bridge built to the logarithmic pattern has advantages over one built to the helicoidal pattern, especially so in the case of full-centred designs.
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In computational complexity theory, NL-complete is a complexity class containing the languages that are complete for NL, the class of decision problems that can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space. The NL-complete languages are the most "difficult" or "expressive" problems in NL. If a method exists for solving any one of the NL-complete problems in logarithmic memory space, then NL = L.
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Logarithmic conformal field theories are two-dimensional CFTs such that the action of the Virasoro algebra generator L_0 on the spectrum is not diagonalizable. In particular, the spectrum cannot be built solely from lowest weight representations. As a consequence, the dependence of correlation functions on the positions of the fields can be logarithmic. This contrasts with the power-like dependence of the two- and three-point functions that are associated to lowest weight representations.
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Binary search trees are searched using an algorithm similar to binary search. A binary search tree is a binary tree data structure that works based on the principle of binary search. The records of the tree are arranged in sorted order, and each record in the tree can be searched using an algorithm similar to binary search, taking on average logarithmic time. Insertion and deletion also require on average logarithmic time in binary search trees.
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In 1998, Broadhurst gave a series representation that allows arbitrary binary digits to be computed, and thus, for the constant to be obtained in nearly linear time, and logarithmic space.
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Bipartite maximum matchings can be approximated arbitrarily accurately in constant time by distributed algorithms; in contrast, approximating the minimum vertex cover of a bipartite graph requires at least logarithmic time..
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Linear or logarithmic axes may be used and multiple curves can be plotted on each graph. Graphs may be copied and pasted into other documents or programs for further editing.
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Thus, overall, the insertion procedure consists of a search, the creation of a constant number of new nodes, a logarithmic number of rank changes, and a constant number of tree rotations.
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Most mathematical functions commonly used by engineers, scientists and navigators, including logarithmic and trigonometric functions, can be approximated by polynomials, so a difference engine can compute many useful tables of numbers.
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According to the Erdős–Pósa theorem, the size of a minimum feedback vertex set is within a logarithmic factor of the maximum number of vertex-disjoint cycles in the given graph.
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The sixfold crystalline order is easy to detect on a local scale, since the logarithmic increase of displacements is rather slow. The deviations from the (red) lattice axis are easy to detect, too, here shown as green arrows. The deviations are basically given by the elastic lattice vibrations (acoustic phonons). A direct experimental proof of Mermin-Wagner-Hohenberg fluctuations would be, if the displacements increase logarithmic with the distance of a locally fitted coordinate frame (blue).
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The three line graphs below give a detailed overview of the current and historical case, recovery, and death counts throughout the Pakistan. The first two show the exponential growth of the pandemic in the country by using a linear scale for their Y-Axes. The third plot uses a Logarithmic scale for its Y-Axis to show relationships between the trends. On a Logarithmic Scale, data that shows exponential growth will plot as a straight line.
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The log-linear type of a semi-log graph, defined by a logarithmic scale on the y-axis, and a linear scale on the x-axis. Plotted lines are: y = 10x (red), y = x (green), y = log(x) (blue). The linear-log type of a semi-log graph, defined by a logarithmic scale on the x axis, and a linear scale on the y axis. Plotted lines are: y = 10x (red), y = x (green), y = log(x) (blue).
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This is a line with slope \gamma and \log_a \lambda vertical intercept. The logarithmic scale is usually labeled in base 10; occasionally in base 2: :\log (y) = (\gamma \log (a)) x + \log (\lambda). A log-linear (sometimes log-lin) plot has the logarithmic scale on the y-axis, and a linear scale on the x-axis; a linear-log (sometimes lin-log) is the opposite. The naming is output-input (y-x), the opposite order from (x, y).
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The Harbisson's Sonochromatic Music Scale Harbisson's Sonochromatic Music Scale (2003) is a microtonal and logarithmic scale with 360 notes in an octave. Each note corresponds to a specific degree of the color wheel. The scale was introduced to the first antenna in 2004. Modern Painters, The International Contemporary Art Magazine pp 70-73 (New York, June 2008) Harbisson's Pure Sonochromatic Scale (2005) is a non-logarithmic scale based on the transposition of light frequencies to sound frequencies.
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The ISO system defines both an arithmetic and a logarithmic scale. The arithmetic ISO scale corresponds to the arithmetic ASA system, where a doubling of film sensitivity is represented by a doubling of the numerical film speed value. In the logarithmic ISO scale, which corresponds to the DIN scale, adding 3° to the numerical value constitutes a doubling of sensitivity. For example, a film rated ISO 200/24° is twice as sensitive as one rated ISO 100/21°.
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For the active pressure coefficient, the logarithmic spiral rupture surface provides a negligible difference compared to Muller-Breslau. These equations are too complex to use, so tables or computers are used instead.
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It remains zero at all higher temperatures: this is the ferromagnetic phase. In four and higher dimensions the phase transition has mean field theory critical exponents (with logarithmic corrections in four dimensions).
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In potential theory, a branch of mathematics, Cartan's lemma, named after Henri Cartan, is a bound on the measure and complexity of the set on which a logarithmic Newtonian potential is small.
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The general idea, with small logarithmic modifications, is explained in quantum chromodynamics by "asymptotic freedom". Bjorken co-authored, with Sidney Drell, a classic companion volume textbook on relativistic quantum mechanics and quantum fields.
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Cumulative number of deaths per million inhabitants for European Union countries, over time. The legend is sorted in descending order of these values. Countries without COVID-19 deaths are omitted. Logarithmic vertical axis.
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This is analogous to the notations pH and pKa for an acid dissociation constant, where the symbol p denotes a cologarithm. The logarithmic form of the equilibrium constant equation is pKw = pH + pOH.
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The product of probabilities x \cdot y corresponds to addition in logarithmic space. : \log(x \cdot y) = \log(x) + \log(y) = x' + y'. The sum of probabilities x + y is a bit more involved to compute in logarithmic space, requiring the computation of one exponent and one logarithm. However, in many applications a multiplication of probabilities (giving the probability of all independent events occurring) is used more often than their addition (giving the probability of at least one of them occurring).
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Lethal doses LD50-values in a logarithmic scale The LD50 values have a very wide range. The botulinum toxin as the most toxic substance known has an LD50 value of 1 ng / kg, while the most non-toxic substance water has an LD50 value of more than 90 g / kg. That's a difference of about 1 in 100 billion or 11 orders of magnitude. As with all measured values that differ by many orders of magnitude, a logarithmic view is advisable.
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It is open if directed st-connectivity is in SC, although it is known to be in P ∩ PolyL (because of a DFS algorithm and Savitch's theorem). This question is equivalent to NL ⊆ SC. RL and BPL are classes of problems acceptable by probabilistic Turing machines in logarithmic space and polynomial time. Noam Nisan showed in 1992 the weak derandomization result that both are contained in SC.. In other words, given polylogarithmic space, a deterministic machine can simulate logarithmic space probabilistic algorithms.
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The simple multicellular eukaryote Volvox carteri undergoes sex in response to oxidative stress or stress from heat shock. These examples, and others, suggest that, in simple single-celled and multicellular eukaryotes, meiosis is an adaptation to respond to stress. Prokaryotic sex also appears to be an adaptation to stress. For instance, transformation occurs near the end of logarithmic growth, when amino acids become limiting in Bacillus subtilis, or in Haemophilus influenzae when cells are grown to the end of logarithmic phase.
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In computational complexity theory, a log-space reduction is a reduction computable by a deterministic Turing machine using logarithmic space. Conceptually, this means it can keep a constant number of pointers into the input, along with a logarithmic number of fixed-size integers.Arora & Barak (2009) p. 88 It is possible that such a machine may not have space to write down its own output, so the only requirement is that any given bit of the output be computable in log-space.
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The standard system for comparing interval sizes is with cents. The cent is a logarithmic unit of measurement. If frequency is expressed in a logarithmic scale, and along that scale the distance between a given frequency and its double (also called octave) is divided into 1200 equal parts, each of these parts is one cent. In twelve-tone equal temperament (12-TET), a tuning system in which all semitones have the same size, the size of one semitone is exactly 100 cents.
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In fade- ins the shape of the perceived volume curve looks like the level in decibels pulled towards the middle part of the line to the bottom right corner. The fade-out in turn looks like it has been pulled toward the bottom-left corner. A logarithmic fade takes a line that has already been curved and straightens it out [15]. The logarithmic fade sounds consistent and smooth since the perceived volume is increased over the whole duration of the fade.
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The problem of characterizing these sequence was described as "very difficult" by Paul Erdős in 1979. These sequences were named "Behrend sequences" in 1990 by Richard R. Hall, with a definition using logarithmic density in place of natural density. Hall chose their name in honor of Felix Behrend, who proved that for a Behrend sequence A, the sum of reciprocals of A must diverge. Later, Hall and Gérald Tenenbaum used natural density to define Behrend sequences in place of logarithmic density.
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It includes Behrend's theorem that such a sequence must have logarithmic density zero, and the seemingly-contradictory construction by Abram Samoilovitch Besicovitch of primitive sequences with natural density close to 1/2. It also discusses the sequences that contain all integer multiples of their members, the Davenport–Erdős theorem according to which the lower natural and logarithmic density exist and are equal for such sequences, and a related construction of Besicovitch of a sequence of multiples that has no natural density.
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The book has 15 chapters, roughly grouped into five chapters on first-order logic, three on second-order logic, and seven independent chapters on advanced topics. The first two chapters provide background material in first-order logic (including first-order arithmetic, the BIT predicate, and the notion of a first-order query) and complexity theory (including formal languages, resource-bounded complexity classes, and complete problems). Chapter three begins the connection between logic and complexity, with a proof that the first-order-recognizable languages can be recognized in logarithmic space, and the construction of complete languages for logarithmic space, nondeterministic logarithmic space, and polynomial time. The fourth chapter concerns inductive definitions, fixed-point operators, and the characterization of polynomial time in terms of first-order logic with the least fixed-point operator.
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The main use for log-space computable functions is in log-space reductions. This is a means of transforming an instance of one problem into an instance of another problem, using only logarithmic space.
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BEIC digital library.) William Gardiner (died 1752) was an English mathematician.Gardiner, William His logarithmic tables of sines and tangents (Tables of logarithms, 1742) had various reprints and saw use by scientists and other mathematicians.
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In 1908, Lawrence Joseph Henderson derived an equation to calculate the pH of a buffer solution. In 1917, Karl Albert Hasselbalch re-expressed that formula in logarithmic terms, resulting in the Henderson–Hasselbalch equation.
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Their positions are used to determine the fiber tilt angle \beta. The image has been recorded on a CCD detector. It shows the logarithmic intensitity in pseudo-color representation. Here bright colors represent high intensity.
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Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
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This variation in definitions makes no difference in which sequences are Behrend sequences, because the Davenport–Erdős theorem shows that, for sets of multiples, having natural density one and having logarithmic density one are equivalent.
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The individual band sound pressure levels are converted to "noy" valuesEnvironmental Technical Manual, Volume I. Procedures for the Noise Certification of Aircraft. (2012) International Civil Aviation Committee on Aviation Environmental Protection. which are then summed in the manner of Stevens' MKVI loudness to yield a total noy value. Noy is a linear unit of noisiness like sone is for loudness, and is then converted into PNL or PNdB (the terms are interchangeable) which is a logarithmic unit like phon which is the logarithmic unit for loudness.
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Supported common mathematical functions (unary, binary and variable number of arguments), including: trigonometric functions, inverse trigonometric functions, logarithm functions, exponential function, hyperbolic functions, Inverse hyperbolic functions, Bell numbers, Lucas numbers, Stirling numbers, prime-counting function, exponential integral function, logarithmic integral function, offset logarithmic integral , binomial coefficient and others. Expression e = new Expression("sin(0)+ln(2)+log(3,9)"); double v = e.calculate(); Expression e = new Expression("min(1,2,3,4)+gcd(1000,100,10)"); double v = e.calculate(); Expression e = new Expression("if(2<1, 3, 4)"); double v = e.
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Smith, Foreign influence shows itself indirectly some of his published work.Smith, Sakabe's Sampo Tenzan Shinan-roku (Treatise on Tenzan Algebra) in 1810 was the first published work in Japan proposing the use of logarithmic tables. He explained that "these tables save much labor, [but] they are but little known for the reason that they have never been printed in our country."Smith, Sakabe's proposal would not be realized until twenty years after his death when the first extensive logarithmic table was published in 1844 by Koide Shuke.
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Implantable colour-to-sound chips, software, mobile apps and cyborg antennas use Harbisson's Sonochromatic Scales as a standard transposition of colour frequencies to sound frequencies. The Sonochromatic Music Scale is a microtonal and logarithmic scale with 360 notes in an octave. Each note corresponds to a specific degree of the color wheel. Modern Painters, The International Contemporary Art Magazine pp 70–73 (New York City, June 2008) Harbisson's Pure Sonochromatic Scale is a non- logarithmic scale based on the transposition of light frequencies to sound frequencies.
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Perceived "loudness" varies approximately logarithmically with acoustical output power. The change in perceived loudness as a function of change in acoustical power is dependent on the reference power level. It is both useful and technically accurate to express perceived loudness in the logarithmic decibel (dB) scale that is independent of the reference power, with a somewhat straight-line relationship between 10 dB changes and doublings of perceived loudness. The approximately logarithmic relationship between power and perceived loudness is an important factor in audio system design.
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Logarithms occur in several laws describing human perception:, pp. 355–56, p. 48 Hick's law proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have., p.
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In 2013, Timothy M. Chan developed a simpler algorithm that avoids the need for dynamic data structures and eliminates the logarithmic factor, lowering the best known running time for d ≥ 3 to O(n^{d/2}).
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In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne.Deligne, Pierre. Equations différentielles à points singuliers réguliers.
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Oymurania is an organophosphatic Cambrian small shelly fossil interpreted as a stem-group Brachiopod. It consists of a pair of Micrina-like shells that broadly follow a logarithmic coiling trajectory with a high rate of expansion.
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147, pp. 63–65, 1978. and others. Prior to the development of digital computers, lens optimization was a hand- calculation task using trigonometric and logarithmic tables to plot 2-D cuts through the multi-dimensional space.
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A common format is a graph with two geometric dimensions: one axis represents time, and the other axis represents frequency; a third dimension indicating the amplitude of a particular frequency at a particular time is represented by the intensity or color of each point in the image. There are many variations of format: sometimes the vertical and horizontal axes are switched, so time runs up and down; sometimes as a waterfall plot where the amplitude is represented by height of a 3D surface instead of color or intensity. The frequency and amplitude axes can be either linear or logarithmic, depending on what the graph is being used for. Audio would usually be represented with a logarithmic amplitude axis (probably in decibels, or dB), and frequency would be linear to emphasize harmonic relationships, or logarithmic to emphasize musical, tonal relationships.
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Approximate logarithmic spirals can occur in nature, for example the arms of spiral galaxies \- golden spirals are one special case of these logarithmic spirals, although there is no evidence that there is any general tendency towards this case appearing. Phyllotaxis is connected with the golden ratio because it involves successive leaves or petals being separated by the golden angle; it also results in the emergence of spirals, although again none of them are (necessarily) golden spirals. It is sometimes stated that spiral galaxies and nautilus shells get wider in the pattern of a golden spiral, and hence are related to both and the Fibonacci series. For example, these books: , , , , , In truth, spiral galaxies and nautilus shells (and many mollusk shells) exhibit logarithmic spiral growth, but at a variety of angles usually distinctly different from that of the golden spiral.
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SLIM was developed by Embrey et al. [1] for use within the US nuclear industry. By use of this method, relative success likelihoods are established for a range of tasks, and then calibrated using a logarithmic transformation.
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All of these methods used theoretical, a priori restrictions. According to an article by Carl F. Christ, the Cowles approach was grounded on certain assumptions: :1. simultaneous economic behavior; :2. linear or logarithmic equations and disturbances; :3.
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Jacob Bernoulli's tombstone in Basel Münster Bernoulli wanted a logarithmic spiral and the motto Eadem mutata resurgo ('Although changed, I rise again the same') engraved on his tombstone. He wrote that the self- similar spiral "may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self." Bernoulli died in 1705, but an Archimedean spiral was engraved rather than a logarithmic one. Translation of Latin inscription: :Jacob Bernoulli, the incomparable mathematician.
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For purely functional languages, the worst- case slowdown is logarithmic in the number of memory cells used, because mutable memory can be represented by a purely functional data structure with logarithmic access time (such as a balanced tree). However, such slowdowns are not universal. For programs that perform intensive numerical computations, functional languages such as OCaml and Clean are only slightly slower than C according to The Computer Language Benchmarks Game. For programs that handle large matrices and multidimensional databases, array functional languages (such as J and K) were designed with speed optimizations.
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A logarithmic timeline is a timeline laid out according to a logarithmic scale. This necessarily implies a zero point and an infinity point, neither of which can be displayed. The most natural zero point is the Big Bang, looking forward, but the most common is the ever-changing present, looking backward. (Also possible is a zero point in the present, looking forward to the infinite future.) The idea of presenting history logarithmically goes back at least to 1932, when John B. Sparks copyrighted his chart "Histomap of Evolution".
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Works by Barenblatt and others have shown that besides the logarithmic law of the wall — the limit for infinite Reynolds numbers — there exist power-law solutions, which are dependent on the Reynolds number. In 1996, Cipra submitted experimental evidence in support of these power-law descriptions. This evidence itself has not been fully accepted by other experts. In 2001, Oberlack claimed to have derived both the logarithmic law of the wall, as well as power laws, directly from the Reynolds-averaged Navier–Stokes equations, exploiting the symmetries in a Lie group approach.
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In the area of game theory, more specifically of non-cooperative games, Lipton together with E. Markakis and A. Mehta provedRichard Lipton, Evangelos Markakis, Aranyak Mehta (2007) "Playing Games with Simple Strategies", "EC '03: Proceedings of the 4th ACM conference on Electronic commerce", "ACM" the existence of epsilon- equilibrium strategies with support logarithmic in the number of pure strategies. Furthermore, the payoff of such strategies can epsilon-approximate the payoffs of exact Nash equilibria. The limited (logarithmic) size of the support provides a natural quasi-polynomial algorithm to compute epsilon- equilibria.
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In constrained optimization, a field of mathematics, a barrier function is a continuous function whose value on a point increases to infinity as the point approaches the boundary of the feasible region of an optimization problem. Such functions are used to replace inequality constraints by a penalizing term in the objective function that is easier to handle. The two most common types of barrier functions are inverse barrier functions and logarithmic barrier functions. Resumption of interest in logarithmic barrier functions was motivated by their connection with primal-dual interior point methods.
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By 1800, Routledge had become Manager at the Round Foundry. Somehow, along the way, he learned the value of logarithms and thereby had the means of developing a method of measuring "all kinds of metals and other bodies" British Library, Call 717.a.18 needed for engineering purposes. Using the principles of Edmund Gunter's (1581–1626) logarithmic scales and William Oughtred's (1574–1660) sliding rule, Routledge combined a 12-inch brass slide containing the logarithmic scales with an ordinary 2-foot ruler to which he added a table of commonly used references called gauge points.
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Changing the shape of a logarithmic fade will change how soon the sound will rise above 50%, and then how long it takes for the end of the fade-out to drop below 50% once again. With exponential fades the shape change will affect the shape in reverse, to the shape of the logarithmic fade. In the S-curve's traditional form the shape determines how quickly the change can occur and in the type 2 curve the change can determine the time it takes for both the sounds to get to a nearly equal level.
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For this reason, the use of mantissa for significand is discouraged by some including the creator of the standard, William Kahan and prominent computer programmer and author of The Art of Computer Programming, Donald E. Knuth The confusion is because scientific notation and floating- point representation are log-linear, not logarithmic. To multiply two numbers, given their logarithms, one just adds the characteristic (integer part) and the mantissa (fractional part). By contrast, to multiply two floating-point numbers, one adds the exponent (which is logarithmic) and multiplies the significand (which is linear).
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Conical spiral with an archimedean spiral as floor plan floor plan: Fermat's spiral floor plan: logarithmic spiral floor plan: hyperbolic spiral In mathematics, a conical spiral is a curve on a right circular cone, whose floor plan is a plane spiral. If the floor plan is a logarithmic spiral, it is called conchospiral (from conch). Conchospirals are used in biology for modelling snail shells, and flight paths of insects New ScientistConchospirals in the Flight of Insects and in electrical engineering for the construction of antennas.John D. Dyson: The Equiangular Spiral Antenna.
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The overall morphology of Thermoplasma volcanium isolates take on different shapes depending on their placement within the growth curve. During early logarithmic growth, the isolates take on forms of all shapes including, but not limited to, coccoid-, disc-, and club-shaped of around 0.2-0.5 micrometers. During stationary and late logarithmic growth phases, the isolates primarily take on a spherical (coccoid) shape and can produce buds around 0.3 micrometers in width that are thought to contain DNA. A single flagella is present on the organism, emerging from one polar end of the cell.
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American researcher Don P. Mitchell has processed the color images from Venera 13 and 14 using the raw original data. The new images are based on a more accurate linearization of the original 9-bit logarithmic pixel encoding.
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A cosmological decade (CÐ) is a division of the lifetime of the cosmos. The divisions are logarithmic in size, with base 10. Each successive cosmological decade represents a ten-fold increase in the total age of the universe.
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Logarithmic gap penalty was invented to modify the affine gap so that long gaps are desirable. However, in contrast to this, it has been found that using logarithmatic models had produced poor alignments when compared to affine models.
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There are complex design trade-offs involving lookup performance, index size, and index-update performance. Many index designs exhibit logarithmic (O(log(N))) lookup performance and in some applications it is possible to achieve flat (O(1)) performance.
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Retrieved March 3, 2011. RMS power detectors and logarithmic amplifiers; PLL and DDS synthesizers; RF prescalers; variable gain amplifiers;Janine Love, EE Times. "Variable gain amplifier takes aim at wireless infrastructure." January 16, 2006. Retrieved July 15, 2011.
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White noise spectrum. Flat power spectrum. (logarithmic frequency axis) White noise is a signal (or process), named by analogy to white light, with a flat frequency spectrum when plotted as a linear function of frequency (e.g., in Hz).
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Nasdaq Composite 1970–2012, logarithmic scale The NASDAQ Composite index spiked in the late 1990s and then fell sharply as a result of the dot-com bubble. The index was launched in 1971, with a starting value of 100.
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Concentrations of colony-forming units can be expressed using logarithmic notation, where the value shown is the base 10 logarithm of the concentration. This allows the log reduction of a decontamination process to be computed as a simple subtraction.
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Logarithmic chart of the hearing ranges of some animalsD Warfield. 1973. The study of hearing in animals. In: W Gay, ed., Methods of Animal Experimentation, IV. Academic Press, London, pp 43-143.RR Fay and AN Popper, eds. 1994.
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Grain size varies from clay in shales and claystones; through silt in siltstones; sand in sandstones; and gravel, cobble, to boulder sized fragments in conglomerates and breccias. The Krumbein phi (φ) scale numerically orders these terms in a logarithmic size scale.
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The two inverses of tetration are called the super-root and the super-logarithm, analogous to the nth root and the logarithmic functions. None of the three functions are elementary. Tetration is used for the notation of very large numbers.
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In applications involving sound, the particle velocity is usually measured using a logarithmic decibel scale called particle velocity level. Mostly pressure sensors (microphones) are used to measure sound pressure which is then propagated to the velocity field using Green's function.
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This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on is demanded additionally. The case must be treated differently because is not normalizable at infinity (the sum of the reciprocals doesn't converge).
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It is important to note the exponential growth of the limits (and thus the range that a bucket holds). In this way the logarithmic dependence of the field quantities is of value C, the maximum difference between two key values.
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Thus a single table of common logarithms can be used for the entire range of positive decimal numbers.E. R. Hedrick, Logarithmic and Trigonometric Tables (Macmillan, New York, 1913). See common logarithm for details on the use of characteristics and mantissas.
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The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity.
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Sang worked for many years on trigonometric and logarithmic tables. Summaries of his tables were published by Alex Craik. Sang's 1871 table and his project for a 9-place million table were (re)constructed as part of the LOCOMAT project.
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Fortnow also examined the logarithmic market scoring rule (LMSR) with market makers. He helped to show that LMSR pricing is #P-hard and propose an approximation technique for pricing permutation markets.Y. Chen, L. Fortnow, N. Lambert, D. Pennock, and J. Wortman.
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The labeled magnitude scale (LMS) is a scaling technique which uses quasi- logarithmic spacing. The scale consists of different intensities and subjects are asked to put a mark on the line where they think they intensity of the sensation fits.
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Lists can be manipulated using iteration or recursion. The former is often preferred in imperative programming languages, while the latter is the norm in functional languages. Lists can be implemented as self-balancing binary search trees holding index-value pairs, providing equal-time access to any element (e.g. all residing in the fringe, and internal nodes storing the right-most child's index, used to guide the search), taking the time logarithmic in the list's size, but as long as it doesn't change much will provide the illusion of random access and enable swap, prefix and append operations in logarithmic time as well.
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Around the same time it was also explored by the cyberneticist Heinz von Foerster, who used it to propose that memories naturally fade in an exponential manner. Logarithmic timelines have also been used in future studies to justify the idea of a technological singularity. A logarithmic scale enables events throughout time to be presented accurately, but enables more events to be included closer to one end. Sparks explained this by stating: : As we travel forward in geological time the more complex is the evolution of life forms and the more are the changes to be recorded.
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Exposure to noise can cause vibrations able to cause permanent damage to the ear. Both the volume of the noise and the duration of exposure can influence the likelihood of damage. Sound is measured in units called decibels, which is a logarithmic scale of sound levels that corresponds to the level of loudness that an individual's ear would perceive. Because it is a logarithmic scale, even small incremental increases in decibels correlate to large increases in loudness, and an increase in the risk of hearing loss. Sounds above 80 dB have the potential to cause permanent hearing loss.
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A logarithmic transformation (of order m with center p) of an elliptic surface or fibration turns a fiber of multiplicity 1 over a point p of the base space into a fiber of multiplicity m. It can be reversed, so fibers of high multiplicity can all be turned into fibers of multiplicity 1, and this can be used to eliminate all multiple fibers. Logarithmic transformations can be quite violent: they can change the Kodaira dimension, and can turn algebraic surfaces into non-algebraic surfaces. Example: Let L be the lattice Z+iZ of C, and let E be the elliptic curve C/L.
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Nautilus shell's logarithmic growth spiral According to Stephen Skinner, the study of sacred geometry has its roots in the study of nature, and the mathematical principles at work therein. Many forms observed in nature can be related to geometry; for example, the chambered nautilus grows at a constant rate and so its shell forms a logarithmic spiral to accommodate that growth without changing shape. Also, honeybees construct hexagonal cells to hold their honey. These and other correspondences are sometimes interpreted in terms of sacred geometry and considered to be further proof of the natural significance of geometric forms.
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Trendlines are local regressions, with polls weighted by proximity in time and a logarithmic function of sample size. 95% confidence ribbons represent uncertainty about the trendlines, not the likelihood that actual election results would fall within the intervals. Evolution of voting intentions during the pre-campaign period of the 43rd Canadian federal election. Trendlines are local regressions, with polls weighted by proximity in time and a logarithmic function of sample size. 95% confidence ribbons represent uncertainty about the regressions, not the likelihood that actual election results would fall within the intervals. Source code for plot generation is available here.
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This amplitude modulation occurs with a frequency equal to the difference in frequencies of the two tones and is known as beating. The semitone scale used in Western musical notation is not a linear frequency scale but logarithmic. Other scales have been derived directly from experiments on human hearing perception, such as the mel scale and Bark scale (these are used in studying perception, but not usually in musical composition), and these are approximately logarithmic in frequency at the high-frequency end, but nearly linear at the low-frequency end. The intensity range of audible sounds is enormous.
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The logarithmic mean temperature difference (also known as log mean temperature difference, LMTD) is used to determine the temperature driving force for heat transfer in flow systems, most notably in heat exchangers. The LMTD is a logarithmic average of the temperature difference between the hot and cold feeds at each end of the double pipe exchanger. For a given heat exchanger with constant area and heat transfer coefficient, the larger the LMTD, the more heat is transferred. The use of the LMTD arises straightforwardly from the analysis of a heat exchanger with constant flow rate and fluid thermal properties.
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APEX made exposure computation a relatively simple matter; the foreword of ASA PH2.5-1960 recommended that exposure meters, exposure calculators, and exposure tables be modified to incorporate the logarithmic values that APEX required. In many instances, this was done: the 1973 and 1986 ANSI exposure guides, ANSI PH2.7-1973 and ANSI PH2.7-1986, eliminated exposure calculator dials in favor of tabulated APEX values. However, the logarithmic markings for aperture and shutter speed required to set the computed exposure were never incorporated in consumer cameras. Accordingly, no reference to APEX was made in ANSI PH3.49-1971 (though it was included in the Appendix).
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This logarithmic relationship means that if a stimulus varies as a geometric progression (i.e., multiplied by a fixed factor), the corresponding perception is altered in an arithmetic progression (i.e., in additive constant amounts). For example, if a stimulus is tripled in strength (i.e.
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From a descriptive complexity viewpoint, DLOGTIME-uniform AC0 is equal to the descriptive class FO+BIT of all languages describable in first- order logic with the addition of the BIT predicate, or alternatively by FO(+, ×), or by Turing machine in the logarithmic hierarchy.
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In computational complexity theory, the strict definition of in-place algorithms includes all algorithms with O(1) space complexity, the class DSPACE(1). This class is very limited; it equals the regular languages.Maciej Liśkiewicz and Rüdiger Reischuk. The Complexity World below Logarithmic Space.
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First developed by Angus Deaton and John Muellbauer, the AIDS system is derived from the "Price Invariant Generalized Logarithmic" (PIGLOG) modelThe Piglog Model USDA Web site which allows researchers to treat aggregate consumer behavior as if it were the outcome of a single maximizing consumer.
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The interaction is approximately logarithmic at short range and of Coulomb 1/r form at long range. The diffusion Monte Carlo method has been used to obtain numerically exact results for the binding energies of trions in 2D semiconductors within the effective mass approximation.
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Schröder's equation and Abel equation. On a logarithmic scale, this reduces to the nesting property of Chebyshev polynomials, , since . The relation also holds, analogous to the property of exponentiation that . The sequence of functions is called a Picard sequence, named after Charles Émile Picard.
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Illusie has received the Langevin Prize of the French Academy of Sciences in 1977 and, in 2012, the Émile Picard Medal of the French Academy of Sciences for "his fundamental work on the cotangent complex, the Picard–Lefschetz formula, Hodge theory and logarithmic geometry".
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This is the modern magnitude system, which measures the brightness, not the apparent size, of stars. Using this logarithmic scale, it is possible for a star to be brighter than “first class”, so Arcturus or Vega are magnitude 0, and Sirius is magnitude −1.46.
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An example of logarithmic time is given by dictionary search. Consider a dictionary which contains n entries, sorted by alphabetical order. We suppose that, for , one may access to the th entry of the dictionary in a constant time. Let denote this -th entry.
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As the law is logarithmic, the percentage loss of capacitance will twice between 1 h and 100 h and 3 times between 1 h and 1000 h and so on. So aging is fastest near the beginning, and the capacitance value effectively stabilizes over time.
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This logarithmic divergence goes along with an algebraic (slow) decay of positional correlations. The spatial order of a 2D crystal is called quasi long range (see also auch hexatic phase for the phase behaviour of 2D ensembles). Interestingly, significant signatures of Mermin- Wagner-Hohenberg fluctuations have not been found in crystals but in disordered amorphous systems This work did not investigate the logarithmic displacements of lattice sites (which are difficult to quantify for a finite system size), but the magnitude of the mean squared displacement of the particles as function of time. This way, the displacements are not analysed in space but in the time domain.
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The selected representation of the convex hull may influence on the computational complexity of further operations of the overall algorithm. For example, the point in polygon query for a convex polygon represented by the ordered set of its vertices may be answered in logarithmic time, which would be impossible for convex hulls reported by the set of it vertices without any additional information. Therefore, some research of dynamic convex hull algorithms involves the computational complexity of various geometric search problems with convex hulls stored in specific kinds of data structures. The mentioned approach of Overmars and van Leeuwen allows for logarithmic complexity of various common queries.
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While ten is the most common base, there are times when other bases are more appropriate, as in this example: A semi-logarithmic plot of cases and deaths in the 2009 outbreak of influenza A (H1N1). Notice that while the horizontal (time) axis is linear, with the dates evenly spaced, the vertical (cases) axis is logarithmic, with the evenly spaced divisions being labelled with successive powers of two. The semi-log plot makes it easier to see when the infection has stopped spreading at its maximum rate ie. the straight line on this exponential plot, and starts to curve to indicate a slower rate.
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A fade can be constructed so that the motion of the control (linear or rotary) from its start to end points affects the level of the signal in a different manner at different points in its travel. If there are no overlapping regions on the same track, regular fade (Pre-fade / Post-fade) should be used. A smooth fade is one that changes according to the logarithmic scale, as faders are logarithmic over much of their working range of 30-40 dB. If the engineer requires one region to gradually fade into another on the same track, a cross-fade would be more suitable.
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Using a logarithmic spiral shape resulted in a uniform angle between the rock and each lobe of the cam; this constant angle is designed to always provide the necessary friction to hold a cam in equilibrium.Duke SLCD research retrieved (2009-08-05) Designed so that a load produces a rotational force, the logarithmic cam shape allowed for a single device to fit securely in a range of crack sizes. Modern SLCDs were invented by Ray Jardine in 1978 (US patent 4,184,657) and sold under the brand name of "Friends". Ray designed a spring-loaded opposing multiple cam unit with a more stable 13.75 degree camming angle and an innovative triggering mechanism.
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The Romanesco superficially resembles a cauliflower, but it has a visually striking fractal form Romanesco cauliflower (or broccoli) Romanesco broccoli in a field Romanesco superficially resembles a cauliflower, but it is chartreuse in color, with the form of a natural fractal. The inflorescence (the bud) is self-similar in character, with the branched meristems making up a logarithmic spiral, giving a form approximating a natural fractal; each bud is composed of a series of smaller buds, all arranged in yet another logarithmic spiral. This self-similar pattern continues at smaller levels. The pattern is only an approximate fractal since the pattern eventually terminates when the feature size becomes sufficiently small.
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In computer science, a min-max heap is a complete binary tree data structure which combines the usefulness of both a min-heap and a max-heap, that is, it provides constant time retrieval and logarithmic time removal of both the minimum and maximum elements in it. This makes the min-max heap a very useful data structure to implement a double-ended priority queue. Like binary min- heaps and max-heaps, min-max heaps support logarithmic insertion and deletion and can be built in linear time. Min-max heaps are often represented implicitly in an array; hence it's referred to as an implicit data structure.
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In its logarithmic form it is the following conjecture. Let λ1, λ2, and λ3 be any three logarithms of algebraic numbers and γ be a non-zero algebraic number, and suppose that λ1λ2 = γλ3. Then λ1λ2 = γλ3 = 0\. The exponential form of this conjecture is the following.
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Lossy cavities are usually placed at the back to eliminate back lobes because a unidirectional pattern is usually preferred in such antennas. Spiral antennas are classified into different types; Archimedean spiral, logarithmic spiral, square spiral, and star spiral, etc. Archimedean spiral is the most popular configuration.
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To address this issue, a kit for Southern Blot analysis was developed in 1990, providing the first marker to combine target DNA and probe DNA. This technique took advantage of logarithmic spacing, and could be used to identify target bands ranging over a length of 20,000 nucleotides.
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The Ramanujan-Soldner constant and the Soldner coordinate system are named for him. The latter was used until the middle of the 20th century in Germany. In 1809, Soldner calculated the Euler–Mascheroni constant's value to 24 decimal places. He also published on the logarithmic integral function.
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Satellite antenna aperture is closely related to quality factor (G/T value) of earth station. G/T value and satellite power demand, i.e. equivalent rent bandwidth, show logarithmic linear relationship. In other words, the value of equivalent rent bandwidth increases with the narrowing of antenna aperture.
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Compared to Domain Authority which determines the ranking strength of an entire domain or subdomain, Page Authority measures the strength of an individual page.Moz - Page Authority It's a score developed by Moz on a 100-point logarithmic scale. Unlike TrustFlow, domain authority does not account for spam.
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Plot of CPU transistor counts against dates of introduction, illustrating Moore's Law. Note the logarithmic scale; the fitted line corresponds to exponential growth, with transistor count doubling every two years. Reynolds divides the book into two distinct sections. The first focuses on trends currently taking place.
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In a similar example, with the phrase "He had a seven-figure income", the order of magnitude is the number of figures minus one, so it is very easily determined without a calculator to 6. An order of magnitude is an approximate position on a logarithmic scale.
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There is also no need to preserve balance conditions and no satellite information within the nodes is necessary. Lastly, this structure has good worst-case time efficiency. The execution time of each individual operation is at most logarithmic with high probability.A. Gambin and A. Malinowski. 1998.
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The x- and y-axes are scaled non-linearly by their standard normal deviates (or just by logarithmic transformation), yielding tradeoff curves that are more linear than ROC curves, and use most of the image area to highlight the differences of importance in the critical operating region.
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Edward Sang (1805–1890) Bridge number 74A carrying the Bolton and Preston Railway over the Leeds and Liverpool Canal The development of the intrados of a skew arch built to the logarithmic pattern Detailed view of the intrados of bridge 74A The search for a technically pure orthogonal method of constructing a skew arch led to the proposal of the logarithmic method by Edward Sang, a mathematician living in Edinburgh, in his presentation in three parts to the Society for the Encouragement of the Useful Arts between 18 November 1835 and 27 January 1836, during which time he was elected vice-president of the Society, though his work was not published until 1840. The logarithmic method is based on the principle of laying the voussoirs in "equilibrated" courses in which they follow lines that run truly perpendicular to the arch faces at all elevations, while the header joints between the stones within each course are truly parallel with the arch face. Hyde, 1899, op. cit., pp. 40–41.
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Both are defined via Taylor series analogous to the real case., section II.5. In the context of differential geometry, the exponential map maps the tangent space at a point of a manifold to a neighborhood of that point. Its inverse is also called the logarithmic (or log) map.
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The scale discards color as being part of a color wheel and ignores musical/logarithmic perception so it can overstep the limits of human perception. The introduction of the new scale to the eyeborg in 2010, allows users to decide whether they want to perceive colors logarithmically or not.
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Since the MNREAD charts use logarithmic pattern of letters, near visual acuity is usually measured at a distance of 40 cm from eyes. For low vision patients, chart can also be used at closer distances. After distance vision correction, near vision is measured with and without near vision correction.
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Elementary Functions- a study of the elementary functions (power functions, polynomials, rational, exponential, logarithmic and trigonometric) with an emphasis on their behavior and applications. Some analytic geometry and elements of the calculus as well as the application of matrices to the solution of linear systems is also included.
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Detected active cases (blue), hospitalized patients, incl. ICU (red) and patients in ICU (yellow). The horizontal dashed red and yellow lines are the beds reserved for CoViD-19 patients in the hospitals. The graphs are smoothed by 3-day central moving average and plotted in semi-logarithmic scale.
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There are some Public Key encryption schemes that allow keyword search, however these schemes all require search time linear in the database size. If the database entries were encrypted with a deterministic scheme and sorted, then a specific field of the database could be retrieved in logarithmic time.
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The scale discards color as being part of a color wheel and ignores musical/logarithmic perception so it can overstep the limits of human perception. The introduction of the new scale to the eyeborg in 2010, allows users to decide whether they want to perceive colors logarithmically or not.
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Steels with pearlitic (eutectoid composition) or near-pearlitic microstructure (near-eutectoid composition) can be drawn into thin wires. Such wires, often bundled into ropes, are commercially used as piano wires, ropes for suspension bridges, and as steel cord for tire reinforcement. High degrees of wire drawing (logarithmic strain above 3) leads to pearlitic wires with yield strengths of several gigapascals. It makes pearlite one of the strongest structural bulk materials on earth.. Some hypereutectoid pearlitic steel wires, when cold wire drawn to true (logarithmic) strains above 5, can even show a maximal tensile strength above 6 GPa.. Although pearlite is used in many engineering applications, the origin of its extreme strength is not well understood.
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The most historically significant nonlinear compositing system was the Cineon, which operated in a logarithmic color space, which more closely mimics the natural light response of film emulsions (the Cineon system, made by Kodak, is no longer in production). Due to the limitations of processing speed and memory, compositing artists did not usually have the luxury of having the system make intermediate conversions to linear space for the compositing steps. Over time, the limitations have become much less significant, and now most compositing is done in a linear color space, even in cases where the source imagery is in a logarithmic color space. Compositing often also includes scaling, retouching and colour correction of images.
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Note: this is an intuitive description of how autonomous convergence theorems guarantee stability, not a strictly mathematical description. The key point in the example theorem given above is the existence of a negative logarithmic norm, which is derived from a vector norm. The vector norm effectively measures the distance between points in the vector space on which the differential equation is defined, and the negative logarithmic norm means that distances between points, as measured by the corresponding vector norm, are decreasing with time under the action of f. So long as the trajectories of all points in the phase space are bounded, all trajectories must therefore eventually converge to the same point.
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A log–log plot of y = x (blue), y = x2 (green), and y = x3 (red). Note the logarithmic scale markings on each of the axes, and that the log x and log y axes (where the logarithms are 0) are where x and y themselves are 1. In science and engineering, a log–log graph or log–log plot is a two- dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Monomials – relationships of the form y=ax^k – appear as straight lines in a log–log graph, with the power term corresponding to the slope, and the constant term corresponding to the intercept of the line.
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This is usually done by fitting some kind of functional form to the curve, either by eye or by using non-linear regression techniques. Commonly used functional forms include the logarithmic function and the negative exponential function. The advantage of the negative exponential function is that it tends to an asymptote which equals the number of species that would be discovered if infinite effort is expended. However, some theoretical approaches imply that the logarithmic curve may be more appropriate, implying that though species discovery will slow down with increasing effort, it will never entirely cease, so there is no asymptote, and if infinite effort was expended, an infinite number of species would be discovered.
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Hick's law is similar in form to Fitts's law. Hick's law has a logarithmic form because people subdivide the total collection of choices into categories, eliminating about half of the remaining choices at each step, rather than considering each and every choice one-by-one, which would require linear time.
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Halved shell of Nautilus showing the chambers (camerae) in a logarithmic spiral (1st p. 493 – 2nd p. 748 – Bonner p. 172) ::Thompson observes that there are many spirals in nature, from the horns of ruminants to the shells of molluscs; other spirals are found among the florets of the sunflower.
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Modern lasers have made the production of precise grooves easier and more affordable, but not all lasers nor laser companies have the required technology. A good supplier will produce precise constant depth grooves in ceramic or metal parts to within fractions of a micrometer, including proper logarithmic grooves for thrust bearings.
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Heidelberger, M. (2004)Nature from within: Gustav Theodor Fechner and his psychophysical worldview. Transl. C. Klohr. Pittsburgh, USA: University of Pittsburgh Press. Following the work of S. S. Stevens, many researchers came to believe in the 1960s that the power law was a more general psychophysical principle than Fechner's logarithmic law.
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Logarithmic scale of reported human cases of guinea worm by year, 1989–2017. Data from Guinea Worm Eradication Program. Dracunculiasis or Guinea-worm disease, is an infection by the Guinea worm. In 1986, there were an estimated 3.5 million cases of Guinea worm in 20 endemic nations in Asia and Africa.
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In algebraic geometry, a log structure provides an abstract context to study semistable schemes, and in particular the notion of logarithmic differential form and the related Hodge-theoretic concepts. This idea has applications in the theory of moduli spaces, in deformation theory and Fontaine's p-adic Hodge theory, among others.
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The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (polar slope angle about 17.03239 degrees). It can be approximated by a "Fibonacci spiral", made of a sequence of quarter circles with radii proportional to Fibonacci numbers.
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The linear plot shows the total number of tests performed and the number of people tested as a function of time (by date) since 7 April 2020, the date of the first testing data given in Singapore. Number of cases (blue) and number of deaths (red) on a logarithmic scale.
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July 21, 2014. Jones created the Nautilus machine, then called the Blue Monster, in the late 1960s, with the purpose of developing a fitness machine that accommodates human movement. The company's name was changed to Nautilus because the logarithmic-spiral cam, which made the machine a success, resembled a nautilus.
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This information is often exploited in contour integration. In the field of Nevanlinna Theory, an important lemma states that the proximity function of a logarithmic derivative is small with respect to the Nevanlinna Characteristic of the original function, for instance m(r,h'/h) = S(r,h) = o(T(r,h)).
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Luc Illusie (; born 1940) is a French mathematician, specializing in algebraic geometry. His most important work concerns the theory of the cotangent complex and deformations, crystalline cohomology and the De Rham–Witt complex, and logarithmic geometry. In 2012, he was awarded the Émile Picard Medal of the French Academy of Sciences.
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The chambered nautilus is often used as an example of the golden spiral. While nautiluses show logarithmic spirals, their ratios range from about 1.24 to 1.43, with an average ratio of about 1.33 to 1. The golden spiral's ratio is 1.618. This is actually visible when the cut nautilus is inspected.
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The shape of a regular fade and a cross-fade can be shaped by an audio engineer. Shape implies that you can change the rate at which the level change occurs over the length of the fade. Different types of preset fades shapes include linear, logarithmic, exponential and S-curve.
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An order-of-magnitude difference between two values is a factor of 10. For example, the mass of the planet Saturn is 95 times that of Earth, so Saturn is two orders of magnitude more massive than Earth. Order-of-magnitude differences are called decades when measured on a logarithmic scale.
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Both the radar reflectivity factor and its logarithmic version are commonly referred to as reflectivity when the context is clear. In short, the higher the dBZ value, the more likely it is for severe weather to occur in the form of precipitation. Values above 20 dBZ usually indicate falling precipitation.
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The expression of SraJ was experimentally confirmed by Northern blotting. This ncRNA is expressed in early logarithmic phase, but its level decreases into stationary phase. Northern blot analysis also indicated this RNA undergoes specific cleavage processing. The GlmZ sRNA has been shown to positively control the synthesis of GlmS mRNA.
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This invariant ensures that the height of the tree (length of the longest path from the root to some leaf) is always logarithmic in the number of elements in the tree. Concatenation is implemented as follows: def concat(xs: Conc[T], ys: Conc[T]) { val diff = ys.level - xs.level if (math.
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There are many exit variations of spins.Petkevich, pp. 129–130 A difficult exit is any jump or movement a skater performs that makes the exit significantly more difficult. The 3 turn, a movement used in many spins The entry phase produces a logarithmic curve with an indefinite number of radii.
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In 2004, Omer Reingold proved that SL=L by showing a deterministic algorithm for USTCON running in logarithmic space, for which he received the 2005 Grace Murray Hopper Award and (together with Avi Wigderson and Salil Vadhan) the 2009 Gödel Prize. The proof uses the zig-zag product to efficiently construct expander graphs.
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Hinkelmann and Kempthorne (2008, Volume 1, Section 6.10: Completely randomized design; Transformations) Also, a statistician may specify that logarithmic transforms be applied to the responses, which are believed to follow a multiplicative model.Bailey (2008) According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.
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The Shamos–Hoey algorithm Chapter: "Geometric intersection problems" applies this principle to solve the line segment intersection detection problem, as stated above, of determining whether or not a set of line segments has an intersection; the Bentley–Ottmann algorithm works by the same principle to list all intersections in logarithmic time per intersection.
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The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions,Katz, V. J. 1995. "Ideas of Calculus in Islam and India." Mathematics Magazine (Mathematical Association of America), 68(3):163-174. though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.
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In addition, these subpatterns only need to be evaluated once, not once per copy as in other Life algorithms. This itself leads to significant improvements in resource requirements; for example a generation of the various breeders and spacefillers, which grow at polynomial speeds, can be evaluated in Hashlife using logarithmic space and time.
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The phase axis is in either degrees or radians. The frequency axes are in a [logarithmic scale]. These are useful because for sinusoidal inputs, the output is the input multiplied by the value of the magnitude plot at the frequency and shifted by the value of the phase plot at the frequency.
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Associated with such singularities, renormalon contributions are discussed in the context of quantum chromodynamics (QCD) and usually have the power-like form \left(\Lambda/Q\right)^p as functions of the momentum Q (here \Lambda is the momentum cut-off). They are cited against the usual logarithmic effects like \ln\left(\Lambda/Q\right).
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At the test frequency each element or S-parameter is represented by a unitless complex number that represents magnitude and angle, i.e. amplitude and phase. The complex number may either be expressed in rectangular form or, more commonly, in polar form. The S-parameter magnitude may be expressed in linear form or logarithmic form.
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Logarithmic amplifiers are used in many ways, such as: # To perform mathematical operations like multiplication, division and exponentiation. Multiplication is also sometimes called mixing. This is similar to operation of a slide rule, and is used in analog computers, audio synthesis methods, and some measurement instruments (i.e. power as multiplication of current and voltage).
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At equivalent molecular weights, RNA will migrate faster than DNA. However, both RNA and DNA have a negative linear slope between their migration distance and logarithmic molecular weight. That is, samples of less weight are able to migrate a greater distance. This relationship is a consideration when choosing RNA or DNA markers as a standard.
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In arithmetic combinatorics, Behrend's theorem states that the subsets of the integers from 1 to n in which no member of the set is a multiple of any other must have a logarithmic density that goes to zero as n becomes large. The theorem is named after Felix Behrend, who published it in 1935.
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Animal Behaviour 109: 133–140. doi:10.1016/j.anbehav.2015.08.013 The relationship between the two measured quantities is often expressed as a power law equation which expresses a remarkable scale symmetry: : y = kx^a \,\\! or in a logarithmic form: : \log y = a \log x + \log k\,\\! where a is the scaling exponent of the law.
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For the majority of problems in statistical physics one can hardly solve the partition function due to the enormous amount of particles and degrees of freedoms. This is different in KTHNY theory due to the logarithmic energy functions of dislocations H_{loc} and the e-function from the Boltzmann factor as inverse which can be solved easily.
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Peak signal-to-noise ratio, often abbreviated PSNR, is an engineering term for the ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation. Because many signals have a very wide dynamic range, PSNR is usually expressed in terms of the logarithmic decibel scale.
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Fluids 27, 025112. C. VerHulst & C. Meneveau: “Altering kinetic energy entrainment in LES of large wind farms using unconventional wind turbine actuator forcing” (2015), Energies 8, 370-386. R.J.A.M. Stevens, M. Wilczek & C. Meneveau, “Large-eddy simulation study of the logarithmic law for second and higher-order moments in turbulent wall-bounded flow” (2014), J. Fluid Mech.
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Human hearing is directly sensitive to sound pressure which is related to sound intensity. In consumer audio electronics, the level differences are called "intensity" differences, but sound intensity is a specifically defined quantity and cannot be sensed by a simple microphone. Sound intensity level is a logarithmic expression of sound intensity relative to a reference intensity.
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The run-time of the algorithm described above is polynomial in the triangulation size. This is considered bad, since the triangulations might be very large. It would be desirable to find an algorithm which is logarithmic in the triangulation size. However, the problem of finding a complementary edge is PPA-complete even for n=2 dimensions.
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Approximate and true golden spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of one square divided by that of the next smaller square is the golden ratio.
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Simply rescaling units (e.g., to thousand square kilometers, or to millions of people) will not change this. However, following logarithmic transformations of both area and population, the points will be spread more uniformly in the graph. Another reason for applying data transformation is to improve interpretability, even if no formal statistical analysis or visualization is to be performed.
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The correlation between teledensity and per capita GDP could be represented by a straight line in a logarithmic graph. This relation was first mentioned by A.G.W. Jipp. a German engineer, in his book published in 1962. The graph is helpful to compare the telephone infrastructure development of different countries or regions, on the basis of teledensity.
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This gives rise to a logarithmic spiral. Benford's law on the distribution of leading digits can also be explained by scale invariance., chapter 6, section 64 Logarithms are also linked to self-similarity. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions.
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The book contains a double scale, logarithmic on one side, tabular on the other. In 1630, William Oughtred of Cambridge invented a circular slide rule, and in 1632 combined two handheld Gunter rules to make a device that is recognizably the modern slide rule. Like his contemporary at Cambridge, Isaac Newton, Oughtred taught his ideas privately to his students.
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OMIR exchange rates 1 Jan 2001 to 2 Feb 2009. Note the logarithmic scale. Prices in shops and restaurants were still quoted in Zimbabwean dollars, but were adjusted several times a day. Any Zimbabwean dollars acquired needed to be exchanged for foreign currency on the parallel market immediately, or the holder would suffer a significant loss of value.
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Jostel's TSH index (TSHI or JTI), also referred to as Jostel's thyrotropin index or Thyroid Function index (TFI) is a method for estimating the thyrotropic (i.e. thyroid stimulating) function of the anterior pituitary lobe in a quantitative way. The equation has been derived from the logarithmic standard model of thyroid homeostasis.Cohen, J. L., Thyroid-stimulation hormone and its disorders.
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Number of cases (blue) and number of deaths (red) on a logarithmic scale As of March 16, President López Obrador continued to downplay the impact of coronavirus. "Pandemics ... won't do anything to us," and accused the press and the opposition for its reportage."As Mexican peso collapses over coronavirus threat, criticism falls on president Lopez Obrador". Los Angeles Times.
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The logarithmic spiral continually appears in nature, such as with the curves of the Nautilus shell.Eli Maor, E: Story of a Number ( Princeton University Press, 2009: ), p. 127. The College of St Hild and St Bede at the University of Durham adopted this phrase for its signatory logo.The College of St Hild and St Bede (pdf).
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Graph showing ratio of the prime-counting function to two of its approximations, and . As increases (note axis is logarithmic), both ratios tend towards 1. The ratio for converges from above very slowly, while the ratio for converges more quickly from below. Log-log plot showing absolute error of and , two approximations to the prime-counting function .
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Modified duration can be extended to instruments with non-fixed cash flows, while Macaulay duration applies only to fixed cash flow instruments. Modified duration is defined as the logarithmic derivative of price with respect to yield, and such a definition will apply to instruments that depend on yields, whether or not the cash flows are fixed.
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Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base e) to transform products into sums and divisions into subtractions. The principle can be implemented, at least in part, in the differentiation of almost all differentiable functions, providing that these functions are non-zero.
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The underside is almost completely white, making the animal indistinguishable from brighter waters near the surface. This mode of camouflage is called countershading. The nautilus shell presents one of the finest natural examples of a logarithmic spiral, although it is not a golden spiral. The use of nautilus shells in art and literature is covered at nautilus shell.
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In fact, happiness . Common market health measures such as GDP and GNP have been used as a measure of successful policy. On average richer nations tend to be happier than poorer nations, but this effect seems to diminish with wealth. This has been explained by the fact that the dependency is not linear but logarithmic, i.e.
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201:457, 1953. Unlike the case with drugs, the initial amount of tissue or tissue protein is not zero because daily synthesis offsets daily elimination. In this case, the model is also said to approach a steady state with exponential or logarithmic kinetics. Constituents that change in this manner are said to have a biological half- life.
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Non-linear element. Besides impedance, Miller arrangement can modify the IV characteristic of an arbitrary element. The circuit of a diode log converter is an example of a non-linear virtually zeroed resistance where the logarithmic forward IV curve of a diode is transformed to a vertical straight line overlapping the Y axis. Not constant coefficient.
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The brightness of a star, as seen from Earth, is measured with a standardized, logarithmic scale. This apparent magnitude is a numerical value that decreases in value with increasing brightness of the star. The faintest stars visible to the unaided eye are sixth magnitude, while the brightest in the night sky, Sirius, is of magnitude −1.46.
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The logarithmic solubility product Ksp for leonite is −9.562 at 25 °C. table 7 on page 716 The equilibrium constant log K at 25 °C is −3.979. The chemical potential of leonite is μj°/RT = −1403.97. Thermodynamic properties include ΔfGok = −3480.79 kJ mol−1; ΔfHok = −3942.55 kJ mol−1; and ΔCop,k = 191.32 J K−1 mol−1.
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The semicubical parabola was discovered in 1657 by William Neile who computed its arc length. Although the lengths of some other non-algebraic curves including the logarithmic spiral and cycloid had already been computed (that is, those curves had been rectified), the semicubical parabola was the first algebraic curve (excluding the line and circle) to be rectified.
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After continuing this process for an arbitrary number of steps, the result will be an almost complete partitioning of the rectangle into squares. The corners of these squares can be connected by quarter-circles. The result, though not a true logarithmic spiral, closely approximates a golden spiral. Another approximation is a Fibonacci spiral, which is constructed slightly differently.
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In the asymmetric case with triangle inequality, only logarithmic performance guarantees are known, the best current algorithm achieves performance ratio 0.814 log(n); it is an open question if a constant factor approximation exists. The best known inapproximability bound is 75/74. The corresponding maximization problem of finding the longest travelling salesman tour is approximable within 63/38.
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'K' is the quasi-logarithmic counterpart of the 'a' index. Kp and ap are the average of K and a over 13 geomagnetic observatories to represent planetary- wide geomagnetic disturbances. The Kp/ap indexHelmholtz Centre PotsdamGFZ German Research Centre for Geosciences indicates both geomagnetic storms and substorms (auroral disturbance). Kp/ap is available from 1932 onward.
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Cell populations detected by flow cytometry are often described as having approximately log-normal expression. As such, they have traditionally been transformed to a logarithmic scale. In early cytometers, this was often accomplished even before data acquisition by use of a log amplifier. On modern instruments, data is usually stored in linear form, and transformed digitally prior to analysis.
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Sound is measured based on the amplitude and frequency of a sound wave. Amplitude measures how forceful the wave is. The energy in a sound wave is measured in decibels (dB), the measure of loudness, or intensity of a sound; this measurement describes the amplitude of a sound wave. Decibels are expressed in a logarithmic scale.
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The number of binary trees with n nodes is a Catalan number: for these numbers of trees are : . Thus, if one of these trees is selected uniformly at random, its probability is the reciprocal of a Catalan number. Trees in this model have expected depth proportional to the square root of , rather than to the logarithm;, p. 15. however, the Strahler number of a uniformly random binary tree, a more sensitive measure of the distance from a leaf in which a node has Strahler number whenever it has either a child with that number or two children with number , is with high probability logarithmic.. That it is at most logarithmic is trivial, because the Strahler number of every tree is bounded by the logarithm of the number of its nodes.
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Hybrid Log-Gamma (HLG) is a royalty-free HDR standard jointly developed by the BBC and NHK. HLG is designed to be better-suited for television broadcasting, where the metadata required for other HDR formats is not backward compatible with non-HDR displays, consumes additional bandwidth, and may also become out-of-sync or damaged in transmission. HLG defines a non- linear optical-electro transfer function, in which the lower half of the signal values use a gamma curve and the upper half of the signal values use a logarithmic curve. In practice, the signal is interpreted as normal by standard-dynamic-range displays (albeit capable of displaying more detail in highlights), but HLG-compatible displays can correctly interpret the logarithmic portion of the signal curve to provide a wider dynamic range.
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This counter translates into a semi-exponential growth in newly-created enemies' attack strength, a semi-logarithmic growth in these enemies' health points, and a logarithmic growth in the rate that the player's health increases with character level. Drummond and Morse found this created favorable gameplay that created moments of "highs and lows" and keep the player on edge, having times where the player may feel overpowered to the enemies and moments later find themselves in a struggle to stay alive. Another mechanic they had explored with the "difficulty = time" approach was to incorporate the speed at which the player defeated enemies into the difficulty counter, but found this removed the "highs and lows" in the game. Another element of the difficulty approach was the rate which enemies are generated.
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For example, representing an m × n array as a single list of length m·n, together with the numbers m and n (instead of as a 1-dimensional array of pointers to each 1-dimensional subarray). The elements need not be of the same type, and a table of data (a list of records) may similarly be represented implicitly as a flat (1-dimensional) list, together with the length of each field, so long as each field has uniform size (so a single size can be used per field, not per record). A less trivial example is representing a sorted list by a sorted array, which allows search in logarithmic time by binary search. Contrast with a search tree, specifically a binary search tree, which also allows logarithmic-time search, but requires pointers.
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When rectified, the curve gives a straight line segment with the same length as the curve's arc length. Arc length s of a logarithmic spiral as a function of its parameter θ. Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment is also called of a curve.
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Furthermore, the species discovery curve is also decelerating; the more samples taken, the fewer unseen species are expected to be discovered. The species discovery curve will also never asymptote, as it is assumed that although the discovery rate might become infinitely slow, it will never actually stop. Two common models for a species discovery curve are the logarithmic and the exponential function.
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In cylindrical coordinates, the conchospiral is described by the parametric equations: :r=\mu^t a :\theta=t :z=\mu^t c. The projection of a conchospiral on the (r,\theta) plane is a logarithmic spiral. The parameter \mu controls the opening angle of the projected spiral, while the parameter c controls the slope of the cone on which the curve lies.
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Also, as increases, the incremental warming is less, as the effect is logarithmic so the more , the less warming it produces. # has been totally uncorrelated with temperature over the last decade, and significantly negative since 2002. # is not a pollutant, but a naturally occurring gas. Together with chlorophyll and sunlight, it is an essential ingredient in photosynthesis and is, accordingly, plant food.
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The VIX In finance, volatility (usually denoted by σ) is the degree of variation of a trading price series over time, usually measured by the standard deviation of logarithmic returns. Historic volatility measures a time series of past market prices. Implied volatility looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option).
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Summatory Liouville function L(n) up to n = 104. The readily visible oscillations are due to the first non-trivial zero of the Riemann zeta function. Summatory Liouville function L(n) up to n = 107. Note the apparent scale invariance of the oscillations. Logarithmic graph of the negative of the summatory Liouville function L(n) up to n = 2 × 109.
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The ratio between t and T is the geometric index of reduction. In theory this ratio shall range between 0 and 1. The bigger the number is the larger amount of lost weight from lithic flake. By using a logarithmic scale, a linear relationship between the geometric index and the percentage of original flake weight lost through retouch is confirmed.
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However, the strong and sharp three exponentials conjectures are implied by their four exponentials counterparts, bucking the usual trend. And the three exponentials conjecture is neither implied by nor implies the four exponentials conjecture. The three exponentials conjecture, like the sharp five exponentials conjecture, would imply the transcendence of eπ² by letting (in the logarithmic version) λ1 = iπ, λ2 = −iπ, and γ = 1\.
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Moreover, there is only one solution to this equation, because the function f is strictly increasing (for ), or strictly decreasing (for ). The unique solution is the logarithm of to base , . The function that assigns to its logarithm is called logarithm function or logarithmic function (or just logarithm). The function is essentially characterized by the product formula :\log_b(xy) = \log_b x + \log_b y.
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Landau kept a list of names of physicists which he ranked on a logarithmic scale of productivity ranging from 0 to 5. The highest ranking, 0, was assigned to Isaac Newton. Albert Einstein was ranked 0.5. A rank of 1 was awarded to the founding fathers of quantum mechanics, Niels Bohr, Werner Heisenberg, Satyen Bose, Paul Dirac and Erwin Schrödinger, and others.
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Commission One of the International Society of Soil Science (ISSS) recommended its use at the first International Congress of Soil Science in Washington in 1927.Davis ROE, Bennett HH (1927) "Grouping of soils on the basis of mechanical analysis." United States Department of Agriculture Departmental Circulation No. 419. Australia adopted this system, and its equal logarithmic intervals are an attractive feature worth maintaining.
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Power spectrum of the Sun using data from instruments aboard the Solar and Heliospheric Observatory on double-logarithmic axes. The three passbands of the VIRGO/SPM instrument show nearly the same power spectrum. The line-of-sight velocity observations from GOLF are less sensitive to the red noise produced by granulation. All the datasets clearly show the oscillation modes around 3mHz.
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In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass. The relation between the sigma, zeta, and \wp functions is analogous to that between the sine, cotangent, and squared cosecant functions: the logarithmic derivative of the sine is the cotangent, whose derivative is negative the squared cosecant.
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The double-logarithmic SNR curve [2b] is a nice overall graphical representation of all camera performance parameters except for the dark current. The absolute sensitivity threshold is marked as well as the saturation capacity. In addition, the maximum signal-to-noise ratio and the dynamic range can be read from the graph. The total SNR is plotted as a dashed line.
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RFC 2627 describes one Group Membership Management protocol that allows selective key updates to members to efficiently remove a member from the group. "Efficiency" is evaluated in terms of space, time and message complexity. RFC 2627 and other algorithms such as "subset-difference" are logarithmic in space, time and message complexity. Thus, RFC 2627 supports efficient group "membership management" for GDOI.
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The Earth's orbit approximates an ellipse. Eccentricity measures the departure of this ellipse from circularity. The shape of the Earth's orbit varies between nearly circular (with the lowest eccentricity of 0.000055) and mildly elliptical (highest eccentricity of 0.0679). Its geometric or logarithmic mean is 0.0019. The major component of these variations occurs with a period of 413,000 years (eccentricity variation of ±0.012).
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Note that right ascension, as used by astronomers, is 360° minus the sidereal hour angle. The final characteristic provided in the tables and star charts is the star's brightness, expressed in terms of apparent magnitude. Magnitude is a logarithmic scale of brightness, designed so that a body of one magnitude is approximately 2.512 times brighter than a body of the next magnitude.
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In Scandinavia a variant of the DIN PPM known as 'Nordic' is used. It has the same integration and return times but a different scale, with 'TEST' corresponding to Alignment Level (0 dBu) and +9 corresponding to Permitted Maximum Level (+9 dBu). Compared to the DIN scale, the Nordic scale is more logarithmic and covers a somewhat smaller dynamic range.
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This was the first meter with white markings on a black background. It was driven by a circuit that gave a roughly logarithmic transfer characteristic, so it could be calibrated in decibels. The overall characteristics were the product of the driver circuit and the movement's ballistics. The first of the PPMs was designed by C. G. Mayo, also of the BBC's Research Department.
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"EL" stands both for "Exponential-Logarithmic" and as an abbreviation for "elementary". Whether a number is a closed-form number is related to whether a number is transcendental. Formally, Liouville numbers and elementary numbers contain the algebraic numbers, and they include some but not all transcendental numbers. In contrast, EL numbers do not contain all algebraic numbers, but do include some transcendental numbers.
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This can be of use in fitting distributions to empirical data. However, some further well-known distributions are available if the recursion above need only hold for a restricted range of values of k: for example the logarithmic distribution and the discrete uniform distribution. The (a, b, 0) class of distributions has important applications in actuarial science in the context of loss models.
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To fit a model to one or more data sets, the corresponding model-data-couples are combined into a fitting-assembly. Parameters like initial values, rate constants, and scaling factors can be fitted in an experiment-wise or global fashion. The user may select from several numerical integrators, optimization algorithms, and calibration strategies like fitting in normal or logarithmic parameter space.
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The system contains both a leaf set of neighbor nodes, which provides fault tolerance and a probabilistic invariant of constant routing progress, and a PRR-style routing table to improve routing time to a logarithmic factor of network size. Chimera is currently being used in industry labs, as part of research done by the U.S. Department of Defense, and by startup companies.
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The trains of pulses were passed through quasi-logarithmic data processors and then to the radio telemetry system of the spacecraft. Angular distributions were measured as the spacecraft rotated. This telemetry data was transmitted to the earth by an 8 watt S band transmitter within the Pioneer probe at one of eight data rates (from 16 to 2048 bits per second).
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The translog production function is an approximation of the CES function by a second-order Taylor polynomial about \gamma = 0, i.e. the Cobb–Douglas case. The name translog stands for 'transcendental logarithmic'. It is often used in econometrics for the fact that it is linear in the parameters, which means ordinary least squares could be used if inputs could be assumed exogenous.
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This image compares the results from Dynamical Energy Analysis (DEA) with that of frequency averaged FEM. Shown is the kinetic energy distribution resulting from a point excitation on a carfloor panel on a logarithmic color scale. As an example application, a simulation of a carfloor panel is shown here. A point excitation at 2500 Hz with 0.04 hysteretic damping was applied.
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The table below is a brief chronology of computed numerical values of, or bounds on, the mathematical constant pi (). For more detailed explanations for some of these calculations, see Approximations of . Graph showing how the record precision of numerical approximations to pi measured in decimal places (depicted on a logarithmic scale), evolved in human history. The time before 1400 is compressed.
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Manufacturers of transistors and integrated circuits often divide their product lines into 'linear' and 'digital' lines. "Linear" here means "analog"; the linear line includes integrated circuits designed to process signals linearly, such as op-amps, audio amplifiers, and active filters, as well as a variety of signal processing circuits that implement nonlinear analog functions such as logarithmic amplifiers, analog multipliers, and peak detectors.
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The shape of the central dense overcast is also considered. The farther the center is tucked into the CDO, the stronger it is deemed. Banding features can be utilized to objectively determine the tropical cyclone's center, using a ten degree logarithmic spiral. Using the 85–92 GHz channels of polar-orbiting microwave satellite imagery can definitively locate the center within the CDO.
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Stafford Beer defines variety as "the total number of possible states of a system, or of an element of a system",Beer (1981) cf. Ludwig Boltzmann's Wahrscheinlichkeit. Beer restates the Law of Requisite Variety as "Variety absorbs variety".Beer (1979) p286 Stated more simply the logarithmic measure of variety represents the minimum number of choices (by binary chop) needed to resolve uncertainty.
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In SI units, luminosity is measured in joules per second, or watts. In astronomy, values for luminosity are often given in the terms of the luminosity of the Sun, L⊙. Luminosity can also be given in terms of the astronomical magnitude system: the absolute bolometric magnitude (Mbol) of an object is a logarithmic measure of its total energy emission rate, while absolute magnitude is a logarithmic measure of the luminosity within some specific wavelength range or filter band. In contrast, the term brightness in astronomy is generally used to refer to an object's apparent brightness: that is, how bright an object appears to an observer. Apparent brightness depends on both the luminosity of the object and the distance between the object and observer, and also on any absorption of light along the path from object to observer.
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The primary calculation in GRAPE hardware is a summation of the forces between a particular star and every other star in the simulation. Several versions (GRAPE-1, GRAPE-3 and GRAPE-5) use the Logarithmic Number System (LNS) in the pipeline to calculate the approximate force between two stars, and take the antilogarithms of the x, y and z components before adding them to their corresponding total. The GRAPE-2, GRAPE-4 and GRAPE-6 use floating point arithmetic for more accurate calculation of such forces. The advantage of the logarithmic-arithmetic versions is they allow more and faster parallel pipes for a given hardware cost because all but the sum portion of the GRAPE algorithm (1.5 power of the sum of the squares of the input data divided by the input data) is easy to perform with LNS.
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A linear chart of the S&P; 500 daily closing values from January 3, 1950 to February 19, 2016 A logarithmic chart of the S&P; 500 index daily closing values from January 3, 1950 to February 19, 2016. A daily volume chart of the S&P; 500 index from January 3, 1950 to February 19, 2016 Logarithmic graphs of S&P; 500 index with and without inflation and with best fit lines The S&P; 500, or simply the S&P;, is a stock market index that measures the stock performance of 500 large companies listed on stock exchanges in the United States. It is one of the most commonly followed equity indices. The S&P; 500 index is a capitalization-weighted index and the 10 largest companies in the index account for 26% of the market capitalization of the index.
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This is called the Distribution Stochastic Shortest Path Problem (d-SSPPR or R-SSPPR) and is NP-complete. The first variant is harder than the second because the former can represent in logarithmic space some distributions that the latter represents in linear space. The fourth and final parameter is how the graph changes over time. In CTP and SSPPR, the realization is fixed but not known.
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The quantitative representation of the photon structure function introduced above is strictly valid only for asymptotically high resolution , i.e. the logarithm of being much larger than the logarithm of the quark masses. However, the asymptotic behavior is approached steadily with increasing for away from zero as demonstrated next. In this asymptotic regime the photon structure function is predicted uniquely in QCD to logarithmic accuracy.
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In computer science, a skew binomial heap (or skew binomial queue) is a variant of the binomial heap that supports constant-time insertion operations in the worst case, rather than the logarithmic worst case and constant amortized time of the original binomial heap. Just as binomial heaps are based on the binary number system, skew binary heaps are based on the skew binary number system.
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During the late 1930s he developed and exhibited a style of painting based on a logarithmic form of anamorphic projection which he called "siderealism". This work appears to have been well received. In 1947 he exhibited at the Archer Gallery, producing over 200 works for the show. It was a very successful show and led to something of a post-war renaissance of interest.
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In 1966 and 1967, Liston developed a Digital- Alpha circuit to calculate the logarithmic decay of a capacitor-resistor. The circuit was patented by Liston Scientific in 1967. It was used in bichromatic analyzers which Liston designed for Abbott Laboratories, beginning with the ABA-100. The ABA-100 was a single-reagent double-channel kinetic analyzer for ultra-micro chemical analysis and simultaneous bichromatic spectrophotometry.
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The scale does not have to be straight; it can be curved, as on a protractor, but the graduations must be spaced at a constant length apart. Graduations can also be spaced at non-linear intervals, such as logarithmic, where the interval between them varies. Volumetric graduations, for instance on a measuring cup, can vary in scale due to the container's conical or abstract shape.
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Since the common logarithm of a power of is exactly the exponent, the characteristic is an integer number, which makes the common logarithm exceptionally useful in dealing with decimal numbers. For numbers less than the characteristic makes the resulting logarithm negative, as required.E. R. Hedrick, Logarithmic and Trigonometric Tables (Macmillan, New York, 1913). See common logarithm for details on the use of characteristics and mantissas.
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The game system in DC Heroes is sometimes called the Mayfair Exponential Game System (or MEGS). DC Heroes uses a logarithmic scale for character attributes. For example, a value of 3 is double a value of 2 and four times a value of 1. The scale allows characters of wildly different power levels to co-exist within the same game without one completely dominating a given area.
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The 2D histogram of SDSS SFGs is shown in logarithmic scale and their best likelihood fit is shown by a black solid line. The subset of 62 GPs are indicated by circles and their best linear fit is shown by a dashed line. For comparison we also show the quadratic fit presented in Amorin et al. 2010 for the full sample of 80 GPs.
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The spectral series of hydrogen, on a logarithmic scale. The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. These observed spectral lines are due to the electron making transitions between two energy levels in an atom. The classification of the series by the Rydberg formula was important in the development of quantum mechanics.
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A Method of Determining the Regression Curve When the Marginal Distribution is of the Normal Logarithmic Type, Annals of Mathematical Statistics, 7:196-201, 1936. Second moment and of the Correlation Coefficient in Samples from Populations of Type A, The Statistical Institute at the University of Lund. Lund, C. W. K. Gleerup/Leipzig, Otto Harrassowitz, 1938. Lärobok i den teoretiska statistikens grunder, Lund 1944.
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Rao is a prolific scientific researcher and author of hundreds of peer reviewed international journal articles. Several research articles have been cited extensively throughout the world scientific community. Most of his research work has been focused on Chromatin biology and Cancer biology. Rao has an h-index of above 26 (as per June 2011) and it's still in logarithmic phase and heading up continuously.
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Located south of Umina Beach, being separated from it by a ridge upon which sits Mount Ettalong at a height of . It is bounded on the west and south by Brisbane Water National Park, and on the east by Broken Bay. Green Point, with Paul Landa Reserve, adjoins the southern end of the beach. The bay provides an example of a logarithmic spiral beach.
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Size scaled 10k and 100k pots that combine traditional mountings and knob shafts with newer and smaller electrical assemblies. The "B" designates a linear (USA/Asian style) taper. The relationship between slider position and resistance, known as the "taper" or "law", is controlled by the manufacturer. In principle any relationship is possible, but for most purposes linear or logarithmic (aka "audio taper") potentiometers are sufficient.
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This pathway involves exchange of information between a damaged chromosome and another homologous chromosome in the same cell. It depends on the RecA protein that catalyzes strand exchange and the ADN protein that acts as a presynaptic nuclease. HR is an accurate repair process and is the preferred pathway during logarithmic growth. The NHEJ pathway for repairing double-strand breaks involves the rejoining of the broken ends.
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Kayamar has a so-called logarithmic or Hertz-pitch better than the absolute pitch that is widely considered to be the highest level of musical hearing. He is able not just to identify and re-create musical notes without using a reference but he can tell how many Hertz he hears or sings. This way Kayamar is able to use much smaller intervals than quarter tones.
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However, these tests generally include a control (negative and/or positive), a geometric dilution series or other appropriate logarithmic dilution series, test chambers and equal numbers of replicates, and a test organism. Exact exposure time and test duration will depend on type of test (acute vs. chronic) and organism type. Temperature, water quality parameters and light will depend on regulator requirements and organism type.
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See also for various other examples in degree 5. Évariste Galois introduced a criterion allowing one to decide which equations are solvable in radicals. See Radical extension for the precise formulation of his result. Algebraic solutions form a subset of closed-form expressions, because the latter permit transcendental functions (non-algebraic functions) such as the exponential function, the logarithmic function, and the trigonometric functions and their inverses.
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The numerical renormalization group is an iterative procedure, which is an example of a renormalization group technique. The technique consists of first dividing the conduction band into logarithmic intervals (i.e. intervals which get smaller exponentially as you move closer to the Fermi energy). One conduction band state from each interval is retained, this being the totally symmetric combination of all the states in that interval.
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In many cases, a statistician may specify that logarithmic transforms be applied to the responses, which are believed to follow a multiplicative model. Pre-publication chapters are available on-line. The assumption of unit treatment additivity was enunciated in experimental design by Kempthorne and Cox. Kempthorne's use of unit treatment additivity and randomization is similar to the design-based analysis of finite population survey sampling.
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The splicing of percussive samples results in a more attention-grabbing sound than it would with a single sustained pitch. Stutters also often reduce notes within bars, beginning with 32nd notes, then reducing to 64th and 128th or something similar. There are instances of stutter edits that use logarithmic curves rather than relying on musically locked timings giving the impression of a "speed up" or "slow down".
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Since sound attenuates at a set rate, extremely high outputs are required to cover the vast distances needed. Acoustic hailing devices have an output of 135 decibels (dB) or greater. The acoustic level of the source is commonly expressed in terms of Sound Pressure Level or SPL. SPL is a logarithmic measure of the rms sound pressure of a sound relative to a reference value.
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Statistical inference might be thought of as gambling theory applied to the world around us. The myriad applications for logarithmic information measures tell us precisely how to take the best guess in the face of partial information.Jaynes, E.T. (1998/2003) Probability Theory: The Logic of Science (Cambridge U. Press, New York). In that sense, information theory might be considered a formal expression of the theory of gambling.
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Göttingen 1863. In 1838 Peter Gustav Lejeune Dirichlet came up with his own approximating function, the logarithmic integral (under the slightly different form of a series, which he communicated to Gauss). Both Legendre's and Dirichlet's formulas imply the same conjectured asymptotic equivalence of and stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.
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This calculation uses a logarithmic function. It does so because the incremental increase in the skill set of the field theoretically diminishes as the buy-in amount increases. For example, in the GPI formula the percentage increase of GPI's Buy-In Factor between a $1500 event and an $2000 event is much greater than its percentage increase between $19,500 and a $20,000 buy-in events.
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A logarithmic equation is an equation of the form log_a(x) = b for a > 0, which has solution : X = a^b. For example, if : 4\log_5(x - 3) - 2 = 6 then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get : \log_5(x - 3) = 2 whence : x - 3 = 5^2 = 25 from which we obtain : x = 28.
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Moog established standards for control interfacing, using a logarithmic 1-volt-per-octave for pitch control and a separate triggering signal. This standardization allowed synthesizers from different manufacturers to operate simultaneously. Pitch control was usually performed either with an organ-style keyboard or a music sequencer producing a timed series of control voltages. During the late 1960s hundreds of popular recordings used Moog synthesizers.
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248 (1961) One definition of the term is "...a frequency scale on which equal distances correspond with perceptually equal distances. Above about 500 Hz this scale is more or less equal to a logarithmic frequency axis. Below 500 Hz the Bark scale becomes more and more linear." The scale ranges from 1 to 24 and corresponds to the first 24 critical bands of hearing.
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Martin Wiberg improved Scheutz's construction (c. 1859, his machine has the same capacity as Scheutz's - 15-digit and fourth-order) but used his device only for producing and publishing printed tables (interest tables in 1860, and logarithmic tables in 1875).Raymond Clare Archibald: Martin Wiberg, his Table and Difference Engine, Mathematical Tables and Other Aids to Computation, 1947(2:20) 371–374. (online review) (PDF; 561 kB).
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The logarithmic scale nature of the decibel means that a very large range of ratios can be represented by a convenient number, in a manner similar to scientific notation. This allows one to clearly visualize huge changes of some quantity. See Bode plot and Semi-log plot. For example, 120 dB SPL may be clearer than "a trillion times more intense than the threshold of hearing".
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Just as P and FP are closely related, NP is closely related to FNP. Because a machine that uses logarithmic space has at most polynomially many configurations, FL, the set of function problems which can be calculated in logspace, is contained in FP. It is not known whether FL = FP; this is analogous to the problem of determining whether the decision classes P and L are equal.
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Competence in B. subtilis is induced toward the end of logarithmic growth, especially under conditions of amino acid limitation. Similarly, in Micrococcus luteus (a representative of the less well studied Actinobacteria phylum), competence develops during the mid-late exponential growth phase and is also triggered by amino acids starvation. By releasing intact host and plasmid DNA, certain bacteriophages are thought to contribute to transformation.
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Since the unit is logarithmic on a base of 2, for beauty to increase by 1 Ha, a woman must be the most beautiful of twice as many women, so one helen is equivalent to 25.6 Ha. The most beautiful woman who ever lived would score 34.2 Ha, the equivalent of 1.34 helen, the pick of a dozen women would be 3.6 Ha, or 0.14 helen.
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For a number written in scientific notation, this logarithmic rounding scale requires rounding up to the next power of ten when the multiplier is greater than the square root of ten (about 3.162). For example, the nearest order of magnitude for is 8, whereas the nearest order of magnitude for is 9. An order-of- magnitude estimate is sometimes also called a zeroth order approximation.
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64 kbit/s CVSD is one of the options to encode voice signals in telephony-related Bluetooth service profiles; e.g., between mobile phones and wireless headsets. The other options are PCM with logarithmic a-law or μ-law quantization. Numerous arcade games, such as Sinistar and Smash TV, and pinball machines, such as Gorgar or Space Shuttle, play pre-recorded speech through an HC-55516 CVSD decoder.
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He wrote the Henderson equation in 1908 to describe the use of carbonic acid as a buffer solution. Karl Albert Hasselbalch later expressed the equation in logarithmic terms, creating the Henderson–Hasselbalch equation. In addition, he described blood gas transport and the general physiology of blood as physico-chemical system (1920–1932). He invented and constructed new charts, nomograms, with the help of Maurice d'Ocagne.
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The backscattering method is also employed in fiber optics applications to detect optical faults. Light propagating through a fiber optic cable gradually attenuates due to Rayleigh scattering. Faults are thus detected by monitoring the variation of part of the Rayleigh backscattered light. Since the backscattered light attenuates exponentially as it travels along the optical fiber cable, the attenuation characteristic is represented in a logarithmic scale graph.
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The estimate is a specific value of a functional approximation to f(x) = over the interval. Obtaining a better estimate involves either obtaining tighter bounds on the interval, or finding a better functional approximation to f(x). The latter usually means using a higher order polynomial in the approximation, though not all approximations are polynomial. Common methods of estimating include scalar, linear, hyperbolic and logarithmic.
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The log wind profile is a semi-empirical relationship commonly used to describe the vertical distribution of horizontal mean wind speeds within the lowest portion of the planetary boundary layer. The relationship is well described in the literature. The logarithmic profile of wind speeds is generally limited to the lowest 100 m of the atmosphere (i.e., the surface layer of the atmospheric boundary layer).
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Insoluble solids, such as granite, diamond, and platinum, are diluted by grinding them with lactose ("trituration"). Fluxion, which dilutes the substance by continuously passing water through the vial, and radionic preparation methods of preparation do not require succussion. Three main logarithmic dilution scales are in regular use in homeopathy. Hahnemann created the "centesimal" or "C scale", diluting a substance by a factor of 100 at each stage.
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Tequila Fermentation Vessel in City of Tequila Museum Organoleptic compounds enhance flavor and aroma. These include fusel oil, methanol, aldehydes, organic acids and esters. Production of isoamyl and isobutyl alcohols begins after the sugar level is lowered substantially and continues for several hours after the alcoholic fermentation ends. In contrast, ethanol production begins in the first hours of the fermentation and ends with logarithmic yeast growth.
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A skip graph is a distributed data structure based on skip lists designed to resemble a balanced search tree. They are one of several methods to implement a distributed hash table, which are used to locate resources stored in different locations across a network, given the name (or key) of the resource. Skip graphs offer several benefits over other distributed hash table schemes such as Chord (peer-to-peer) and Tapestry (DHT), including addition and deletion in expected logarithmic time, logarithmic space per resource to store indexing information, no required knowledge of the number of nodes in a set and support for complex range queries. A major distinction from Chord and Tapestry is that there is no hashing of search keys of resources, which allows related resources to be near each other in the skip graph; this property makes searches for values within a given range feasible.
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Original image of a logistic curve, contrasted with a logarithmic curve The logistic function was introduced in a series of three papers by Pierre François Verhulst between 1838 and 1847, who devised it as a model of population growth by adjusting the exponential growth model, under the guidance of Adolphe Quetelet. Verhulst first devised the function in the mid 1830s, publishing a brief note in 1838, then presented an expanded analysis and named the function in 1844 (published 1845); the third paper adjusted the correction term in his model of Belgian population growth. The initial stage of growth is approximately exponential (geometric); then, as saturation begins, the growth slows to linear (arithmetic), and at maturity, growth stops. Verhulst did not explain the choice of the term "logistic" (), but it is presumably in contrast to the logarithmic curve, and by analogy with arithmetic and geometric.
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In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are defined by conjugation: exponentiate the real numbers, obtaining a positive (or zero) number, add or multiply these numbers with the ordinary "linear" operations on real numbers, and then take the logarithm to reverse the initial exponentiation. As usual in tropical analysis, the operations are denoted by ⊕ and ⊗ to distinguish them from the usual addition + and multiplication × (or ⋅). These operations depend on the choice of base for the exponent and logarithm ( is a choice of logarithmic unit), which corresponds to a scale factor, and are well-defined for any positive base other than 1; using a base is equivalent to using a negative sign and using the inverse .
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In a stepped sweep, one variable input parameter (frequency or amplitude) is incremented or decremented in discrete steps. After each change, the analyzer waits until a stable reading is detected before switching to the next step. The scaling of the steps is linear or logarithmic. Since the settling time of different test objects cannot be predicted, the duration of a stepped sweep cannot be determined exactly in advance.
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In 1948, Albert Caquot (1881–1976) and Jean Kerisel (1908–2005) developed an advanced theory that modified Muller-Breslau's equations to account for a non-planar rupture surface. They used a logarithmic spiral to represent the rupture surface instead. This modification is extremely important for passive earth pressure where there is soil-wall friction. Mayniel and Muller-Breslau's equations are unconservative in this situation and are dangerous to apply.
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For example, to find a given word (e.g. the name of a command) in a randomly ordered word list (e.g. a menu), scanning of each word in the list is required, consuming linear time, so Hick's law does not apply. However, if the list is alphabetical and the user knows the name of the command, he or she may be able to use a subdividing strategy that works in logarithmic time.
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Charles Francis and Erik Anderson showed from observations of motions of over 20,000 local stars (within 300 parsecs) that stars do move along spiral arms, and described how mutual gravity between stars causes orbits to align on logarithmic spirals. When the theory is applied to gas, collisions between gas clouds generate the molecular clouds in which new stars form, and evolution towards grand-design bisymmetric spirals is explained.
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8.7: "Logarithmic quantities and units: level, neper, bel", Guide for the Use of the International System of Units (SI) 2008 Edition, NIST Special Publication 811, 2nd printing (November 2008), SP811 PDF. Most sound-level measurements will be made relative to this reference, meaning will equal an SPL of . In other media, such as underwater, a reference level of is used. These references are defined in ANSI S1.1-2013.
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This method is used when more accurate and more uniform grooves are required. The grooves are cut by an electrical diameter cutter, The disc surface is rotated, and the cutter it is steered by a guider ring, so that the spirals have the required logarithmic shape. One disadvantage of this method is that more specialized equipment is required to accurately cut smaller grooves. (approximately 6 cm and less).
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Another way of stating this is that (after logarithmic overhead for the first insertion in a sequence) each successive insert has an amortized time of O(1) (i.e. constant) per insertion. A variant of the binomial heap, the skew binomial heap, achieves constant worst case insertion time by using forests whose tree sizes are based on the skew binary number system rather than on the binary number system.
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Material damping or internal friction is characterized by the decay of the vibration amplitude of the sample in free vibration as the logarithmic decrement. The damping behaviour originates from anelastic processes occurring in a strained solid i.e. thermoelastic damping, magnetic damping, viscous damping, defect damping, ... For example, different materials defects (dislocations, vacancies, ...) can contribute to an increase in the internal friction between the vibrating defects and the neighboring regions.
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In 1838 Peter Gustav Lejeune Dirichlet came up with his own approximating function, the logarithmic integral li(x) (under the slightly different form of a series, which he communicated to Gauss). Both Legendre's and Dirichlet's formulas imply the same conjectured asymptotic equivalence of π(x) and x / ln(x) stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.
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The original of this figure has y axis of the length 8 cm and spans the interval (2.5, 3.8), so if the n axis would be plotted in the linear scale instead of logarithmic, then it should be 5.33(3) \times 10^9 km long — that is the size of the Solar System. The value of M is approximately :M ≈ 0.2614972128476427837554268386086958590516... . Mertens' second theorem establishes that the limit exists.
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Originally portrayed solely in black and white, Broweleit requested that the pieces be portrayed in color, and Logg altered the game accordingly prior to the next Consumer Electronics Show. When asked which version of Tetris he liked the most, Logg stated the Nintendo version of Tetris for the NES "wasn't tuned right", citing a lack of logarithmic speed adjustment as the source of that version's overly steep increases in difficulty.
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As Atneosen already observed, if edges may instead pass from one page to another across the spine of the book, then every graph may be embedded into a three-page book. For such a three-page topological book embedding in which spine crossings are allowed, every graph can be embedded with at most a logarithmic number of spine crossings per edge,. and some graphs need this many spine crossings..
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NIAflow supports sorting by density, color, shape, metal content, etc.NIAflow Feature Operation Mode Crushing, Milling: Crusher and mill products are calculated assuming a linear behavior of the product PSD in either a double logarithmic or RRSB grid. Each type of crushing or grinding machine creates its own specific inclination of the curve. The inclination combined with the maximum particle size leaving the machine is used for product forecast.
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The study of lengths of the lives of organisms, devices, materials, etc., is of major importance in the biological and engineering sciences. In general, the lifetime of a device is expected to exhibit decreasing failure rate (DFR) when its behavior over time is characterized by 'work-hardening' (in engineering terms) or 'immunity' (in biological terms). The exponential-logarithmic model, together with its various properties, are studied by Tahmasbi and Rezaei (2008).
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Another technique that is utilized to reduce or completely eliminate Z-fighting is switching to a logarithmic Z-buffer, reversing Z. This technique is seen in the game Grand Theft Auto V. Due to the way they are encoded, floating point numbers have much more precision when closer to 0. Here, reversing Z leads to more precision when storing the depth of very distant objects, hence greatly reducing Z-fighting.
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His idea was discussed later as the Limiting Factor by BLACKMAN and again by MITSCHERLICH as the Law of Physiological Relations. The latter was expressed as a logarithmic function between yield and the quantity of plant food constituents, which is virtually the Law of Diminishing Returns.Howard S. Reed (1942) A Short History of the Plant Sciences, page 247, Chronica Botanica Company The relation was reviewed by Hans Schneeberger in 2009.
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Jacques Ozanam was born in Sainte- Olive, Ain, France. In 1670, he published trigonometric and logarithmic tables more accurate than the existing ones of Ulacq, Pitiscus, and Briggs. An act of kindness in lending money to two strangers brought him to the attention of M. d'Aguesseau, father of the chancellor, and he secured an invitation to settle in Paris. There he enjoyed prosperity and contentment for many years.
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According to Courcelle's theorem, every fixed MSO2 property can be tested in linear time on graphs of bounded treewidth, and every fixed MSO1 property can be tested in linear time on graphs of bounded clique-width. The version of this result for graphs of bounded treewidth can also be implemented in logarithmic space. Applications of this result include a fixed-parameter tractable algorithm for computing the crossing number of a graph.; .
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On a semi-log plot the spacing of the scale on the y-axis (or x-axis) is proportional to the logarithm of the number, not the number itself. It is equivalent to converting the y values (or x values) to their log, and plotting the data on linear scales. A log-log plot uses the logarithmic scale for both axes, and hence is not a semi-log plot.
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The first phase of growth is the lag phase, a period of slow growth when the cells are adapting to the high- nutrient environment and preparing for fast growth. The lag phase has high biosynthesis rates, as proteins necessary for rapid growth are produced. The second phase of growth is the logarithmic phase, also known as the exponential phase. The log phase is marked by rapid exponential growth.
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Devlin, Keith. Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996 Visual patterns in nature find explanations in chaos theory, fractals, logarithmic spirals, topology and other mathematical patterns. For example, L-systems form convincing models of different patterns of tree growth. The laws of physics apply the abstractions of mathematics to the real world, often as if it were perfect.
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In 1874 the Boston Thursday Club raised a subscription for the construction of a large-scale model, which was built in 1876. It could be expanded to enhance precision and weighed about . Christel Hamann built one machine (16-digit numbers and second-order differences) in 1909 for the "Tables of Bauschinger and Peters" ("Logarithmic-Trigonometrical Tables with eight decimal places"), which was first published in Leipzig in 1910. It weighed about .
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The design became standardised as DIN 45406. It evolved into the Type I meter in IEC 60268-10 and it is still known colloquially as a DIN PPM. Compared to the Type II designs it has faster integration and return times, a much wider dynamic range and a semi- logarithmic scale, and is calibrated in dB relative to Permitted Maximum Level. It remains in use in much of northern Europe.
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Miramar Beach is located along the shore of the bay opposite the peninsula. Marine species include flatfish, the commercially important English sole, rockfish, surfperch, Pacific herring, lingcod; and abundant winter species including starry flounder and top- smelt.Environmental Impact Report for the Pillar Point East Harbor Master Plan, Earth Metrics Inc., prepared for the San Mateo County Harbor District, February 1989 The bay provides an example of a logarithmic spiral beach.
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The regulator, a calculation of volume in 'logarithmic space' as divided by the logarithms of the units of the cyclotomic field, can be set against the quantities from the L(1) recognisable as logarithms of cyclotomic units. There result formulae stating that the class number is determined by the index of the cyclotomic units in the whole group of units. In Iwasawa theory, these ideas are further combined with Stickelberger's theorem.
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He was a pioneer in the use of pH measurement in medicine (with Christian Bohr, father of Niels Bohr), and he described how the affinity of blood for oxygen was dependent on the concentration of carbon dioxide. He was also first to determine the pH of blood. In 1916, he converted the 1908 equation of Lawrence Joseph Henderson to logarithmic form, which is now known as the Henderson–Hasselbalch equation.
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This is the special advantage of the form of table first introduced by Professor Inman, of the Portsmouth Royal Navy College, nearly a century ago.W. W. Sheppard and C. C. Soule, Practical navigation (World Technical Institute: Jersey City, 1922).E. R. Hedrick, Logarithmic and Trigonometric Tables (Macmillan, New York, 1913). These days, the haversine form is also convenient in that it has no coefficient in front of the function.
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Random binary search trees had been studied for much longer, and are known to behave well as search trees (they have logarithmic depth with high probability); the same good behavior carries over to treaps. It is also possible, as suggested by Aragon and Seidel, to reprioritize frequently-accessed nodes, causing them to move towards the root of the treap and speeding up future accesses for the same keys.
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A value of 0 is given for non-explosive eruptions, defined as less than of tephra ejected; and 8 representing a mega-colossal explosive eruption that can eject (240 cubic miles) of tephra and have a cloud column height of over . The scale is logarithmic, with each interval on the scale representing a tenfold increase in observed ejecta criteria, with the exception of between VEI-0, VEI-1 and VEI-2.
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A Queap Q with k = 6 and n = 9 In computer science, a queap is a priority queue data structure. The data structure allows insertions and deletions of arbitrary elements, as well as retrieval of the highest-priority element. Each deletion takes amortized time logarithmic in the number of items that have been in the structure for a longer time than the removed item. Insertions take constant amortized time.
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For all practical purposes extreme accuracy is not required (mechanical shutter speeds were notoriously inaccurate as wear and lubrication varied, with no effect on exposure). It is not significant that aperture areas and shutter speeds do not vary by a factor of precisely two. Photographers sometimes express other exposure ratios in terms of 'stops'. Ignoring the f-number markings, the f-stops make a logarithmic scale of exposure intensity.
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CODALEMA (Cosmic ray Detection Array with Logarithmic ElectroMagnetic Antennas) is a set of instruments to try and detect ultra-high energy cosmic rays, which cause cascades of particles in the atmosphere. These air showers generate very brief electromagnetic signals that are measured in a wide frequency band from 20 MHz to 200 MHz. An array of about 50 antennas is spread over a large area of the site.
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Approximate and true golden spirals: the green spiral is made from quarter- circles tangent to the interior of each square, while the red spiral is a golden spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of a larger square to the next smaller square is in the golden ratio. For a square with side length 1, the next smaller square is wide.
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He extracted the 77th root of a 148-digit number in 18 seconds, while it took about 10 minutes to program the related operation for computer. Shelushkov was a postgraduate at Gorki Polytechnic Institute (now Nizhny Novgorod State Technical University). According to Shelushkov, he used the memorized logarithmic table for calculations. His abilities were mentioned by Russian mathematician Vladimir Tvorogov, who attended one of his performances, and by psychologist Artur Petrovsky.
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Delete-min (max) can simply look up the minimum (maximum), then delete it. This way, insertion and deletion both take logarithmic time, just as they do in a binary heap, but unlike a binary heap and most other priority queue implementations, a single tree can support all of find-min, find-max, delete-min and delete-max at the same time, making binary search trees suitable as double-ended priority queues.
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The rhinoceros and rhinoceros horn shapes began to proliferate in Dalí's work from the mid-1950s. According to Dalí, the rhinoceros horn signifies divine geometry because it grows in a logarithmic spiral. He linked the rhinoceros to themes of chastity and to the Virgin Mary. However, he also used it as an obvious phallic symbol as in Young Virgin Auto-Sodomized by the Horns of Her Own Chastity.
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The LPDA normally consists of a series of half wave dipole "elements" each consisting of a pair of metal rods, positioned along a support boom lying along the antenna axis. The elements are spaced at intervals following a logarithmic function of the frequency, known as d or sigma. The successive elements gradually decrease in length along the boom. The relationship between the lengths is a function known as tau.
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When Robertson died in 1777 William Wales decided to revise the book and under the same title an edition was published in 1780 attributed to Robertson and Wales. In 1750 Robertson published A Translation of De La Caille's Elements of Astronomy and he published nine papers in the Philosophical Transactions between 1750 and 1772. These were On Logarithmic Tangents, On Logarithmic Lines on Gunter's Scale, On Extraordinary Phenomena in Portsmouth Harbour, On the Specific Gravity of Living Men, On the Fall of Water under Bridges, On Circulating Decimals, On the Motion of a Body deflected by Forces from Two Fixed Points, and On Twenty Cases of Compound Interest After losing his position at the Royal Naval Academy in 1766 Robertson was appointed as a clerk and librarian to the Royal Society, positions which he held until his death. He continued his scientific practice and was the first to be show that stereographic projection from the sphere is a conformal map projection.
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In computational complexity theory, the PCP theorem (also known as the PCP characterization theorem) states that every decision problem in the NP complexity class has probabilistically checkable proofs (proofs that can be checked by a randomized algorithm) of constant query complexity and logarithmic randomness complexity (uses a logarithmic number of random bits). The PCP theorem says that for some universal constant K, for every n, any mathematical proof of length n can be rewritten as a different proof of length poly(n) that is formally verifiable with 99% accuracy by a randomized algorithm that inspects only K letters of that proof. The PCP theorem is the cornerstone of the theory of computational hardness of approximation, which investigates the inherent difficulty in designing efficient approximation algorithms for various optimization problems. It has been described by Ingo Wegener as "the most important result in complexity theory since Cook's theorem" and by Oded Goldreich as "a culmination of a sequence of impressive works […] rich in innovative ideas".
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MetaSynth will then convert the spectrogram to digital sound and "play" the picture. According to an article on the website Wired News, photographs run through the program tend to produce "a kind of discordant, metallic scratching". A logarithmic spectrogram of "ΔMi−1 = −αΣn=1NDi[n] [Σj∈C[i]Fji[n − 1] +Fexti[n−1 " (commonly known as 'Equation' or 'Formula') reveals a portrait of James' face near the end of the track, grinning.
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Thus, genetic algorithms have been adopted by the software of practically all X-ray diffractometer manufacturers and also by open source fitting software. Fitting a curve requires a function usually called fitness function, cost function, fitting error function or figure of merit (FOM). It measures the difference between measured curve and simulated curve, and therefore, lower values are better. When fitting, the measurement and the best simulation are typically represented in logarithmic space.
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From there, the two channels are represented with a sequence of inductors and resistors for fluid flow within each channel with the two channels joined with a sequence of series resonant RLC circuits. Voltages across capacitances represent basilar membrane displacements. Element values along the cochlea are tapered in a logarithmic fashion to represent lowering frequency responses with distance. The pattern of voltages along the basilar membrane can be viewed on an oscilloscope.
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The probabilistic Turing machines in the definition of BPL may only accept or reject incorrectly less than 1/3 of the time; this is called two-sided error. The constant 1/3 is arbitrary; any x with 0 ≤ x < 1/2 would suffice. This error can be made 2−p(x) times smaller for any polynomial p(x) without using more than polynomial time or logarithmic space by running the algorithm repeatedly.
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The logarithmic scale can compactly represent the relationship among variously sized numbers. This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity and probabilities. Each number is given a name in the short scale, which is used in English- speaking countries, as well as a name in the long scale, which is used in some of the countries that do not have English as their national language.
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The success of the Church–Turing thesis prompted variations of the thesis to be proposed. For example, the physical Church–Turing thesis states: "All physically computable functions are Turing-computable." The Church–Turing thesis says nothing about the efficiency with which one model of computation can simulate another. It has been proved for instance that a (multi-tape) universal Turing machine only suffers a logarithmic slowdown factor in simulating any Turing machine.
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Rapid growth in the atmospheric concentration of SF6, NF3, and several widely used HFCs and PFCs between years 1978 and 2015 (right graph). Note the logarithmic scale. F-gases are ozone-friendly, enable energy efficiency, and are relatively safe for use by the public due to their low levels of toxicity and flammability. However, most F-gases have a high global warming potential (GWP), and some are nearly inert to removal by chemical processes.
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The result is read off the unknown scale at the point where the line intersects that scale. The scales include 'tick marks' to indicate exact number locations, and they may also include labeled reference values. These scales may be linear, logarithmic, or have some more complex relationship. The sample isopleth shown in red on the nomogram at the top of this article calculates the value of T when S = 7.30 and R = 1.17.
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The arithmetic Riemann–Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties. A complete proof of this theorem was only published recently by Gillet, Rössler and Soulé. Arakelov's intersection theory for arithmetic surfaces was developed further by . The theory of Bost is based on the use of Green functions which, up to logarithmic singularities, belong to the Sobolev space L^2_1.
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The probabilistic Turing machines in the definition of RL never accept incorrectly but are allowed to reject incorrectly less than 1/3 of the time; this is called one-sided error. The constant 1/3 is arbitrary; any x with 0 < x < 1 would suffice. This error can be made 2−p(x) times smaller for any polynomial p(x) without using more than polynomial time or logarithmic space by running the algorithm repeatedly.
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Lewi Tonks (1897–1971) was an American quantum physicist noted for his discovery (with Marvin D. Girardeau) of the Tonks-Girardeau gas. Tonks was employed by General Electric for most of his working life, researching microwaves and ferromagnetism. He worked under Irving Langmuir on plasma physics, with a special interest in ball lightning, nuclear fusion, tungsten filament light bulbs, and lasers. Tonks advocated a logarithmic pressure scale for vacuum technology to replace the torr.
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Based on the International Waterfall Classification System, Cline Falls is rated at 2.83, making it a Class-3 waterfall. The International Waterfall Classification System is a logarithmic scale that groups waterfalls into ten classes based on the height and pitch of the waterfall as well as the average volume of water flowing over the falls.Beisel, Richard H., International Waterfall Classification System, Outskirts Press, Parker, Colorado, 2006. Cline Falls is west of Redmond, Oregon.
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The two charts below display historical COVID-19 testing data since 3 April, when reliable testing data became available in Pakistan. The first chart covers raw data of numbers of cumulative tests, new tests, and cumulative confirmed cases and new confirmed case counts for comparison with testing numbers. It can be viewed on a linear or logarithmic scale. The second chart shows different types of test positivity rates in Pakistan since the same date.
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The list ranking problem was posed by , who solved it with a parallel algorithm using logarithmic time and O(n log n) total steps (that is, O(n) processors). Over a sequence of many subsequent papers, this was eventually improved to linearly many steps (O(n/log n) processors), on the most restrictive model of synchronous shared-memory parallel computation, the exclusive read exclusive write PRAM (; ;). This number of steps matches the sequential algorithm.
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His growth model is preceded by a discussion of arithmetic growth and geometric growth (whose curve he calls a logarithmic curve, instead of the modern term exponential curve), and thus "logistic growth" is presumably named by analogy, logistic being from , a traditional division of Greek mathematics. The term is unrelated to the military and management term logistics, which is instead from "lodgings", though some believe the Greek term also influenced logistics; see for details.
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The S-curve shape is interesting since it has qualities that correlate with the previously mentioned curves. The level of the sound is 50% at the midpoint, but before and after the midpoint the shape is not linear. There are also two types of S-curves. Traditional S-curve fade-in has attributes of the exponential curve can be seen at the beginning; at the midpoint to the end it is more logarithmic in nature.
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The x-axis is the ratio τ1 / τ2 on a logarithmic scale. An increase in this variable means the higher pole is further above the corner frequency. The y-axis is the ratio of the OCTC (open-circuit time constant) estimate to the true time constant. For the lowest pole use curve T_1; this curve refers to the corner frequency; and for the higher pole use curve T_2. The worst agreement is for τ1 = τ2.
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Several algorithms exist that run faster than the presented dynamic programming approach. One of them is Hunt–Szymanski algorithm, which typically runs in O((n + r)\log(n)) time (for n > m), where r is the number of matches between the two sequences. For problems with a bounded alphabet size, the Method of Four Russians can be used to reduce the running time of the dynamic programming algorithm by a logarithmic factor..
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For a graph with vertices and degeneracy , the Grundy number is . In particular, for graphs of bounded degeneracy (such as planar graphs) or graphs for which the chromatic number and degeneracy are bounded within constant factors of each other (such as chordal graphs) the Grundy number and chromatic number are within a logarithmic factor of each other.. For interval graphs, the chromatic number and Grundy number are within a factor of 8 of each other..
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However, in most cases the curve bends strongly, making it difficult to plot a projection accurately. This problem can be overcome by plotting the discharge and/or recurrence interval data on logarithmic graph paper. Once the plot is straightened, a line can be ruled drawn through the points. A projection can then be made by extending the line beyond the points and then reading the appropriate discharge for the recurrence interval in question.
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Susceptible cells are inoculated with serial logarithmic dilutions of samples in a 96-well plate. After viral growth, viral detection by IPA yields the infectious virus titer, expressed as tissue culture infectious dose (TCID50). This represents the dilution of a virus-containing sample at which half of a series of laboratory wells contain replicating viruses. This technique is a reliable method for the titration of human coronaviruses (HCoV) in biological samples (cells, tissues, or fluids).
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Although many references assert that the instrument scroll closely follows the golden spiral (a specific form of the logarithmic spiral) this assertion is demonstrably false. Scrollwork is a common feature of Baroque ornament, the period when string instrument design became essentially fixed. Carved lion's head on a Stainer violin Below the scroll is a hollowed- out compartment (the pegbox) through which the tuning pegs pass. The instrument's strings are wound around these pegs.
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For instance, following this transformation, the Held–Karp algorithm could be used to solve the bottleneck TSP in time . Alternatively, the problem can be solved by performing a binary search or sequential search for the smallest such that the subgraph of edges of weight at most has a Hamiltonian cycle. This method leads to solutions whose running time is only a logarithmic factor larger than the time to find a Hamiltonian cycle.
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In algebraic geometry, the Kawamata–Viehweg vanishing theorem is an extension of the Kodaira vanishing theorem, on the vanishing of coherent cohomology groups, to logarithmic pairs, proved independently by Viehweg and Kawamata in 1982. The theorem states that if L is a big nef line bundle (for example, an ample line bundle) on a complex projective manifold with canonical line bundle K, then the coherent cohomology groups Hi(L⊗K) vanish for all positive i.
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In plotting an animal's basal metabolic rate (BMR) against the animal's own body mass, a logarithmic straight line is obtained, indicating a power-law dependence. Overall metabolic rate in animals is generally accepted to show negative allometry, scaling to mass to a power of ≈ 0.75, known as Kleiber's law, 1932. This means that larger-bodied species (e.g., elephants) have lower mass- specific metabolic rates and lower heart rates, as compared with smaller- bodied species (e.g.
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The segment tree can be generalized to higher dimension spaces, in the form of multi-level segment trees. In higher dimensional versions, the segment tree stores a collection of axis-parallel (hyper-)rectangles, and can retrieve the rectangles that contain a given query point. The structure uses O(n logd n) storage, and answers queries in O(logd n). The use of fractional cascading lowers the query time bound by a logarithmic factor.
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The nested osculating circles of an Archimedean spiral. The spiral itself is not shown, but is visible where the circles are more dense. In differential geometry, the Tait–Kneser theorem states that, if a smooth plane curve has monotonic curvature, then the osculating circles of the curve are disjoint and nested within each other. The logarithmic spiral or the pictured Archimedean spiral provide examples of curves whose curvature is monotonic for the entire curve.
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CORDIC uses simple shift-add operations for several computing tasks such as the calculation of trigonometric, hyperbolic and logarithmic functions, real and complex multiplications, division, square-root calculation, solution of linear systems, eigenvalue estimation, singular value decomposition, QR factorization and many others. As a consequence, CORDIC has been used for applications in diverse areas such as signal and image processing, communication systems, robotics and 3D graphics apart from general scientific and technical computation.
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For example, a novice player might reason that allocating 20% of one's income to mining a particular planet would yield twice as much metal as allocating 10%; in fact, it yields only 40% more metal. A pseudo-logarithmic bar graph displays the player's spending allocations. The player manipulates the bars with the mouse to allocate spending. As one bar is lengthened, the lengths of the other bars are automatically shortened (and vice versa).
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Binary search runs in logarithmic time in the worst case, making O(\log n) comparisons, where n is the number of elements in the array. Binary search is faster than linear search except for small arrays. However, the array must be sorted first to be able to apply binary search. There are specialized data structures designed for fast searching, such as hash tables, that can be searched more efficiently than binary search.
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How much space does this algorithm use? Within each invocation of the algorithm, it needs to store the intermediate results of computing A and B. Every recursive call takes off one quantifier, so the total recursive depth is linear in the number of quantifiers. Formulas that lack quantifiers can be evaluated in space logarithmic in the number of variables. The initial QBF was fully quantified, so there are at least as many quantifiers as variables.
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The size of the neighborhood must be sufficient to accommodate a logarithmic number of items in the worst case (i.e. it must accommodate log(n) items), but only a constant number on average. If some bucket's neighborhood is filled, the table is resized. In hopscotch hashing, as in cuckoo hashing, and unlike in linear probing, a given item will always be inserted-into and found-in the neighborhood of its hashed bucket.
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Frequently Questioned Answers about Gamma. Gamma encoding of floating-point images is not required (and may be counterproductive), because the floating- point format already provides a piecewise linear approximation of a logarithmic curve. Although gamma encoding was developed originally to compensate for the input–output characteristic of cathode ray tube (CRT) displays, that is not its main purpose or advantage in modern systems. In CRT displays, the light intensity varies nonlinearly with the electron-gun voltage.
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In 1696 Bernoulli solved the equation, now called the Bernoulli differential equation, : y' = p(x)y + q(x)y^n. Jacob Bernoulli also discovered a general method to determine evolutes of a curve as the envelope of its circles of curvature. He also investigated caustic curves and in particular he studied these associated curves of the parabola, the logarithmic spiral and epicycloids around 1692. The lemniscate of Bernoulli was first conceived by Jacob Bernoulli in 1694.
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David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. CRC Press. Boca Raton, Florida, 2003; Section 10, Atomic, Molecular, and Optical Physics; Ionization Potentials of Atoms and Atomic Ions The ionic radius of hexacoordinate Md3+ had been preliminarily estimated in 1978 to be around 91.2 pm; 1988 calculations based on the logarithmic trend between distribution coefficients and ionic radius produced a value of 89.6 pm, as well as an enthalpy of hydration of .
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In electronics, an octave (symbol oct) is a logarithmic unit for ratios between frequencies, with one octave corresponding to a doubling of frequency. For example, the frequency one octave above 40 Hz is 80 Hz. The term is derived from the Western musical scale where an octave is a doubling in frequency. Specification in terms of octaves is therefore common in audio electronics. Along with the decade, it is a unit used to describe frequency bands or frequency ratios.
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After graduating, Gabriele went to Rome at the end of 1702, where he became librarian to Cardinal Pietro Ottoboni, a historian, antiquarian and astronomer. He helped Ottoboni build a sundial at Santa Maria degli Angeli e dei Martiri and helped in the work of reforming the Gregorian calendar. He continued to study mathematics, including differential and integral calculus and logarithmic curves. In 1707 he returned to Bologna where he published his best known work on first-order differential equations.
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KCalc is the software calculator integrated with the KDE Software Compilation. In the default view it includes a number pad, buttons for adding, subtracting, multiplying, and dividing, brackets, memory keys, percent, reciprocal, factorial, square, and x to the power of y buttons. Additional buttons for scientific and engineering (trigonometric and logarithmic functions), statistics and logic functions can be enabled as needed. 6 additional buttons can be predefined with mathematical constants and physical constants or custom values.
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The 9100A was the first scientific calculator by the modern definition, i.e., capable of trigonometric, logarithmic (log/ln), and exponential functions, and was the beginning of Hewlett-Packard's long history of using Reverse Polish notation (RPN) entry on their calculators. Due to the similarities of the machines, Hewlett-Packard was ordered to pay about $900,000 in royalties to Olivetti after copying some of the solutions adopted in the Programma 101, like the magnetic card and the architecture.
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Extracellular DNA is a major structural component of many different microbial biofilms. Enzymatic degradation of extracellular DNA can weaken the biofilm structure and release microbial cells from the surface. However, biofilms are not always less susceptible to antibiotics. For instance, the biofilm form of Pseudomonas aeruginosa has no greater resistance to antimicrobials than do stationary- phase planktonic cells, although when the biofilm is compared to logarithmic- phase planktonic cells, the biofilm does have greater resistance to antimicrobials.
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In the preface of the latter, he wrote: "There is in the world a great deal of brilliant, witty political discussion which leads to no settled convictions. My aim has been different: namely to examine a few notions by quantitative techniques in the hope of reaching a reliable answer." In Statistics of Deadly Quarrels Richardson presented data on virtually every war from 1815 to 1945. As a result, he hypothesized a base 10 logarithmic scale for conflicts.
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It is more expensive in Iceland, where it is bottled, than as an imported product elsewhere. Iceland Spring claim the water is chemically basic with a pH of 8.88, 75 times more base than the 7 pH of pure water, but testing by the American Council on Science and Health in 2015 resulted in a pH of 7.64, only 4 times more base than pure water (the pH scale is logarithmic). It has a metallic taste.
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This is a natural inverse of the linear approximation to tetration. Authors like Holmes recognize that the super- logarithm would be a great use to the next evolution of computer floating- point arithmetic, but for this purpose, the function need not be infinitely differentiable. Thus, for the purpose of representing large numbers, the linear approximation approach provides enough continuity (C^0 continuity) to ensure that all real numbers can be represented on a super-logarithmic scale.
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Mean-field-type filters, which are filters that depend on the distribution of the state, were first proposed by L&G; Lab members. They provided explicit solutions to a class of mean-field-type games with non- linear state dynamics and or non-quadratic cost functions. The non-linearity includes trigonometric functions, hyperbolic functions, logarithmic functions and power (polynomial) cost functions. Tembine has worked on game theory with small, medium and large number of interacting agents.
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The Wang LOCI-2 Logarithmic Computing Instrument desktop calculator (the earlier LOCI-1 in September 1964 was not a real product) was introduced in January 1965. Using factor combining it was probably the first desktop calculator capable of computing logarithms, quite an achievement for a machine without any integrated circuits. The electronics included 1,275 discrete transistors. It actually performed multiplication by adding logarithms, and roundoff in the display conversion was noticeable: 2 times 2 yielded 3.999999999.
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In geometry, a W-curve is a curve in projective n-space that is invariant under a 1-parameter group of projective transformations. W-curves were first investigated by Felix Klein and Sophus Lie in 1871, who also named them. W-curves in the real projective plane can be constructed with straightedge alone. Many well-known curves are W-curves, among them conics, logarithmic spirals, powers (like y = x3), logarithms and the helix, but not e.g.
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Arratia developed the ideas of interlace polynomials with Béla Bollobás and Gregory Sorkin,. found an equivalent formulation of the Stanley–Wilf conjecture as the convergence of a limit, and was the first to investigate the lengths of superpatterns of permutations. He has also written highly cited papers on the Chen–Stein method on distances between probability distributions,.. on random walks with exclusion,. and on sequence alignment... He is a coauthor of the book Logarithmic Combinatorial Structures: A Probabilistic Approach....
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The Bode plot of a first-order Butterworth low-pass filter The frequency response of the Butterworth filter is maximally flat (i.e. has no ripples) in the passband and rolls off towards zero in the stopband. When viewed on a logarithmic Bode plot, the response slopes off linearly towards negative infinity. A first-order filter's response rolls off at −6 dB per octave (−20 dB per decade) (all first-order lowpass filters have the same normalized frequency response).
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Power spectrum of the Sun around where the modes have maximum power, using data from the GOLF and VIRGO/SPM instruments aboard the Solar and Heliospheric Observatory. The low-degree modes (l<4) show a clear comb-like pattern with a regular spacing. SOHO. The colour scale is logarithmic and saturated at one hundredth the maximum power in the signal, to make the modes more visible. The low-frequency region is dominated by the signal of granulation.
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1660)', Oxford Dictionary of National Biography (2004). After John Napier invented logarithms and Edmund Gunter created the logarithmic scales (lines, or rules) upon which slide rules are based, Oughtred was the first to use two such scales sliding by one another to perform direct multiplication and division. He is credited with inventing the slide rule in about 1622. He also introduced the "×" symbol for multiplication and the abbreviations "sin" and "cos" for the sine and cosine functions.
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Macintyre developed a first-order model theory for intersection theory and showed connections to Alexander Grothendieck's standard conjectures on algebraic cycles. Macintyre has proved many results on the model theory of real and complex exponentiation. With Alex Wilkie he proved the decidability of real exponential fields (solving a problem of Alfred Tarski) modulo Schanuel's conjecture from transcendental number theory. With Lou van den Dries he initiated and studied the model theory of logarithmic-exponential series and Hardy fields.
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Using the affine gap penalty requires the assigning of fixed penalty values for both opening and extending a gap. This can be too rigid for use in a biological context. The logarithmic gap takes the form G(L)=A+C\ln L and was proposed as studies had shown the distribution of indel sizes obey a power law. Another proposed issue with the use of affine gaps is the favoritism of aligning sequences with shorter gaps.
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A prefix hash tree (PHT) is a distributed data structure that enables more sophisticated queries over a distributed hash table (DHT). The prefix hash tree uses the lookup interface of a DHT to construct a trie-based data structure that is both efficient (updates are doubly logarithmic in the size of the domain being indexed), and resilient (the failure of any given node in a prefix hash tree does not affect the availability of data stored at other nodes).
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Bayes' theorem is applied successively to all evidence presented, with the posterior from one stage becoming the prior for the next. The benefit of a Bayesian approach is that it gives the juror an unbiased, rational mechanism for combining evidence. It may be appropriate to explain Bayes' theorem to jurors in odds form, as betting odds are more widely understood than probabilities. Alternatively, a logarithmic approach, replacing multiplication with addition, might be easier for a jury to handle.
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The largest known prime number () is , a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018. logarithmic. A prime number is a positive integer with no divisors other than 1 and itself, excluding 1. Euclid recorded a proof that there is no largest prime number, and many mathematicians and hobbyists continue to search for large prime numbers.
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This was later revised and renamed the local magnitude scale, denoted as ML or . Because of various shortcomings of the scale, most seismological authorities now use other scales, such as the moment magnitude scale (), to report earthquake magnitudes, but much of the news media still refers to these as "Richter" magnitudes. All magnitude scales retain the logarithmic character of the original and are scaled to have roughly comparable numeric values (typically in the middle of the scale).
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They discovered the Kinetic Alfvén wave, which resolves the logarithmic singularity of magnetohydrodynamic shear Alfvén waves and plays important roles in the heating, acceleration and transport of charged particles in solar, magnetospheric, and laboratory plasmas. Before joining the faculty at UCI, Chen was a professor at the Princeton University. Chen was the Deputy Head of the Theory Division of the Princeton Plasma Physics Laboratory. From 1993 till 2012, Chen was a professor of physics at UCI.
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While many organisms are competent only under certain environmental conditions, such as starvation, H. pylori is competent throughout logarithmic growth. All organisms encode genetic programs for response to stressful conditions including those that cause DNA damage. In H. pylori, homologous recombination is required for repairing DNA double-strand breaks (DSBs). The AddAB helicase-nuclease complex resects DSBs and loads RecA onto single-strand DNA (ssDNA), which then mediates strand exchange, leading to homologous recombination and repair.
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The Sun has absolute magnitude MV=+4.83. Highly luminous objects can have negative absolute magnitudes: for example, the Milky Way galaxy has an absolute B magnitude of about −20.8. An object's absolute bolometric magnitude (Mbol) represents its total luminosity over all wavelengths, rather than in a single filter band, as expressed on a logarithmic magnitude scale. To convert from an absolute magnitude in a specific filter band to absolute bolometric magnitude, a bolometric correction (BC) is applied.
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Newer files have a header that consists of six unsigned 32-bit words, an optional information chunk and then the data (in big endian format). Although the format now supports many audio encoding formats, it remains associated with the μ-law logarithmic encoding. This encoding was native to the SPARCstation 1 hardware, where SunOS exposed the encoding to application programs through the /dev/audio interface. This encoding and interface became a de facto standard for Unix sound.
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In the history of electronic analog computers, there were some special high-speed types. Nonlinear functions and calculations can be constructed to a limited precision (three or four digits) by designing function generators—special circuits of various combinations of resistors and diodes to provide the nonlinearity. Typically, as the input voltage increases, progressively more diodes conduct. When compensated for temperature, the forward voltage drop of a transistor's base-emitter junction can provide a usably accurate logarithmic or exponential function.
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Rosemary Varley, 'Substance or Scafforld? The role of language in thought', in Victoria Joffe, Madeleine Cruice, Shula Chiat (eds.)Language Disorders in Children and Adults: New Issues in Research and Practice, Wiley-Blackwell, 2008 pp.20-38, p.27. Furthermore, the Mundurucu use logarithmic mapping of numbers to assess scales, a point cited as possible evidence for the notion that this kind of numbering is innate, whereas the linear mode has to be acquired by study.
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Nagra double modulometer The 'modulometer' is a proprietary type of quasi-PPM found on Nagra products. It has an integration time (−2 dB) of 7.5 ms, and a semi- logarithmic scale with an appearance between that of a VU meter and a DIN-type PPM. A stereo version ("double modulometer") uses a meter movement with two coaxial needles. In typical practice for Nagra analogue tape recorders, Alignment Level is regarded as −8 and maximum level 0.
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JSBSim uses a coefficient build-up method for modeling the aerodynamic characteristics of aircraft. Any number of forces and moments (or none at all) can be defined for each of the axes. Each force/moment specification includes a definition comment, and a specification of the function that calculates the force or moment. The function definition can be a simple value, or a complicated function that includes trigonometric and logarithmic functions, and a one-, two-, or three-dimensional table lookup.
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It can be CRCW, CREW, or EREW. See PRAM for descriptions of those models. Equivalently, NC can be defined as those decision problems decidable by a uniform Boolean circuit (which can be calculated from the length of the input, for NC, we suppose we can compute the Boolean circuit of size n in logarithmic space in n) with polylogarithmic depth and a polynomial number of gates. RNC is a class extending NC with access to randomness.
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The "particles" in this application are clusters of protein subunits arranged on a shell. Other realizations include regular arrangements of colloid particles in colloidosomes, proposed for encapsulation of active ingredients such as drugs, nutrients or living cells, fullerene patterns of carbon atoms, and VSEPR theory. An example with long-range logarithmic interactions is provided by the Abrikosov vortices which would form at low temperatures in a superconducting metal shell with a large monopole at the center.
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Each element consists of three bars which form a minimal Ronchi ruling. These 54 elements are provided in a standardized series of logarithmic steps in the spatial frequency range from 0.250 to 912.3 line pairs per millimeter (lp/mm). The series of elements spans the range of resolution of the unaided eye, down to the diffraction limits of conventional light microscopy. Commercially produced devices typically consist of a transparent square glass slide, 2 inches or 50mm in dimension.
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When the reference value is ten, the order of magnitude can be understood as the number of digits in the base-10 representation of the value. Similarly, if the reference value is one of certain powers of two, the magnitude can be understood as the amount of computer memory needed to store the exact integer value. Differences in order of magnitude can be measured on a base-10 logarithmic scale in “decades” (i.e., factors of ten).
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This is usually related to resolution and noise floor. Other measurements, such as phase distortion and jitter, can also be very important for some applications, some of which (e.g. wireless data transmission, composite video) may even rely on accurate production of phase-adjusted signals. Non-linear PCM encodings (A-law / μ-law, ADPCM, NICAM) attempt to improve their effective dynamic ranges by using logarithmic step sizes between the output signal strengths represented by each data bit.
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The hartley (symbol Hart), also called a ban, or a dit (short for decimal digit), is a logarithmic unit which measures information or entropy, based on base 10 logarithms and powers of 10. One hartley is the information content of an event if the probability of that event occurring is . It is therefore equal to the information contained in one decimal digit (or dit), assuming a priori equiprobability of each possible value. It is named after Ralph Hartley.
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The decrease in zero-point energy due to isotopic substitution is therefore less important in D3O+ than in DA so that KD < KH, and the deuterated acid in D2O is weaker than the non-deuterated acid in H2O. In many cases the difference of logarithmic constants pKD – pKH is about 0.6, so that the pD corresponding to 50% dissociation of the deuterated acid is about 0.6 units higher than the pH for 50% dissociation of the non- deuterated acid.
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The square is named for the psychiatrist Philippe Pinel (1745 - 1826), "benefactor of strangers", because of its proximity to the Hôpital de la Salpêtrière where he worked. In 2012, the square was completely redeveloped by the Direction de la Voirie et des Déplacements de la Mairie de Paris, the City of Paris transport section. At this time, the central circle was recovered in granite paving. Its design represents a pine cone, represented with logarithmic spirals based on Fibonacci numbers.
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An ivory set of Napier's Bones, an early calculating device invented by John Napier John Napier introduced logarithms as a powerful mathematical tool. With the help of the prominent mathematician Henry Briggs their logarithmic tables embodied a computational advance that made calculations by hand much quicker. His Napier's bones used a set of numbered rods as a multiplication tool using the system of lattice multiplication. The way was opened to later scientific advances, particularly in astronomy and dynamics.
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In both digital and film photography, the reduction of exposure corresponding to use of higher sensitivities generally leads to reduced image quality (via coarser film grain or higher image noise of other types). In short, the higher the sensitivity, the grainier the image will be. Ultimately sensitivity is limited by the quantum efficiency of the film or sensor. This film container denotes its speed as ISO 100/21°, including both arithmetic (100 ASA) and logarithmic (21 DIN) components.
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Three decades: 1, 10, 100, 1000 (100, 101, 102, 103) Three decades: One thousand 0.001s, one-hundred 0.01s, ten 0.1s, one 1. One decade (symbol decISO 80000-3:2006 Quantities and Units – Space and time) is a unit for measuring frequency ratios on a logarithmic scale, with one decade corresponding to a ratio of 10 between two frequencies (an order of magnitude difference).Levine, William S. (2010). The Control Handbook: Control System Fundamentals, p. 9-29. .
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Laboratory data is used to construct a plot of strain or void ratio versus effective stress where the effective stress axis is on a logarithmic scale. The plot's slope is the compression index or recompression index. The equation for consolidation settlement of a normally consolidated soil can then be determined to be: The soil which had its load removed is considered to be "overconsolidated". This is the case for soils that have previously had glaciers on them.
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The use of Greek letter names is presumably by extension from the common finance terms alpha and beta, and the use of sigma (the standard deviation of logarithmic returns) and tau (time to expiry) in the Black–Scholes option pricing model. Several names such as 'vega' and 'zomma' are invented, but sound similar to Greek letters. The names 'color' and 'charm' presumably derive from the use of these terms for exotic properties of quarks in particle physics.
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Dalí's keen interest in natural science and mathematics was further manifested by the proliferation of images of DNA and rhinoceros horn shapes in works from the mid-1950s. According to Dalí, the rhinoceros horn signifies divine geometry because it grows in a logarithmic spiral.Elliott H. King in Dawn Ades (ed.), Dalí, Bompiani Arte, Milan, 2004, p. 456. Dalí was also fascinated by the tesseract (a four-dimensional cube), using it, for example, in Crucifixion (Corpus Hypercubus).
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This algorithm takes linear time, but because it needs to refer back to earlier positions in the sequence it needs to store the whole sequence, taking linear space. An alternative algorithm that generates multiple copies of the sequence at different speeds, with each copy of the sequence using the output of the previous copy to determine what to do at each step, can be used to generate the sequence in linear time and only logarithmic space.
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Whenever another record is hashed to anywhere within the cluster, it grows in size by one cell. Because of this phenomenon, it is likely that a linear-probing hash table with a constant load factor (that is, with the size of the table proportional to the number of items it stores) will have some clusters of logarithmic length, and will take logarithmic time to search for the keys within that cluster.. A related phenomenon, secondary clustering, occurs more generally with open addressing modes including linear probing and quadratic probing in which the probe sequence is independent of the key, as well as in hash chaining. In this phenomenon, a low-quality hash function may cause many keys to hash to the same location, after which they all follow the same probe sequence or are placed in the same hash chain as each other, causing them to have slow access times. Both types of clustering may be reduced by using a higher-quality hash function, or by using a hashing method such as double hashing that is less susceptible to clustering.
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Milin’s research mostly deals with an important part of complex analysis: theory of regular and meromorphic univalent functions including problems for Taylor and Loran coefficients. Milin's area theorem and coefficient estimates, as well as Milin’s functionals, Milin’s Tauberian theorem, Milin’s constant, Lebedev–Milin inequalities are widely known. In 1949 I.M. Milin and Nikolai Andreevich Lebedev proved a notable Rogozinskij's conjecture (1939) on coefficients of Bieberbach-Eilenberg functions. In 1964 exploring the famous Bieberbach conjecture (1916) Milin seriously improved the known coefficient estimate for univalent functions. Milin’s monograph “Univalent functions and orthonormal systems” (1971) includes the author’s results and thoroughly covers all the achievements on systems of regular functions orthonormal with respect to area obtained by then. There Milin also constructed a sequence of logarithmic functionals (Milin’s functionals) on the basic class of univalent functions S, conjecturing them to be non-positive for any function of this class and showed that his conjecture implied Bieberbach’s. In 1984 Louis de Branges proved Milin’s conjecture and, therefore, the Bieberbach conjecture. The second Milin’s conjecture on logarithmic coefficients published in 1983 is still an open problem.
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Aging of different Class 2 ceramic capacitors compared with NP0-Class 1 ceramic capacitor In ferroelectric Class 2 ceramic capacitors, capacitance decreases over time. This behavior is called "aging". This aging occurs in ferroelectric dielectrics, where domains of polarization in the dielectric contribute to the total polarization. Degradation of polarized domains in the dielectric decreases permittivity and therefore capacitance over time.Takaaki Tsurumi & Motohiro Shono & Hirofumi Kakemoto & Satoshi Wada & Kenji Saito & Hirokazu Chazono, Mechanism of capacitance aging under DC-bias field in X7R-MLCCs Published online: 23 March 2007, # Springer Science + Business Media, LLC 2007 The aging follows a logarithmic law. This defines the decrease of capacitance as constant percentage for a time decade after the soldering recovery time at a defined temperature, for example, in the period from 1 to 10 hours at 20 °C. As the law is logarithmic, the percentage loss of capacitance will twice between 1 h and 100 h and 3 times between 1 h and 1,000 h and so on. Aging is fastest near the beginning, and the absolute capacitance value stabilizes over time.
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The participant performed the task the first two times with the instruction to perform the task as accurately as possible. For the last task, the participant was asked to perform the task as quickly as possible. While Hick was stating that the relationship between reaction time and the number of choices was logarithmic, Hyman wanted to better understand the relationship between the reaction time and the mean number of choices. In Hyman’s experiment, he had eight different lights arranged in a 6x6 matrix.
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John Hershberger has been a significant contributor to computational geometry and the algorithms community since the mid-1980s. His earliest work focused on shortest paths and visibility. With Leonidas Guibas and by himself, he devised optimal linear-time algorithms to compute visibility polygons, shortest path trees, visibility graphs, and data structures for logarithmic-time shortest path queries in simple polygons. With Jack Snoeyink he extended the algorithms for simple polygons to compute homotopic shortest paths among polygonal obstacles in the plane.
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The term has also long been used in fields such as geophysics and astronomy to characterise the properties of regions through which radiation passes, such as the ionosphere.The Relation of Radio Sky-Wave Transmission to Ionosphere Measurements, N Smith, Proceedings of the I.R.E., May 1939; discusses linear and logarithmic transmission curves of the ionosphere Radiation transmission data for radionuclides and materials relevant to brachytherapy facility shielding, P. Papagiannis et al., 2008, American Association of Physicists in Medicine. DOI:10.1118/1.2986153 .
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In contrast, for random graphs in the Erdős–Rényi model with edge probability 1/2, both the maximum clique and the maximum independent set are much smaller: their size is proportional to the logarithm of n, rather than growing polynomially. Ramsey's theorem proves that no graph has both its maximum clique size and maximum independent set size smaller than logarithmic. Ramsey's theorem also implies the special case of the Erdős–Hajnal conjecture when H itself is a clique or independent set.
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Simulation of ternary computers using binary computers, or interfacing between ternary and binary computers, can involve use of binary-coded ternary (BCT) numbers, with two bits used to encode each trit. BCT encoding is analogous to binary-coded decimal (BCD) encoding. If the trit values 0, 1 and 2 are encoded 00, 01 and 10, conversion in either direction between binary-coded ternary and binary can be done in logarithmic time. A library of C code supporting BCT arithmetic is available.
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The signal is divided into a luma (Y') component and two color difference components (chroma). A variety of filtering methods can be used to arrive at the resolution-reduced chroma values. Luma (Y') is differentiated from luminance (Y) by the presence of gamma correction in its calculation, hence the prime symbol added here. A gamma-corrected signal has the advantage of emulating the logarithmic sensitivity of human vision, with more levels dedicated to the darker levels than the lighter ones.
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Phylogeny and evolution of body size in Asterophryinae. Colours of branches correspond to maximum male snout-vent length (Paedophryne) or average snout-vent length within each clade on a logarithmic scale. Microhylid frogs are generally small. A few species such as Callulops robustus and Asterophrys turpicola attain snout-vent lengths (SVL) in excess of , whereas frogs in genus Paedophryne are particularly small, and Paedophryne amauensis is the world's smallest known vertebrate, attaining an average body size of only (range 7.0–8.0 mm).
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Other studies of willingness-to-pay to prevent harm have found a logarithmic relationship or no relationship to scope size. Daniel Kahneman explains scope neglect in terms of judgment by prototype, a refinement of the representativeness heuristic. "The story [...] probably evokes for many readers a mental representation of a prototypical incident, perhaps an image of an exhausted bird, its feathers soaked in black oil, unable to escape,"p. 212 and subjects based their willingness-to-pay mostly on that mental image.
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The first single-database computational PIR scheme to achieve communication complexity less than n was created in 1997 by Kushilevitz and Ostrovsky and achieved communication complexity of n^\epsilon for any \epsilon, where n is the number of bits in the database. The security of their scheme was based on the well-studied Quadratic residuosity problem. In 1999, Christian Cachin, Silvio Micali and Markus Stadler achieved poly-logarithmic communication complexity. The security of their system is based on the Phi-hiding assumption.
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A slide rule Since real numbers can be represented as distances or intervals on a line, the slide rule was invented in the 1620s, shortly after Napier's work, to allow multiplication and division operations to be carried out significantly faster than was previously possible. Edmund Gunter built a calculating device with a single logarithmic scale at the University of Oxford. His device greatly simplified arithmetic calculations, including multiplication and division. William Oughtred greatly improved this in 1630 with his circular slide rule.
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In mathematics, log-polar coordinates (or logarithmic polar coordinates) is a coordinate system in two dimensions, where a point is identified by two numbers, one for the logarithm of the distance to a certain point, and one for an angle. Log-polar coordinates are closely connected to polar coordinates, which are usually used to describe domains in the plane with some sort of rotational symmetry. In areas like harmonic and complex analysis, the log- polar coordinates are more canonical than polar coordinates.
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HLG is defined in ATSC 3.0, Digital Video Broadcasting (DVB) UHD-1 Phase 2, and International Telecommunication Union (ITU) Rec. 2100. HLG is supported by HDMI 2.0b, HEVC, VP9, and H.264/MPEG-4 AVC, and is used by video services such as BBC iPlayer, DirecTV, Freeview Play, and YouTube. Chart showing a conventional SDR gamma curve and Hybrid Log-Gamma (HLG). HLG uses a logarithmic curve for the upper half of the signal values which allows for a larger dynamic range.
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In circuit complexity, FO(ARB) where ARB is the set of all predicates, the logic where we can use arbitrary predicates, can be shown to be equal to AC0, the first class in the AC hierarchy. Indeed, there is a natural translation from FO's symbols to nodes of circuits, with \forall, \exists being \land and \lor of size . FO(BIT) is the restriction of AC0 family of circuit constructible in alternating logarithmic time. FO(<) is the set of star-free languages.
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In the data stream model, some or all of the input is represented as a finite sequence of integers (from some finite domain) which is generally not available for random access, but instead arrives one at a time in a "stream". If the stream has length and the domain has size , algorithms are generally constrained to use space that is logarithmic in and . They can generally make only some small constant number of passes over the stream, sometimes just one.
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Pointer jumping or path doubling is a design technique for parallel algorithms that operate on pointer structures, such as linked lists and directed graphs. Pointer jumping allows an algorithm to follow paths with a time complexity that is logarithmic with respect to the length of the longest path. It does this by "jumping" to the end of the path computed by neighbors. The basic operation of pointer jumping is to replace each neighbor in a pointer structure with its neighbor's neighbor.
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A given ONU may have several so-called transmission containers (T-CONTs), each with its own priority or traffic class. The ONU reports each T-CONT separately to the OLT. The report message contains a logarithmic measure of the backlog in the T-CONT queue. By knowledge of the service level agreement for each T-CONT across the entire PON, as well as the size of each T-CONT's backlog, the OLT can optimize allocation of the spare bandwidth on the PON.
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In terms of power at a constant bandwidth, pink noise falls off at 3 dB per octave. At high enough frequencies pink noise is never dominant. (White noise has equal energy per frequency interval.) The human auditory system, which processes frequencies in a roughly logarithmic fashion approximated by the Bark scale, does not perceive different frequencies with equal sensitivity; signals around 1–4 kHz sound loudest for a given intensity. However, humans still differentiate between white noise and pink noise with ease.
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The point separating the integers from the decimal fractions seems to be the invention of Bartholomaeus Pitiscus, in whose trigonometrical tables (1612) it occurs and it was accepted by John Napier in his logarithmic papers (1614 and 1619). File:Stevin-decimal notation.svg Stevin printed little circles around the exponents of the different powers of one-tenth. That Stevin intended these encircled numerals to denote mere exponents is clear from the fact that he employed the very same symbol for powers of algebraic quantities.
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Appendix no. 21: 200-224 He chose the term normal because of its frequent occurrence in naturally occurring variables. Lagrange also suggested in 1781 two other distributions for errors - a raised cosine distribution and a logarithmic distribution. Laplace gave (1781) a formula for the law of facility of error (a term due to Joseph Louis Lagrange, 1774), but one which led to unmanageable equations. Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
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For logarithmic barrier functions, g(x,b) is defined as -\log(b-x) when x < b and \infty otherwise (in 1 dimension. See below for a definition in higher dimensions). This essentially relies on the fact that \log(t) tends to negative infinity as t tends to 0. This introduces a gradient to the function being optimized which favors less extreme values of x (in this case values lower than b), while having relatively low impact on the function away from these extremes.
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Such a design leads to a matrix: columns represent increments in calculator functionality, and rows represent different presentation front-ends. Such a matrix M is shown to the right: columns allow one to pair basic calculator functionality (base) with optional logarithmic/exponentiation (lx) and trigonometric (td) features. Rows allow one to pair core functionality with no front-end (core), with optional GUI (gui) and web-based (web) front-ends. An element Mij implements the interaction of column feature i and row feature j.
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In the Circles of Proportion and the Horizontal Instrument, Oughtred introduces the abbreviations for trigonometric functions. This book was originally in manuscript before it eventually became published. Also, the slide rule is discussed, an invention that was made by Oughtred which provided a mechanical method of finding logarithmic results. It is mentioned in this book that John Napier was the first person to ever use to the decimal point and comma, however Bartholomaeus Pitiscus was actually the first to do so.
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Illustration of Haitz's law. Light output per LED package as a function of time, note the logarithmic scale on the vertical axis. Haitz's law is an observation and forecast about the steady improvement, over many years, of light-emitting diodes (LEDs). It claims that every decade, the cost per lumen (unit of useful light emitted) falls by a factor of 10, and the amount of light generated per LED package increases by a factor of 20, for a given wavelength (color) of light.
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However, because the resulting "fundamental durations" are not small enough for use in the musical detail, subdivisions corresponding to the transposition of the overtones of a pitch's harmonic spectrum are used . The twelve logarithmic metronomic tempos used in Gruppen, covering a tempo "octave" (doubling in speed) from = 60 to 120 are : The composer recalled that, when Igor Stravinsky saw the score for the first time, he wrote that such fractional metronomic values as 63.5 and 113.5 were "a sign of German thoroughness" .
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Gunter's scale or Gunter's rule, generally called the "Gunter" by seamen, is a large plane scale, usually long by about 1½ inches broad (40 mm), engraved with various scales, or lines. On one side are placed the natural lines (as the line of chords, the line of sines, tangents, rhumbs, etc.), and on the other side the corresponding artificial or logarithmic ones. By means of this instrument questions in navigation, trigonometry, etc., are solved with the aid of a pair of compasses.
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Accessed July 21, 2020. doi:10.2307/2373688. Submitted June 21, 1973 established these inequalities for the Bosonic and Fermionic cases. The inequalities were named by Gross, who established the inequalities in dimension-independent form, a key feature especially in the context of applications to infinite-dimensional settings such as for quantum field theories. Gross's logarithmic Sobolev inequalities proved to be of great significance well beyond their original intended scope of application, for example in the proof of the Poincaré conjecture by Grigori Perelman.
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A log profile, or logarithmic profile, is a shooting profile, or gamma curve, found on some digital video cameras that gives a wide dynamic and tonal range, allowing more latitude to apply colour and style choices. The resulting image appears washed out, requiring color grading in post-production, but retains shadow and highlight detail that would otherwise be lost if a regular linear profile had been used that clipped shadow and highlight detail. The feature is mostly used in filmmaking and videography.
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While the image is opened for editing, the user is provided with a preview window with pan and zoom capabilities. A color histogram is also present offering linear and logarithmic scales and separate R, G, B and L channels. All adjustments are reflected in the history queue and the user can revert any of the changes at any time. There is also the possibility of taking multiple snapshots of the history queue allowing for various versions of the image being shown.
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A spectrogram of a violin waveform, with linear frequency on the vertical axis and time on the horizontal axis. The bright lines show how the spectral components change over time. The intensity colouring is logarithmic (black is −120 dBFS). Music theory has no axiomatic foundation in modern mathematics, although some interesting work has recently been done in this direction (see the External Links), yet the basis of musical sound can be described mathematically (in acoustics) and exhibits "a remarkable array of number properties".
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The exponential curve shape is in many ways the precise opposite of the logarithmic curves. The fade-in works as follows: it increases in volume slowly and then it shoots up very quickly at the end of the fade. The fade-out drops very quickly (from the maximum volume) and then declines slowly again over the duration of the fade. Simply stated a linear fade could thus be seen as an exaggerated version of an exponential fade in terms of the apparent volume.
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As was normal at the time, memory was not preserved on power-down. Its principal functions were (1) time value of money (TVM) calculations, where the user could enter any three of the variables and the fourth would be calculated, and (2) statistics calculations, including linear regression. Basic logarithmic and exponential functions were also provided. For TVM calculations, a physical slider switch labelled "begin" and "end" could be used to specify whether payments would be applied at the beginning or end of periods.
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Ragone plot showing specific energy versus specific power for various energy- storing devices A Ragone plot ( ) is a plot used for comparing the energy density of various energy-storing devices. On such a chart the values of specific energy (in W·h/kg) are plotted versus specific power (in W/kg). Both axes are logarithmic, which allows comparing performance of very different devices. Ragone plots can reveal information about gravimetric energy density, but do not convey details about volumetric energy density.
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The filter spacing is chosen to be logarithmic above 1 kHz > and the filter bandwidths are increased there as well. We will, therefore, > call these the mel-based cepstral parameters. Sometimes both early originators are cited. Many authors, including Davis and Mermelstein, have commented that the spectral basis functions of the cosine transform in the MFC are very similar to the principal components of the log spectra, which were applied to speech representation and recognition much earlier by Pols and his colleagues.
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The modern scale was mathematically defined in a way to closely match this historical system. The scale is reverse logarithmic: the brighter an object is, the lower its magnitude number. A difference of 1.0 in magnitude corresponds to a brightness ratio of , or about 2.512. For example, a star of magnitude 2.0 is 2.512 times brighter than a star of magnitude 3.0, 6.31 times brighter than a star of magnitude 4.0, and 100 times brighter than one of magnitude 7.0.
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According to Mitschke, "The advantage of using a logarithmic measure is that in a transmission chain, there are many elements concatenated, and each has its own gain or attenuation. To obtain the total, addition of decibel values is much more convenient than multiplication of the individual factors." However, for the same reason that humans excel at additive operation over multiplication, decibels are awkward in inherently additive operations:R. J. Peters, Acoustics and Noise Control, Routledge, 12 November 2013, 400 pages, p.
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Charpentier et al. suggested that competence for transformation probably evolved as a DNA damage response. Logarithmically growing bacteria differ from stationary phase bacteria with respect to the number of genome copies present in the cell, and this has implications for the capability to carry out an important DNA repair process. During logarithmic growth, two or more copies of any particular region of the chromosome may be present in a bacterial cell, as cell division is not precisely matched with chromosome replication.
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The scroll of a double bass A scroll is the decoratively carved beginning of the neck of certain stringed instruments, mainly members of the violin family. The scroll is typically carved in the shape of a volute (a rolled-up spiral) according to a canonical pattern, although some violins are adorned with carved heads, human and animal. The quality of a scroll is one of the things used to judge the luthier's skill. Instrument scrolls usually approximate a logarithmic spiral.
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Accurate sound level measurement devices were not invented until after the microphone or for that matter after the proliferation of the SI system in physics and internationally. Hence, although certain information on sound pressure can theoretically be evaluated in terms of pounds per square inch (PSI), this is virtually never done. Instead, the internationally used decibel (dB) scale is most common. In some cases where there is a desire for a non-logarithmic scale, the decibel-related sone scale is used.
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Sound field of a non focusing 4 MHz ultrasonic transducer with a near field length of N = 67 mm in water. The plot shows the sound pressure at a logarithmic db-scale. Sound pressure field of the same ultrasonic transducer (4 MHz, N = 67 mm) with the transducer surface having a spherical curvature with the curvature radius R = 30 mm Ultrasonic transducers convert AC into ultrasound, as well as the reverse. Ultrasonics, typically refers to piezoelectric transducers or capacitive transducers.
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Pre-Calculus served as the introductory mathematics class at TGA for those who had not taken the course elsewhere or those who are not prepared to take Calculus upon arrival at TGA. The course was similar in content to the Math 130 course offered at the University of Tennessee at Knoxville. Generally speaking, the course content serves to prepare juniors to take Calculus during the Spring semester of the junior year. Content includes a review of algebraic, logarithmic, exponential, and trigonometric functions.
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In fact, it can be traced all the way back to the Hellenistic Civilization. While people have devised such machines over the centuries, mathematicians continued to perform calculations by hand, a machines offered little advantage in speed. For complicated calculations, they employed tables, especially of logarithmic and trigonometric functions, which were computed by hand. But right in the middle of the Industrial Revolution in England, Charles Babbage thought of using the all-important steam engine to power a mechanical computer, the Difference Engine.
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The acceleration structures permitted by the Monte Carlo Polarization consist mainly in BVH and EBVH hierarchies. The logical subdivision of the kernel space leads to a logarithmic complexity, which is key to the scalability of the sentient analysis tools. A key application is the direct targeting of hidden nodes in neural networks. By applying a Monte Carlo Polarization filter to the input layer of the neural system, hidden layers will be systematically and dynamically selected based on user-defined characteristics.
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Poly-APX-complete, consisting of the hardest problems that can be approximated efficiently to within a factor polynomial in the input size, includes max independent set in the general case. There also exist problems that are exp-APX-complete, where the approximation ratio is exponential in the input size. This may occur when the approximation is dependent on the value of numbers within the problem instance; these numbers may be expressed in space logarithmic in their value, hence the exponential factor.
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The two-tree broadcast (abbreviated 2tree-broadcast or 23-broadcast) is an algorithm that implements a broadcast communication pattern on a distributed system using message passing. A broadcast is a commonly used collective operation that sends data from one processor to all other processors. The two- tree broadcast communicates concurrently over two binary trees that span all processors. This achieves full usage of the bandwidth in the full-duplex communication model while having a startup latency logarithmic in the number of partaking processors.
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In computer science, a B-tree is a self-balancing tree data structure that maintains sorted data and allows searches, sequential access, insertions, and deletions in logarithmic time. The B-tree generalizes the binary search tree, allowing for nodes with more than two children. Unlike other self-balancing binary search trees, the B-tree is well suited for storage systems that read and write relatively large blocks of data, such as disks. It is commonly used in databases and file systems.
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The perceptual strength of the guman visual model quantizer generation process is calibrated in visiBels (vB), a logarithmic scale roughly corresponding to clarity of the visual as measured in screen heights. As the eye moves further from the screen, it becomes less able to perceive details in the image. The ZPEG model also includes a temporal component, and thus is not fully described by viewing distance. In terms of viewing distance, the visiBel strength increases by six as the screen distance halves.
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It had a complete instruction set for controlling its arithmetic units. The algorithms for trigonometric and logarithmic algorithms took advantage of this instruction set which has put the slide rule out of business. But it also opened the door to a new genre of pocket calculators. Using the same hardware as the HP-35, France designed the HP-80 business calculator, that replaced reams of tables used to compute mortgages, returns on investments and other business transactions with a dedicated keyboard.
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In the 17th century, the method of exhaustion led to the rectification by geometrical methods of several transcendental curves: the logarithmic spiral by Evangelista Torricelli in 1645 (some sources say John Wallis in the 1650s), the cycloid by Christopher Wren in 1658, and the catenary by Gottfried Leibniz in 1691. In 1659, Wallis credited William Neile's discovery of the first rectification of a nontrivial algebraic curve, the semicubical parabola. The accompanying figures appear on page 145. On page 91, William Neile is mentioned as Gulielmus Nelius.
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In probability theory and computer science, a log probability is simply a logarithm of a probability. The use of log probabilities means representing probabilities on a logarithmic scale, instead of the standard [0, 1] unit interval. Since the probability of independent events multiply, and logarithms convert multiplication to addition, log probabilities of independent events add. Log probabilities are thus practical for computations, and have an intuitive interpretation in terms of information theory: the negative of the log probability is the information content of an event.
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Activation of neurons by sensory stimuli in many parts of the brain is by a proportional law: neurons change their spike rate by about 10–30%, when a stimulus (e.g. a natural scene for vision) has been applied. However, as Scheler (2017) showed, the population distribution of the intrinsic excitability or gain of a neuron is a heavy tail distribution, more precisely a lognormal shape, which is equivalent to a logarithmic coding scheme. Neurons may therefore spike with 5–10 fold different mean rates.
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Well-known examples are the indication of the earthquake strength using the Richter scale, the pH value, as a measure for the acidic or basic character of an aqueous solution or of loudness in decibels . In this case, the negative decimal logarithm of the LD50 values, which is standardized in kg per kg body weight, is considered. : − log10LD50 (kg/kg) = value The dimensionless value found can be entered in a toxin scale. Water as the baseline substance is neatly 1 in the negative logarithmic toxin scale.
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Many exercises are not strictly isotonic because the force on the muscle varies as the joint moves through its range of motion. Movements can become easier or harder depending on the angle of muscular force relative to gravity; for example, a standard biceps curl becomes easier as the hand approaches the shoulder as more of the load is taken by the structure of the elbow. Originating from Nautilus, Inc., some machines use a logarithmic-spiral cam to keep resistance constant irrespective of the joint angle.
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The slide rule, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century, and who also introduced the letter as the base of natural logarithms. Logarithmic scales reduce wide-ranging quantities to tiny scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example).
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From the perspective of group theory, the identity expresses a group isomorphism between positive reals under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups., section V.4.1 By means of that isomorphism, the Haar measure (Lebesgue measure) dx on the reals corresponds to the Haar measure on the positive reals., section 1.4 The non- negative reals not only have a multiplication, but also have addition, and form a semiring, called the probability semiring; this is in fact a semifield.
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The space hierarchy theorem guarantees that DSPACE(logd n) ⊊ DSPACE(logd + 1 n) for all integers d > 0. If polyL had a complete problem, call it A, it would be an element of DSPACE(logk n) for some integer k > 0. Suppose problem B is an element of DSPACE(logk + 1 n) \ DSPACE(logk n). The assumption that A is complete implies the following O(logk n) space algorithm for B: reduce B to A in logarithmic space, then decide A in O(logk n) space.
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The logarithmic rule An example of probabilistic forecasting is in meteorology where a weather forecaster may give the probability of rain on the next day. One could note the number of times that a 25% probability was quoted, over a long period, and compare this with the actual proportion of times that rain fell. If the actual percentage was substantially different from the stated probability we say that the forecaster is poorly calibrated. A poorly calibrated forecaster might be encouraged to do better by a bonus system.
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A set of John Napier's calculating tables from around 1680 Scottish mathematician and physicist John Napier discovered that the multiplication and division of numbers could be performed by the addition and subtraction, respectively, of the logarithms of those numbers. While producing the first logarithmic tables, Napier needed to perform many tedious multiplications. It was at this point that he designed his 'Napier's bones', an abacus-like device that greatly simplified calculations that involved multiplication and division.A Spanish implementation of Napier's bones (1617), is documented in .
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The algorithm can be implemented using a pattern matching machine. The algorithm can also be implemented to run on a nondeterministic Turing machine that uses only logarithmic space; the problem of testing unique decipherability is NL-complete, so this space bound is optimal. proves that the complementary problem, of testing for the existence of a string with two decodings, is NL-complete, and therefore that unique decipherability is co-NL-complete. The equivalence of NL-completeness and co-NL-completeness follows from the Immerman–Szelepcsényi theorem.
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Hipparchus is only conjectured to have ranked the apparent magnitudes of stars on a numerical scale from 1, the brightest, to 6, the faintest. Quote by Toomer, not Ptolemy. Nevertheless, this system certainly precedes Ptolemy, who used it extensively about 150. This system was made more precise and extended by N. R. Pogson in 1856, who placed the magnitudes on a logarithmic scale, making magnitude 1 stars 100 times brighter than magnitude 6 stars, thus each magnitude is or 2.512 times brighter than the next faintest magnitude.
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The equation is valid for a range of idle currents over , allowing wide tuning opportunity. The circuit has fast attack and slow decay, which are locked to each other and cannot be adjusted separately. Logarithmic output voltage is proportional to the mean of the square at a rate of around 3 mV/dB, and proportional to RMS at around 6 mV/dB. When the crude test circuit was built, Blackmer and his associates did not expect it to work as a true RMS detector, but it did.
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Already at the end of the 1970s, applications for the discrete spiral coordinate system were given in image analysis. To represent an image in this coordinate system rather than in Cartesian coordinates, gives computational advantages when rotating or zooming in an image. Also, the photo receptors in the retina in the human eye are distributed in a way that has big similarities with the spiral coordinate system.Weiman, Chaikin, Logarithmic Spiral Grids for Image Processing and Display, Computer Graphics and Image Processing 11, 197-226 (1979).
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In most cases, the creep modulus, defined as the ratio of applied stress to the time-dependent strain, decreases with increasing temperature. Generally speaking, an increase in temperature correlates to a logarithmic decrease in the time required to impart equal strain under a constant stress. In other words, it takes less work to stretch a viscoelastic material an equal distance at a higher temperature than it does at a lower temperature. More detailed effect of temperature on the viscoelastic behavior of polymer can be plotted as shown.
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Indus Valley Civilization weights and measures - National Museum, New Delhi. Hindu units of time—largely of mythological and ritual importance—displayed on a logarithmic scale. Standard weights and measures were developed by the Indus Valley Civilization. The centralised weight and measure system served the commercial interest of Indus merchants as smaller weight measures were used to measure luxury goods while larger weights were employed for buying bulkier items, such as food grains etc.Kenoyer, 265 Weights existed in multiples of a standard weight and in categories.
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The output from a colorimeter may be displayed by an analogue or digital meter and may be shown as transmittance (a linear scale from 0-100%) or as absorbance (a logarithmic scale from zero to infinity). The useful range of the absorbance scale is from 0-2 but it is desirable to keep within the range 0-1 because, above 1, the results become unreliable due to scattering of light. In addition, the output may be sent to a chart recorder, data logger, or computer.
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On a random-access Turing machine, there is a special pointer tape of logarithmic space accepting a binary vocabulary. The Turing machine has a special state such that when the binary number on the pointer tape is 'p', the Turing machine will write on the working tape the pth symbol of the input. This lets the Turing machine read any letter of the input without taking time to move over the entire input. This is mandatory for complexity classes using less than linear time.
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The logarithmic flow profile has long been observed in the ocean, but recent, highly sensitive measurements reveal a sublayer within the surface layer in which turbulent eddies are enhanced by the action of surface waves. It is becoming clear that the surface layer of the ocean is only poorly modeled as being up against the "wall" of the air-sea interaction. Observations of turbulence in Lake Ontario reveal under wave-breaking conditions the traditional theory significantly underestimates the production of turbulent kinetic energy within the surface layer.
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Total radiated power (TRP) is the sum of all RF power radiated by the antenna when the source power is included in the measurement. TRP is expressed in watts or the corresponding logarithmic expressions, often dBm or dBW. When testing mobile devices, TRP can be measured while in close proximity of power- absorbing losses such as the body and hand of the user.Mobile Broadband Multimedia Networks: Techniques, Models and Tools for 4G by Luís M. Correia The TRP can be used to determine body loss (BoL).
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William Froude (1810–1879) Skew arch at Cowley Bridge Junction showing the complex brickwork The corne de vache or "cow's horn" method is another way of laying courses such that they meet the face of the arch orthogonally at all elevations. Hyde, 1899, op. cit., pp. 74–101. Unlike the helicoidal and logarithmic methods, in which the intrados of the arch barrel is cylindrical, the corne de vache method results in a warped hyperbolic paraboloid surface that dips in the middle, rather like a saddle.
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Spectrogram of a young girl saying "oh, no" In fluids such as air and water, sound waves propagate as disturbances in the ambient pressure level. While this disturbance is usually small, it is still noticeable to the human ear. The smallest sound that a person can hear, known as the threshold of hearing, is nine orders of magnitude smaller than the ambient pressure. The loudness of these disturbances is related to the sound pressure level (SPL) which is measured on a logarithmic scale in decibels.
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A solution of a strong alkali, such as sodium hydroxide, at concentration 1 mol dm−3, has a pH of 14. Thus, measured pH values will lie mostly in the range 0 to 14, though negative pH values and values above 14 are entirely possible. Since pH is a logarithmic scale, a difference of one pH unit is equivalent to a tenfold difference in hydrogen ion concentration. The pH of neutrality is not exactly 7 (25 °C), although this is a good approximation in most cases.
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A classical molecular dynamics computer simulation of a collision cascade in Au induced by a 10 keV Au self-recoil. This is a typical case of a collision cascade in the heat spike regime. Each small sphere illustrates the position of an atom, in a 2-atom-layer-thick cross section of a three-dimensional simulation cell. The colors show (on a logarithmic scale) the kinetic energy of the atoms, with white and red being high kinetic energy from 10 keV downwards, and blue being low.
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Ralph Hartley suggested the use of a logarithmic measure of information in 1928. Claude E. Shannon first used the word "bit" in his seminal 1948 paper "A Mathematical Theory of Communication". He attributed its origin to John W. Tukey, who had written a Bell Labs memo on 9 January 1947 in which he contracted "binary information digit" to simply "bit". Vannevar Bush had written in 1936 of "bits of information" that could be stored on the punched cards used in the mechanical computers of that time.
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A simple voltage/current regulator can be made from a resistor in series with a diode (or series of diodes). Due to the logarithmic shape of diode V-I curves, the voltage across the diode changes only slightly due to changes in current drawn or changes in the input. When precise voltage control and efficiency are not important, this design may be fine. Since the forward voltage of a diode is small, this kind of voltage regulator is only suitable for low voltage regulated output.
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A real implementation can skip M[0] and adjust the indices accordingly. Note that, at any point in the algorithm, the sequence :X[M[1 , X[M[2 , ..., X[M[L is increasing. For, if there is an increasing subsequence of length j ≥ 2 ending at X[M[j , then there is also a subsequence of length j-1 ending at a smaller value: namely the one ending at X[P[M[j ]. Thus, we may do binary searches in this sequence in logarithmic time.
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Complexity classes arising from other definitions of acceptance include RP, co-RP, and ZPP. If the machine is restricted to logarithmic space instead of polynomial time, the analogous RL, co-RL, and ZPL complexity classes are obtained. By enforcing both restrictions, RLP, co-RLP, BPLP, and ZPLP are yielded. Probabilistic computation is also critical for the definition of most classes of interactive proof systems, in which the verifier machine depends on randomness to avoid being predicted and tricked by the all-powerful prover machine.
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A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties (or a set of algebraic varieties) to the real numbers. For instance, the classical or naive height over the rational numbers is typically defined to be the maximum of the numerators and denominators of the coordinates (e.g. 3 for the coordinates ), but in a logarithmic scale.
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Estimates of how much processing power is needed to emulate a human brain at various levels (from Ray Kurzweil, and Anders Sandberg and Nick Bostrom), along with the fastest supercomputer from TOP500 mapped by year. Note the logarithmic scale and exponential trendline, which assumes the computational capacity doubles every 1.1 years. Kurzweil believes that mind uploading will be possible at neural simulation, while the Sandberg, Bostrom report is less certain about where consciousness arises. For low-level brain simulation, an extremely powerful computer would be required.
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If the coefficient of the variable is not zero (), then a linear function is represented by a degree 1 polynomial (also called a linear polynomial), otherwise it is a constant function – also a polynomial function, but of zero degree. A straight line, when drawn in a different kind of coordinate system may represent other functions. For example, it may represent an exponential function when its values are expressed in the logarithmic scale. It means that when is a linear function of , the function is exponential.
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The GDI is often considered a "gender-sensitive extension of the HDI" (Klasen 245). It addresses gender-gaps in life expectancy, education, and incomes. It uses an "inequality aversion" penalty, which creates a development score penalty for gender gaps in any of the categories of the Human Development Index which include life expectancy, adult literacy, school enrollment, and logarithmic transformations of per-capita income. In terms of life expectancy, the GDI assumes that women will live an average of five years longer than men.
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Schematic representation of the energy levels (Jabłoński diagrams) of the fluorescence process, example of a fluorescent dye that emits light at 460 nm. One (purple, 1PEF), two (light red, 2PEF) or three (dark red, 3PEF) photons are absorbed to emit a photon of fluorescence (turquoise). Optical response from a point source. From left to right: calculated intensity distributions xy (top) and rz (bottom), with logarithmic scale, for a point source imaged by means of a wide field (a), 2PEF (b) and confocal microscopy (c).
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Under these hypotheses, the test if a word is in the dictionary may be done in logarithmic time: consider D(\lfloor n/2 \rfloor), where \lfloor\;\rfloor denotes the floor function. If w=D(\lfloor n/2 \rfloor), then we are done. Else, if w continue the search in the same way in the left half of the dictionary, otherwise continue similarly with the right half of the dictionary. This algorithm is similar to the method often used to find an entry in a paper dictionary.
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It incorporated the first multiplier-accumulator (MAC), and was the first to exploit a MAC to perform division (using multiplication seeded by reciprocal, via the convergent series ). Ludgate's engine used a mechanism similar to slide rules, but employed his unique discrete Logarithmic Indexes (now known as Irish logarithms (Boys, 1909)), and provided a very novel memory using concentric cylinders, storing numbers as displacements of rods in shuttles. His design featured several other novel features, including for program control (e.g. preemption and subroutines – or microcode, depending on viewpoint).
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Self-similarity means that a pattern is non- trivially similar to itself, e.g., the set of numbers of the form where ranges over all integers. When this set is plotted on a logarithmic scale it has one- dimensional translational symmetry: adding or subtracting the logarithm of two to the logarithm of one of these numbers produces the logarithm of another of these numbers. In the given set of numbers themselves, this corresponds to a similarity transformation in which the numbers are multiplied or divided by two.
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When a photographic film is exposed to light, the result of the exposure can be represented on a graph showing log of exposure on the horizontal axis, and density, or log of transmittance, on the vertical axis. For a given film formulation and processing method, this curve is its characteristic or Hurter–Driffield curve.Kodak, "Basic sensitometry and characteristics of film" : "A characteristic curve is like a film’s fingerprint." Since both axes use logarithmic units, the slope of the linear section of the curve is called the gamma of the film.
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A Cartesian grid is a special case where the elements are unit squares or unit cubes, and the vertices are points on the integer lattice. A rectilinear grid is a tessellation by rectangles or rectangular cuboids (also known as rectangular parallelepipeds) that are not, in general, all congruent to each other. The cells may still be indexed by integers as above, but the mapping from indexes to vertex coordinates is less uniform than in a regular grid. An example of a rectilinear grid that is not regular appears on logarithmic scale graph paper.
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There a new driver was supplied by West Coast Railway Company (WCRC) who drove the train to , where the train was terminated. The incident was rated the most serious SPAD since December 2010, rating 25 out of 28 on Network Rail's scale. Any SPAD rated at 20 or more leads to a mandatory investigation by the Office of Rail and Road (ORR). The scale is logarithmic, with each increment rated twice as serious as the previous; thus the incident was rated as nominally over thirty times more serious than this threshold.
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He notes that the mathematics of these are similar but the biology differs. He describes the spiral of Archimedes before moving on to the logarithmic spiral, which has the property of never changing its shape: it is equiangular and is continually self- similar. Shells as diverse as Haliotis, Triton, Terebra and Nautilus (illustrated with a halved shell and a radiograph) have this property; different shapes are generated by sweeping out curves (or arbitrary shapes) by rotation, and if desired also by moving downwards. Thompson analyses both living molluscs and fossils such as ammonites.
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The Carlsberg Laboratory was known for isolating Saccharomyces carlsbergensis, the species of yeast responsible for lager fermentation, as well as introducing the concept of pH in acid-base chemistry. The Danish chemist Søren Peder Lauritz Sørensen introduced the concept of pH, a scale for measuring acidity and basicity of substances. While working at the Carlsberg Laboratory, he studied the effect of ion concentration on proteins, and understood the concentration of hydrogen ions was particularly important. To express the hydronium ion (H3O+) concentration in a solution, he devised a logarithmic scale known as the pH scale.
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Further research into crackling noise was done in the late 1940s by Charles Francis Richter and Beno Gutenberg who examined earthquakes analytically. Before the invention of the well-known Richter scale, the Mercalli intensity scale was used; this is a subjective measurement of how damaging an earthquake was to property, i.e. II would be small vibrations and objects moving, while XII would be wide spread destruction of all buildings. The Richter scale is a logarithmic scale which measures the energy and amplitude of vibrations dissipated from the epicentre of the earthquake, i.e.
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He moved on to the Universität Hamburg in 2008, and in 2011, he became the head of the Graduiertenkolleg Mathematics Inspired by String Theory and QFT. In 2018, Siebert joined the faculty of the University of Texas at Austin as a professor of mathematics and holds the Sid W. Richardson Foundation Regents Chair in Mathematics #4. In his research, Bernd Siebert contributed substantially to the theory of Gromov–Witten invariants. Around 2002 by his insights in logarithmic geometry, he entered into an ongoing joint research program with Mark Gross.
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A Doyle spiral of type (8,16) printed in 1911 in Popular Science as an illustration of phyllotaxis. One of its spiral arms is shaded. Coxeter's loxodromic sequence of tangent circles, a Doyle spiral of type (1,3) In the mathematics of circle packing, a Doyle spiral is a pattern of non-crossing circles in the plane, each tangent to six others. The sequences of circles linked to each other through opposite points of tangency lie on logarithmic spirals (or, in degenerate cases, circles or lines) having, in general, three different shapes of spirals.
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A nondeterministic algorithm for determining whether a 2-satisfiability instance is not satisfiable, using only a logarithmic amount of writable memory, is easy to describe: simply choose (nondeterministically) a variable v and search (nondeterministically) for a chain of implications leading from v to its negation and then back to v. If such a chain is found, the instance cannot be satisfiable. By the Immerman–Szelepcsényi theorem, it is also possible in nondeterministic logspace to verify that a satisfiable 2-satisfiability instance is satisfiable. 2-satisfiability is NL-complete,.
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Radiation pattern of a German parabolic antenna. The main lobe (top) is only a few degrees wide. The sidelobes are all at least 20 dB below (1/100 the power density of) the main lobe, and most are 30 dB below. (If this pattern was drawn with linear power levels instead of logarithmic dB levels, all lobes other than the main lobe would be much too small to see.) In parabolic antennas, virtually all the power radiated is concentrated in a narrow main lobe along the antenna's axis.
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One end of each manometer is connected to its respective plenum chamber while the other is open to the atmosphere. Ordinarily all flow bench manometers measure in inches of water although the inclined manometer's scale is usually replaced with a logarithmic scale reading in percentage of total flow of the selected metering element which makes flow calculation simpler. Temperature must also be accounted for because the air pump will heat the air passing through it making the air down stream of it less dense and more viscous. This difference must be corrected for.
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Abundance of elements in Earth's crust per million Si atoms (y axis is logarithmic) Rare-earth element cerium is actually the 25th most abundant element in Earth's crust, having 68 parts per million (about as common as copper). Only the highly unstable and radioactive promethium "rare earth" is quite scarce. The rare-earth elements are often found together. The longest-lived isotope of promethium has a half-life of 17.7 years, so the element exists in nature in only negligible amounts (approximately 572 g in the entire Earth's crust).
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World population, 10,000 BCE – 2,000 CE (vertical population scale is logarithmic) Human history, also known as world history, is the description of humanity's past. It is informed by archaeology, anthropology, genetics, linguistics, and other disciplines; and, for periods since the invention of writing, by recorded history and by secondary sources and studies. Humanity's written history was preceded by its prehistory, beginning with the Palaeolithic Era ("Old Stone Age"), followed by the Neolithic Era ("New Stone Age"). The Neolithic saw the Agricultural Revolution begin, between 10,000 and 5000 BCE, in the Near East's Fertile Crescent.
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In many cases, the scale used is linear, not logarithmic. According to Aldershof, the inverted scale is merely accidental. During an early development session, Dr. Zakon sketched the two- by-two matrix on a chalkboard, and labeled the sections as High and Low from top to bottom, and High and Low from left to right, simply for convenience. The format persisted through later development sessions, without anyone considering that in the future the axis might become calibrated— where a low- to-high scale from left to right would be more typical and easier to produce.
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The Bohr–Mollerup theorem is useful because it is relatively easy to prove logarithmic convexity for any of the different formulas used to define the gamma function. Taking things further, instead of defining the gamma function by any particular formula, we can choose the conditions of the Bohr–Mollerup theorem as the definition, and then pick any formula we like that satisfies the conditions as a starting point for studying the gamma function. This approach was used by the Bourbaki group. Borwein & Corless review three centuries of work on the gamma function.
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The proper EV was determined by the scene luminance and film speed; it was intended that the system also include adjustment for filters, exposure compensation, and other variables. With all of these elements included, the camera would be set by transferring the single number thus determined. Exposure value has been indicated in various ways. The ASA and ANSI standards used the quantity symbol Ev, with the subscript v indicating the logarithmic value; this symbol continues to be used in ISO standards, but the acronym EV is more common elsewhere.
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He contributed to p-adic Hodge theory, logarithmic geometry (he was one of its creators together with Jean-Marc Fontaine and Luc Illusie), comparison conjectures, special values of zeta functions including the Birch-Swinnerton- Dyer conjecture and Bloch-Kato conjecture on Tamagawa numbers, Iwasawa theory. A special volume of Documenta Mathematica was published in honor of his 50th birthday, together with research papers written by leading number theorists and former students it contains Kato's song on Prime Numbers. In 2005 Kato received the Imperial Prize of the Japan Academy for "Research on Arithmetic Geometry".
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In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function, whether applied to real numbers or complex numbers.
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The colors as shown on the logarithmic color scale indicate the irradiance (a,c) and spectral density (b,d) normalized to the maximum value. Although one typically thinks of an image as planar, or two- dimensional, the imaging system will produce a three-dimensional intensity distribution in image space that in principle can be measured. e.g. a two- dimensional sensor could be translated to capture a three-dimensional intensity distribution. The image of a point source is also a three dimensional (3D) intensity distribution which can be represented by a 3D point-spread function.
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The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (equal to 2πb if θ is measured in radians), hence the name "arithmetic spiral". In contrast to this, in a logarithmic spiral these distances, as well as the distances of the intersection points measured from the origin, form a geometric progression.207x207px The Archimedean spiral has two arms, one for θ > 0 and one for θ < 0\. The two arms are smoothly connected at the origin.
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It is indicated by log(x), log10(x), or sometimes Log(x) with a capital L (however, this notation is ambiguous, since it can also mean the complex natural logarithmic multi-valued function). On calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when they write "log". To mitigate this ambiguity, the ISO 80000 specification recommends that log10(x) should be written lg(x), and loge(x) should be ln(x). Page from a table of common logarithms.
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Niemeyer began to elaborate his big scale project "20 steps around the world" which would be installed in 1997 in the City of Ropinsalmi in Finland. In this project, he explained, the earth replaced the canvas. According to him, earth is the carrier of his artistic work being integrated into the creative process only by minimal changes. In the context of this work an arbitrarily defined route around the earth is divided systematically and exactly into 20 segments which develop to a dynamic, logarithmic progression according to the 'Golden Section'.
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Acid strength is commonly measured by two methods. One measurement, based on the Arrhenius definition of acidity, is pH, which is a measurement of the hydronium ion concentration in a solution, as expressed on a negative logarithmic scale. Thus, solutions that have a low pH have a high hydronium ion concentration and can be said to be more acidic. The other measurement, based on the Brønsted–Lowry definition, is the acid dissociation constant (Ka), which measures the relative ability of a substance to act as an acid under the Brønsted–Lowry definition of an acid.
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One of the most severe challenges for inflation arises from the need for fine tuning. In new inflation, the slow-roll conditions must be satisfied for inflation to occur. The slow-roll conditions say that the inflaton potential must be flat (compared to the large vacuum energy) and that the inflaton particles must have a small mass.Technically, these conditions are that the logarithmic derivative of the potential, \epsilon=(1/2)(V'/V)^2 and second derivative \eta=V/V are small, where V is the potential and the equations are written in reduced Planck units.
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In 2005, Christian Schindelhauer and Gunnar Schomaker described a logarithmic method for re-weighting hash scores in a way that does not require relative scaling of load factors when a node's weight changes or when nodes are added or removed. This enabled the dual benefits of perfect precision when weighting nodes, along with perfect stability, as only a minimum number of keys needed to be remapped to new nodes. A similar logarithm-based hashing strategy is used to assign data to storage nodes in Cleversafe's data storage system, now IBM Cloud Object Storage.
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NEAs are generally well known, though a few have been lost. However, large numbers of smaller NEAs have highly uncertain orbits The uncertainty parameter U is a parameter introduced by the Minor Planet Center (MPC) to quantify concisely the uncertainty of a perturbed orbital solution for a minor planet. The parameter is a logarithmic scale from 0 to 9 that measures the anticipated longitudinal uncertainty in the minor planet's mean anomaly after 10 years. The uncertainty parameter is also known as condition code in JPL's Small-Body Database Browser.
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Some hyoliths had helens, long structures that taper as they coil gently in a logarithmic spiral in a ventral direction. The helens had an organic-rich central core surrounded by concentric laminae of calcite. They grew by the addition of new material at their base, on the cavity side, leaving growth lines. They were originally described by Walcott as separate fossils under the genus name Helenia, (Walcott's wife was named Helena and his daughter Helen); Bruce Runnegar adopted the name helen when they were recognized as part of the hyolith organism.
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Jean Marguin, p. 93-94 (1994) Poleni described his machine in his Miscellanea in 1709, but it was also described by Jacob Leupold in his Theatrum Machinarum Generale ("The General Theory of Machines") which was published in 1727. In 1729, he also built a tractional device that enabled logarithmic functions to be drawn. Poleni's observations on the impact of falling weights (similar to Willem 's Gravesande's) led to a controversy with Samuel Clarke and other Newtonians that became a part of the so-called "vis viva dispute" in the history of classical mechanics.
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Note that potential of the working or active electrode is space charge sensitive and this is often used. Further, the label-free and direct electrical detection of small peptides and proteins is possible by their intrinsic charges using biofunctionalized ion-sensitive field-effect transistors. Another example, the potentiometric biosensor, (potential produced at zero current) gives a logarithmic response with a high dynamic range. Such biosensors are often made by screen printing the electrode patterns on a plastic substrate, coated with a conducting polymer and then some protein (enzyme or antibody) is attached.
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A diagram of a bitcoin transfer Number of bitcoin transactions per month (logarithmic scale) The bitcoin network is a peer-to-peer payment network that operates on a cryptographic protocol. Users send and receive bitcoins, the units of currency, by broadcasting digitally signed messages to the network using bitcoin cryptocurrency wallet software. Transactions are recorded into a distributed, replicated public database known as the blockchain, with consensus achieved by a proof-of-work system called mining. Satoshi Nakamoto, the designer of bitcoin, claimed that design and coding of bitcoin began in 2007.
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Semi-logarithmic graph for the determination of z-value "F0" is defined as the number of equivalent minutes of steam sterilization at temperature 121.1 °C (250 °F) delivered to a container or unit of product calculated using a z-value of 10 °C. The term F-value or "FTref/z" is defined as the equivalent number of minutes to a certain reference temperature (Tref) for a certain control microorganism with an established Z-value.Stumbo C.R., Thermobacteriology in Food Processing, 1973. , 9780080886473 Z-value is a term used in microbial thermal death time calculations.
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All magnitude scales retain the logarithmic scale as devised by Charles Richter, and are adjusted so the mid-range approximately correlates with the original "Richter" scale.. Most magnitude scales are based on measurements of only part of an earthquake's seismic wave-train, and therefore are incomplete. This results in systematic underestimation of magnitude in certain cases, a condition called saturation.. Since 2005 the International Association of Seismology and Physics of the Earth's Interior (IASPEI) has standardized the measurement procedures and equations for the principal magnitude scales, , , , and .IASPEI .
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The infrared sensor was developed by Hughes Research Laboratories. The sensor used a strip detector where four strips of Indium Bismuth were arranged in a cross and four strips were arranged as logarithmic spirals. As the detector was spun, the infrared target's position could be measured as it crossed the strips in the sensor's field of view. The MHV infrared detector was cooled by liquid helium from a dewar installed in place of the F-15's gun ammunition drum and from a smaller dewar located in the second stage of the ASM-135.
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In computer science, streaming algorithms are algorithms for processing data streams in which the input is presented as a sequence of items and can be examined in only a few passes (typically just one). In most models, these algorithms have access to limited memory (generally logarithmic in the size of and/or the maximum value in the stream). They may also have limited processing time per item. These constraints may mean that an algorithm produces an approximate answer based on a summary or "sketch" of the data stream.
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Following a path in a graph is an inherently serial operation, but pointer jumping reduces the total amount of work by following all paths simultaneously and sharing results among dependent operations. Pointer jumping iterates and finds a successor — a vertex closer to the tree root — each time. By following successors computed for other vertices, the traversal down each path can be doubled every iteration, which means that the tree roots can be found in logarithmic time. Pointer doubling operates on an array `successor` with an entry for every vertex in the graph.
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The first one to theorize about the marginal value of money was Daniel Bernoulli in 1738. He assumed that the value of an additional amount is inversely proportional to the pecuniary possessions which a person already owns. Since Bernoulli tacitly assumed that an interpersonal measure for the utility reaction of different persons can be discovered, he was then inadvertedly using an early conception of cardinality. Bernoulli's imaginary logarithmic utility function and Gabriel Cramer's function were conceived at the time not for a theory of demand but to solve the St. Petersburg's game.
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Walter John Savitch (born February 21, 1943) is best known for defining the complexity class NL (nondeterministic logarithmic space), and for Savitch's theorem, which defines a relationship between the NSPACE and DSPACE complexity classes. His work in establishing complexity classes has helped to create the background against which non-deterministic and probabilistic reasoning can be performed. He has also done extensive work in the field of natural language processing and mathematical linguistics. He has been focused on computational complexity as it applies to genetics and biology for over 10 years.
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Number of cases (blue) and number of deaths (red) on a logarithmic scale. By early April, the health system in Guayas Province was overwhelmed, and many of Ecuador's COVID-19 deaths were reported there. Corpses were abandoned on the street as local funeral homes were incapable of handling so much work.Corpses on the streets of Guayaquil, after the collapse of the Ecuadorian health system EFE/Diario de Yucatan, 1 Abril 2020 On 2 April president Lenín Moreno said that the government was building a "special camp" for the victims of the coronavirus in Guayaquil.
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There are no known general point location data structures with linear space and logarithmic query time for dimensions greater than 2. Therefore, we need to sacrifice either query time, or storage space, or restrict ourselves to some less general type of subdivision. In three-dimensional space, it is possible to answer point location queries in O(log² n) using O(n log n) space. The general idea is to maintain several planar point location data structures, corresponding to the intersection of the subdivision with n parallel planes that contain each subdivision vertex.
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The Second Fundamental Theorem allows to give an upper bound for the characteristic function in terms of N(r,a). For example, if f is a transcendental entire function, using the Second Fundamental theorem with k = 3 and a3 = ∞, we obtain that f takes every value infinitely often, with at most two exceptions, proving Picard's Theorem. Nevanlinna's original proof of the Second Fundamental Theorem was based on the so-called Lemma on the logarithmic derivative, which says that m(r,f'/f) = S(r,f). A similar proof also applies to many multi-dimensional generalizations.
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Johan 't Hart (1981) found that JND for speech averaged between 1 and 2 STs but concluded that "only differences of more than 3 semitones play a part in communicative situations" ('t Hart, 1981, page 811). Note that, given the logarithmic characteristics of Hz, for both music and speech perception results should not be reported in Hz but either as percentages or in STs (5 Hz between 20 and 25 Hz is very different from 5 Hz between 2000 and 2005 Hz, but the same when reported as a percentage or in STs).
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Number of bitcoin transactions per month (logarithmic scale) Bitcoin is a cryptocurrency, a digital asset designed to work as a medium of exchange that uses cryptography to control its creation and management, rather than relying on central authorities. It was invented and implemented by the presumed pseudonymous Satoshi Nakamoto, who integrated many existing ideas from the cypherpunk community. Over the course of bitcoin's history, it has undergone rapid growth to become a significant currency both on- and offline. From the mid 2010s, some businesses began accepting bitcoin in addition to traditional currencies.
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Instructions for ratio calculations and wind problems are printed on either side of the computer for reference and are also found in a booklet sold with the computer. Also, many computers have Fahrenheit to Celsius conversion charts and various reference tables. The front side of the flight computer is a logarithmic slide rule that performs multiplication and division. Throughout the wheel, unit names are marked (such as gallons, miles, kilometers, pounds, minutes, seconds, etc.) at locations that correspond to the constants that are used when going from one unit to another in various calculations.
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While not included as a SI Unit in the International System of Quantities, several ratio measures are included by the International Committee for Weights and Measures (CIPM) as acceptable in the "non-SI unit" category. The level of a quantity is a logarithmic quantification of the ratio of the quantity with a stated reference value of that quantity. It is differently defined for a root-power quantity (also known by the deprecated term field quantity) and for a power quantity. It is not defined for ratios of quantities of other kinds.
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HE1327-2326, discovered in 2005 by Anna Frebel and collaborators, was the star with the lowest known iron abundance until SMSS J031300.36−670839.3 was discovered. The star is a member of Population II stars, with a solar- standardised iron to hydrogen index (Fe:H), or metallicity, of −5.6. The scale being logarithmic, this number indicates that its iron content is 1/400,000 that of the Earth's sun. However, it has a carbon abundance of roughly one- tenth solar ([C/H] = −1.0), and it is not known how these two abundances can have been produced/exist simultaneously.
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In computer science, a finger tree is a purely functional data structure that can be used to efficiently implement other functional data structures. A finger tree gives amortized constant time access to the "fingers" (leaves) of the tree, which is where data is stored, and concatenation and splitting logarithmic time in the size of the smaller piece. It also stores in each internal node the result of applying some associative operation to its descendants. This "summary" data stored in the internal nodes can be used to provide the functionality of data structures other than trees.
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The majority problem, or density classification task, is the problem of finding one-dimensional cellular automaton rules that accurately perform majority voting. Using local transition rules, cells cannot know the total count of all the ones in system. In order to count the number of ones (or, by symmetry, the number of zeros), the system requires a logarithmic number of bits in the total size of the system. It also requires the system send messages over a distance linear in the size of the system and for the system to recognize a non-regular language.
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This reflects that the increase in the Gini coefficient of the US in this time period is due to gains by upper income earners (relative to the median), rather than by losses by lower income earners (relative to the median). This graph shows the income of the given percentiles from 1947 to 2010 in 2010 dollars. The 2 columns of numbers in the right margin are the cumulative growth 1970-2010 and the annual growth rate over that period. The vertical scale is logarithmic, which makes constant percentage growth appear as a straight line.
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The hues of the Munsell color system, at varying values, and maximum chroma to stay in the sRGB gamut. a similar shade to the cloth. The color defined as purple in the Munsell color system (Munsell 5P) is shown at right. The Munsell color system is a color space that specifies colors based on three color dimensions: hue, value (lightness), and chroma (color purity), spaced uniformly in three dimensions in the elongated oval at an angle shaped Munsell color solid according to the logarithmic scale which governs human perception.
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The calculations are weighted in favor of the UV wavelengths to which human skin is most sensitive, according to the CIE-standard McKinlay–Diffey erythemal action spectrum. The resulting UV index cannot be expressed in pure physical units, but is a good indicator of likely sunburn damage. Because the index scale is linear (and not logarithmic, as is often the case when measuring things such as brightness or sound level), it is reasonable to assume that one hour of exposure at index 5 is approximately equivalent to a half-hour at index 10.
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Their names were changed in the 1980s to be the same in any language. I-time-weighting is no longer in the body of the standard because it has little real correlation with the impulsive character of noise events. The output of the RMS circuit is linear in voltage and is passed through a logarithmic circuit to give a readout linear in decibels (dB). This is 20 times the base 10 logarithm of the ratio of a given root-mean-square sound pressure to the reference sound pressure.
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The TI-32 Math Explorer Plus is a calculator by Texas Instruments specifically designed for middle school students. The Math Explorer Plus was offered as a more advanced version of the TI-12 Math Explorer. The TI-32 Math Explorer Plus offered trigonometric, exponential, logarithmic, and probability functions, and thus can be considered a true scientific calculator unlike the TI-12 Math Explorer. The Math Explorer Plus was eventually replaced by the TI-34 II Explorer Plus, which combined features of the TI-32 and TI-34, as well as incorporating a two-line display.
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Here the quantity that's measured in bits is the logarithmic information measure mentioned above. Hence there are N bits of surprisal in landing all heads on one's first toss of N coins. The additive nature of surprisals, and one's ability to get a feel for their meaning with a handful of coins, can help one put improbable events (like winning the lottery, or having an accident) into context. For example if one out of 17 million tickets is a winner, then the surprisal of winning from a single random selection is about 24 bits.
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If the largest competitor had a share of 60 percent, however, the ratio would be 1:3, implying that the organization's brand was in a relatively weak position. If the largest competitor only had a share of 5 percent, the ratio would be 4:1, implying that the brand owned was in a relatively strong position, which might be reflected in profits and cash flows. If this technique is used in practice, this scale is logarithmic, not linear. On the other hand, exactly what is a high relative share is a matter of some debate.
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The dimension of power is energy divided by time. In the International System of Units (SI), the unit of power is the watt (W), which is equal to one joule per second. Other common and traditional measures are horsepower (hp), comparing to the power of a horse; one mechanical horsepower equals about 745.7 watts. Other units of power include ergs per second (erg/s), foot-pounds per minute, dBm, a logarithmic measure relative to a reference of 1 milliwatt, calories per hour, BTU per hour (BTU/h), and tons of refrigeration.
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A traditional S-curve fade-out: is logarithmic from the beginning up to the midpoint, then its attributes are based on the exponential curve from the midpoint to the end. This is true for the situation in reverse as well (for both fade-in and fade-out). Cross-fading S-curves works as follows; it diminishes the amount of time that both sounds are playing simultaneously. This ensures that the edits sound like a direct cut when the two edits meet - this adds an extra smoothness to the edited regions.
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Nevertheless, the main limitation of CAFM preamplifiers is their narrow current dynamic range, which usually allows collecting electrical signals only within three or four orders of magnitude (or even less). To solve this problem preamplifiers with an adjustable gain can be used to focus on specific ranges. A more sophisticated solution for this problem is to combine the CAFM with a sourcemeter, semiconductor parameter analyzer or with a logarithmic preamplifier, which can capture the currents flowing through the tip/sample system at any range and with a high resolution.
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Uffe Haagerup was born in Kolding, but grew up on the island of Funen, in the small town of Fåborg. The field of mathematics had his interest from early on, encouraged and inspired by his older brother. In fourth grade Uffe was doing trigonometric and logarithmic calculations. He graduated as a student from Svendborg Gymnasium in 1968, whereupon he relocated to Copenhagen and immediately began his studies of mathematics and physics at the University of Copenhagen, again inspired by his older brother who also studied the same subjects at the same university.
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Photometric measurements are made in the ultraviolet, visible, or infrared wavelength bands using standard passband filters belonging to photometric systems such as the UBV system or the Strömgren uvbyβ system. Absolute magnitude is a measure of the intrinsic luminosity of a celestial object rather than its apparent brightness and is expressed on the same reverse logarithmic scale. Absolute magnitude is defined as the apparent magnitude that a star or object would have if it were observed from a distance of . When referring to just "magnitude", apparent magnitude rather than absolute magnitude is normally intended.
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This allows smaller space classes, such as L (logarithmic space), to be defined in terms of the amount of space used by all of the work tapes (excluding the special input and output tapes). Since many symbols might be packed into one by taking a suitable power of the alphabet, for all c ≥ 1 and f such that f(n) ≥ 1, the class of languages recognizable in c f(n) space is the same as the class of languages recognizable in f(n) space. This justifies usage of big O notation in the definition.
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The process of homologous recombinational repair (HRR) is a key DNA repair process that is especially effective for repairing double- strand damages, such as double-strand breaks. This process depends on a second homologous chromosome in addition to the damaged chromosome. During logarithmic growth, a DNA damage in one chromosome may be repaired by HRR using sequence information from the other homologous chromosome. Once cells approach stationary phase, however, they typically have just one copy of the chromosome, and HRR requires input of homologous template from outside the cell by transformation.
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The logarithmic spiral had been rectified by Evangelista Torricelli and was the first curved line (other than the circle) whose length was determined, but the extension by Neile and Wallis to an algebraic curve was novel. The cycloid was the next curve rectified; this was done by Christopher Wren in 1658. Early in 1658 a similar discovery, independent of that of Neile, was made by van Heuraët, and this was published by van Schooten in his edition of Descartes's Geometria in 1659. Van Heuraët's method is as follows.
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The 3 in column 4 means that three species, species "b", "c", and "d", have abundance four. The final 1 in column 10 means that one species, species "a", has abundance 10. This type of dataset is typical in biodiversity studies. Observe how more than half the biodiversity (as measured by species count) is due to singletons. For real datasets, the species abundances are binned into logarithmic categories, usually using base 2, which gives bins of abundance 0-1, abundance 1-2, abundance 2-4, abundance 4-8, etc.
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Even earlier (1709), Nicolas Bernoulli studies problems related to savings and interest in the Ars Conjectandi. In 1730, Daniel Bernoulli studied "moral probability" in his book Mensura Sortis, where he introduced what would today be called "logarithmic utility of money" and applied it to gambling and insurance problems, including a solution of the paradoxical Saint Petersburg problem. All of these developments were summarized by Laplace in his Analytical Theory of Probabilities (1812). Clearly, by the time David Ricardo came along he had a lot of well-established math to draw from.
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Since computational complexity measures difficulty with respect to the length of the (encoded) input, this naive algorithm is actually exponential. It is, however, pseudo-polynomial time. Contrast this algorithm with a true polynomial numeric algorithm — say, the straightforward algorithm for addition: Adding two 9-digit numbers takes around 9 simple steps, and in general the algorithm is truly linear in the length of the input. Compared with the actual numbers being added (in the billions), the algorithm could be called "pseudo-logarithmic time", though such a term is not standard.
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For species of mammals, larger brains (in absolute terms, not just in relation to body size) tend to have thicker cortices. The smallest mammals, such as shrews, have a neocortical thickness of about 0.5 mm; the ones with the largest brains, such as humans and fin whales, have thicknesses of 2–4 mm. There is an approximately logarithmic relationship between brain weight and cortical thickness. Magnetic resonance imaging of the brain (MRI) makes it possible to get a measure for the thickness of the human cerebral cortex and relate it to other measures.
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This profile is used to allow car hands-free kits to communicate with mobile phones in the car. It commonly uses Synchronous Connection Oriented link (SCO) to carry a monaural audio channel with continuously variable slope delta modulation or pulse-code modulation, and with logarithmic a-law or μ-law quantization. Version 1.6 adds optional support for wide band speech with the mSBC codec, a 16 kHz monaural configuration of the SBC codec mandated by the A2DP profile. Version 1.7 adds indicator support to report such things as headset battery level.
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On-line sodium measurement in ultrapure water most commonly uses a glass membrane sodium ion-selective electrode and a reference electrode in an analyzer measuring a small continuously flowing side-stream sample. The voltage measured between the electrodes is proportional to the logarithm of the sodium ion activity or concentration, according to the Nernst equation. Because of the logarithmic response, low concentrations in sub-parts per billion ranges can be measured routinely. To prevent interference from hydrogen ion, the sample pH is raised by the continuous addition of a pure amine before measurement.
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He converted the frequency intervals into a logarithmic musical scale, using the concept of cents from Ellis (1884) and Hornbostel (1920) and Reiner's concept of musical rule. In 1933, the colonial government commissioned Koesoemadinata to form a Sundanese music education system for all-indigenous schools in West Java. After the independence of Indonesia, from 1945 until 1947, he taught science, history and English for high school teachers in Bandung. The rest of his professional career was spent mainly as an expert for the Department of Culture of West Java in Bandung.
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The sentience quotient concept was introduced by Robert A. Freitas Jr. in the late 1970s. It defines sentience as the relationship between the information processing rate of each individual processing unit (neuron), the weight/size of a single unit, and the total number of processing units (expressed as mass). It was proposed as a measure for the sentience of all living beings and computers from a single neuron up to a hypothetical being at the theoretical computational limit of the entire universe. On a logarithmic scale it runs from −70 up to +50.
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But there are three feet in a yard, so the probability that the first digit of a length in yards is 1 must be the same as the probability that the first digit of a length in feet is 3, 4, or 5; similarly the probability that the first digit of a length in yards is 2 must be the same as the probability that the first digit of a length in feet is 6, 7, or 8\. Applying this to all possible measurement scales gives the logarithmic distribution of Benford's law.
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When Wang Laboratories found that the hp 9100A used an approach similar to the factor combining method in their earlier LOCI-1 (September 1964) and LOCI-2 (January 1965) Logarithmic Computing Instrument desktop calculators, they unsuccessfully accused Hewlett-Packard of infringement of one of An Wang's patents in 1968. John Stephen Walther at Hewlett-Packard generalized the algorithm into the Unified CORDIC algorithm in 1971, allowing it to calculate hyperbolic functions, natural exponentials, natural logarithms, multiplications, divisions, and square roots. The CORDIC subroutines for trigonometric and hyperbolic functions could share most of their code.
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It has been suggested that one integral facet of brain dynamics underlying conscious thought is the brain’s ability to convert seemingly noisy or chaotic signals into predictable oscillatory patterns. In EEG oscillations of neural networks, neighboring waveform frequencies are correlated on a logarithmic scale rather than a linear scale. As a result, mean frequencies in oscillatory bands cannot link together according to linearity of their mean frequencies. Instead, phase transitions are linked according to their ability to couple with adjacent phase shifts in a constant state of transition between unstable and stable phase synchronization.
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In particle physics, dimensional transmutation is a physical mechanism providing a linkage between a dimensionless parameter and a dimensionful parameter. In classical field theory, such as gauge theory in four-dimensional spacetime, the coupling constant is a dimensionless constant. However, upon quantization, logarithmic divergences in one-loop diagrams of perturbation theory imply that this "constant" actually depends on the typical energy scale of the processes under considerations, called the renormalization group (RG) scale. This "running" of the coupling is specified by the beta-function of the renormalization group.
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The magnitude of consolidation can be predicted by many different methods. In the classical method developed by Terzaghi, soils are tested with an oedometer test to determine their compressibility. In most theoretical formulations, a logarithmic relationship is assumed between the volume of the soil sample and the effective stress carried by the soil particles. The constant of proportionality (change in void ratio per order of magnitude change in effective stress) is known as the compression index, given the symbol \lambda when calculated in natural logarithm and C_C when calculated in base-10 logarithm.
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One might wish to measure the harmonic content of a musical note from a particular instrument or the harmonic distortion of an amplifier at a given frequency. Referring again to Figure 2, we can observe that there is no leakage at a discrete set of harmonically-related frequencies sampled by the DFT. (The spectral nulls are actually zero-crossings, which cannot be shown on a logarithmic scale such as this.) This property is unique to the rectangular window, and it must be appropriately configured for the signal frequency, as described above.
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Everything else is leakage, exaggerated by the use of a logarithmic presentation. The unit of frequency is "DFT bins"; that is, the integer values on the frequency axis correspond to the frequencies sampled by the DFT. So the figure depicts a case where the actual frequency of the sinusoid coincides with a DFT sample, and the maximum value of the spectrum is accurately measured by that sample. In row 4, it misses the maximum value by ½ bin, and the resultant measurement error is referred to as scalloping loss (inspired by the shape of the peak).
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Based on these samples, which are evaluated by the solver similarly as in the sensitivity analysis, the statistical properties of the model responses as mean value, standard deviation, quantiles and higher order stochastic moments are estimated. Reliability analysis: Within the framework of probabilistic safety assessment or reliability analysis, the scattering influences are modelled as random variables, which are defined by distribution type, stochastic moments and mutual correlations. The result of the analysis is the complementary of reliability, the probability of failure, which can be represented on a logarithmic scale.
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In computer science, the treap and the randomized binary search tree are two closely related forms of binary search tree data structures that maintain a dynamic set of ordered keys and allow binary searches among the keys. After any sequence of insertions and deletions of keys, the shape of the tree is a random variable with the same probability distribution as a random binary tree; in particular, with high probability its height is proportional to the logarithm of the number of keys, so that each search, insertion, or deletion operation takes logarithmic time to perform.
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For implementing associative arrays, hash tables, a data structure that maps keys to records using a hash function, are generally faster than binary search on a sorted array of records. Most hash table implementations require only amortized constant time on average. However, hashing is not useful for approximate matches, such as computing the next-smallest, next-largest, and nearest key, as the only information given on a failed search is that the target is not present in any record. Binary search is ideal for such matches, performing them in logarithmic time.
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This technique was proposed by Cernohouz and SolcCernohouz, J, and I Solc (1966) Use of sandstone wanes and weathered basaltic crust in absolute chronology. Nature 212:806–807. who first argued that the relationship between the thickness of a weathering-rind thickness and the time it took to form is expressed by a logarithmic function. This is done by determining the absolute age of sedimentary deposits containing either gravel-size rocks or artifacts using absolute dating methods such as C14 and measuring the weathering-rind thickness of rocks of similar lithology.
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For, in a forest, one can always find a constant number of vertices the removal of which leaves a forest that can be partitioned into two smaller subforests with at most 2n/3 vertices each. A linear arrangement formed by recursively partitioning each of these two subforests, placing the separating vertices between them, has logarithmic vertex searching number. The same technique, applied to a tree-decomposition of a graph, shows that, if the treewidth of an n-vertex graph G is t, then the pathwidth of G is O(t log n)., Theorem 6, p.
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The Munsell value has long been used as a perceptually uniform lightness scale. A question of interest is the relationship between the Munsell value scale and the relative luminance. Aware of the Weber–Fechner law, Munsell remarked "Should we use a logarithmic curve or curve of squares?" Neither option turned out to be quite correct; scientists eventually converged on a roughly cube-root curve, consistent with the Stevens's power law for brightness perception, reflecting the fact that lightness is proportional to the number of nerve impulses per nerve fiber per unit time.
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However, compensated flow cytometry data frequently contains negative values due to compensation, and cell populations do occur which have low means and normal distributions. Logarithmic transformations cannot properly handle negative values, and poorly display normally distributed cell types. Alternative transformations which address this issue include the log-linear hybrid transformations Logicle and Hyperlog, as well as the hyperbolic arcsine and the Box-Cox. A comparison of commonly used transformations concluded that the biexponential and Box-Cox transformations, when optimally parameterized, provided the clearest visualization and least variance of cell populations across samples.
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This is also the practice of the University Corporation for Atmospheric Research, which operates the National Center for Atmospheric Research. Use of the degree symbol to refer to temperatures measured in kelvins (symbol: K) was abolished in 1967 by the 13th General Conference on Weights and Measures (CGPM). Therefore, the triple point of water, for instance, is written simply as 273.16 K. The name of the SI unit of temperature is now "kelvin", in lower case, and no longer "degrees Kelvin". In photography, the symbol is used to denote logarithmic film speed grades.
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His major work was Thesaurus Logarithmorum Completus (Treasury of all Logarithms) that was first published 1794 in Leipzig (its 90th edition was published in 1924). This mathematical table was actually based on Adriaan Vlacq's tables, but corrected a number of errors and extended the logarithms of trigonometric functions for the small angles. An engineer, Franc Allmer, honourable senator of the Graz University of Technology, has found Vega's logarithmic tables with 10 decimal places in the Museum of Carl Friedrich Gauss in Göttingen. Gauss used this work frequently and he has written in it several calculations.
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580-584, April 1965. (in Japanese) The broadband property of log-periodic antennas comes from its self-similarity. A planar log-periodic antenna can also be made self-complementary, such as logarithmic spiral antennas (which are not classified as log-periodic per se but among the frequency independent antennas that are also self-similar) or the log-periodic toothed design. Y. Mushiake found, for what he termed "the simplest self- complementary planar antenna," a driving point impedance of η0/2=188.4 Ω at frequencies well within its bandwidth limits.
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This method, a variant of the Cyrus–Beck algorithm, takes time linear in the number of face planes of the input polyhedron. Alternatively, by testing the line against each of the polygonal facets of the given polyhedron, it is possible to stop the search early when a facet pierced by the line is found. If a single polyhedron is to be intersected with many lines, it is possible to preprocess the polyhedron into a hierarchical data structure in such a way that intersections with each query line can be determined in logarithmic time per query..
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Another meaning for generalized continued fraction is a generalization to higher dimensions. For example, there is a close relationship between the simple continued fraction in canonical form for the irrational real number α, and the way lattice points in two dimensions lie to either side of the line y = αx. Generalizing this idea, one might ask about something related to lattice points in three or more dimensions. One reason to study this area is to quantify the mathematical coincidence idea; for example, for monomials in several real numbers, take the logarithmic form and consider how small it can be.
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Once he explained to a superior why a rate-of-climb indicator had a logarithmic scale (by formulating a first order differential equation to describe the operation of the device and then solving it). Also, for amusement, he built a digital clock from various basic components such as transistors, capacitors and LEDs that were lying unused in lockers in the laboratory. This clock was the model for an illustration on the front page of the Defence Forces Journal on the occasion of the 50th anniversary of the foundation of the Apprentice School. He determined to study for a degree in Electrical Engineering.
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Purely functional data structures are often represented in a different way than their imperative counterparts.Purely functional data structures by Chris Okasaki, Cambridge University Press, 1998, For example, the array with constant access and update times is a basic component of most imperative languages, and many imperative data-structures, such as the hash table and binary heap, are based on arrays. Arrays can be replaced by maps or random access lists, which admit purely functional implementation, but have logarithmic access and update times. Purely functional data structures have persistence, a property of keeping previous versions of the data structure unmodified.
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A linear time algorithm is known to report all directions in which a given simple polygon is monotone.. It was generalized to report all ways to decompose a simple polygon into two monotone chains (possibly monotone in different directions.). Point in polygon queries with respect to a monotone polygon may be answered in logarithmic time after linear time preprocessing (to find the leftmost and rightmost vertices). A monotone polygon may be easily triangulated in linear time. For a given set of points in the plane, a bitonic tour is a monotone polygon that connects the points.
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The exact process that leads to the formation of horizontal rolls is complicated. The basic stress mechanism in the PBL is turbulent flux of momentum, and this term must be approximated in the fluid dynamic equations of motion in order to model the Ekman layer flow and fluxes. The linear approximation, the eddy diffusivity equation with an eddy diffusion coefficient K, allowed Ekman to obtain a simple logarithmic spiral solution. However the frequent presence of the horizontal roll vortices in the PBL, which represent an organization of the turbulence (coherent structures), indicate that the diffusivity approximation is not adequate.
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The eye senses brightness approximately logarithmically over a moderate range and stellar magnitude is measured on a logarithmic scale. This magnitude scale was invented by the ancient Greek astronomer Hipparchus in about 150 B.C. He ranked the stars he could see in terms of their brightness, with 1 representing the brightest down to 6 representing the faintest, though now the scale has been extended beyond these limits; an increase in 5 magnitudes corresponds to a decrease in brightness by a factor of 100. Modern researchers have attempted to incorporate such perceptual effects into mathematical models of vision.
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All measurements in MEGS are done using a logarithmic scale. The units on this scale are called "Attribute Points" or "APs" in the superhero games and simply "Units" in Underground, with each unit on the scale represents exponentially increasing values for length, weight, time, etc. Because of the nature of logarithms and exponents, 0 APs/Units is a meaningful, positive value. Indeed, even negative APs/Units still represent positive values, though exponentially smaller, down to -100 APs, which is defined as absolute zero for all units. In the superhero games, 1 AP corresponds to 8 seconds, , , , $50, or a typed page of information.
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Multiplication and division of raw values are simplified to addition and subtraction on a logarithmic scale, so the MEGS scale functions essentially the same way that slide rules do. For example, raw distance travelled is normally calculated by multiplying raw speed by raw time. In MEGS, speed and time in APs/Units are simply added together to yield the distance travelled in APs/Units. So a car traveling at a speed of 5 APs (about 55 MPH) for 9 APs of time (about 34 minutes) will travel 5+9=14 APs of distance (about 31 miles).
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Only they could provide sufficient detail.The University of Alabama Telescopic Tracking of the Apollo Lunar Missions An image orthicon camera can take television pictures by candlelight because of the more ordered light-sensitive area and the presence of an electron multiplier at the base of the tube, which operated as a high- efficiency amplifier. It also has a logarithmic light sensitivity curve similar to the human eye. However, it tends to flare in bright light, causing a dark halo to be seen around the object; this anomaly was referred to as blooming in the broadcast industry when image orthicon tubes were in operation.dtic.
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A 43×43×43-megaparsec cube shows the evolution of the large-scale structure over a logarithmic period starting from a redshift of 30 and ending at redshift 0. The model makes it clear to see how the matter-dense regions contract under the collective gravitational force while simultaneously aiding in the expansion of cosmic voids as the matter flees to the walls and filaments. Cosmic voids contain a mix of galaxies and matter that is slightly different than other regions in the universe. This unique mix supports the biased galaxy formation picture predicted in Gaussian adiabatic cold dark matter models.
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One application of log structures is the ability to define logarithmic forms on any log scheme. From this, one can for instance define corresponding notions of log-smoothness and log-étaleness which parallel the usual definitions for schemes. This then allows the study of deformation theory. In addition, log structures serve to define the mixed Hodge structure on any smooth variety X, by taking a compactification with boundary a normal crossings divisor D, and writing down the Hodge–De Rham complex associated to X with the standard log structure defined by D.Chris A.M. Peters; Joseph H.M. Steenbrink (2007).
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The speed in which a galaxy rotates is thought to correlate with the flatness of the disc as some spiral galaxies have thick bulges, while others are thin and dense."Fat or flat: Getting galaxies into shape". phys.org. February 2014 NGC 1300, an example of a barred spiral galaxy In spiral galaxies, the spiral arms do have the shape of approximate logarithmic spirals, a pattern that can be theoretically shown to result from a disturbance in a uniformly rotating mass of stars. Like the stars, the spiral arms rotate around the center, but they do so with constant angular velocity.
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The pair of nomograms at the top of the image determine the probability of occurrence and the availability, which are then incorporated into the bottom multistage nomogram. Lines 8 and 10 are ‘tie lines’ or ‘pivot lines’ and are used for the transition between the stages of the compound nomogram. The final pair of parallel logarithmic scales (12) are not nomograms as such, but reading-off scales to translate the risk score (11, remote to extremely high) into a sampling frequency to address safety aspects and other ‘consumer protection’ aspects respectively. This stage requires political ‘buy in’ balancing cost against risk.
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William Oughtred (1575–1660), inventor of the circular slide rule. A collection of slide rules at the Museum of the History of Science, Oxford The slide rule was invented around 1620–1630, shortly after John Napier's publication of the concept of the logarithm. Edmund Gunter of Oxford developed a calculating device with a single logarithmic scale; with additional measuring tools it could be used to multiply and divide. The first description of this scale was published in Paris in 1624 by Edmund Wingate (c.1593–1656), an English mathematician, in a book entitled L'usage de la reigle de proportion en l'arithmetique & geometrie.
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US period life table for 2003, showing considerable acceleration rather than deceleration; note the logarithmic scale. In gerontology, late-life mortality deceleration is the disputed theory that hazard rate increases at a decreasing rate in late life rather than increasing exponentially as in the Gompertz law. Late-life mortality deceleration is a well-established phenomenon in insects, which often spend much of their lives in a constant hazard rate region, but it is much more controversial in mammals. Rodent studies have found varying conclusions, with some finding short-term periods of mortality deceleration in mice, others not finding such.
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Sometimes the name RL is reserved for the class of problems solvable by logarithmic-space probabilistic machines in unbounded time. However, this class can be shown to be equal to NL using a probabilistic counter, and so is usually referred to as NL instead; this also shows that RL is contained in NL. RL is contained in BPL, which is similar but allows two-sided error (incorrect accepts). RL contains L, the problems solvable by deterministic Turing machines in log space, since its definition is just more general. Noam Nisan showed in 1992 the weak derandomization result that RL is contained in SC,.
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Factors that can cause a small frequency drift over time are stress relief in the mounting structure, loss of hermetic seal, contaminations contained in the crystal lattice, moisture absorption, changes in or on the quartz crystal, severe shock and vibrations effects, exposure to very high temperatures.Introduction to Quartz Frequency Standards - Aging Crystal aging tends to be logarithmic meaning the maximum rate of change of frequency occurs immediately after manufacture and decays thereafter. Most of the aging will occur within the first year of the crystals service life. Crystals do eventually stop aging (asymptotically), but it can take many years.
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Grapher is a graphing calculator capable of creating both 2D graphs including classic (linear- linear), polar coordinates, linear-logarithmic, log-log, and polar log, as well as 3D graphs including standard system, cylindrical system, and spherical system. Grapher is a Cocoa application which takes advantage of Mac OS X APIs. It also supports multiple equations in one graph, exporting equations to LaTeX format, and comes with several pre-made equation examples. It is one of the few sophisticated graphing programs available capable of easily exporting clean vector art for use in printed documents (although exporting 3D graphs to vector is not possible).
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The compact disc packaging for 10,000 Days consists of a thick cardboard-bound booklet partly covered by a flap holding a pair of stereoscopic eyeglasses, which can be used to view a series of images inside. Viewed with the glasses, the artwork produces an illusion of depth and three-dimensionality. Alex Grey, who created a majority of the album art for Lateralus and its accompanying video "Parabola", reprised his role for 10,000 Days. The CD face itself is decorated with stylized eyes, arranged in a seemingly logarithmic spiral toward the center (adapted from a previous Alex Grey painting, "Collective Vision").
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In computational complexity theory, DLOGTIME is the complexity class of all computational problems solvable in a logarithmic amount of computation time on a deterministic Turing machine. It must be defined on a random-access Turing machine, since otherwise the input tape is longer than the range of cells that can be accessed by the machine. It is a very weak model of time complexity: no random-access Turing machine with a smaller deterministic time bound can access the whole input.. See in particular p. 140. DLOGTIME-uniformity is important in circuit complexity.. See in particular p. 23.
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The data generated by flow-cytometers can be plotted in a single dimension, to produce a histogram, or in two-dimensional dot plots, or even in three dimensions. The regions on these plots can be sequentially separated, based on fluorescence intensity, by creating a series of subset extractions, termed "gates." Specific gating protocols exist for diagnostic and clinical purposes, especially in relation to hematology. Individual single cells are often distinguished from cell doublets or higher aggregates by their "time-of-flight" (denoted also as a "pulse-width") through the narrowly focused laser beam The plots are often made on logarithmic scales.
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It is almost always used in a ganged configuration with a logarithmic potentiometer, for instance, in an audio balance control. Potentiometers used in combination with filter networks act as tone controls or equalizers. In audio systems, the word linear, is sometimes applied in a confusing way to describe slide potentiometers because of the straight line nature of the physical sliding motion. The word linear when applied to a potentiometer regardless of being a slide or rotary type, describes a linear relationship of the pot's position versus the measured value of the pot's tap (wiper or electrical output) pin.
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Dickerson et al use simulations to check under what conditions an envy-free assignment of discrete items is likely to exist. They generate instances by sampling the value of each item to each agent from two probability distributions: uniform and correlated. In the correlated sampling, they first sample an intrinsic value for each good, and then assign a random value to each agent drawn from a truncated nonnegative normal distribution around that intrinsic value. Their simulations show that, when the number of goods is larger than the number of agents by a logarithmic factor, envy-free allocations exist with high probability.
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Although the power law exponent approximation is convenient, it has no theoretical basis. When the temperature profile is adiabatic, the wind speed should vary logarithmically with height, Measurements over open terrain in 1961 showed good agreement with the logarithmic fit up to 100 m or so, with near constant average wind speed up through 1000 m. The shearing of the wind is usually three-dimensional, that is, there is also a change in direction between the 'free' pressure-driven geostrophic wind and the wind close to the ground. This is related to the Ekman spiral effect.
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To see how, notice that, as the segments do not intersect and completely cross the slab, the segments can be sorted vertically inside each slab. While this algorithm allows point location in logarithmic time and is easy to implement, the space required to build the slabs and the regions contained within the slabs can be as high as O(n²), since each slab can cross a significant fraction of the segments. Several authors noticed that the segments that cross two adjacent slabs are mostly the same. Therefore, the size of the data structure can be significantly reduced.
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Official poster explaining hygiene rules and correct reaction in case of symptoms (version of 5 March 2020) Number of cases (blue) and number of deaths (red) on a logarithmic scale On 27 February, following the confirmation of COVID-19 cases in the region, Graubünden cancelled the Engadine Skimarathon. On 28 February, the Federal Council banned events involving more than 1,000 people in an effort to curb the spread of the infection. Multiple events such as carnivals and fairs were either postponed or cancelled. Geneva Motor Show, Baselworld, Bern Carnival and the Carnival of Basel were cancelled.
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Number of cases (blue) and number of deaths (red) on a logarithmic scale. An example of a self- certification form used in Romania's coronavirus lockdown The Bucharest Mega Mall has temporarily closed to help stop the spread of COVID-19 On 14 March, after over 101 people had been diagnosed with coronavirus, Romania entered the third COVID-19 scenario. The third scenario goes from 101 to 2,000 cases. In the third scenario the doctors will perform epidemiological screening in the tents installed in the hospitals' yards, and the hospitals of infectious diseases will treat only cases of SARS-CoV-2 infection.
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The pathwidth of a graph has a very similar definition to treewidth via tree decompositions, but is restricted to tree decompositions in which the underlying tree of the decomposition is a path graph. Alternatively, the pathwidth may be defined from interval graphs analogously to the definition of treewidth from chordal graphs. As a consequence, the pathwidth of a graph is always at least as large as its treewidth, but it can only be larger by a logarithmic factor. Another parameter, the graph bandwidth, has an analogous definition from proper interval graphs, and is at least as large as the pathwidth.
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Number of cases (blue) and number of deaths (red) on a logarithmic scale. On 21 March, the Ministry of Interior announced a total curfew for those who are over the age 65 or chronically ill. M1 metro line explaining 14 rules to avoid the coronavirus infection On 23 March, at a press conference, Koca announced that a drug called Favipiravir, which was reported by Chinese authorities to be effective in treating the disease, was imported and started to be administered to intensive care patients. Koca also announced that healthcare workers would be paid an additional fee on their paychecks for 3 months.
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This outcome revealed a previously unknown and already existing natural link between general relativity and quantum mechanics. There lacks clarity in the generalization of this theory to 3+1 dimensions. However, a recent derivation in 3+1 dimensions under the right coordinate conditions yields a formulation similar to the earlier 1+1, a dilaton field governed by the logarithmic Schrödinger equation that is seen in condensed matter physics and superfluids. The field equations are amenable to such a generalization, as shown with the inclusion of a one-graviton process, and yield the correct Newtonian limit in d dimensions, but only with a dilaton.
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The invention of the logarithmic tables by John Napier (1550-1617) allowed the development of the sciences, while significant contributions to science were made by other Scots such as James Gregory (1638-75), Robert Sibbald (1641-1722) and Archibald Pitcairne (1652-1713). In the second half of the 17th century, Scottish universities developed their own form of Cartesianism, influence in large part by Reformed Scholasticism of the first half of the 17th century. Mention of Descartes first appeared in the graduation theses by regent Andrew Cant for Marischal College, the University of Aberdeen in 1654. Cartesianism was very successful in Scottish universities.
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Although the value is relative to the standards against which it is compared, the unit used to measure the times changes the score (see examples 1 and 2). This is a consequence of the requirement that the argument of the logarithmic function must be dimensionless. The multiplier also can't have a numeric value of 1 or less, because the logarithm of 1 is 0 (examples 3 and 4), and the logarithm of any value less than 1 is negative (examples 5 and 6); that would result in scores of value 0 (even with changes), undefined, or negative (even if better than positive).
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Any n-vertex forest has tree- depth O(log n). For, in a forest, one can always find a constant number of vertices the removal of which leaves a forest that can be partitioned into two smaller subforests with at most 2n/3 vertices each. By recursively partitioning each of these two subforests, we can easily derive a logarithmic upper bound on the tree-depth. The same technique, applied to a tree decomposition of a graph, shows that, if the treewidth of an n-vertex graph G is t, then the tree-depth of G is O(t log n).
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He is best known for the soundtracks to the Turrican series of games. He also created a music replay routine for the Amiga called TFMX — "The Final Musicsystem eXtended", which features more musically-oriented features than rival Soundtracker, such as logarithmic pitch-bends, sound macros and individual tempos for each track. His music from Apidya, Turrican 2, Turrican 3 and The Great Giana Sisters was performed live at the Symphonic Game Music Concert series in Leipzig, Germany between 2003–2007, conducted by Andy Brick. On 23 August 2008 his music was performed at Symphonic Shades, a concert dedicated to his work exclusively.
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In a weak max-heap, the maximum value can be found (in constant time) as the value associated with the root node; similarly, in a weak min-heap, the minimum value can be found at the root. As with binary heaps, weak heaps can support the typical operations of a priority queue data structure: insert, delete-min, delete, or decrease- key, in logarithmic time per operation. Sifting up is done using the same process as in binary heaps. The new node is added at the leaf level, then compared with its distinguished ancestor and swapped if necessary (the merge operation).
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Where both current and voltage are plotted on logarithmic scales, the borders of the SOA are straight lines: # IC = ICmax — current limit # VCE = VCEmax — voltage limit # IC VCE = Pmax — dissipation limit, thermal breakdown # IC VCEα = const — this is the limit given by the secondary breakdown (bipolar junction transistors only) SOA specifications are useful to the design engineer working on power circuits such as amplifiers and power supplies as they allow quick assessment of the limits of device performance, the design of appropriate protection circuitry, or selection of a more capable device. SOA curves are also important in the design of foldback circuits.
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The Helicoidal Skyscraper can be considered a sustainable building or an example of the green architecture concept due to several aspects of its design. As a tall building, for instance, it addresses the energy problem by minimising the quantities of materials needed for its construction. Also, its unique logarithmic spiral would have reacted to the wind with a vertical force that drives it upwards, taking with the air pollution - out from the streets below. It avoids the capability of most tall buildings to serve as obstructions that affect the atmospheric circulation and the dispersion of pollutants.
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Logarithmic plot of mass against mean density (with solar values as origin) showing possible kinds of stellar equilibrium state. For a configuration in the shaded region, beyond the black hole limit line, no equilibrium is possible, so runaway collapse will be inevitable. According to Einstein's theory, for even larger stars, above the Landau–Oppenheimer–Volkoff limit, also known as the Tolman–Oppenheimer–Volkoff limit (roughly double the mass of our Sun) no known form of cold matter can provide the force needed to oppose gravity in a new dynamical equilibrium. Hence, the collapse continues with nothing to stop it.
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In telephony, a standard audio signal for a single phone call is encoded as 8000 analog samples per second, of 8 bits each, giving a 64 kbit/s digital signal known as DS0. The default signal compression encoding on a DS0 is either μ-law (mu-law) PCM (North America and Japan) or A-law PCM (Europe and most of the rest of the world). These are logarithmic compression systems where a 13 or 14 bit linear PCM sample number is mapped into an 8 bit value. This system is described by international standard G.711.
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In mathematics, the conformal radius is a way to measure the size of a simply connected planar domain D viewed from a point z in it. As opposed to notions using Euclidean distance (say, the radius of the largest inscribed disk with center z), this notion is well-suited to use in complex analysis, in particular in conformal maps and conformal geometry. A closely related notion is the transfinite diameter or (logarithmic) capacity of a compact simply connected set D, which can be considered as the inverse of the conformal radius of the complement E = Dc viewed from infinity.
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Her doctoral dissertation, Über eine singuläre Intergralgleichung 1. Art mit logarithmischer Unstetigkeit [On a singular integral equation of the 1st kind with logarithmic discontinuity], was supervised by Hans Schubert; her habilitation thesis was Über eine Hillsche Differentialgleichung [On Hill's differential equation]. She worked as a professor of algebra at TU Dresden from 1954 until her 1981 retirement. The Rostock CPR gives the date of her start at Dresden as 1964, but this would leave a ten-year gap in her work life, and Voss is clear that she arrived before the 1962 start of Lieselott Herforth.
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If a WAVL tree is created using only insertions, without deletions, then it has the same small height bound that an AVL tree has. On the other hand, red–black trees have the advantage over AVL trees in lesser restructuring of their trees. In AVL trees, each deletion may require a logarithmic number of tree rotation operations, while red–black trees have simpler deletion operations that use only a constant number of tree rotations. WAVL trees, like red–black trees, use only a constant number of tree rotations, and the constant is even better than for red–black trees.
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Mann has written many astrology books based on the concept of a logarithmic time scale derived from the work of G. I. Gurdjieff, P. D. Ouspensky and Rodney Collin, and contributing to an application of astrology called Life Time Astrology. Mann lived in England from 1973 to 1991 and in Copenhagen from 1991 to 1999. He regularly taught at the UK Astrological Association in London, and the Unicorn School of Astro- psychology and the Scandinavian Astrologi Skole in Copenhagen. Mann is one of the more prolific astrological authors, and he designed and illustrated most of his own books.
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CTK won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003, the Encyclopædia Britannicas Internet Guide Award, and Sciences NetWatch award. The site was remarkably prolific and contained extensive analysis of many of the classic problems in recreational mathematics including the Apollonian gasket, Napoleon's theorem, logarithmic spirals, The Prisoner of Benda, the Pitot theorem, and the monkey and the coconuts problem. Once, in a remarkable tour de force, CTK published 122 proofs of the Pythagorean theorem.Cut-the-Knot: Proofs of the Pythagorean Theorem Bogomolny did indeed entertain but his deeper goal was to educate.
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The classical resolution of the paradox involved the explicit introduction of a utility function, an expected utility hypothesis, and the presumption of diminishing marginal utility of money. In Daniel Bernoulli's own words: > The determination of the value of an item must not be based on the price, > but rather on the utility it yields…. There is no doubt that a gain of one > thousand ducats is more significant to the pauper than to a rich man though > both gain the same amount. A common utility model, suggested by Bernoulli himself, is the logarithmic function U(w) = ln(w) (known as log utility).
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Spiral optimization (SPO) algorithm The spiral optimization (SPO) algorithm is an uncomplicated search concept inspired by spiral phenomena in nature. The motivation for focusing on spiral phenomena was due to the insight that the dynamics that generate logarithmic spirals share the diversification and intensification behavior. The diversification behavior can work for a global search (exploration) and the intensification behavior enables an intensive search around a current found good solution (exploitation). The SPO algorithm is a multipoint search algorithm that has no objective function gradient, which uses multiple spiral models that can be described as deterministic dynamical systems.
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According to Seitz [1985]: :It was a premise of the Cosmic Cube experiment that the internode communication should scale well to very large numbers of nodes. A direct network like the hypercube satisfies this requirement, with respect to both the aggregate bandwidth achieved across the many concurrent communication channels and the feasibility of the implementation. The hypercube is actually a distributed variant of an indirect logarithmic switching network like the Omega or banyan networks: the kind that might be used in shared-storage organizations. With the hypercube, however, communication paths traverse different numbers of channels and so exhibit different latencies.
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If we more thoroughly studied the > distances and proportions of the stars we'd probably find certain > relationships of multiples based on some logarithmic scale or whatever the > scale may be . Stravinsky's adoption of twelve-tone serial techniques offers an example of the level of influence serialism had after the Second World War. Previously Stravinsky had used series of notes without rhythmic or harmonic implications . Because many of the basic techniques of serial composition have analogs in traditional counterpoint, uses of inversion, retrograde, and retrograde inversion from before the war do not necessarily indicate Stravinsky was adopting Schoenbergian techniques.
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Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Gottfried Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular.Struik (1969), 367 In the 18th century, "function" lost these geometrical associations. Leibniz realized that the coefficients of a system of linear equations could be arranged into an array, now called a matrix, which can be manipulated to find the solution of the system, if any. This method was later called Gaussian elimination.
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A greedy algorithm is used: The new key is inserted in one of its two possible locations, "kicking out", that is, displacing, any key that might already reside in this location. This displaced key is then inserted in its alternative location, again kicking out any key that might reside there. The process continues in the same way until an empty position is found, completing the algorithm. However, it is possible for this insertion process to fail, by entering an infinite loop or by finding a very long chain (longer than a preset threshold that is logarithmic in the table size).
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The air-water interface is now endowed with a surface roughness due to the capillary-gravity waves, and a second phase of wave growth takes place. A wave established on the surface either spontaneously as described above, or in laboratory conditions, interacts with the turbulent mean flow in a manner described by Miles. This is the so-called critical-layer mechanism. A critical layer forms at a height where the wave speed c equals the mean turbulent flow U. As the flow is turbulent, its mean profile is logarithmic, and its second derivative is thus negative.
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In 1544, Michael Stifel published Arithmetica integra, which contains a table of integers and powers of 2 that has been considered an early version of a logarithmic table. The method of logarithms was publicly propounded by John Napier in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms). The book contained fifty-seven pages of explanatory matter and ninety pages of tables related to natural logarithms. The English mathematician Henry Briggs visited Napier in 1615, and proposed a re-scaling of Napier's logarithms to form what is now known as the common or base-10 logarithms.
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Chapter nine concerns the complementation of languages and the transitive closure operator, including the Immerman–Szelepcsényi theorem that nondeterministic logarithmic space is closed under complementation. Chapter ten provides complete problems and a second-order logical characterization of polynomial space. Chapter eleven concerns uniformity in circuit complexity (the distinction between the existence of circuits for solving a problem, and their algorithmic constructibility), and chapter twelve concerns the role of ordering and counting predicates in logical characterizations of complexity classes. Chapter thirteen uses the switching lemma for lower bounds, and chapter fourteen concerns applications to databases and model checking.
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A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a scale is nonlinear: the numbers 10 and 20, and 60 and 70, are not the same distance apart on a log scale. Rather, the numbers 10 and 100, and 60 and 600 are equally spaced. Thus moving a unit of distance along the scale means the number has been multiplied by 10 (or some other fixed factor).
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A family of examples proving that the approximation ratio is \Omega(\sqrt n) was given by , and the matching upper bound is by . As with the approximation ratio for Delaunay triangulation, a weaker bound was also given by . Nevertheless, for randomly distributed point sets, both the Delaunay and greedy triangulations are within a logarithmic factor of the minimum weight.. The hardness result of Mulzer and Rote also implies the NP-hardness of finding an approximate solution with relative approximation error at most O(1/n2). Thus, a fully polynomial approximation scheme for minimum weight triangulation is unlikely.
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Sharp was born in Horton Hall in Little Horton, Bradford, the son of well-to-do merchant John Sharp and Mary (née Clarkson) Sharp and was educated at Bradford Grammar School. Abraham Sharp's wooden telescope In 1669 he became a merchant's apprentice before becoming a schoolmaster in Liverpool and subsequently a bookkeeper in London. His wide knowledge of mathematics and astronomy attracted Flamsteed's attention and it was through Flamsteed that Sharp was invited, in 1688, to enter the Greenwich Royal Observatory. There he did notable work, improving instruments and showing great skill as a calculator, publishing Geometry Improved and logarithmic tables.
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Finished lumber, writing paper, capacitors, and many other products are usually sold in only a few standard sizes. Many design procedures describe how to calculate an approximate value, and then "round" to some standard size using phrases such as "round down to nearest standard value", "round up to nearest standard value", or "round to nearest standard value". "Zener Diode Voltage Regulators" "Build a Mirror Tester" When a set of preferred values is equally spaced on a logarithmic scale, choosing the closest preferred value to any given value can be seen as a form of scaled rounding. Such rounded values can be directly calculated.
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A time-delay fuse (also known as an anti-surge or slow-blow fuse) is designed to allow a current which is above the rated value of the fuse to flow for a short period of time without the fuse blowing. These types of fuse are used on equipment such as motors, which can draw larger than normal currents for up to several seconds while coming up to speed. Manufacturers can provide a plot of current vs time, often plotted on logarithmic scales, to characterize the device and to allow comparison with the characteristics of protective devices upstream and downstream of the fuse.
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A daxophone being played by the inventor, Hans Reichel Normally, the daxophone is played by bowing the free end with a bow, most commonly a double bass bow, but it can also be struck or plucked, which propagates sound in the same way a ruler halfway off a table does. Vibrations then continue to the wooden-block base, which in turn is amplified by the contact microphone(s) therein. The timbre is adjusted by where it is bowed, and where along its length it is stopped with the dax. One side of the dax is fretted according to a randomly chosen logarithmic succession.
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Number of Bitcoin Cash transactions per month (logarithmic scale) Bitcoin Cash trades on digital currency exchanges including Bitstamp, Coinbase, Gemini, Kraken, Bitfinex, and ShapeShift using the Bitcoin Cash name and the BCH ticker symbol for the cryptocurrency. On 26 March 2018, OKEx removed all Bitcoin Cash trading pairs except for BCH/BTC, BCH/ETH and BCH/USDT due to "inadequate liquidity". , daily transaction numbers for Bitcoin Cash are about one-tenth of those of bitcoin. Coinbase listed Bitcoin Cash on December 19, 2017 and the coinbase platform experienced price abnormalities that led to an insider trading investigation.
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For number theorists his main fame is the series for the Riemann zeta function (the leading function in Riemann's exact prime- counting function). Instead of using a series of logarithmic integrals, Gram's function uses logarithm powers and the zeta function of positive integers. It has recently been supplanted by a formula of Ramanujan that uses the Bernoulli numbers directly instead of the zeta function. Gram was the first mathematician to provide a systematic theory of the development of skew frequency curves, showing that the normal symmetric Gaussian error curve was but one special case of a more general class of frequency curves.
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Per Georg Scheutz's third difference engine, on which Gravatt's replica was based When Per Georg Scheutz brought his Difference engine to London in 1854, Gravatt engaged its inventor in conversation. His knowledge allowed Gravatt to commission a copy from Donkin, which was sent to Somerset House. From 1855 he gave lectures on the machine to professional audiences, including the Royal Society attended by Prince Albert, and followed this with lectures at the International Exposition in Paris. Gravatt then worked with the Registrar- General to establish public faith in the machine, by quickly calculating specimens of logarithmic and other tables.
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The right- hand column is a plot of the measure of effect (e.g. an odds ratio) for each of these studies (often represented by a square) incorporating confidence intervals represented by horizontal lines. The graph may be plotted on a natural logarithmic scale when using odds ratios or other ratio-based effect measures, so that the confidence intervals are symmetrical about the means from each study and to ensure undue emphasis is not given to odds ratios greater than 1 when compared to those less than 1. The area of each square is proportional to the study's weight in the meta-analysis.
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Audiograms are set out with frequency in hertz (Hz) on the horizontal axis, most commonly on a logarithmic scale, and a linear dBHL scale on the vertical axis. For humans, normal hearing is between −10 dB(HL) and 15 dB(HL), although 0 dB from 250 Hz to 8 kHz is deemed to be 'average' normal hearing. Hearing thresholds of humans and other mammals can be found with behavioural hearing tests or physiological tests used in audiometry. For adults, a behavioural hearing test involves a tester who presents tones at specific frequencies (pitches) and intensities (loudnesses).
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For a while, he was a member of Nicolas Malebranche's circle in Paris and it was there that in 1691 he met young Johann Bernoulli, who was visiting France and agreed to supplement his Paris talks on infinitesimal calculus with private lectures to l'Hôpital at his estate at Oucques. In 1693, l'Hôpital was elected to the French academy of sciences and even served twice as its vice- president.Yushkevich, p. 270. Among his accomplishments were the determination of the arc length of the logarithmic graph,Unbeknownst to him, a solution had already been obtained by James Gregory in letters to Collins (1670–1671), ibid.
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Although this a very recent cutting-edge technique, it has seen a couple of variations upon the basic algorithm in the last couple of month, most notably JSON driven resolution methods. the basic idea is that instead of supplying the algorithm with n records, it is more useful to provide the algorithm with emotional meta-data to guide its search and improve its complexity beyond the usual logarithmic bounds and this by a factor of log(n)/2. It allows to select intermediate m values for the search index and skew them towards the wanted emotional value in the initial records.
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In order for a recipient bacterium to bind, take up exogenous DNA from another bacterium of the same species and recombine it into its chromosome, it must enter a special physiological state called competence. Competence in B. subtilis is induced toward the end of logarithmic growth, especially under conditions of amino-acid limitation. Under these stressful conditions of semistarvation, cells typically have just one copy of their chromosome and likely have increased DNA damage. To test whether transformation is an adaptive function for B. subtilis to repair its DNA damage, experiments were conducted using UV light as the damaging agent.
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Each of the preceding algorithms runs in time. However, the former takes exactly steps, while the latter requires steps. For the 16-input examples illustrated, Algorithm 1 is 12-way parallel (49 units of work divided by a span of 4) while Algorithm 2 is only 4-way parallel (26 units of work divided by a span of 6). However, Algorithm 2 is work-efficient--it performs only a constant factor (2) of the amount of work required by the sequential algorithm --while Algorithm 1 is work-inefficient--it performs asymptotically more work (a logarithmic factor) than is required sequentially.
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When a data set may be updated dynamically, it may be stored in a Fenwick tree data structure. This structure allows both the lookup of any individual prefix sum value and the modification of any array value in logarithmic time per operation. However, an earlier 1982 paper presents a data structure called Partial Sums Tree (see Section 5.1) that appears to overlap Fenwick trees; in 1982 the term prefix-sum was not yet as common as it is today. For higher-dimensional arrays, the summed area table provides a data structure based on prefix sums for computing sums of arbitrary rectangular subarrays.
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In the 1730s, he first established and used what was later to be known as Catalan numbers.The 18th century Chinese discovery of the Catalan numbers The Jesuit missionaries' influence can be seen by many traces of European mathematics in his works, including the use of Euclidean notions of continuous proportions, series addition, subtraction, multiplication and division, series reversion, and the binomial theorem. Minggatu's work is remarkable in that expansions in series, trigonometric and logarithmic were apprehended algebraically and inductively without the aid of differential and integral calculus. In 1742 he participated in the revision of the Compendium of Observational and Computational Astronomy.
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Ctries have logarithmic complexity bounds of the basic operations, albeit with a low constant factor due to the high branching level (usually 32). Ctries support a lock-free, linearizable, constant-time snapshot operation, based on the insight obtained from persistent data structures. This is a breakthrough in concurrent data- structure design, since existing concurrent data-structures do not support snapshots. The snapshot operation allows implementing lock-free, linearizable iterator, size and clear operations - existing concurrent data-structures have implementations which either use global locks or are correct only given that there are no concurrent modifications to the data-structure.
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Buchmann's achievements include scientific essays on algorithms in algebraic number theory, the construction of new cryptographic methods and the use of cryptographic methods in practice. Due to his collaboration with Kálmán Győry he has the Erdős number 2. Buchmann dealt with algorithms in algebraic number theory and their application in cryptography. In 1988, he proposed with Hugh C. Williams a cryptographic system based on the discrete logarithmic problem in the ideal class group of imaginary-square number bodies (which, according to Carl Friedrich Gauss, is related to the theory of binary-square forms), which triggered further developments in cryptography with number bodies.
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A distinctive feature of this shape is that when a square section is added—or removed—the product is another golden rectangle, having the same aspect ratio as the first. Square addition or removal can be repeated infinitely, in which case corresponding corners of the squares form an infinite sequence of points on the golden spiral, the unique logarithmic spiral with this property. Diagonal lines drawn between the first two orders of embedded golden rectangles will define the intersection point of the diagonals of all the embedded golden rectangles; Clifford A. Pickover referred to this point as "the Eye of God".
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S-N curve for a brittle aluminium with an ultimate tensile strength of 320 MPa Materials fatigue performance is commonly characterized by an S-N curve, also known as a Wöhler curve. This is often plotted with the cyclic stress (S) against the cycles to failure (N) on a logarithmic scale. S-N curves are derived from tests on samples of the material to be characterized (often called coupons or specimens) where a regular sinusoidal stress is applied by a testing machine which also counts the number of cycles to failure. This process is sometimes known as coupon testing.
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Shown is the out-of-plane acceleration on a logarithmic color scale for a frequency of 1000 Hz. As a more applied example, the result of a DEA simulation on a Yanmar tractor model (body in blue: chassis/cabin steel frame and windows) is shown here to the left. In the cited work, the numerical DEA results are compared with experimental measurements at frequencies between 400 Hz and 4000 Hz for an excitation on the back of the gear casing. Both results agree favorably. The DEA simulation can be extended to predict the sound pressure level at driver's ear.
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The second son of Roger Wingate of Sharpenhoe in Bedfordshire and of his wife Jane, daughter of Henry Birch, he was born at Flamborough in Yorkshire in 1596 and baptised there on 11 June. He matriculated from The Queen's College, Oxford, on 12 October 1610, graduated B.A. on 30 June 1614, and was admitted to Gray's Inn on 24 May. Before 1624 he went to Paris, where he became teacher of the English language to the Princess Henrietta Maria. He had learned in England the "rule of proportion" (logarithmic scale) recently invented by Edmund Gunter which he communicated to mathematicians in Paris.
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Graph showing the historical evolution of the record precision of numerical approximations to pi, measured in decimal places (depicted on a logarithmic scale; time before 1400 is not shown to scale). Approximations for the mathematical constant pi () in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era (Archimedes). In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century. Further progress was not made until the 15th century (through the efforts of Jamshīd al-Kāshī).
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Augmented chord in the chromatic circle A one-third octave is a logarithmic unit of frequency ratio equal to either one third of an octave (1200/3 = 400 cents: major third) or one tenth of a decade (3986.31/10 = 398.631 cents: M3 ).Malcolm J. Crocker, Handbook of Acoustics (1997) An alternative (unambiguous) term for one tenth of a decade is a decidecade.von Benda- Beckmann, A. M., Aarts, G., Sertlek, H. Ö., Lucke, K., Verboom, W. C., Kastelein, R. A., ... & Ainslie, M. A. (2015). Assessing the impact of underwater clearance of unexploded ordnance on harbour porpoises (Phocoena phocoena) in the Southern North Sea.
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The K-scale is quasi-logarithmic. The conversion table from maximum fluctuation R (in units of nanoteslas, nT) to K-index, varies from observatory to observatory in such a way that the historical rate of occurrence of certain levels of K are about the same at all observatories. In practice this means that observatories at higher geomagnetic latitude require higher levels of fluctuation for a given K-index. For example, at Godhavn, Greenland, a value of K = 9 is derived with R = 1500 nT, while in Honolulu, Hawaii, a fluctuation of only 300 nT is recorded as K = 9\.
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Similarly, in a unit disk graph (with a known geometric representation), there is a polynomial time algorithm for maximum cliques based on applying the algorithm for complements of bipartite graphs to shared neighborhoods of pairs of vertices. The algorithmic problem of finding a maximum clique in a random graph drawn from the Erdős–Rényi model (in which each edge appears with probability , independently from the other edges) was suggested by . Because the maximum clique in a random graph has logarithmic size with high probability, it can be found by a brute force search in expected time . This is a quasi-polynomial time bound.
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D. J. Broadhurst provides a generalization of the BBP algorithm that may be used to compute a number of other constants in nearly linear time and logarithmic space.D. J. Broadhurst, "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)", (1998) arXiv math.CA/9803067 Explicit results are given for Catalan's constant, \pi^3, \pi^4, Apéry's constant \zeta(3), \zeta(5), (where \zeta(x) is the Riemann zeta function), \log^32, \log^42, \log^52, and various products of powers of \pi and \log2. These results are obtained primarily by the use of polylogarithm ladders.
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Rather than bounding the time complexity of an algorithm that recognizes an MSO property on bounded-treewidth graphs, it is also possible to analyze the space complexity of such an algorithm; that is, the amount of memory needed above and beyond the size of the input itself (which is assumed to be represented in a read-only way so that its space requirements cannot be put to other purposes). In particular, it is possible to recognize the graphs of bounded treewidth, and any MSO property on these graphs, by a deterministic Turing machine that uses only logarithmic space..
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However, parametric search leads to an increase in time complexity (compared to the decision algorithm) that may be larger than logarithmic. Because they are strongly rather than weakly polynomial, algorithms based on parametric search are more satisfactory from a theoretical point of view. In practice, binary search is fast and often much simpler to implement, so algorithm engineering efforts are needed to make parametric search practical. Nevertheless, write that "while a simple binary-search approach is often advocated as a practical replacement for parametric search, it is outperformed by our [parametric search] algorithm" in the experimental comparisons that they performed.
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On a stereographic projection map, a loxodrome is an equiangular spiral whose center is the north or south pole. All loxodromes spiral from one pole to the other. Near the poles, they are close to being logarithmic spirals (which they are exactly on a stereographic projection, see below), so they wind around each pole an infinite number of times but reach the pole in a finite distance. The pole-to-pole length of a loxodrome (assuming a perfect sphere) is the length of the meridian divided by the cosine of the bearing away from true north.
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Corbino disc – dashed curves represent logarithmic spiral paths of deflected electrons The Corbino effect is a phenomenon involving the Hall effect, but a disc-shaped metal sample is used in place of a rectangular one. Because of its shape the Corbino disc allows the observation of Hall effect–based magnetoresistance without the associated Hall voltage. A radial current through a circular disc, subjected to a magnetic field perpendicular to the plane of the disc, produces a "circular" current through the disc. The absence of the free transverse boundaries renders the interpretation of the Corbino effect simpler than that of the Hall effect.
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Thus, search is limited to the number of entries in this neighborhood, which is logarithmic in the worst case, constant on average, and with proper alignment of the neighborhood typically requires one cache miss. When inserting an entry, one first attempts to add it to a bucket in the neighborhood. However, if all buckets in this neighborhood are occupied, the algorithm traverses buckets in sequence until an open slot (an unoccupied bucket) is found (as in linear probing). At that point, since the empty bucket is outside the neighborhood, items are repeatedly displaced in a sequence of hops.
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The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy channel in the limit of long block lengths, when the channel statistics are determined by the joint distribution. The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. A common unit of information is the bit, based on the binary logarithm. Other units include the nat, which is based on the natural logarithm, and the decimal digit, which is based on the common logarithm.
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The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As is the case for the decibel and bel, the neper is a unit defined in the international standard ISO 80000. It is not part of the International System of Units (SI), but is accepted for use alongside the SI.Bureau International des Poids et Mesures (2006), The International System of Units (SI) Brochure, 8th edition, pp. 127–128.
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In the cases above, gain will be a dimensionless quantity, as it is the ratio of like units (decibels are not used as units, but rather as a method of indicating a logarithmic relationship). In the bipolar transistor example, it is the ratio of the output current to the input current, both measured in amperes. In the case of other devices, the gain will have a value in SI units. Such is the case with the operational transconductance amplifier, which has an open-loop gain (transconductance) in siemens (mhos), because the gain is a ratio of the output current to the input voltage.
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It was thus more suitable for some of the newer analysis techniques being invented by the United States Air Force. Such a diagram has pressure plotted on the vertical axis, with a logarithmic scale (thus the "log-P" part of the name), and the temperature plotted skewed, with isothermal lines at 45° to the plot (thus the "skew-T" part of the name). Plotting a hypothetical set of measurements with constant temperature for all altitudes would result in a line angled 45° to the right. In practice, since temperature usually drops with altitude, the graphs are usually mostly vertical (see examples linked to below).
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A representation of the INES levels The International Nuclear and Radiological Event Scale (INES) was introduced in 1990 by the International Atomic Energy Agency (IAEA) in order to enable prompt communication of safety significant information in case of nuclear accidents. The scale is intended to be logarithmic, similar to the moment magnitude scale that is used to describe the comparative magnitude of earthquakes. Each increasing level represents an accident approximately ten times as severe as the previous level. Compared to earthquakes, where the event intensity can be quantitatively evaluated, the level of severity of a man-made disaster, such as a nuclear accident, is more subject to interpretation.
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G.711 is a narrowband audio codec originally designed for use in telephony that provides toll-quality audio at 64 kbit/s. G.711 passes audio signals in the range of 300–3400 Hz and samples them at the rate of 8,000 samples per second, with the tolerance on that rate of 50 parts per million (ppm). Non- uniform (logarithmic) quantization with 8 bits is used to represent each sample, resulting in a 64 kbit/s bit rate. There are two slightly different versions: μ-law, which is used primarily in North America and Japan, and A-law, which is in use in most other countries outside North America.
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In cryptography, key size or key length is the number of bits in a key used by a cryptographic algorithm (such as a cipher). Key length defines the upper- bound on an algorithm's security (i.e. a logarithmic measure of the fastest known attack against an algorithm), since the security of all algorithms can be violated by brute-force attacks. Ideally, the lower-bound on an algorithm's security is by design equal to the key length (that is, the security is determined entirely by the keylength, or in other words, the algorithm's design doesn't detract from the degree of security inherent in the key length).
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Dury's model is an extension of the Bradshaw model which shows how river characteristics will change from the upper course to the lower course of a river. Dury used a line graph and a logarithmic scale, plotting the discharge along the x-axis and the other stream variables along the y-axis. Other 'stream variables' included depth, width, velocity, slope (gradient) and friction (Manning formula). Dury's work on quantitative fluvial geomorphology led to a temporary period with the United States Geological Survey in 1960–61 and to co-authorship of the influential study Fluvial Processes in Geomorphology (with Luna B. Leopold, M. Gordon Wolman, and John Miller, 1964).
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It was based on the SR16 design from Kinpo Electronics. Power sources come from smaller solar cells than the 1994 TI-34, and CR2025 battery. Feature set was based on TI-36X II, but without unit conversions and constants, base calculations, boolean algebra, complex value functions (abs now only works in real numbers), integral calculation, engineering notation display modes, gradian angle mode, percentage sign, three 2-variable statistic modes (logarithmic, exponent, power), hyperbolic trigonometry. Functions added over TI-36X II (primarily coming from the TI-32 Math Explorer Plus) include rounding by digits, min/max, lcm/gcd, cube/cubic root, remainder, integer division, percentage conversion, unsimplified fraction, adjustable fraction simplification factor.
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The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus. Calculus is used in every branch of the physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, demography, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired. It allows one to go from (non-constant) rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other. Physics makes particular use of calculus; all concepts in classical mechanics and electromagnetism are related through calculus.
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Thus we should have a macro-theory of > remembering rather than of memory, to say nothing of short-term memory, > proactive inhibition release, or memory scanning. To take another example, > we should have a macro-theory of attending, rather than a mini-theory of > attention, or micro-theories of limited channel capacities or logarithmic > dependencies in disjunctive reaction times. This would ease the dependence > on the information processing analogy, but not necessarily lead to an > abandonment of the information processing terminology, the Flowchart, or the > concept of control structures. The meta-technical sciences can contribute to > a psychology of cognition as well as to cognitive psychology.
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We make X into a fiber space over C by mapping (c,s) to s2. We construct an isomorphism from X minus the fiber over 0 to E×C minus the fiber over 0 by mapping (c,s) to (c-log(s)/2πi,s2). (The two fibers over 0 are non-isomorphic elliptic curves, so the fibration X is certainly not isomorphic to the fibration E×C over all of C.) Then the fibration X has a fiber of multiplicity 2 over 0, and otherwise looks like E×C. We say that X is obtained by applying a logarithmic transformation of order 2 to E×C with center 0.
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The notation is sometimes used in the context of meantone temperament, and does not always assume equal temperament nor the standard concert A4 of 440 Hz; this is particularly the case in connection with earlier music. The standard proposed to the Acoustical Society of America explicitly states a logarithmic scale for frequency, which excludes meantone temperament, and the base frequency it uses gives A4 a frequency of exactly 440 Hz. However, when dealing with earlier music that did not use equal temperament, it is understandably easier to simply refer to notes by their closest modern equivalent, as opposed to specifying the difference using cents every time.
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Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy. Despite the algebraic branch point, the function w is well-defined as a multiple-valued function and, in an appropriate sense, is continuous at the origin. This is in contrast to transcendental and logarithmic branch points, that is, points at which a multiple-valued function has nontrivial monodromy and an essential singularity. In geometric function theory, unqualified use of the term branch point typically means the former more restrictive kind: the algebraic branch points.
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In late February, multiple cases appeared in France, notably within three new clusters, in Oise, Haute-Savoie, and Morbihan. Number of cases (blue) and number of deaths (red) on a logarithmic scale. On 25 February, a French teacher from Crépy-en-Valois died; on the same day, a Chinese man who had returned from China was confirmed as a carrier of SARS-CoV-2, but showed signs of recent recovery. A 64-year-old man from La Balme-de-Sillingy, who returned from a trip to Lombardy on 15 February, tested positive for SARS-CoV-2 and was treated in Centre Hospitalier Annecy-Genevois, Épagny-Metz-Tessy.
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Number of cases (blue) and number of deaths (red) on a logarithmic scale. From January until February 2020, Indonesia reported zero cases of COVID-19, despite being surrounded by infected countries such as Malaysia, Singapore, the Philippines, and Australia. Flights from countries with high infection rate, including South Korea and Thailand, also continued to operate. Health experts and researchers at Harvard University in the United States expressed their concerns, saying that Indonesia is ill-prepared for an outbreak and there could be undetected COVID-19 cases. On 2 March 2020, Indonesian president Joko Widodo announced the first cases in the country: a dance instructor and her mother in Depok, West Java.
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In other words, it is one tactic among many in standardization, whether within a company or within an industry, and it is usually desirable in industrial contexts (unless the goal is vendor lock-in or planned obsolescence) # They are chosen such that when a product is manufactured in many different sizes, these will end up roughly equally spaced on a logarithmic scale. They therefore help to minimize the number of different sizes that need to be manufactured or kept in stock. Preferred numbers represent preferences of simple numbers (such as 1, 2, and 5) multiplied by the powers of a convenient basis, usually 10.
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The results followed Weber's Law, with accuracy decreasing as the ratio between numbers became smaller. This supports the findings made by Neider in macaque monkeys and shows definitive evidence for an approximate number logarithmic scale in humans. With an established mechanism for approximating non-symbolic number in both humans and primates, a necessary further investigation is needed to determine if this mechanism is innate and present in children, which would suggest an inborn ability to process numerical stimuli much like humans are born ready to process language. Cantlon and colleagues set out to investigate this in 4 year old healthy, normally developing children in parallel with adults.
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Games are played between two agents: a machine and its environment, where the machine is required to follow only effective strategies. This way, games are seen as interactive computational problems, and the machine's winning strategies for them as solutions to those problems. It has been established that computability logic is robust with respect to reasonable variations in the complexity of allowed strategies, which can be brought down as low as logarithmic space and polynomial time (one does not imply the other in interactive computations) without affecting the logic. All this explains the name “computability logic” and determines applicability in various areas of computer science.
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While circles are still expanding, a graph that compares the quantities or concentrations of the antigen on a logarithmic scale with the diameters or areas of the circles on a linear scale may be a straight line (kinetic method). . However, circles of the precipitate are smaller and less distinct during expansion than they are after expansion has ended. Further, temperature affects the rate of expansion, but does not affect the size of a circle at its end point. In addition, the range of circle diameters for the same quantities or concentrations of antigen is smaller while some circles are enlarging than they are after all circles have reached their end points.
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Computing tree-depth is computationally hard: the corresponding decision problem is NP-complete.. The problem remains NP-complete for bipartite graphs , as well as for chordal graphs.. On the positive side, tree-depth can be computed in polynomial time on interval graphs,. as well as on permutation, trapezoid, circular-arc, circular permutation graphs, and cocomparability graphs of bounded dimension.. For undirected trees, tree-depth can be computed in linear time.; . give an approximation algorithm for tree-depth with an approximation ratio of O((\log n)^2), based on the fact that tree-depth is always within a logarithmic factor of the treewidth of a graph.
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Those abundances, when plotted on a graph as a function of atomic number, have a jagged sawtooth structure that varies by factors up to ten million. A very influential stimulus to nucleosynthesis research was an abundance table created by Hans Suess and Harold Urey that was based on the unfractionated abundances of the non-volatile elements found within unevolved meteorites. Such a graph of the abundances is displayed on a logarithmic scale below, where the dramatically jagged structure is visually suppressed by the many powers of ten spanned in the vertical scale of this graph. Abundances of the chemical elements in the Solar System.
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Hydrogen and helium are most common, residuals within the paradigm of the Big Bang. The next three elements (Li, Be, B) are rare because they are poorly synthesized in the Big Bang and also in stars. The two general trends in the remaining stellar-produced elements are: (1) an alternation of abundance of elements according to whether they have even or odd atomic numbers, and (2) a general decrease in abundance, as elements become heavier. Within this trend is a peak at abundances of iron and nickel, which is especially visible on a logarithmic graph spanning fewer powers of ten, say between logA=2 (A=100) and logA=6 (A=1,000,000).
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For example, he showed that Charles Elton's observation that the species-to-genus ratio was lower on islands than on mainlands could be expected from chance alone and hence that Elton's interpretation (competitive exclusion) was redundant (which had already been shown three decades earlier by Alvar Palmgren and Arthur Maillefer). In his 1964 book, "Patterns in the Balance of Nature", Williams gave a still valuable overview of this discipline. With Fisher, Williams was able to establish patterns in the diversity and numbers of insects caught in light traps. He noticed that logarithmic patterns were widespread, an idea which was later developed by other ecologists like Frank W. Preston.
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He also designed mechanical devices called "logarithmic dials", for converting traditional and metric measures, that were quickly abandoned in favour of slide rules A member of the Lycée des Arts (School of Arts) and employed in the Prints Department of the Royal Library, the Bibliothèque Royale, he was particularly interested in what was known as "education through the eyes", the use of visual methods for teaching. This led to his collaboration with his friend, the botanist Antoine Nicolas Duchesne, a pioneer of entertaining education, on the forerunner of illustrated magazines, a popular work entitled Portefeuille des enfants (1783–1791), under the direction of Cochin.
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A second disc was marked with a circular scale all around the edge labelled in degrees, with radial lines from the centre of the disc stretching out every ten degrees. The disc also had concentric circles which coincided with the marked scale on the other disc. To aid clarity, markings on the second disc were in a different colour to those on the first. The rear of the device had a separate rotating calculator, where if the ship's speed was set against the 60-minute guide mark, then the distance travelled at any time 0–60 minutes could be read off against the logarithmic time scale.
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A bipolar transistor can be used as the simplest current-to-current converter but its transfer ratio would highly depend on temperature variations, β tolerances, etc. To eliminate these undesired disturbances, a current mirror is composed of two cascaded current-to-voltage and voltage-to- current converters placed at the same conditions and having reverse characteristics. It is not obligatory for them to be linear; the only requirement is their characteristics to be mirrorlike (for example, in the BJT current mirror below, they are logarithmic and exponential). Usually, two identical converters are used but the characteristic of the first one is reversed by applying a negative feedback.
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Reviewing the book for the Telegraph's Family Book Club, the writer Christopher Middleton encapsulates the work as a "7-foot, six-inch-long chart, which starts out some four billion years ago, with the explosion that triggered the Earth’s birth, and ends just a matter of months ago, with the election of Barack Obama and the financial crisis of 2007–2008". The What on Earth? Wallbook is notable for its use of a logarithmic timescale. At the beginning of the timeline 1 cm represents the passage of 1 billion years but by the end of the timeline the same space accounts for just five years.
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In equal temperament, the octave is divided into equal parts on the logarithmic scale. While it is possible to construct equal temperament scale with any number of notes (for example, the 24-tone Arab tone system), the most common number is 12, which makes up the equal-temperament chromatic scale. In western music, a division into twelve intervals is commonly assumed unless it is specified otherwise. For the chromatic scale, the octave is divided into twelve equal parts, each semitone (half-step) is an interval of the twelfth root of two so that twelve of these equal half steps add up to exactly an octave.
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The program can produce generalizations of the normal, logistic, and other distributions by transforming the data using an exponent that is optimized to obtain the best fit. This feature is not common in other distribution-fitting software which normally include only a logarithmic transformation of data obtaining distributions like the lognormal and loglogistic. Generalization of symmetrical distributions (like the normal and the logistic) makes them applicable to data obeying a distribution that is skewed to the right (using an exponent <1) as well as to data obeying a distribution that is skewed to the left (using an exponent >1). This enhances the versatility of symmetrical distributions.
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Because of this inhibition the antibiotics are most effective when the bacteria are in the logarithmic phase of growth, were then they are synthesizing the cell wall. If the bacteria are in the stationary phase of growth then there is no wall synthesizing in progress and the antibiotics have much lower effect. Although the mechanism of action for β-lactam antibiotics is not completely known they are believed to exert their mechanism of action by mimicking the structure of the transition state of the chemical reaction when the transpeptidase is bound to the D-alanyl-D-alanine sequence. These proteins are often referred to as penicillin binding proteins (PBP).
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A problem he published in The Lady's and Gentleman's Diary was the inspiration for Thomas Kirkman to publish his first mathematical work, on Kirkman's schoolgirl problem, beginning the mathematical study of combinatorial designs.. His book, Essays on Musical Intervals, Harmonics, and the Temperament of the Musical Scale, advocated 19-tone equal temperament and used a division of the octave into 730 parts, now designated as Woolhouse units,Logarithmic musical intervals for measuring musical intervals. He is credited with a formula for numerical integration.Woolhouse's Formulas in the MathWorld encyclopedia In 1848 he was a co-founder of the Institute of Actuaries. He died on 12 August 1893.
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The alt=The Ulam spiral Euler noted that the function :n^2 - n + 41 yields prime numbers for 1\le n\le 40, although composite numbers appear among its later values.The sequence of these primes, starting at n=1 rather than n=0, is listed by The search for an explanation for this phenomenon led to the deep algebraic number theory of Heegner numbers and the class number problem. The Hardy-Littlewood conjecture F predicts the density of primes among the values of quadratic polynomials with integer coefficients in terms of the logarithmic integral and the polynomial coefficients. No quadratic polynomial has been proven to take infinitely many prime values.
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Thus, the fundamental solution may be found by performing the continued fraction expansion and testing each successive convergent until a solution to Pell's equation is found. The time for finding the fundamental solution using the continued fraction method, with the aid of the Schönhage–Strassen algorithm for fast integer multiplication, is within a logarithmic factor of the solution size, the number of digits in the pair (x1,y1). However, this is not a polynomial time algorithm because the number of digits in the solution may be as large as , far larger than a polynomial in the number of digits in the input value n.
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At high school level, in most of the U.S., algebra, geometry and analysis (pre-calculus and calculus) are taught as separate courses in different years. Mathematics in most other countries (and in a few U.S. states) is integrated, with topics from all branches of mathematics studied every year. Students in many countries choose an option or pre-defined course of study rather than choosing courses à la carte as in the United States. Students in science-oriented curricula typically study differential calculus and trigonometry at age 16–17 and integral calculus, complex numbers, analytic geometry, exponential and logarithmic functions, and infinite series in their final year of secondary school.
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The works of Adriaan de Groot, William Chase, Herbert A. Simon, and Fernand Gobet have established that knowledge, more than the ability to anticipate moves, plays an essential role in chess-playing. Linearly arranged board games have been shown to improve children's spatial numerical understanding. This is because the game is similar to a number line in that they promote a linear understanding of numbers rather than the innate logarithmic one. Research studies show that board games such as Snakes and Ladders result in children showing significant improvements in aspects of basic number skills such as counting, recognizing numbers, numerical estimation and number comprehension.
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Description of contents of general audio signals usually requires extraction of features that capture specific aspects of the audio signal. Generally speaking, one could divide the features into signal or mathematical descriptors such as energy, description of spectral shape etc., statistical characterization such as change or novelty detection, special representations that are better adapted to the nature of musical signals or the auditory system, such as logarithmic growth of sensitivity (bandwidth) in frequency or octave invariance (chroma). Since parametric models in audio usually require very many parameters, the features are used to summarize properties of multiple parameters in a more compact or salient representation.
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As a result of the error introduced by utilizing probabilistic coin tosses, the notion of acceptance of a string by a probabilistic Turing machine can be defined in different ways. One such notion that includes several important complexity classes is allowing for an error probability of 1/3. For instance, the complexity class BPP is defined as the class of languages recognized by a probabilistic Turing machine in polynomial time with an error probability of 1/3. Another class defined using this notion of acceptance is BPL, which is the same as BPP but places the additional restriction that languages must be solvable in logarithmic space.
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Typically, the geometric mean is used in systems based on certain transformations of lowpass filter designs, where the frequency response is constructed to be symmetric on a logarithmic frequency scale. The geometric center frequency corresponds to a mapping of the DC response of the prototype lowpass filter, which is a resonant frequency sometimes equal to the peak frequency of such systems, for example as in a Butterworth filter. The arithmetic definition is used in more general situations, such as in describing passband telecommunication systems, where filters are not necessarily symmetric but are treated on a linear frequency scale for applications such as frequency-division multiplexing.
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As a result, at the point x\rightarrow \infty where the accuracy of the approximation may be the worst in the ordinary Pade approximation, Good accuracy of the 2-point Pade approximant is guaranteed. Therefore, the 2-point Pade approximant can be a method that gives a good approximation globally for x=0\sim \infty. In cases that f_0(x),f_\infty(x) are expressed by Polynomials or series of negative powers,exponential function,logarithmic function or x\ln x, we can apply 2-point Padé approximant to f(x). There is a method of using this to give an approximate solution of a differential equation with high accuracy.
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Since true range and ATR are calculated by subtracting prices, the volatility they compute does not change when historical prices are back-adjusted by adding or subtracting a constant to every price. Back-adjustments are often employed when splicing together individual monthly futures contracts to form a continuous futures contract spanning a long period of time. However the standard procedures used to compute volatility of stock prices, such as the standard deviation of logarithmic price ratios, are not invariant (to addition of a constant). Thus futures traders and analysts typically use one method (ATR) to calculate volatility, while stock traders and analysts typically use standard deviation of log price ratios.
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The class P, typically taken to consist of all the "tractable" problems for a sequential computer, contains the class NC, which consists of those problems which can be efficiently solved on a parallel computer. This is because parallel computers can be simulated on a sequential machine. It is not known whether NC = P. In other words, it is not known whether there are any tractable problems that are inherently sequential. Just as it is widely suspected that P does not equal NP, so it is widely suspected that NC does not equal P. Similarly, the class L contains all problems that can be solved by a sequential computer in logarithmic space.
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In the first chapter, "Economy", when writing about how indispensable it is to cultivate the habits of a businessman in anything one does, Thoreau describes these habits in a very long list, including :... taking advantage of the results of all exploring expeditions, using new passages and all improvements in navigation;—charts to be studied, the position of reefs and new lights and buoys to be ascertained, and ever, and ever, the logarithmic tables to be corrected, for by the error of some calculator the vessel often splits upon a rock that should have reached a friendly pier—there is the untold fate of La Perouse.
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The development of calculus was at the forefront of 18th century mathematical research, and the Bernoullis--family friends of Euler--were responsible for much of the early progress in the field. Understanding the infinite was the major focus of Euler's research. While some of Euler's proofs may not have been acceptable under modern standards of rigor, his ideas were responsible for many great advances. First of all, Euler introduced the concept of a function, and introduced the use of the exponential function and logarithms in analytic proofs Euler frequently used the logarithmic functions as a tool in analysis problems, and discovered new ways by which they could be used.
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At Oxford University, Edmund Gunter built the first analog device to aid computation. The 'Gunter's scale' was a large plane scale, engraved with various scales, or lines. Natural lines, such as the line of chords, the line of sines and tangents are placed on one side of the scale and the corresponding artificial or logarithmic ones were on the other side. This calculating aid was a predecessor of the slide rule. It was William Oughtred (1575–1660) who first used two such scales sliding by one another to perform direct multiplication and division, and thus is credited as the inventor of the slide rule in 1622.
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Estimated abundances of the 83 primordial elements in the Solar system, plotted on a logarithmic scale. Thorium, at atomic number 90, is one of the rarest elements. In the universe, thorium is among the rarest of the primordial elements, because it is one of the two elements that can be produced only in the r-process (the other being uranium), and also because it has slowly been decaying away from the moment it formed. The only primordial elements rarer than thorium are thulium, lutetium, tantalum, and rhenium, the odd-numbered elements just before the third peak of r-process abundances around the heavy platinum group metals, as well as uranium.
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Intertemporal portfolio choice is the allocation of funds to various assets repeatedly over time, with the amount of investable funds at any future time depending on the portfolio returns at any prior time. Thus the future decisions may depend on the results of current decisions. In general this dependence on prior decisions implies that current decisions must take into account their probabilistic effect on future portfolio constraints. There are some exceptions to this, however: with a logarithmic utility function, or with a HARA utility function and serial independence of returns, it is optimal to act with (rational) myopia, ignoring the effects of current decisions on the future decisions.
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Against this, photographic film can be made with a higher spatial resolution than any other type of imaging detector, and, because of its logarithmic response to light, has a wider dynamic range than most digital detectors. For example, Agfa 10E56 holographic film has a resolution of over 4,000 lines/mm—equivalent to a pixel size of 0.125 micrometers—and an active dynamic range of over five orders of magnitude in brightness, compared to typical scientific CCDs that might have pixels of about 10 micrometers and a dynamic range of 3–4 orders of magnitude. Special films are used for the long exposures required by astrophotography.
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The Human Development Index takes into consideration a number of development and well-being factors that are not taken into account in the calculation of GDP and GNP. The Human Development Index is calculated using the indicators of life expectancy, adult literacy, school enrollment, and logarithmic transformations of per-capita income. Moreover, it is noted that the HDI "is a weighted average of income adjusted for distributions and purchasing power, life expectancy, literacy and health" (p. 16) The HDI is calculated for individual countries with a value between 0 and 1 and is "interpreted...as the ultimate development that has been attained by that nation" (p. 17).
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The spectral power density, compared with white noise, decreases by 3 dB per octave (density proportional to 1/f ). For this reason, pink noise is often called "1/f noise". Since there are an infinite number of logarithmic bands at both the low frequency (DC) and high frequency ends of the spectrum, any finite energy spectrum must have less energy than pink noise at both ends. Pink noise is the only power-law spectral density that has this property: all steeper power-law spectra are finite if integrated to the high-frequency end, and all flatter power-law spectra are finite if integrated to the DC, low- frequency limit.
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Purely functional data structures are often represented in a different way than their imperative counterparts.Purely functional data structures by Chris Okasaki, Cambridge University Press, 1998, For example, array with constant- time access and update is a basic component of most imperative languages and many imperative data-structures, such as hash table and binary heap, are based on arrays. Arrays can be replaced by map or random access list, which admits purely functional implementation, but the access and update time is logarithmic. Therefore, purely functional data structures can be used in languages which are non-functional, but they may not be the most efficient tool available, especially if persistency is not required.
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In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the middle element of the array. If they are not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array.
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It also has the lowest normal boiling point (−24.2 °C), which is where the vapor pressure curve of methyl chloride (the blue line) intersects the horizontal pressure line of one atmosphere (atm) of absolute vapor pressure. Although the relation between vapor pressure and temperature is non-linear, the chart uses a logarithmic vertical axis to produce slightly curved lines, so one chart can graph many liquids. A nearly straight line is obtained when the logarithm of the vapor pressure is plotted against 1/(T + 230) where T is the temperature in degrees Celsius. The vapor pressure of a liquid at its boiling point equals the pressure of its surrounding environment.
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While the dynamic range compression used in audio recording and the like depends on a variable-gain amplifier, and so is a locally linear process (linear for short regions, but not globally), companding is non-linear. The dynamic range of a signal is compressed before transmission and is expanded to the original value at the receiver. The electronic circuit that does this is called a compander and works by compressing or expanding the dynamic range of an analog electronic signal such as sound recorded by a microphone. One variety is a triplet of amplifiers: a logarithmic amplifier, followed by a variable-gain linear amplifier and an exponential amplifier.
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The response of the human eye to light is logarithmic. That is, while the human eye is highly sensitive to changes in the intensity of faint light sources, it is less sensitive to changes in the intensity of brighter light sources since the pupils compensate by dilating or constricting. So, presuming the illumination provided by the lamp was ample at the beginning of its life, and the light output of a bulb gradually decreases by 25%, viewers will perceive a much smaller change in light intensity. Fluorescent lamps get dimmer over their lifetime, so what starts out as an adequate luminosity may become inadequate.
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Another way of looking at it is that these are numbers whose hexadecimal representation contains only the digits 0, 1, 4, 5. For instance, 69 = 10114 = 4516. Equivalently, they are the numbers whose binary and negabinary representations are equal.. Plot of the number of sequence elements up to n divided by \sqrt n, on a logarithmic horizontal scale It follows from either the binary or base-4 definitions of these numbers that they grow roughly in proportion to the square numbers. The number of elements in the Moser–de Bruijn sequence that are below any given threshold n is proportional to \sqrt n, a fact which is also true of the square numbers.
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A world map showing global variations in fertility rate per woman according to the CIA World Factbook's 2016 data Estimates of population evolution in different continents between 1950 and 2050 according to the United Nations. The vertical axis is logarithmic and is in millions of people. World population growth rates between 1950–2050 In 2017, the estimated annual growth rate was 1.1%. The CIA World Factbook gives the world annual birthrate, mortality rate, and growth rate as 1.86%, 0.78%, and 1.08% respectively. The last 100 years have seen a massive fourfold increase in the population, due to medical advances, lower mortality rates, and an increase in agricultural productivity made possible by the Green Revolution.
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Other discrimination tasks, such as detecting changes in brightness, or in tone height (pure tone frequency), or in the length of a line shown on a screen, may have different Weber fractions, but they all obey Weber's law in that observed values need to change by at least some small but constant proportion of the current value to ensure human observers will reliably be able to detect that change. Fechner did not conduct any experiments on how perceived heaviness increased with the mass of the stimulus. Instead, he assumed that all JNDs are subjectively equal, and argued mathematically that this would produce a logarithmic relation between the stimulus intensity and the sensation. These assumptions have both been questioned.
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This means that starting from an empty data structure, any sequence of a insert and decrease key operations and b delete operations would take O(a + b log n) worst case time, where n is the maximum heap size. In a binary or binomial heap such a sequence of operations would take O((a + b) log n) time. A Fibonacci heap is thus better than a binary or binomial heap when b is smaller than a by a non-constant factor. It is also possible to merge two Fibonacci heaps in constant amortized time, improving on the logarithmic merge time of a binomial heap, and improving on binary heaps which cannot handle merges efficiently.
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It consisted of a brass disk on which a number of circular logarithmic scales were inscribed, with two radial wires that could each be locked to a point on the circumference. Using this instrument, a sailor could perform various trigonometric calculations by setting the wire to the position of the argument on one of the circular scales and reading the result from another of the circular scales. Ayres made fine, large Azimuth compasses, used in determining how much the magnetic compass deviated from true north. A brass mariner's compass in gimbals set in mahogany box, made by Ayres in Amsterdam around 1775, is said to have been the property of Sir Isaac Newton.
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When using the Siacci method for different G models, the formula used to compute the trajectories is the same. What differs is retardation factors found through testing of actual projectiles that are similar in shape to the standard project reference. This creates slightly different set of retardation factors between differing G models. When the correct G model retardation factors are applied within the Siacci mathematical formula for the same G model BC, a corrected trajectory can be calculated for any G model. Another method of determining trajectory and ballistic coefficient was developed and published by Wallace H. Coxe and Edgar Beugless of DuPont in 1936. This method is by shape comparison an logarithmic scale as drawn on 10 charts.
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From this, Fechner derived his well-known logarithmic scale, now known as Fechner scale. Weber's and Fechner's work formed one of the bases of psychology as a science, with Wilhelm Wundt founding the first laboratory for psychological research in Leipzig (Institut für experimentelle Psychologie). Fechner's work systematised the introspectionist approach (psychology as the science of consciousness), that had to contend with the Behaviorist approach in which even verbal responses are as physical as the stimuli. During the 1930s, when psychological research in Nazi Germany essentially came to a halt, both approaches eventually began to be replaced by use of stimulus-response relationships as evidence for conscious or unconscious processing in the mind.
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Some operating systems such as OS X express hard drive capacity or file size using decimal multipliers, while others such as Microsoft Windows report size using binary multipliers. This discrepancy causes confusion, as a disk with an advertised capacity of, for example, (meaning ) might be reported by the operating system as , meaning 372 GiB. The JEDEC memory standards use IEEE 100 nomenclature which quote the gigabyte as (230 bytes). The difference between units based on decimal and binary prefixes increases as a semi-logarithmic (linear-log) function—for example, the decimal kilobyte value is nearly 98% of the kibibyte, a megabyte is under 96% of a mebibyte, and a gigabyte is just over 93% of a gibibyte value.
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Gribov noted that this is crucial for gluon confinement, since a mass gap precisely means that the field fluctuations are of a bounded size. This insight played a crucial role in Feynman's semi-quantitative explanation for the confinement phenomenon in 2+1 dimensional nonabelian gauge theory, a method which was recently extended by Karbali and Nair into a fully quantitative description of the 2+1 dimensional nonabelian gauge vacuum. In collaboration with Lev Lipatov, he developed in 1971 an influential theory of logarithmic corrections to deep-inelastic lepton–hadron scattering and electron-positron hadron- production, using evolution equations for the structure functions of the hadrons, the quark–gluon distribution functions. This was a foundational advance in perturbative QCD.
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A bonus system designed around a proper scoring rule will incentivize the forecaster to report probabilities equal to his personal beliefs. In addition to the simple case of a binary decision, such as assigning probabilities to 'rain' or 'no rain', scoring rules may be used for multiple classes, such as 'rain', 'snow', or 'clear'. The image to the right shows an example of a scoring rule, the logarithmic scoring rule, as a function of the probability reported for the event that actually occurred. One way to use this rule would be as a cost based on the probability that a forecaster or algorithm assigns, then checking to see which event actually occurs.
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For health care settings like hospitals and clinics, optimum alcohol concentration to kill bacteria is 70% to 95%. Products with alcohol concentrations as low as 40% are available in American stores, according to researchers at East Tennessee State University. Alcohol rub sanitizers kill most bacteria, and fungi, and stop some viruses. Alcohol rub sanitizers containing at least 70% alcohol (mainly ethyl alcohol) kill 99.9% of the bacteria on hands 30 seconds after application and 99.99% to 99.999%Medical research papers sometimes use "n-log" to mean a reduction of n on a (base 10) logarithmic scale graphing the number of bacteria, thus "5-log" means a reduction by a factor of 105, or 99.999% in one minute.
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Taurinus corresponded with his uncle Ferdinand Karl Schweikart (1780-1859), who was a law professor in Königsberg, among other things about mathematics. Schweikart examined a model (after Giovanni Girolamo Saccheri and Johann Heinrich Lambert) in which the parallel postulate is not satisfied, and in which the sum of three angles of a triangle is less than two right angles (which is now called hyperbolic geometry). While Schweikart never published his work (which he called "astral geometry"), he sent a short summary of its main principles by letter to Carl Friedrich Gauß. Motivated by the work of Schweikart, Taurinus examined the model of geometry on a "sphere" of imaginary radius, which he called "logarithmic-spherical" (now called hyperbolic geometry).
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While he noticed that no contradictions can be found in his logarithmic-spherical geometry, he remained convinced of the special role of Euclidean geometry. According to Paul Stäckel and Friedrich Engel, as well as Zacharias, Taurinus must be given credit as a founder of non-Euclidean trigonometry (together with Gauss), but his contributions cannot be considered as being on the same level as those of the main founders of non-Euclidean geometry, Nikolai Lobachevsky and János Bolyai. Taurinus corresponded with Gauss about his ideas in 1824. In his reply, Gauss mentioned some of his own ideas on the subject, and encouraged Taurinus to further investigate this topic, but he also told Taurinus not to publicly cite Gauss.
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Hartley did not work out exactly how the number M should depend on the noise statistics of the channel, or how the communication could be made reliable even when individual symbol pulses could not be reliably distinguished to M levels; with Gaussian noise statistics, system designers had to choose a very conservative value of M to achieve a low error rate. The concept of an error-free capacity awaited Claude Shannon, who built on Hartley's observations about a logarithmic measure of information and Nyquist's observations about the effect of bandwidth limitations. Hartley's rate result can be viewed as the capacity of an errorless M-ary channel of 2B symbols per second. Some authors refer to it as a capacity.
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A student of Carl Friedrich Gauss and an assistant to Gauss at the university observatory, Goldschmidt frequently collaborated with Gauss on various mathematical and scientific works. Goldschmidt was in turn a professor of Gauss's protegé Bernhard Riemann. Data gathered by Gauss and Goldschmidt on the growth of the logarithmic integral compared to the distribution of prime numbers was cited by Riemann in "On the Number of Primes Less Than a Given Magnitude", Riemann's seminal paper on the prime-counting function. In 1831, Goldschmidt wrote a mathematical treatise in Latin, "Determinatio superficiei minimae rotatione curvae data duo puncta jungentis circa datum axem ortae" ("Determination of the surface-minimal rotation curve given two joined points about a given axis of origin").
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Electrochemical biosensors are normally based on enzymatic catalysis of a reaction that produces or consumes electrons (such enzymes are rightly called redox enzymes). The sensor substrate usually contains three electrodes; a reference electrode, a working electrode and a counter electrode. The target analyte is involved in the reaction that takes place on the active electrode surface, and the reaction may cause either electron transfer across the double layer (producing a current) or can contribute to the double layer potential (producing a voltage). We can either measure the current (rate of flow of electrons is now proportional to the analyte concentration) at a fixed potential or the potential can be measured at zero current (this gives a logarithmic response).
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Bounding volume hierarchies are used to support several operations on sets of geometric objects efficiently, such as in collision detection and ray tracing. Although wrapping objects in bounding volumes and performing collision tests on them before testing the object geometry itself simplifies the tests and can result in significant performance improvements, the same number of pairwise tests between bounding volumes are still being performed. By arranging the bounding volumes into a bounding volume hierarchy, the time complexity (the number of tests performed) can be reduced to logarithmic in the number of objects. With such a hierarchy in place, during collision testing, children volumes do not have to be examined if their parent volumes are not intersected.
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Unlike end point PCR (conventional PCR), real time PCR allows monitoring of the desired product at any point in the amplification process by measuring fluorescence (in real time frame, measurement is made of its level over a given threshold). A commonly employed method of DNA quantification by real- time PCR relies on plotting fluorescence against the number of cycles on a logarithmic scale. A threshold for detection of DNA-based fluorescence is set 3–5 times of the standard deviation of the signal noise above background. The number of cycles at which the fluorescence exceeds the threshold is called the threshold cycle (Ct) or, according to the MIQE guidelines, quantification cycle (Cq).
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Then the photon flux density (watts per metre squared usually) of the transmitted or reflected light is measured with a photodiode, charge coupled device or other light sensor. The transmittance or reflectance value for each wavelength of the test sample is then compared with the transmission or reflectance values from the reference sample. Most instruments will apply a logarithmic function to the linear transmittance ratio to calculate the 'absorbency' of the sample, a value which is proportional to the 'concentration' of the chemical being measured. In short, the sequence of events in a scanning spectrophotometer is as follows: # The light source is shone into a monochromator, diffracted into a rainbow, and split into two beams.
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In the mathematical field of complex analysis, a branch point of a multi- valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept. Branch points fall into three broad categories: algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation w2 = z for w as a function of z.
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This notation gives the logarithm of the ratio of metal elements (M) to hydrogen (H), minus the logarithm of the Sun's metal ratio. (Thus if the star matches the metal abundance of the Sun, this value will be zero.) A logarithmic value of 0.07 is equivalent to an actual metallicity ratio of 1.17, so the star is about 17% richer in metallic elements than the Sun. However the margin of error for this result is relatively large. The spectrum of A-class stars such as IK Pegasi A show strong Balmer lines of hydrogen along with absorption lines of ionized metals, including the K line of ionized calcium (Ca II) at a wavelength of 393.3 nm.
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There were some differences in the activations, with adults displaying more robust bilateral activation, where the 4 year olds primarily showed activation in their right IPS and activated 112 less voxels than the adults. This suggests that at age 4, children have an established mechanism of neurons in the IPS tuned for processing non-symbolic numerosities. Other studies have gone deeper into this mechanism in children and discovered that children do also represent approximate numbers on a logarithmic scale, aligning with the claims made by Piazza in adults. A study by Izard and colleagues investigated abstract number representations in infants using a different paradigm than the previous researchers because of the nature and developmental stage of the infants.
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Then, in a joint work with Eric Carlen, he addressed the stability analysis of some Gagliardo–Nirenberg and logarithmic Hardy–Littlewood–Sobolev inequalities to obtain a quantitative rate of convergence for the critical mass Keller–Segel equation. He also worked on Hamilton–Jacobi equations and their connections to weak Kolmogorov–Arnold–Moser theory. In a paper with Gonzalo Contreras and Ludovic Rifford, he proved generic hyperbolicity of Aubry sets on compact surfaces. In addition, he has given several contributions to the Di Perna–Lions' theory, applying it both to the understanding of semiclassical limits of the Schrödinger equation with very rough potentials, and to study the Lagrangian structure of weak solutions to the Vlasov–Poisson equation.
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With these algorithms, recipients can find their respective entrypoints into the PURB with only a logarithmic number of trial decryptions using symmetric-key cryptography and only one expensive public-key operation per cipher suite. A third technical challenge is representing the public-key cryptographic material that needs to be encoded into each entrypoint in a PURB, such as the ephemeral Diffie- Hellman public key a recipient needs to derive the shared secret, in an encoding indistinguishable from uniformly random bits. Because the standard encodings of elliptic-curve points are readily distinguishable from random bits, for example, special indistinguishable encoding algorithms must be used for this purpose, such as Elligator and its successors.
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It can be proved by direct analysis of the doubling of a point on E. Some years later André Weil took up the subject, producing the generalisation to Jacobians of higher genus curves over arbitrary number fields in his doctoral dissertation published in 1928. More abstract methods were required, to carry out a proof with the same basic structure. The second half of the proof needs some type of height function, in terms of which to bound the 'size' of points of A(K). Some measure of the co-ordinates will do; heights are logarithmic, so that (roughly speaking) it is a question of how many digits are required to write down a set of homogeneous coordinates.
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Brown, 2006, op. cit., p. 26. The brickwork is considerably more complex than in a helicoidal design and, in order to ensure that the courses of bricks meet the faces of the arch at right angles, many had to be cut to produce tapers. The corne de vache approach tends to result in a structure that is almost as strong as one built to the logarithmic pattern and considerably stronger than one built to the helicoidal pattern but, again, the extra complexity has meant that the method has not seen widespread adoption, especially since the simpler helicoidal structure can be built much stronger if a segmental design is chosen, rather than a full-centred one.
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Spaced responding and choice: A preliminary analysis. Journal of the Experimental Analysis of Behavior, 11, 669-682. and was generalized by (Baum, 1974)Baum, W. M. (1974) On two types of deviation from the matching law: bias and undermatching. J. Exp. Anal. Behav. 22(1), 231-242.. It has been found to fit a wide variety of matching data. The power law was shown by Mackay (1963) MacKay, D. M. Psychophysics of perceived intensity:A theoretical basis for Fechner's and Stevens' laws. Science, 1963, 139, 1213-1216. to be derivable from input and output logarithmic function and psychophysical and other behavioral data fitting this model were described by Staddon (1975). Staddon, J. E. R. (1978).
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Furthermore, all tournaments with a buy- in larger than $20,000 are treated as though the buy-in was only $20,000 and all tournaments with a buy-in smaller than $400 are treated as though the buy- in was $400. The GPI also includes a Finishing Factor, which measures how players perform relative to the rest of the field they compete within in any given poker tournament. This Finishing Factor is calculated initially by dividing the field size by the player's finishing position and is also done using a logarithmic function. Furthermore, for any tournaments with a field size larger than 2,700 players will be treated as though their field was only 2,700.
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The advantage of direct measurement is the ability to evaluate trace levels in the gaseous composition.Oppenheimer, C. Fischer, T., Scaillet, B., 2014, Volcanic Degassing: Process and Impact, In Treatise on Geochemistry (Second Edition), edited by H. D. Holland and K. K. Turekian, Elsevier, Oxford, pp. 111–179, Volcanic gasses can be indirectly measured using Total Ozone Mapping Spectrometry (TOMS), a satellite-remote sensing tool which evaluates SO2 clouds in the atmosphere.[11][14] TOMS’ disadvantage is that its high detection limit can only measure large amounts of exuded gases, such as those emitted by an eruption with a Volcanic Explosivity Index (VEI) of 3, on a logarithmic scale of 0 to 7.
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Many researchers in the field challenged the report, because they asserted it was not biologically possible for the bat's sonar system to discriminate such small time differences at ultrasonic frequencies. Simmons continues to work on this problem to explore biological processes that could support sensitivity to small changes in echo delay. Through behavioral experiments, Simmons demonstrated time-varying gain in the sonar receiver of echolocating bats. The hearing sensitivity of the big brown bat decreases before each sonar pulse is emitted and then recovers in a logarithmic fashion to compensate for the two- way transmission loss of sonar returns, thereby maintaining a constant echo sensation level over a distance of about 1.5 meters.
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Glick based this work on a prior scholar's work (Bulliet). On page 33 of this book, Glick writes that Bulliet said "that the rate of conversion to Islam is logarithmic, and may be illustrated graphically by a logistic curve". horseshoe arches of the grand Mosque of Cordoba, built by Abd al Rahman I. Christians more often converted to Islam than Jews although there were converted Jews among the new followers of Islam. There was a great deal of freedom of interaction among the groups: for example, Sarah, the granddaughter of the Visigoth king Wittiza, married a Muslim man and bore two sons who were later counted among the ranks of the highest Arab nobility.
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Many common reference-based data structures, such as red–black trees, stacks, and treaps, can easily be adapted to create a persistent version. Some others need slightly more effort, for example: queues, dequeues, and extensions including min-deques (which have an additional O(1) operation min returning the minimal element) and random access deques (which have an additional operation of random access with sub-linear, most often logarithmic, complexity). There also exist persistent data structures which use destructive operations, making them impossible to implement efficiently in purely functional languages (like Haskell outside specialized monads like state or IO), but possible in languages like C or Java. These types of data structures can often be avoided with a different design.
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ISO 1683:2015 Use of the decibel in underwater acoustics leads to confusion, in part because of this difference in reference value.C. S. Clay (1999), Underwater sound transmission and SI units, J Acoust Soc Am 106, 3047 The human ear has a large dynamic range in sound reception. The ratio of the sound intensity that causes permanent damage during short exposure to that of the quietest sound that the ear can hear is greater than or equal to 1 trillion (1012). Such large measurement ranges are conveniently expressed in logarithmic scale: the base-10 logarithm of 1012 is 12, which is expressed as a sound pressure level of 120 dB re 20 μPa.
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Between 1933 and 1938 he applied his results to elliptic equations, establishing the majorizing limits for their solutions, generalizing the two- dimensional case of Felix Bernstein. At the same time he studied analytic functions of several complex variables, that is, analytic functions whose domain belongs to the vector space , proving in 1933 the fundamental theorem on normal families of such functions: if a family is normal with respect to every complex variable, it is also normal with respect to the set of the variables. He also proved a logarithmic residue formula for functions of two complex variables in 1949. In 1935 Caccioppoli proved the analyticity of class solutions of elliptic equations with analytic coefficients.
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Digital Picture Exchange (DPX) is a common file format for digital intermediate and visual effects work and is a SMPTE standard (ST 268-1:2014). The file format is most commonly used to represent the density of each colour channel of a scanned negative film in an uncompressed "logarithmic" image where the gamma of the original camera negative is preserved as taken by a film scanner. For this reason, DPX is the worldwide-chosen format for still frames storage in most digital intermediate post-production facilities and film labs. Other common video formats are supported as well (see below), from video to purely digital ones, making DPX a file format suitable for almost any raster digital imaging applications.
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It came into service in 1938. It kept the Smith meter's logarithmic, white-on-black display, and included all the key design features that are still used to this day with only slight modification: full-wave rectification, fast integration and slow return times, and a simple scale calibrated from 1 to 7. Mayo and others determined the integration and return times by a series of experiments. At first, they intended to create a true peak meter to prevent transmitters from exceeding 100% modulation. They created a prototype meter with an integration time of about 1 ms. They found that the ear tolerates distortion of only a few ms, and that a 'registration time' of 4 ms is sufficient.
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A LNS has been used in the Gravity Pipe (GRAPE-5) special- purpose supercomputer that won the Gordon Bell Prize in 1999. A substantial effort to explore the applicability of LNSs as a viable alternative to floating point for general-purpose processing of single-precision real numbers is described in the context of the European Logarithmic Microprocessor (ELM). A fabricated prototype of the processor, which has a 32-bit cotransformation- based LNS arithmetic logic unit (ALU), demonstrated LNSs as a "more accurate alternative to floating-point", with improved speed. Further improvement of the LNS design based on the ELM architecture has shown its capability to offer significantly higher speed and accuracy than floating-point as well.
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Optimizing compilers such as GCC or Clang may compile a switch statement into either a branch table or a binary search through the values in the cases.Vlad Lazarenko. From Switch Statement Down to Machine Code A branch table allows the switch statement to determine with a small, constant number of instructions which branch to execute without having to go through a list of comparisons, while a binary search takes only a logarithmic number of comparisons, measured in the number of cases in the switch statement. Normally, the only method of finding out if this optimization has occurred is by actually looking at the resultant assembly or machine code output that has been generated by the compiler.
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In 1898 he became successor to G. van Overbeek de Meyer, as Professor in Hygiene and Forensic Medicine at Utrecht. His inaugural speech was entitled Over Gezondheid en Ziekten in Tropische Gewesten (On health and diseases in tropical regions). At Utrecht, Eijkman turned to the study of bacteriology, and carried out his well-known fermentation test, by means of which it can be readily established if water has been polluted by human and animal defecation containing coli bacilli. Another research was into the rate of mortality of bacteria as a result of various external factors, whereby he was able to show that this process could not be represented by a logarithmic curve.
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From 1924-1927 Ballantine carried out independent studies of radio propagation in White Haven, Pennsylvania, then was briefly research director at the Radio Frequency Laboratories, and in 1929 collaborated with F. M. Huntoon in studying the effects of high pressure on bacteria. From 1929-1934 he was President of the Boonton Research Laboratories investigating errors in microphones due to diffraction and cavity resonance, and developing new devices including an electrostethoscope, automatic optical recorder for frequency-response measurements, and logarithmic voltmeter. In 1934 he founded Ballantine Laboratories, which he led until his death. There he developed improved techniques for measuring the performance of microphones and loudspeakers, and most notably, the first throat microphone for aircraft pilots.
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The color defined as green in the Munsell color system (Munsell 5G) is shown at right. The Munsell color system is a color space that specifies colors based on three color dimensions: hue, value (lightness), and chroma (color purity), spaced uniformly in three dimensions in the elongated oval at an angle shaped Munsell color solid according to the logarithmic scale which governs human perception. In order for all the colors to be spaced uniformly, it was found necessary to use a color wheel with five primary colors—red, yellow, green, blue, and purple. The hues of the Munsell color system, at varying values, and maximum chroma to stay in the sRGB gamut.
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The scale of dBZ values can be seen along the bottom of the image. dBZ stands for decibel relative to Z. It is a logarithmic dimensionless technical unit used in radar, mostly in weather radar, to compare the equivalent reflectivity factor (Z) of a remote object (in mm6 per m3) to the return of a droplet of rain with a diameter of 1 mm (1 mm6 per m3). It is proportional to the number of drops per unit volume and the sixth power of drops' diameter and is thus used to estimate the rain or snow intensity. With other variables analyzed from the radar returns it helps to determine the type of precipitation.
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For some while, ASA grades were also printed on film boxes, and they saw life in the form of the APEX speed value Sv (without degree symbol) as well. ASA PH2.5-1960 was revised as ANSI PH2.5-1979, without the logarithmic speeds, and later replaced by NAPM IT2.5–1986 of the National Association of Photographic Manufacturers, which represented the US adoption of the international standard ISO 6\. The latest issue of ANSI/NAPM IT2.5 was published in 1993. The standard for color negative film was introduced as ASA PH2.27-1965 and saw a string of revisions in 1971, 1976, 1979 and 1981, before it finally became ANSI IT2.27–1988 prior to its withdrawal.
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Space partitioning is particularly important in computer graphics, especially heavily used in ray tracing, where it is frequently used to organize the objects in a virtual scene. A typical scene may contain millions of polygons. Performing a ray/polygon intersection test with each would be a very computationally expensive task. Storing objects in a space-partitioning data structure (k-d tree or BSP tree for example) makes it easy and fast to perform certain kinds of geometry queries—for example in determining whether a ray intersects an object, space partitioning can reduce the number of intersection test to just a few per primary ray, yielding a logarithmic time complexity with respect to the number of polygons.
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From 1632 Gregoire resided with The Society in Ghent and served as a mathematics teacher.Herman van Looy (1984) "A Chronology and Historical Analysis of the mathematical Manuscripts of Gregorius a Sancto Vincentio (1584–1667)", Historia Mathematica 11: 57–75 :The mathematical thinking of Sancto Vincentio underwent a clear evolution during his stay in Antwerp. Starting from the problem of trisection of the angle and the determination of the two mean proportional, he made use of infinite series, the logarithmic property of the hyperbola, limits, and the related method of exhaustion. Sancto Vicentio later applied this last method, in particular to his theory ducere planum in planum, which he developed in Louvain in the years 1621 to 24.
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An algorithm is said to run in sub-linear time (often spelled sublinear time) if T(n) = o(n). In particular this includes algorithms with the time complexities defined above. Typical algorithms that are exact and yet run in sub-linear time use parallel processing (as the NC1 matrix determinant calculation does), or alternatively have guaranteed assumptions on the input structure (as the logarithmic time binary search and many tree maintenance algorithms do). However, formal languages such as the set of all strings that have a 1-bit in the position indicated by the first log(n) bits of the string may depend on every bit of the input and yet be computable in sub-linear time.
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In 1988 Hinnant exhibited work based on the 13th century Fibonacci numbers and Lucas Pacioli's treatise. He began studying Sacred Geometry formally in 1989 with his mentor, inventor and physicist Robert L. Powell, Sr. In the 1980s Hinnant introduced fractal mathematics, the Golden Ratio, the Logarithmic Spiral, Sacred Geometery and most currently STEAM concepts into his work to explore metaphysical ideas. The ancient architectures in Egypt, India, Rome and Greece that use these concepts were his original influences, unpacking "pre-material template of energies," he said. Hinnant also has referenced Buckminster Fuller, Leonardo da Vinci, Frank Lloyd Wright and M.C. Escher as inspirations because of their use of geometry and math in making their artworks.
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The hues of the Munsell color system, at varying values, and maximum chroma to stay in the sRGB gamut. The color defined as yellow in the Munsell color system (Munsell 5Y) is shown at apex of color wheel. The Munsell color system is a color space that specifies colors based on three color dimensions: hue, value (lightness), and chroma (color purity), spaced uniformly in three dimensions in the elongated oval at an angle shaped Munsell color solid according to the logarithmic scale which governs human perception. In order for all the colors to be spaced uniformly, it was found necessary to use a color wheel with five primary colors—red, yellow, green, blue, and purple.
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In her practice Dodd explores “the exhibition as a ritualized space -- in which paintings conceived as characters, mythical and poetic fragments, or totems, are activated and transformed over a period of time.” Her work and its viewers are "cast" as “protagonists in a highly complex theatre of signifiers.” These signifiers are drawn from elemental, art historical, and religious iconographies such as: logarithmic spirals, the bony labyrinth, the Cretan labyrinth, the Georgian dragon, Greek mythology, astrological symbols, Venus and the Divine Feminine, Pablo Picasso’s Guernica, and mid-century modern furniture. Dodd conceived of a single monumental painting the exact size of Picasso's Guernica for an exhibition at the Rubell Family Collection in 2014.
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Amplitude integrated electroencephalography (aEEG) or cerebral function monitoring (CFM) is a technique for monitoring brain function in intensive care settings over longer periods of time than the traditional electroencephalogram (EEG), typically hours to days. By placing electrodes on the scalp of the patient, a trace of electrical activity is produced which is then displayed on a semilogarithmic graph of peak-to-peak amplitude over time; amplitude is logarithmic and time is linear. In this way, trends in electrical activity in the cerebral cortex can be interpreted to inform on events such as seizures or suppressed brain activity.Maynard D, Prior P, Scott D. Device for continuous monitoring of cerebral activity in resuscitated patients.
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Conventionally, the surviving fraction is depicted on a logarithmic scale, and is plotted on the y-axis against dose on the x-axis. The linear quadratic model is now most often used to describe the cell survival curve, assuming that there are two mechanisms to cell death by radiation: A single lethal event or an accumulation of harmful but non-lethal events. Cell survival fractions are exponential functions with a dose-dependent term in the exponent due to the Poisson statistics underlying the stochastic process. Whereas single lethal events lead to an exponent that is linearly related to dose, the survival fraction function for a two-stage mechanism carries an exponent proportional to the square of dose.
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Their gravitational field would deform the horizon of the black hole, and the deformed horizon could produce different outgoing particles than the undeformed horizon. When a particle falls into a black hole, it is boosted relative to an outside observer, and its gravitational field assumes a universal form. 't Hooft showed that this field makes a logarithmic tent-pole shaped bump on the horizon of a black hole, and like a shadow, the bump is an alternative description of the particle's location and mass. For a four- dimensional spherical uncharged black hole, the deformation of the horizon is similar to the type of deformation which describes the emission and absorption of particles on a string-theory world sheet.
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Also, the `siftDown` version of heapify has time complexity, while the `siftUp` version given below has time complexity due to its equivalence with inserting each element, one at a time, into an empty heap. This may seem counter-intuitive since, at a glance, it is apparent that the former only makes half as many calls to its logarithmic-time sifting function as the latter; i.e., they seem to differ only by a constant factor, which never affects asymptotic analysis. To grasp the intuition behind this difference in complexity, note that the number of swaps that may occur during any one siftUp call increases with the depth of the node on which the call is made.
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In electronics, gain is a measure of the ability of a two-port circuit (often an amplifier) to increase the power or amplitude of a signal from the input to the output port by adding energy converted from some power supply to the signal. It is usually defined as the mean ratio of the signal amplitude or power at the output port to the amplitude or power at the input port. It is often expressed using the logarithmic decibel (dB) units ("dB gain"). A gain greater than one (greater than zero dB), that is amplification, is the defining property of an active component or circuit, while a passive circuit will have a gain of less than one.
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In computer science, fractional cascading is a technique to speed up a sequence of binary searches for the same value in a sequence of related data structures. The first binary search in the sequence takes a logarithmic amount of time, as is standard for binary searches, but successive searches in the sequence are faster. The original version of fractional cascading, introduced in two papers by Chazelle and Guibas in 1986 (; ), combined the idea of cascading, originating in range searching data structures of and , with the idea of fractional sampling, which originated in . Later authors introduced more complex forms of fractional cascading that allow the data structure to be maintained as the data changes by a sequence of discrete insertion and deletion events.
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Many modern audio analyzers contain measurement sequences that automate this procedure, and the focus of recent developments has been on quasi-anechoic measurements. These techniques allow loudspeakers to be characterised in a non-ideal (noisy) environment, without the need for an anechoic chamber, which makes them ideally suited for use in high volume production line manufacturing. Most quasi-anechoic measurements are based around an impulse response created from a sine wave whose frequency is swept on a logarithmic scale, with a window function applied to remove any acoustic reflections. The log swept sine method increases signal-to-noise ratio and also allows measurement of individual distortion harmonics up to the Nyquist frequency, something which previously impossible with older analysis techniques such as MLS (Maximum Length Sequence).
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The vertical scale is logarithmic in the number of digits, thus being a \log(\log(y)) function in the value of the prime. The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. Alan Turing searched for them on the Manchester Mark 1 in 1949,Brian Napper, The Mathematics Department and the Mark 1. but the first successful identification of a Mersenne prime, , by this means was achieved at 10:00 pm on January 30, 1952 using the U.S. National Bureau of Standards Western Automatic Computer (SWAC) at the Institute for Numerical Analysis at the University of California, Los Angeles, under the direction of Lehmer, with a computer search program written and run by Prof.
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The Palermo Technical Impact Hazard Scale is a logarithmic scale used by astronomers to rate the potential hazard of impact of a near-earth object (NEO). It combines two types of data--probability of impact and estimated kinetic yield--into a single "hazard" value. A rating of 0 means the hazard is equivalent to the background hazard (defined as the average risk posed by objects of the same size or larger over the years until the date of the potential impact). A rating of +2 would indicate the hazard is 100 times greater than a random background event. Scale values less than −2 reflect events for which there are no likely consequences, while Palermo Scale values between −2 and 0 indicate situations that merit careful monitoring.
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Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular (see History of the function concept).Struik (1969), 367 In the 18th century, "function" lost these geometrical associations. Leibniz also believed that the sum of an infinite number of zeros would equal to one half using the analogy of the creation of the world from nothing. Leibniz was also one of the pioneers in actuarial science, calculating the purchase price of life annuities and the liquidation of a state's debt.
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To produce an audible signal in a pair of headphones requires this signal to be amplified a trillion-fold or more. The magnitudes of the required gain are so great that the logarithmic unit decibel is preferred - a gain of 1 trillion times the power is 120 decibels, which is a value achieved by many common receivers. Gain is provided by one or more amplifier stages in a receiver design; some of the gain is applied at the radio-frequency part of the system, and the rest at the frequencies used by the recovered information (audio, video, or data signals). Selectivity is the ability to "tune in" to just one station of the many that may be transmitting at any given time.
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Monin–Obukhov (M–O) similarity theory describes non-dimensionalized mean flow and mean temperature in the surface layer under non-neutral conditions as a function of the dimensionless height parameter, named after Russian scientists A. S. Monin and A. M. Obukhov. Similarity theory is an empirical method which describes universal relationships between non-dimensionalized variables of fluids based on the Buckingham Pi theorem. Similarity theory is extensively used in boundary layer meteorology, since relations in turbulent processes are not always resolvable from first principles. An idealized vertical profile of the mean flow for a neutral boundary layer is the logarithmic wind profile derived from Prandtl's mixing length theory, which states that the horizontal component of mean flow is proportional to the logarithm of height.
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According to a historiographical tradition widespread in the Arab world, his work would have led to the discovery of the logarithm function around 1591; 23 years before the Scottish John Napier, notoriously known to be the inventor of the function of the natural logarithm. This hypothesis is based initially on the interpretation of Sâlih Zekî of the handwritten copy of the work of Ibn Hamza, interpreted a posteriori in the Arab and Ottoman world as laying the foundations of the logarithmic function. Zekî published in 1913, a two-volume work on the history of mathematical sciences, written in Ottoman Turkish: Âsâr-ı Bâkiye (literally in Turkish: The memories that remain). where his observations on Ibn Hamza's role in the invention of logarithms appear.
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In computational complexity theory, polyL is the complexity class of decision problems that can be solved on a deterministic Turing machine by an algorithm whose space complexity is bounded by a polylogarithmic function in the size of the input. In other words, polyL = DSPACE((log n)O(1)), where n denotes the input size, and O(1) denotes a constant. Just as L ⊆ P, polyL ⊆ QP. However, the only proven relationship between polyL and P is that polyL ≠ P; it is unknown if polyL ⊊ P, if P ⊊ polyL, or if neither is contained in the other. One proof that polyL ≠ P is that P has a complete problem under logarithmic space many-one reductions but polyL does not due to the space hierarchy theorem.
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A proper scoring rule is said to be local if its estimate for the probability of a specific event depends only on the probability of that event. This statement is vague in most descriptions but we can, in most cases, think of this as the optimal solution of the scoring problem "at a specific event" is invariant to all changes in the observation distribution that leave the probability of that event unchanged. All binary scores are local because the probability assigned to the event that did not occur is determined so there is no degree of flexibility to vary over. Affine functions of the logarithmic scoring rule are the only strictly proper local scoring rules on a finite set that is not binary.
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Spectrophotometry is a tool that hinges on the quantitative analysis of molecules depending on how much light is absorbed by colored compounds. Important features of spectrophotometers are spectral bandwidth (the range of colors it can transmit through the test sample), the percentage of sample-transmission, the logarithmic range of sample-absorption, and sometimes a percentage of reflectance measurement. A spectrophotometer is commonly used for the measurement of transmittance or reflectance of solutions, transparent or opaque solids, such as polished glass, or gases. Although many biochemicals are colored, as in, they absorb visible light and therefore can be measured by colorimetric procedures, even colorless biochemicals can often be converted to colored compounds suitable for chromogenic color-forming reactions to yield compounds suitable for colorimetric analysis.
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K (from the Russian word класс, "class", in the sense of a category .) is a measure of earthquake magnitude in the energy class or K-class system, developed in 1955 by Soviet seismologists in the remote Garm (Tadjikistan) region of Central Asia; in revised form it is still used for local and regional quakes in many states formerly aligned with the Soviet Union (including Cuba). Based on seismic energy (K = log ES, in Joules), difficulty in implementing it using the technology of the time led to revisions in 1958 and 1960. Adaptation to local conditions has led to various regional K scales, such as KF and KS.; NMSOP-2 ; . K values are logarithmic, similar to Richter-style magnitudes, but have a different scaling and zero point.
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The Rowland circle geometry ensures that the slits are both in focus, but in order for the Bragg condition to be met at all points, the crystal must first be bent to a radius of 2R (where R is the radius of the Rowland circle), then ground to a radius of R. This arrangement allows higher intensities (typically 8-fold) with higher resolution (typically 4-fold) and lower background. However, the mechanics of keeping Rowland circle geometry in a variable-angle monochromator is extremely difficult. In the case of fixed- angle monochromators (for use in simultaneous spectrometers), crystals bent to a logarithmic spiral shape give the best focusing performance. The manufacture of curved crystals to acceptable tolerances increases their price considerably.
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A novel infectious pathogen to which a population has no immunity will generally spread exponentially in the early stages, while the supply of susceptible individuals is plentiful. The SARS- CoV-2 virus that causes COVID-19 exhibited exponential growth early in the course of infection in several countries in early 2020.Worldometer: COVID-19 CORONAVIRUS PANDEMIC Many factors, ranging from lack of susceptibles (either through the continued spread of infection until it passes the threshold for herd immunity or reduction in the accessibility of susceptibles through physical distancing measures), exponential-looking epidemic curves may first linearize (replicating the "logarithmic" to "logistic" transition first noted by Pierre-François Verhulst, as noted above) and then reach a maximal limit. A logistic function, or related functions (e.g.
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The kth power of a graph with n vertices and m edges may be computed in time O(mn) by performing a breadth first search starting from each vertex to determine the distances to all other vertices, or slightly faster using more sophisticated algorithms. Alternatively, If A is an adjacency matrix for the graph, modified to have nonzero entries on its main diagonal, then the nonzero entries of Ak give the adjacency matrix of the kth power of the graph,. from which it follows that constructing kth powers may be performed in an amount of time that is within a logarithmic factor of the time for matrix multiplication. The kth powers of trees can be recognized in time linear in the size of the input graph. .
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Ralf Hinze and Ross Paterson state a finger tree is a functional representation of persistent sequences that can access the ends in amortized constant time. Concatenation and splitting can be done in logarithmic time in the size of the smaller piece. The structure can also be made into a general purpose data structure by defining the split operation in a general form, allowing it to act as a sequence, priority queue, search tree, or priority search queue, among other varieties of abstract data types.. A finger is a point where one can access part of a data structure; in imperative languages, this is called a pointer. In a finger tree, the fingers are structures that point to the ends of a sequence, or the leaf nodes.
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If a voltage is applied to the BJT base-emitter junction as an input quantity and the collector current is taken as an output quantity, the transistor will act as an exponential voltage-to-current converter. By applying a negative feedback (simply joining the base and collector) the transistor can be "reversed" and it will begin acting as the opposite logarithmic current-to-voltage converter; now it will adjust the "output" base-emitter voltage so as to pass the applied "input" collector current. The simplest bipolar current mirror (shown in Figure 1) implements this idea. It consists of two cascaded transistor stages acting accordingly as a reversed and direct voltage-to-current converters. The emitter of transistor Q1 is connected to ground.
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The hierarchical decision process (HDP) refines the classical analytic hierarchy process (AHP) a step further in eliciting and evaluating subjective judgements. These improvements, proposed initially by Dr. Jang Ra (a student of Dr. Thomas L. Saaty who developed and refined AHP) include the constant-sum measurement scale (1–99 scale) for comparing two elements, the logarithmic least squares method (LLSM) for computing normalized values, the sum of inverse column sums (SICS) for measuring the degree of (in)consistency, and sensitivity analysis of pairwise comparisons matrices. These subtle modifications address issues concerning normal AHP consistency and applicability in the process of constructing hierarchies: generating criteria, classifying/selecting criteria, and screening/selecting decision alternatives.Technology Management: the New International Language, 27–31 Oct 1991, pp.
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The second type of S-curve is more apt for longer cross-fades, since they are smooth and have the ability to have both of the crossfades in the overall level; so that they are audible for as long as possible. There is a short period at the start of each of the cross-fades where the outgoing sound drops toward 50% quickly (with the incoming sound rising just as fast to 50%). This acceleration of sound slows and both sounds will appear as if they are at the same level for most of the cross-fade (in the middle) before the changeover happens. DAW's gives one the ability to change the shape of logarithmic, exponential, and S-curve fades and cross-fades.
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At one end the resistance of the scale is at 0 and at the other side it is infinite. A. Nisbett explains the fader law as follows in his book called The Sound studio:"The ‘law’ of the fader is near-logarithmic over much of its range, which means that a scale of decibels can be made linear (or close to it) over a working range of perhaps 60 dB. If the resistance were to increase according to the same law beyond this, it would be twice as long before reaching a point where the signal is negligible. But the range below -50 dB is of little practical use, so here the rate of fade increases rapidly to the final cut-off".
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Cotes's major original work was in mathematics, especially in the fields of integral calculus, logarithms, and numerical analysis. He published only one scientific paper in his lifetime, titled Logometria, in which he successfully constructs the logarithmic spiral.O'Connor & Robertson (2005)In Logometria, Cotes evaluated e, the base of natural logarithms, to 12 decimal places. See: Roger Cotes (1714) "Logometria," Philosophical Transactions of the Royal Society of London, 29 (338) : 5-45; see especially the bottom of page 10. From page 10: "Porro eadem ratio est inter 2,718281828459 &c; et 1, … " (Furthermore, the same ratio is between 2.718281828459… and 1, … ) After his death, many of Cotes's mathematical papers were hastily edited by his cousin Robert Smith and published in a book, Harmonia mensurarum.
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The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations of percentages, square roots, exponentiation, logarithmic functions, and even trigonometric functions, in the same vein as logarithms (prosthaphaeresis). Arithmetic expressions must be evaluated according to the intended sequence of operations. There are several methods to specify this, either—most common, together with infix notation—explicitly using parentheses and relying on precedence rules, or using a prefix or postfix notation, which uniquely fix the order of execution by themselves. Any set of objects upon which all four arithmetic operations (except division by zero) can be performed, and where these four operations obey the usual laws (including distributivity), is called a field.
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It is believedBoaz Barak's course on Computational Complexity Lecture 2 that NP is not equal to co-NP; however, it has not yet been proven. It is clear that if these two complexity classes are not equal then P is not equal to NP, since P=co-P. Thus if P=NP we would have co-P=co-NP whence NP=P=co-P=co-NP. Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes.
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One of the artist's ongoing projects is "An End to Modernity" (2005), commissioned by the Wexner Center for the Arts at Ohio State University. The piece is a twelve-foot-wide by ten-foot-high chandelier of chrome and transparent glass modeled on the 1960s Lobmeyr design for the chandeliers found in Lincoln Center, and evoking as well the Big Bang theory. "The End of the Dark Ages," again inspired by the Metropolitan Opera House chandeliers and informed by logarithmic equations devised by the cosmologist David H. Weinberg was shown in New York City in 2008. Later that year, the series culminated in a massive installation titled "Island Universe" at White Cube in London"The Big Picture" by Alex Browne, The New York Times, September 26, 2008.
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For proving bounds on this problem, it may be assumed without loss of generality that the inputs are strings over a two-letter alphabet. For, if two strings over a larger alphabet differ then there exists a string homomorphism that maps them to binary strings of the same length that also differ. Any automaton that distinguishes the binary strings can be translated into an automaton that distinguishes the original strings, without any increase in the number of states.. It may also be assumed that the two strings have equal length. For strings of unequal length, there always exists a prime number whose value is logarithmic in the smaller of the two input lengths, such that the two lengths are different modulo .
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The SI does not permit attaching qualifiers to units, whether as suffix or prefix, other than standard SI prefixes. Therefore, even though the decibel is accepted for use alongside SI units, the practice of attaching a suffix to the basic dB unit, forming compound units such as dBm, dBu, dBA, etc., is not.Thompson, A. and Taylor, B. N. sec 8.7, "Logarithmic quantities and units: level, neper, bel", Guide for the Use of the International System of Units (SI) 2008 Edition, NIST Special Publication 811, 2nd printing (November 2008), SP811 PDF The proper way, according to the IEC 60027-3, is either as Lx (re xref) or as Lx/xref, where x is the quantity symbol and xref is the value of the reference quantity, e.g.
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Theoretical physicist Thomas Fink defines beauty in The Man's Book both in terms of ships launched and in terms of the number of women, on average, than whom one woman will be more beautiful. He defines one helen (H) as the quantity of beauty to be more beautiful than 50 million women, the number of women estimated to have been alive in the 12th century BCE. Fink also defines a related, but smaller unit called a “helena” (Ha), based on a logarithmic scale, like the Richter scale, or decibels (dB). Ten helena (Ha) is the beauty sufficient for one oarsman (of which 50 are on a ship) to risk his life, or to be the most beautiful of a thousand women.
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Early photometric measurements (made, for example, by using a light to project an artificial “star” into a telescope's field of view and adjusting it to match real stars in brightness) demonstrated that first magnitude stars are about 100 times brighter than sixth magnitude stars. Thus in 1856 Norman Pogson of Oxford proposed that a logarithmic scale of ≈ 2.512 be adopted between magnitudes, so five magnitude steps corresponded precisely to a factor of 100 in brightness. Every interval of one magnitude equates to a variation in brightness of or roughly 2.512 times. Consequently, a magnitude 1 star is about 2.5 times brighter than a magnitude 2 star, 2.52 brighter than a magnitude 3 star, 2.53 brighter than a magnitude 4 star, and so on.
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In logarithmic form the Bethe ansatz equations can be generated by the Yang action. The square of the norm of Bethe wave function is equal to the determinant of the matrix of second derivatives of the Yang action. The recently developed algebraic Bethe ansatz led to essential progress, stating that The exact solutions of the so-called s-d model (by P.B. Wiegmann in 1980 and independently by N. Andrei, also in 1980) and the Anderson model (by P.B. Wiegmann in 1981, and by N. Kawakami and A. Okiji in 1981) are also both based on the Bethe ansatz. There exist multi-channel generalizations of these two models also amenable to exact solutions (by N. Andrei and C. Destri and by C.J. Bolech and N. Andrei).
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These methods compute the hash function quickly, and can be proven to work well with linear probing. In particular, linear probing has been analyzed from the framework of -independent hashing, a class of hash functions that are initialized from a small random seed and that are equally likely to map any -tuple of distinct keys to any -tuple of indexes. The parameter can be thought of as a measure of hash function quality: the larger is, the more time it will take to compute the hash function but it will behave more similarly to completely random functions. For linear probing, 5-independence is enough to guarantee constant expected time per operation, while some 4-independent hash functions perform badly, taking up to logarithmic time per operation.
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Brenner wrote "A Life In Science",A Life in Science (2001) a paperback published by BioMed Central. Brenner is also noted for his generosity with ideas and the great number of students and colleagues his ideas have stimulated. In 2017, Brenner co-organized a seminal lecture series in Singapore describing ten logarithmic scales of time from the Big Bang to the present, spanning the appearance of multicellular life forms, the evolution of humans, and the emergence of language, culture and technology. Prominent scientists and thinkers, including W. Brian Arthur, Svante Pääbo, Helga Nowotny and Jack Szostak, spoke during the lecture series. In 2018, the lectures were adapted into a popular science book titled Sydney Brenner’s 10-on-10: The Chronicles of Evolution, published by Wildtype Books.
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The traditional terminology also included differentials of the second kind and of the third kind. The idea behind this has been supported by modern theories of algebraic differential forms, both from the side of more Hodge theory, and through the use of morphisms to commutative algebraic groups. The Weierstrass zeta function was called an integral of the second kind in elliptic function theory; it is a logarithmic derivative of a theta function, and therefore has simple poles, with integer residues. The decomposition of a (meromorphic) elliptic function into pieces of 'three kinds' parallels the representation as (i) a constant, plus (ii) a linear combination of translates of the Weierstrass zeta function, plus (iii) a function with arbitrary poles but no residues at them.
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Official, black market, and OMIR exchange rates Jan 1, 2001 to Feb 2, 2009. Note the logarithmic scale. Zimbabwe began experiencing severe foreign exchange shortages, exacerbated by the difference between the official rate and the black market rate in 2000. In 2004 a system of auctioning scarce foreign currency for importers was introduced, which temporarily led to a slight reduction in the foreign currency crisis, but by mid-2005 foreign currency shortages were once again severe. The currency was devalued by the central bank twice, first to 9,000 to the US$, and then to 17,500 to the US$ on 20 July 2005, but at that date it was reported that that was only half the rate available on the black market.
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A logarithmic scale depicting Weimar hyperinflation to 1923. One paper Mark per Gold Mark increased to one trillion paper Marks per Gold Mark. Erik Goldstein wrote that in 1921, the payment of reparations caused a crisis and that the occupation of the Ruhr had a disastrous effect on the German economy, resulting in the German Government printing more money as the currency collapsed. Hyperinflation began and printing presses worked overtime to print Reichsbank notes; by November 1923 one US dollar was worth Ferguson writes that the policy of the Economics Minister Robert Schmidt led Germany to avoid economic collapse from 1919 to 1920, but that reparations accounted for most of Germany's deficit in 1921 and 1922 and that reparations were the cause of the hyperinflation.
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Double hashing is another method of hashing that requires a low degree of independence. It is a form of open addressing that uses two hash functions: one to determine the start of a probe sequence, and the other to determine the step size between positions in the probe sequence. As long as both of these are 2-independent, this method gives constant expected time per operation.. On the other hand, linear probing, a simpler form of open addressing where the step size is always one, requires 5-independence. It can be guaranteed to work in constant expected time per operation with a 5-independent hash function, and there exist 4-independent hash functions for which it takes logarithmic time per operation.
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Given a finite field F, there is, up to isomorphism, just one field Fk with :[ F_k : F ] = k \,, for k = 1, 2, ... . Given a set of polynomial equations -- or an algebraic variety V -- defined over F, we can count the number :N_k \, of solutions in Fk and create the generating function :G(t) = N_1t +N_2t^2/2 + N_3t^3/3 +\cdots \,. The correct definition for Z(t) is to make log Z equal to G, and so :Z= \exp (G(t)) \, we will have Z(0) = 1 since G(0) = 0, and Z(t) is a priori a formal power series. Note that the logarithmic derivative :Z'(t)/Z(t) \, equals the generating function :G'(t) = N_1 +N_2t^1 + N_3t^2 +\cdots \,.
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Cuckoo hashing, another technique for implementing hash tables, guarantees constant time per lookup (regardless of the hash function). Insertions into a cuckoo hash table may fail, causing the entire table to be rebuilt, but such failures are sufficiently unlikely that the expected time per insertion (using either a truly random hash function or a hash function with logarithmic independence) is constant. With tabulation hashing, on the other hand, the best bound known on the failure probability is higher, high enough that insertions cannot be guaranteed to take constant expected time. Nevertheless, tabulation hashing is adequate to ensure the linear-expected- time construction of a cuckoo hash table for a static set of keys that does not change as the table is used.
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Pink noise spectrum. Power density falls off at 10 dB/decade (−3 dB/octave). The frequency spectrum of pink noise is linear in logarithmic scale; it has equal power in bands that are proportionally wide. This means that pink noise would have equal power in the frequency range from 40 to 60 Hz as in the band from 4000 to 6000 Hz. Since humans hear in such a proportional space, where a doubling of frequency (an octave) is perceived the same regardless of actual frequency (40–60 Hz is heard as the same interval and distance as 4000–6000 Hz), every octave contains the same amount of energy and thus pink noise is often used as a reference signal in audio engineering.
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One can also define a family of complexity classes f(n)-APX, where f(n)-APX contains problems with a polynomial time approximation algorithm with a O(f(n)) approximation ratio. One can analogously define f(n)-APX-complete classes; some such classes contain well- known optimization problems. Log-APX-completeness and poly-APX-completeness are defined in terms of AP-reductions rather than PTAS-reductions; this is because PTAS-reductions are not strong enough to preserve membership in Log- APX and Poly-APX, even though they suffice for APX. Log-APX-complete, consisting of the hardest problems that can be approximated efficiently to within a factor logarithmic in the input size, includes min dominating set when degree is unbounded.
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Kikuchi lines in a convergent beam diffraction pattern of single crystal silicon taken with a 300 keV electron beam The figure at left shows the Kikuchi lines leading to a silicon [100] zone, taken with the beam direction approximately 7.9° away from the zone along the (004) Kikuchi band. The dynamic range in the image is so large that only portions of the film are not overexposed. Kikuchi lines are much easier to follow with dark-adapted eyes on a fluorescent screen, than they are to capture unmoving on paper or film, even though eyes and photographic media both have a roughly logarithmic response to illumination intensity. Fully quantitative work on such diffraction features is therefore assisted by the large linear dynamic range of CCD detectors.
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A sound limiter or noise limiter is a digital device fitted with a microphone to measure the sound pressure level of environmental noise, expressed by the decibel logarithmic unit (dB). If the environmental noise level as measured by the microphone exceeds a pre-set level for a certain amount of time (e.g. 5 seconds), the limiter’s circuitry will cut the power supply to the musical equipment and PA system requiring the venue's staff to reset the system.. Sound limiters are commonly installed at live music venues, including private venues and particularly those that host wedding receptions with live wedding bands. The visual indicator on the limiter works most commonly on a “traffic light” system: green = no problem, amber = sound levels approaching the threshold, red = threshold breached.
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Even before the patent was actually granted, it was argued that there might have been prior art that was applicable.Various posts by Matthew Saltzman, Clemson University Mathematicians who specialized in numerical analysis, including Philip Gill and others, claimed that Karmarkar's algorithm is equivalent to a projected Newton barrier method with a logarithmic barrier function, if the parameters are chosen suitably. Legal scholar Andrew Chin opines that Gill's argument was flawed, insofar as the method they describe does not constitute an "algorithm", since it requires choices of parameters that don't follow from the internal logic of the method, but rely on external guidance, essentially from Karmarkar's algorithm. Furthermore, Karmarkar's contributions are considered far from obvious in light of all prior work, including Fiacco-McCormick, Gill and others cited by Saltzman.Mark A. Paley (1995).
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The hues of the Munsell color system, at varying values, and maximum chroma to stay in the sRGB gamut The color defined as red in the Munsell color system (Munsell 5R) is shown at right. The Munsell color system is a color space that specifies colors based on three color dimensions: hue, value (lightness), and chroma (color purity), spaced uniformly in three dimensions in the elongated oval at an angle shaped Munsell color solid according to the logarithmic scale which governs human perception. In order for all the colors to be spaced uniformly, it was found necessary to use a color wheel with five primary colors—red, yellow, green, blue, and purple. The Munsell colors displayed are only approximate as they have been adjusted to fit into the sRGB gamut.
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Per Georg Scheutz (1785–1873) was a Swedish lawyer, publicist and inventor who created the first working programmable difference engine with a printing unit. Martin Wiberg (1826–1905) was a prolific inventor who, among many things, created the first difference engine the size of a sewing machine that could calculate and print logarithmic tables. Johannes Rydberg (1854–1919) was a renowned physicist famous for the Rydberg formula and the Rydberg constant. Carl Charlier (1862–1934) was an internationally acclaimed astronomer who made important contributions to astronomy as well as statistics and was awarded the James Craig Watson Medal in 1924 and the Bruce Medal in 1933. Manne Siegbahn (1886–1978), a student of Rydberg, was awarded the Nobel Prize in Physics 1924 for his discoveries and research in the field of X-ray spectroscopy.
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Rumack Matthew Nomogram with treatment line added at 150 Rumack-Matthew Nomogram The Rumack-Matthew nomogram, also known as Rumack-Matthews nomogram or the acetaminophen nomogram is an acetaminophen toxicity nomogram plotting serum concentration of acetaminophen against the time since ingestion in an attempt to prognosticate possible liver toxicity as well as allowing a clinician to decide whether to proceed with N-Acetylcysteine (NAC) treatment or not. It is a logarithmic graph starting not directly from ingestion, but from 4 hours post ingestion after absorption is considered likely to be complete. In hands of skilled clinicians this nomogram allows for timely management of acetaminophen overdose. Generally, a serum plasma concentration (APAP) of 140–150 microgram/mL (or milligrams/L) at 4 hours post ingestion, indicates the need for NAC treatment.
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The World Bank suggests the usage of Human Development Index (HDI) and the Gross National Happiness Index (NHI). The HDI is a composite index of # life expectancy at birth, as an index of population health and longevity, # knowledge and education as measured by the adult literacy rate and functions of school enrollment rate and # standard of living measured as a logarithmic function of GDP, adjusted to purchasing power parity. The NHI focuses on the spiritual and material development of human beings by focussing on the four pillars of sustainable development, preservation of cultural values, conservation of natural resources and establishment of good governance. The bank also notes suggestions made by President Nicholas Sarkozy for the modification of the definition of GDP that stops the social and cultural damage that the current definitions are leading to.
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The Flajolet–Martin algorithm is an algorithm for approximating the number of distinct elements in a stream with a single pass and space-consumption logarithmic in the maximal number of possible distinct elements in the stream (the count-distinct problem). The algorithm was introduced by Philippe Flajolet and G. Nigel Martin in their 1984 article "Probabilistic Counting Algorithms for Data Base Applications". Later it has been refined in "LogLog counting of large cardinalities" by Marianne Durand and Philippe Flajolet, and "HyperLogLog: The analysis of a near-optimal cardinality estimation algorithm" by Philippe Flajolet et al. In their 2010 article "An optimal algorithm for the distinct elements problem", Daniel M. Kane, Jelani Nelson and David P. Woodruff give an improved algorithm, which uses nearly optimal space and has optimal O(1) update and reporting times.
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As an undergraduate, Whitworth became the founding editor in chief of the Messenger of Mathematics, and he continued as its editor until 1880. He published works about the logarithmic spiral and about trilinear coordinates, but his most famous mathematical publication is the book Choice and Chance: An Elementary Treatise on Permutations, Combinations, and Probability (first published in 1867 and extended over several later editions). The first edition of the book treated the subject primarily from the point of view of arithmetic calculations, but had an appendix on algebra, and was based on lectures he had given at Queen's College. Later editions added material on enumerative combinatorics (the numbers of ways of arranging items into groups with various constraints), derangements, frequentist probability, life expectancy, and the fairness of bets, among other topics.
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Though streaming algorithms had already been studied by Munro and Paterson as early as 1978, as well as Philippe Flajolet and G. Nigel Martin in 1982/83, the field of streaming algorithms was first formalized and popularized in a 1996 paper by Noga Alon, Yossi Matias, and Mario Szegedy. For this paper, the authors later won the Gödel Prize in 2005 "for their foundational contribution to streaming algorithms." There has since been a large body of work centered around data streaming algorithms that spans a diverse spectrum of computer science fields such as theory, databases, networking, and natural language processing. Semi-streaming algorithms were introduced in 2005 as a relaxation of streaming algorithms for graphs , in which the space allowed is linear in the number of vertices , but only logarithmic in the number of edges .
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Although Rosenkrantz et al. prove only a logarithmic approximation ratio for this method, they show that in practice it often works better than other insertion methods with better provable approximation ratios.. Later, the same sequence of points was popularized by , who used it as part of a greedy approximation algorithm for the problem of finding k clusters that minimize the maximum diameter of a cluster. The same algorithm applies also, with the same approximation quality, to the metric k-center problem. This problem is one of several formulations of cluster analysis and facility location, in which the goal is to partition a given set of points into k different clusters, each with a chosen center point, such that the maximum distance from any point to the center of its cluster is minimized.
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Such a system is usually a conservative extension of PA. It typically includes all Peano axioms, and adds to them one or two extra-Peano axioms such as ⊓x⊔y(y=x') expressing the computability of the successor function. Typically it also has one or two non- logical rules of inference, such as constructive versions of induction or comprehension. Through routine variations in such rules one can obtain sound and complete systems characterizing one or another interactive computational complexity class C. This is in the sense that a problem belongs to C if and only if it has a proof in the theory. So, such a theory can be used for finding not merely algorithmic solutions, but also efficient ones on demand, such as solutions that run in polynomial time or logarithmic space.
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It is easy to construct an example for which the convex hull contains all input points, but after the insertion of a single point the convex hull becomes a triangle. And conversely, the deletion of a single point may produce the opposite drastic change of the size of the output. Therefore, if the convex hull is required to be reported in traditional way as a polygon, the lower bound for the worst- case computational complexity of the recomputation of the convex hull is \Omega(N), since this time is required for a mere reporting of the output. This lower bound is attainable, because several general-purpose convex hull algorithms run in linear time when input points are ordered in some way and logarithmic-time methods for dynamic maintenance of ordered data are well- known.
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There are five common variants of the QBD test method: # Linear voltage ramp (V-ramp test procedure) # Constant current stress (CCS) # Exponential current ramp (ECR) or (J-ramp test procedure)Dumin, Nels A., Transformation of Charge-to-Breakdown Obtained from Ramped Current Stresses Into Charge-to- Breakdown and Time-to-Breakdown Domains for Constant Current Stress, # Bounded J-ramp (a variant of the J-ramp procedure, in which the current ramp stops at a defined stress level, and continues as a constant current stress). # Linear current ramp (LCR) For the V-ramp test procedure, the measured current is integrated to obtain QBD. The measured current is also used as a detection criterion for terminating the voltage ramp. One of the defined criteria is the change of logarithmic current slope between successive voltage steps.
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He succeeded in giving the explanation of the phenomenon called "multiple resonance," discovered by Sarasin and De la Rive. Continuing his experiments at the University of Christiania (1891–1892), he proved experimentally the influence which the conductivity and the magnetic properties of the metallic conductors exert upon the electric oscillations, and measured the depth to which the electric oscillations penetrate in metals of different conductivity and magnetic permeability (the "skin effect"). Finally, in 1895 he furnished a complete theory of the phenomenon of electric resonance, involving a method of utilizing resonance experiments for the determination of the wavelengths, and especially of the damping (the logarithmic decrement) of the oscillations in the transmitter and the receiver of the electric oscillations. These methods contributed much to the development of wireless telegraphy.
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It may be given an adjacency labeling scheme in which the points that are endpoints of line segments are numbered from 1 to 2n and each vertex of the graph is represented by the numbers of the two endpoints of its corresponding interval. With this representation, one may check whether two vertices are adjacent by comparing the numbers that represent them and verifying that these numbers define overlapping intervals. The same approach works for other geometric intersection graphs including the graphs of bounded boxicity and the circle graphs, and subfamilies of these families such as the distance-hereditary graphs and cographs. However, a geometric intersection graph representation does not always imply the existence of an adjacency labeling scheme, because it may require more than a logarithmic number of bits to specify each geometric object.
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'Absolute magnitude (') is a measure of the luminosity of a celestial object, on an inverse logarithmic astronomical magnitude scale. An object's absolute magnitude is defined to be equal to the apparent magnitude that the object would have if it were viewed from a distance of exactly , without extinction (or dimming) of its light due to absorption by interstellar matter and cosmic dust. By hypothetically placing all objects at a standard reference distance from the observer, their luminosities can be directly compared on a magnitude scale. As with all astronomical magnitudes, the absolute magnitude can be specified for different wavelength ranges corresponding to specified filter bands or passbands; for stars a commonly quoted absolute magnitude is the absolute visual magnitude, which uses the visual (V) band of the spectrum (in the UBV photometric system).
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A plot of population growth rate vs total fertility rate (logarithmic). Symbol radius reflect population size in each country A population that maintained a TFR of 3.8 over an extended period without a correspondingly high death or emigration rate would increase rapidly (doubling period ~ 32 years), whereas a population that maintained a TFR of 2.0 over a long time would decrease, unless it had a large enough immigration. However, it may take several generations for a change in the total fertility rate to be reflected in birth rate, because the age distribution must reach equilibrium. For example, a population that has recently dropped below replacement-level fertility will continue to grow, because the recent high fertility produced large numbers of young couples who would now be in their childbearing years.
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The ASA standard underwent a major revision in 1960 with ASA PH2.5-1960, when the method to determine film speed was refined and previously applied safety factors against under-exposure were abandoned, effectively doubling the nominal speed of many black-and-white negative films. For example, an Ilford HP3 that had been rated at 200 ASA before 1960 was labeled 400 ASA afterwards without any change to the emulsion. Similar changes were applied to the DIN system with DIN 4512:1961-10 and the BS system with BS 1380:1963 in the following years. In addition to the established arithmetic speed scale, ASA PH2.5-1960 also introduced logarithmic ASA grades (100 ASA = 5° ASA), where a difference of 1° ASA represented a full exposure stop and therefore the doubling of a film speed.
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1,10,100,1k,10k,100k using decades vs. 0,10,20,30,40,50 using linear scale Decades on a logarithmic scale, rather than unit steps (steps of 1) or other linear scale, are commonly used on the horizontal axis when representing the frequency response of electronic circuits in graphical form, such as in Bode plots, since depicting large frequency ranges on a linear scale is often not practical. For example, an audio amplifier will usually have a frequency band ranging from 20 Hz to 20 kHz and representing the entire band using a decade log scale is very convenient. Typically the graph for such a representation would begin at 1 Hz (100) and go up to perhaps 100 kHz (105), to comfortably include the full audio band in a standard-sized graph paper, as shown below.
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Pitiscus supported Frederick's subsequent measures against the Roman Catholic Church. Pitiscus achieved fame with his influential work written in Latin, called Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus (1595, first edition printed in Heidelberg), which introducedGroundbreaking Scientific Experiments, Inventions, and Discoveries the word trigonometry to the English and French languages, translations into which had appeared in 1614 and 1619, respectively. It consists of five books on plane and spherical trigonometry. Pitiscus is sometimes credited with inventing the decimal point, the symbol separating integers from decimal fractions, which appears in his trigonometrical tables and was subsequently accepted by John Napier in his logarithmic papers (1614 and 1619). Pitiscus edited Thesaurus mathematicus (1613) in which he improved the trigonometric tables of Georg Joachim Rheticus and also corrected Rheticus’s Magnus Canon doctrinæ triangulorum.
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In probability and statistics the extended negative binomial distribution is a discrete probability distribution extending the negative binomial distribution. It is a truncated version of the negative binomial distributionJonhnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete Distributions, 2nd edition, Wiley (page 227) for which estimation methods have been studied.Shah S.M. (1971) "The displaced negative binomial distribution", Bulletin of the Calcutta Statistical Association, 20, 143–152 In the context of actuarial science, the distribution appeared in its general form in a paper by K. Hess, A. Liewald and K.D. Schmidt when they characterized all distributions for which the extended Panjer recursion works. For the case , the distribution was already discussed by Willmot and put into a parametrized family with the logarithmic distribution and the negative binomial distribution by H.U. Gerber.
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On Growth and Form is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942. The book covers many topics including the effects of scale on the shape of animals and plants, large ones necessarily being relatively thick in shape; the effects of surface tension in shaping soap films and similar structures such as cells; the logarithmic spiral as seen in mollusc shells and ruminant horns; the arrangement of leaves and other plant parts (phyllotaxis); and Thompson's own method of transformations, showing the changes in shape of animal skulls and other structures on a Cartesian grid. The work is widely admired by biologists, anthropologists and architects among others, but less often read than cited.
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The hues of the Munsell color system, at varying values, and maximum chroma to stay in the sRGB gamut The color defined as blue in the Munsell color system (Munsell 5B) is shown at right. The Munsell color system is a color space that specifies colors based on three color dimensions: hue, value (lightness), and chroma (color purity), spaced uniformly (according to the logarithmic scale which governs human perception) in three dimensions in the Munsell color solid, which is shaped like an elongated oval at an angle. In order for all the colors to be spaced uniformly, it was found necessary to use a color wheel with five primary colors: red, yellow, green, blue, and purple. The Munsell color displayed is only approximate, as these spectral colors have been adjusted to fit into the sRGB gamut.
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791, R2. J. Graham, K. Kanov, X.I.A. Yang, M. K.Lee, N. Malaya, C.C. Lalescu, R. Burns, G. Eyink, A. Szalay, R.D. Moser, and C. Meneveau, “A Web Services-accessible database of turbulent channel flow and its use for testing a new integral wall model for LES” (2015), Journal of Turbulence 17:2, 181-215. D. Yang, B. Chen, M. Chamecki & C. Meneveau: “Oil plumes and dispersion in Langmuir, upper-ocean turbulence: large-eddy simulations and K-profile parameterization” (2015), J. Geophysical Res.-Oceans 120, 4729-4759. M. Wilczek, R. Stevens & C. Meneveau, “Spatio-temporal spectra in the logarithmic layer of wall turbulence: large- eddy simulations and simple models” (2015), J. Fluid Mech. 769, R1. X.I.A. Yang, J. Sadique, R. Mittal & C. Meneveau, “Integral Wall Model for Large Eddy Simulations of wall-bounded turbulent flows” (2015), Phys.
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2-satisfiability may be applied to geometry and visualization problems in which a collection of objects each have two potential locations and the goal is to find a placement for each object that avoids overlaps with other objects. Other applications include clustering data to minimize the sum of the diameters of the clusters, classroom and sports scheduling, and recovering shapes from information about their cross-sections. In computational complexity theory, 2-satisfiability provides an example of an NL-complete problem, one that can be solved non-deterministically using a logarithmic amount of storage and that is among the hardest of the problems solvable in this resource bound. The set of all solutions to a 2-satisfiability instance can be given the structure of a median graph, but counting these solutions is #P-complete and therefore not expected to have a polynomial-time solution.
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In 1902 Victor Henri proposed a quantitative theory of enzyme kinetics, but at the time the experimental significance of the hydrogen ion concentration was not yet recognized. After Peter Lauritz Sørensen had defined the logarithmic pH- scale and introduced the concept of buffering in 1909 the German chemist Leonor Michaelis and Dr. Maud Leonora Menten (a postdoctoral researcher in Michaelis's lab at the time) repeated Henri's experiments and confirmed his equation, which is now generally referred to as Michaelis-Menten kinetics (sometimes also Henri-Michaelis-Menten kinetics).; Their work was further developed by G. E. Briggs and J. B. S. Haldane, who derived kinetic equations that are still widely considered today a starting point in modeling enzymatic activity. The major contribution of the Henri-Michaelis-Menten approach was to think of enzyme reactions in two stages.
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Although the Kelly strategy's promise of doing better than any other strategy in the long run seems compelling, some economists have argued strenuously against it, mainly because an individual's specific investing constraints may override the desire for optimal growth rate. The conventional alternative is expected utility theory which says bets should be sized to maximize the expected utility of the outcome (to an individual with logarithmic utility, the Kelly bet maximizes expected utility, so there is no conflict; moreover, Kelly's original paper clearly states the need for a utility function in the case of gambling games which are played finitely many times). Even Kelly supporters usually argue for fractional Kelly (betting a fixed fraction of the amount recommended by Kelly) for a variety of practical reasons, such as wishing to reduce volatility, or protecting against non-deterministic errors in their advantage (edge) calculations.
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Since two-sided error is more general than one-sided error, RL and its complement co-RL are contained in BPL. BPL is also contained in PL, which is similar except that the error bound is 1/2, instead of a constant less than 1/2; like the class PP, the class PL is less practical because it may require a large number of rounds to reduce the error probability to a small constant. showed the weak derandomization result that BPL is contained in SC. SC is the class of problems solvable in polynomial time and polylogarithmic space on a deterministic Turing machine; in other words, this result shows that, given polylogarithmic space, a deterministic machine can simulate logarithmic space probabilistic algorithms. BPL is contained in NC and in DSPACE(log3/2 n) Complexity theory lecture notes and in L/poly.
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The runtime of the quantum algorithm for solving systems of linear equations originally proposed by Harrow et al. was shown to be O(\kappa^2\log N/\varepsilon), where \varepsilon>0 is the error parameter and \kappa is the condition number of A. This was subsequently improved to O(\kappa \log^3\kappa \log N /\varepsilon^3) by Andris Ambainis and a quantum algorithm with runtime polynomial in \log(1/\varepsilon) was developed by Childs et al. Since the HHL algorithm maintains its logarithmic scaling in N only for sparse or low rank matrices, Wossnig et al. extended the HHL algorithm based on a quantum singular value estimation technique and provided a linear system algorithm for dense matrices which runs in O(\sqrt N \log N \kappa^2) time compared to the O(N \log N \kappa^2) of the standard HHL algorithm.
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This table was later extended by Adriaan Vlacq, but to 10 places, and by Alexander John Thompson to 20 places in 1952. Briggs was one of the first to use finite-difference methods to compute tables of functions. He also completed a table of logarithmic sines and tangents for the hundredth part of every degree to fourteen decimal places, with a table of natural sines to fifteen places and the tangents and secants for the same to ten places, all of which were printed at Gouda in 1631 and published in 1633 under the title of Trigonometria Britannica; this work was probably a successor to his 1617 Logarithmorum Chilias Prima ("The First Thousand Logarithms"), which gave a brief account of logarithms and a long table of the first 1000 integers calculated to the 14th decimal place.
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SigSpec (acronym of SIGnificance SPECtrum) is a statistical technique to provide the reliability of periodicities in a measured (noisy and not necessarily equidistant) time series. It relies on the amplitude spectrum obtained by the Discrete Fourier transform (DFT) and assigns a quantity called the spectral significance (frequently abbreviated by “sig”) to each amplitude. This quantity is a logarithmic measure of the probability that the given amplitude level is due to white noise, in the sense of a type I error. It represents the answer to the question, “What would be the chance to obtain an amplitude like the measured one or higher, if the analysed time series were random?” SigSpec may be considered a formal extension to the Lomb-Scargle periodogram, appropriately incorporating a time series to be averaged to zero before applying the DFT, which is done in many practical applications.
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Number of cases (blue) and number of deaths (red) on a logarithmic scale. On 8 May, Fernández announced that the national lockdown would be "relaxed" throughout the country with the exception of Greater Buenos Aires, where the lockdown was extended first until 24 May, and later until 7 June (on 23 May), due to a big increase in the number of new cases in the previous days of the announcement. May ended with 16,838 confirmed cases, 539 deaths and 5,323 recoveries. On 9 June, The Government of the Province of Formosa reported the first case of COVID-19 in its province, leaving the province of Catamarca as the only province that did not report any cases at the time. Martín Insaurralde, the Mayor of Lomas de Zamora was diagnosed with COVID-19 and isolated on 12 June.
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The logistic function can be used to illustrate the progress of the diffusion of an innovation through its life cycle. In The Laws of Imitation (1890), Gabriel Tarde describes the rise and spread of new ideas through imitative chains. In particular, Tarde identifies three main stages through which innovations spread: the first one corresponds to the difficult beginnings, during which the idea has to struggle within a hostile environment full of opposing habits and beliefs; the second one corresponds to the properly exponential take-off of the idea, with f(x)=2^x; finally, the third stage is logarithmic, with f(x)=\log(x), and corresponds to the time when the impulse of the idea gradually slows down while, simultaneously new opponent ideas appear. The ensuing situation halts or stabilizes the progress of the innovation, which approaches an asymptote.
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It is understood that polynomial running time here means that running time is polynomial in the final block length. The main idea is that if the inner block length is selected to be logarithmic in the size of the outer code then the decoding algorithm for the inner code may run in exponential time of the inner block length, and we can thus use an exponential-time but optimal maximum likelihood decoder (MLD) for the inner code. In detail, let the input to the decoder be the vector y = (y1, ..., yN) ∈ (An)N. Then the decoding algorithm is a two-step process: # Use the MLD of the inner code Cin to reconstruct a set of inner code words y' = (y'1, ..., y'N), with y'i = MLDCin(yi), 1 ≤ i ≤ N. # Run the unique decoding algorithm for Cout on y'.
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Because many receptors are essentially enzymes, the field of pharmakinetics utilizes the Michaelis–Menten equation to describe drug affinity (dissociation constant Kd) and total binding (Bmax). Although Kd and Bmax can be determined pictorally in a normal or logarithmic plot of ligand binding vs drug concentration, Scatchard plots allow for mathematical representation of several ligand binding sites, each with its own Kd. Semi-log plots of two agonists with different Kd. Drug potency is the measure of binding strength between a drug and a specific molecular target, whereas drug efficacy describes the biological effect exerted by the drug itself, at either a cellular or organismal level. Because drugs range widely in their potency and efficacy, drugs have been categorized on the spectrum of agonists and antagonists. Agonists bind to receptors and elicit the same effects as an endogenous neurotransmitter.
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Mathematical PageRanks for a simple network are expressed as percentages. (Google uses a logarithmic scale.) Page C has a higher PageRank than Page E, even though there are fewer links to C; the one link to C comes from an important page and hence is of high value. If web surfers who start on a random page have an 85% likelihood of choosing a random link from the page they are currently visiting, and a 15% likelihood of jumping to a page chosen at random from the entire web, they will reach Page E 8.1% of the time. (The 15% likelihood of jumping to an arbitrary page corresponds to a damping factor of 85%.) Without damping, all web surfers would eventually end up on Pages E, B, or C, and all other pages would have PageRank zero.
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The trigonometric functions can be constructed geometrically in terms of a unit circle centered at O. Historically, the versed sine was considered one of the most important trigonometric functions. As θ goes to zero, versin(θ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation, making separate tables for the latter convenient. Even with a calculator or computer, round-off errors make it advisable to use the sin2 formula for small θ. Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle (θ = 0, 2, …) where it is zero--thus, one could use logarithmic tables for multiplications in formulas involving versines.
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Most chemists, since the discoveries of John Dalton in 1808, and James Clerk Maxwell in Scotland and Josiah Willard Gibbs in the United States, shared Boltzmann's belief in atoms and molecules, but much of the physics establishment did not share this belief until decades later. Boltzmann had a long-running dispute with the editor of the preeminent German physics journal of his day, who refused to let Boltzmann refer to atoms and molecules as anything other than convenient theoretical constructs. Only a couple of years after Boltzmann's death, Perrin's studies of colloidal suspensions (1908–1909), based on Einstein's theoretical studies of 1905, confirmed the values of Avogadro's number and Boltzmann's constant, convincing the world that the tiny particles really exist. To quote Planck, "The logarithmic connection between entropy and probability was first stated by L. Boltzmann in his kinetic theory of gases".
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Although we trim a lot of the tree when we perform this compression, it is still possible to achieve logarithmic-time search, insertion, and deletion by taking advantage of Z-order curves. The Z-order curve maps each cell of the full quadtree (and hence even the compressed quadtree) in O(1) time to a one- dimensional line (and maps it back in O(1) time too), creating a total order on the elements. Therefore, we can store the quadtree in a data structure for ordered sets (in which we store the nodes of the tree). We must state a reasonable assumption before we continue: we assume that given two real numbers \alpha, \beta \in [0, 1) expressed as binary, we can compute in O(1) time the index of the first bit in which they differ.
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Adaptive heap sort is a variant of heap sort that seeks optimality (asymptotically optimal) with respect to the lower bound derived with the measure of presortedness by taking advantage of the existing order in the data. In heap sort, for a data X = , we put all n elements into the heap and then keep extracting the maximum (or minimum) for n times. Since the time of each max-extraction action is the logarithmic in the size of the heap, the total running time of standard heap sort is O(n log n). For adaptive heap sort, instead of putting all the elements into the heap, only the possible maximums of the data (max-candidates) will be put into the heap so that fewer runs are required when each time we try to locate the maximum(or minimum).
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It thus makes sense to define the hyperbolic angle from P0 to an arbitrary point on the curve as a logarithmic function of the point's value of x.Bjørn Felsager, Through the Looking Glass – A glimpse of Euclid's twin geometry, the Minkowski geometry , ICME-10 Copenhagen 2004; p.14. See also example sheets exploring Minkowskian parallels of some standard Euclidean resultsViktor Prasolov and Yuri Solovyev (1997) Elliptic Functions and Elliptic Integrals, page 1, Translations of Mathematical Monographs volume 170, American Mathematical Society Whereas in Euclidean geometry moving steadily in an orthogonal direction to a ray from the origin traces out a circle, in a pseudo-Euclidean plane steadily moving orthogonally to a ray from the origin traces out a hyperbola. In Euclidean space, the multiple of a given angle traces equal distances around a circle while it traces exponential distances upon the hyperbolic line.
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She then used the simplifying assumption that all of the Cepheids within the Small Magellanic Cloud were at approximately the same distance, so that their intrinsic brightness could be deduced from their apparent brightness as registered in the photographic plates, up to a scale factor since the distance to the Magellanic Clouds were as yet unknown. She expressed the hope that parallaxes to some Cepheids would be measured, which soon happened, thereby allowing her period-luminosity scale to be calibrated. This reasoning allowed Leavitt to establish that the logarithm of the period is linearly related to the logarithm of the star's average intrinsic optical luminosity (which is the amount of power radiated by the star in the visible spectrum). Leavitt also developed, and continued to refine, the Harvard Standard for photographic measurements, a logarithmic scale that orders stars by brightness over 17 magnitudes.
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The term hartley is named after Ralph Hartley, who suggested in 1928 to measure information using a logarithmic base equal to the number of distinguishable states in its representation, which would be the base 10 for a decimal digit. The ban and the deciban were invented by Alan Turing with Irving John "Jack" Good in 1940, to measure the amount of information that could be deduced by the codebreakers at Bletchley Park using the Banburismus procedure, towards determining each day's unknown setting of the German naval Enigma cipher machine. The name was inspired by the enormous sheets of card, printed in the town of Banbury about 30 miles away, that were used in the process. Good argued that the sequential summation of decibans to build up a measure of the weight of evidence in favour of a hypothesis, is essentially Bayesian inference.
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With linear functions, increasing the input by one unit causes the output to increase by a fixed amount, which is the slope of the graph of the function. With exponential functions, increasing the input by one unit causes the output to increase by a fixed multiple, which is known as the base of the exponential function. If both arguments and values of a function are in the logarithmic scale (i.e., when is a linear function of ), then the straight line represents a power law: :\log_r y = a \log_r x + b \quad\Rightarrow\quad y = r^b\cdot x^a Archimedean spiral defined by the polar equation r = θ + 2 On the other hand, the graph of a linear function in terms of polar coordinates: :r =f(\theta ) = a\theta + b is an Archimedean spiral if a eq 0 and a circle otherwise.
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Applied Welfare Economics, edited by Richard E. Just et al, Edward Elgar Publishing More generally, Vartia's expertise is axiomatic index numbers, where he is known for his "consistency in aggregation" test and his discovery, along with Kazuo Sato of the "ideal log-change index", which utilised logarithms and logarithmic mean to define the Sato-Vartia or "Vartia II" index (both 1976). He proposed also in his dissertation "Relative Changes and Index Numbers" in 1976 another index known as Montgomery-Vartia (or "Vartia I") index, which satisfies Time and Factor Reversal Tests and is Consistent in Aggregation. But it satisfies only a weaker form of Proportionality Test WPT along with, say, the factor antithesis FA(Törnqvist) of the Törnqvist index. According to WPT, if both prices and quantities change proportionally, the price index must equal the common factor of proportionality.
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Although tabulation hashing as described above ("simple tabulation hashing") is only 3-independent, variations of this method can be used to obtain hash functions with much higher degrees of independence. uses the same idea of using exclusive or operations to combine random values from a table, with a more complicated algorithm based on expander graphs for transforming the key bits into table indices, to define hashing schemes that are k-independent for any constant or even logarithmic value of k. However, the number of table lookups needed to compute each hash value using Siegel's variation of tabulation hashing, while constant, is still too large to be practical, and the use of expanders in Siegel's technique also makes it not fully constructive. provides a scheme based on tabulation hashing that reaches high degrees of independence more quickly, in a more constructive way.
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The min-max heap property is: each node at an even level in the tree is less than all of its descendants, while each node at an odd level in the tree is greater than all of its descendants. The structure can also be generalized to support other order-statistics operations efficiently, such as `find-median`, `delete-median`,`find(k)` (determine the kth smallest value in the structure) and the operation `delete(k)` (delete the kth smallest value in the structure), for any fixed value (or set of values) of k. These last two operations can be implemented in constant and logarithmic time, respectively. The notion of min-max ordering can be extended to other structures based on the max- or min-ordering, such as leftist trees, generating a new (and more powerful) class of data structures.
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Equipotential lines are shown here, which can be compared with the contour lines on a map of a mountainous region: the nearer these lines are to each other, the steeper the slope and the greater the danger, in this case the danger of an electrical breakdown. The equipotential lines can also be compared with the isobars on a weather map: The denser the lines, the more wind and the greater the danger of damage. In order to control the equipotential lines (that is to control the electric field) a device is used that is called a stress cone, see figure 3.Kreuger 1991 Vol. 1, pp. 147-153 The crux of stress relief is to flare the shield end along a logarithmic curve. Before 1960, the stress cones were handmade using tape—after the cable was installed.
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This method leads to a fast method for computing the Thue–Morse sequence: start with , and then, for each n, find the highest-order bit in the binary representation of n that is different from the same bit in the representation of . (This bit can be isolated by letting x be the bitwise exclusive or of n and , shifting x right by one bit, and computing the exclusive or of this shifted value with x.) If this bit is at an even index, tn differs from , and otherwise it is the same as . In pseudo-code form: generateSequence(seqLength): value = 0 for n = 0 to seqLength-1 by 1: x = n ^ (n-1) if ((x ^ (x>>1)) & 0x55555555): value = 1 - value return value The resulting algorithm takes constant time to generate each sequence element, using only a logarithmic number of bits (constant number of words) of memory.
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The beta distribution achieves maximum differential entropy for Beta(1,1): the uniform probability density, for which all values in the domain of the distribution have equal density. This uniform distribution Beta(1,1) was suggested ("with a great deal of doubt") by Thomas Bayes as the prior probability distribution to express ignorance about the correct prior distribution. This prior distribution was adopted (apparently, from his writings, with little sign of doubt) by Pierre-Simon Laplace, and hence it was also known as the "Bayes-Laplace rule" or the "Laplace rule" of "inverse probability" in publications of the first half of the 20th century. In the later part of the 19th century and early part of the 20th century, scientists realized that the assumption of uniform "equal" probability density depended on the actual functions (for example whether a linear or a logarithmic scale was most appropriate) and parametrizations used.
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Roll-off is the steepness of a transfer function with frequency, particularly in electrical network analysis, and most especially in connection with filter circuits in the transition between a passband and a stopband. It is most typically applied to the insertion loss of the network, but can, in principle, be applied to any relevant function of frequency, and any technology, not just electronics. It is usual to measure roll-off as a function of logarithmic frequency; consequently, the units of roll-off are either decibels per decade (dB/decade), where a decade is a tenfold increase in frequency, or decibels per octave (dB/8ve), where an octave is a twofold increase in frequency. The concept of roll-off stems from the fact that in many networks roll-off tends towards a constant gradient at frequencies well away from the cut-off point of the frequency curve.
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Once it has finished, the power light turns and remains green until the amplifier is turned off. The AU-11000 also has its own block diagram printed on the upper-front casing. Sansui designed the AU-11000 with the input, output and speaker terminals on the sides of the unit. The rear of the unit has the power cord and outlets only. The AU-11000 has features such as a logarithmic volume control, a 3-position level-set muting, a -20db mute switch, 3-position high & low filters, 3-Band Linear Bass/Midrange/Treble controls, 3 optional frequency settings for Bass & Treble controls, 3-position Tone Selection, A & B Speakers, Tape Input/Output control, 5-position mono-stereo selector switch, Tuner input, 2 Auxiliary inputs, 2 phono inputs, 2 rear 'always on' power outlets and 1 switched power outlet that is controlled by the units main power switch.
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However, the space of realizations of locally-square spiral packings is infinite-dimensional, unlike the Doyle spirals which can be determined only by a constant number of parameters. It is also possible to describe spiraling systems of overlapping circles that cover the plane, rather than non-crossing circles that pack the plane, with each point of the plane covered by at most two circles except for points where three circles meet at 60^\circ angles, and with each circle surrounded by six others. These have many properties in common with the Doyle spirals. The Doyle spiral, in which the circle centers lie on logarithmic spirals and their radii increase geometrically in proportion to their distance from the central limit point, should be distinguished from a different spiral pattern of disjoint but non-tangent unit circles, also resembling certain forms of plant growth such as the seed heads of sunflowers.
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Briggs was one of the first to use finite-difference methods to compute tables of functions. He also completed a table of logarithmic sines and tangents for the hundredth part of every degree to fourteen decimal places, with a table of natural sines to fifteen places, and the tangents and secants for the same to ten places; all of which were printed at Gouda in 1631 and published in 1633 under the title of Trigonometria Britannica; this work was probably a successor to his 1617 Logarithmorum Chilias Prima ("The First Thousand Logarithms"), which gave a brief account of logarithms and a long table of the first 1000 integers calculated to the 14th decimal place. Briggs discovered, in a somewhat concealed form and without proof, the binomial theorem. English translations of Briggs's Arithmetica and the first part of his Trigonometria Britannica are available on the web.
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Further simplifications were made by and . A variant of the problem is the dynamic LCA problem in which the data structure should be prepared to handle LCA queries intermixed with operations that change the tree (that is, rearrange the tree by adding and removing edges) This variant can be solved using O(logN) time for all modifications and queries. This is done by maintaining the forest using the dynamic trees data structure with partitioning by size; this then maintains a heavy-light decomposition of each tree, and allows LCA queries to be carried out in logarithmic time in the size of the tree. Without preprocessing you can also improve the naïve online algorithm's computation time to O(log h) by storing the paths through the tree using skew-binary random access lists, while still permitting the tree to be extended in constant time (Edward Kmett (2012)).
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In another example, the "center lines" of the arms of a spiral galaxy trace logarithmic spirals. The second definition includes two kinds of 3-dimensional relatives of spirals: # a conical or volute spring (including the spring used to hold and make contact with the negative terminals of AA or AAA batteries in a battery box), and the vortex that is created when water is draining in a sink is often described as a spiral, or as a conical helix. # quite explicitly, definition 2 also includes a cylindrical coil spring and a strand of DNA, both of which are quite helical, so that "helix" is a more useful description than "spiral" for each of them; in general, "spiral" is seldom applied if successive "loops" of a curve have the same diameter. In the side picture, the black curve at the bottom is an Archimedean spiral, while the green curve is a helix.
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