"What turns 1,000 into 3, in base 10" is a LOGARITHM.
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Logarithm Labs: Logarithm Labs is a project management service for chip designers covering data pipelines, scripting interfaces, and portals and dashboards to parse, structure, and analyze data generated in chip design work Snapboard: Snapboard provides software tools to create dashboards, visualizations, and applications without code.
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This is confusing without the math but it's because the pH scale is a logarithm of a negative exponent.
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An economic adviser talked up the country's technological achievements, which include the publication of an early volume of logarithm tables.
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How do you know?" and I'll say, "We have a logarithm" or "We run it through a computer that analyzes it.
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Those were later for developed for the standardized Logarithm of the Minimum Angle of Resolution (LogMAR) tests now used by eye care professionals across the globe.
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The term pH means "potential of hydrogen," and the scale is the negative base 10 logarithm of the concentration of positively charged hydrogen in a solution.
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They are trying to become the logarithm again, but in the meantime they're just another team wondering what's gone wrong, and puzzling over the difference and distance between intention and act.
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Then in 1971 Arnold Schönhage and Volker Strassen published a method capable of multiplying large numbers in n × log n × log(log n) multiplicative steps, where log n is the logarithm of n.
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In an attempt to weight these arguments, I combined the twin improbabilities of each outcome into a single score, by adding the logarithm of the inverse of each team's chance of winning before the second leg and at its lowest point during the game.
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A graph of the common logarithm of numbers from 0.1 to 100. In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use, as well as standard logarithm. Historically, it was known as logarithmus decimalis or logarithmus decadis.
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Graph of as a function of a positive real number In mathematics, the binary logarithm () is the power to which the number must be raised to obtain the value . That is, for any real number , :x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n. For example, the binary logarithm of is , the binary logarithm of is , the binary logarithm of is , and the binary logarithm of is . The binary logarithm is the logarithm to the base .
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In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.
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David Eugene Smith (1925) History of Mathematics, pp. 424,5 v. 1 A. A. de Sarasa interpreted the quadrature as a logarithm and thus the geometrically defined natural logarithm (or "hyperbolic logarithm") is understood as the area under to the right of . As an example of a transcendental function, the logarithm is more familiar than its motivator, the hyperbolic angle.
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The natural logarithm of is the power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the area under the curve from to (with the area being negative when ).
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This specification includes key agreement, signature, and encryption schemes using several mathematical approaches: integer factorization, discrete logarithm, and elliptic curve discrete logarithm.
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Logarithms can be defined for any positive base other than 1, not only . However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter. For instance, the base-2 logarithm (also called the binary logarithm) is equal to the natural logarithm divided by , the natural logarithm of 2. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity.
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In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number is the exponent to which another fixed number, the base , must be raised, to produce that number . In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since , the "logarithm base " of is , or .
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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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The logarithm base (that is ) is called the common logarithm and is commonly used in science and engineering. The natural logarithm has the number (that is ) as its base; its use is widespread in mathematics and physics, because of its simpler integral and derivative. The binary logarithm uses base (that is ) and is commonly used in computer science. Logarithms are examples of concave functions.
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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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Consider the natural logarithm function mapping the real numbers to themselves. The logarithm of a non- positive real is not a real number, so the natural logarithm function doesn't associate any real number in the codomain with any non-positive real number in the domain. Therefore, the natural logarithm function is not a function when viewed as a function from the reals to themselves, but it is a partial function. If the domain is restricted to only include the positive reals (that is, if the natural logarithm function is viewed as a function from the positive reals to the reals), then the natural logarithm is a function.
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For any fixed base, the sum of the digits of a number is proportional to its logarithm; therefore, the additive persistence is proportional to the iterated logarithm.
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For a general positive real number, the binary logarithm may be computed in two parts.. First, one computes the integer part, \lfloor\log_2 x\rfloor (called the characteristic of the logarithm). This reduces the problem to one where the argument of the logarithm is in a restricted range, the interval [1, 2), simplifying the second step of computing the fractional part (the mantissa of the logarithm). For any , there exists a unique integer such that , or equivalently . Now the integer part of the logarithm is simply , and the fractional part is .
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A single branch of the complex logarithm. The hue of the color is used to show the arg (polar coordinate angle) of the complex logarithm. The saturation and value (intensity and brightness) of the color is used to show the modulus of the complex logarithm. In complex analysis, a complex logarithm' of the non- zero complex number , denoted by ', is defined to be any complex number for which .
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Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the logarithm of a matrix is the (multi-valued) inverse function of the matrix exponential., chapter 11. Another example is the p-adic logarithm, the inverse function of the p-adic exponential.
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Hence, plotting the logarithm of the measured in-situ electrical conductivity against the logarithm of the measured in-situ porosity (Pickett plot), according to Archie's law a straight-line relationship is expected with slope equal to the cementation exponent m and intercept equal to the logarithm of the in-situ brine conductivity.
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A plot of the multi-valued imaginary part of the complex logarithm function, which shows the branches. As a complex number z goes around the origin, the imaginary part of the logarithm goes up or down. This makes the origin a branch point of the function. The typical example of a branch cut is the complex logarithm.
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The fractional part of the logarithm can be calculated efficiently.
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The general logarithm, to an arbitrary positive base, Euler presents as the inverse of an exponential function. Then the natural logarithm is obtained by taking as base "the number for which the hyperbolic logarithm is one", sometimes called Euler's number, and written e. This appropriation of the significant number from Gregoire de Saint-Vincent’s calculus suffices to establish the natural logarithm. This part of precalculus prepares the student for integration of the monomial xp in the instance of p = −1\.
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Graph of the natural logarithm function. The function slowly grows to positive infinity as x increases, and slowly goes to negative infinity as x approaches 0 ("slowly" as compared to any power law of x); the y-axis is an asymptote. The natural logarithm of a number is its logarithm to the base of the mathematical constant , where is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply .
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The binary logarithm function may be defined as the inverse function to the power of two function, which is a strictly increasing function over the positive real numbers and therefore has a unique inverse.. Alternatively, it may be defined as , where is the natural logarithm, defined in any of its standard ways. Using the complex logarithm in this definition allows the binary logarithm to be extended to the complex numbers.For instance, Microsoft Excel provides the `IMLOG2` function for complex binary logarithms: see . As with other logarithms, the binary logarithm obeys the following equations, which can be used to simplify formulas that combine binary logarithms with multiplication or exponentiation:.
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These surfaces are glued to each other along the branch cut in the unique way to make the logarithm continuous. Each time the variable goes around the origin, the logarithm moves to a different branch.
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A fast hardware approach for approximate, efficient logarithm and antilogarithm computations.
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The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa before 1649. Their work involved quadrature of the hyperbola with equation , by determination of the area of hyperbolic sectors. Their solution generated the requisite "hyperbolic logarithm" function, which had the properties now associated with the natural logarithm. An early mention of the natural logarithm was by Nicholas Mercator in his work Logarithmotechnia, published in 1668, although the mathematics teacher John Speidell had already compiled a table of what in fact were effectively natural logarithms in 1619.
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The notations and both refer unambiguously to the natural logarithm of , and without an explicit base may also refer to the natural logarithm. This usage is common in mathematics, along with some scientific contexts as well as in many programming languages.Including C, C++, SAS, MATLAB, Mathematica, Fortran, and some BASIC dialects In some other contexts such as chemistry, however, can be used to denote the common (base 10) logarithm. It may also refer to the binary (base 2) logarithm in the context of computer science, particularly in the context of time complexity.
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Many physical and social phenomena exhibit such behavior -- incomes, species populations, galaxy sizes, and rainfall volumes, to name a few. Power transforms, and in particular the logarithm, can often be used to induce symmetry in such data. The logarithm is often favored because it is easy to interpret its result in terms of "fold changes." The logarithm also has a useful effect on ratios.
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The modular discrete logarithm is another variant; it has uses in public-key cryptography.
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If a complex number is represented in polar form z = reiθ, then the logarithm of z is :\ln z = \ln r + i\theta.\, However, there is an obvious ambiguity in defining the angle θ: adding to θ any integer multiple of 2 will yield another possible angle. A branch of the logarithm is a continuous function L(z) giving a logarithm of z for all z in a connected open set in the complex plane. In particular, a branch of the logarithm exists in the complement of any ray from the origin to infinity: a branch cut.
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In order to say something about the security properties of the above explained XTR encryption scheme, first it is important to check the security of the XTR group, which means how hard it is to solve the Discrete Logarithm problem there. The next part will then state the equivalency between the Discrete Logarithm problem in the XTR group and the XTR version of the discrete logarithm problem, using only the traces of elements.
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The logarithm with this special base is called the natural logarithm, and is denoted as ; it behaves well under differentiation since there is no undetermined limit to carry through the calculations. Thus, there are two ways of selecting such special numbers . One way is to set the derivative of the exponential function equal to , and solve for . The other way is to set the derivative of the base logarithm to and solve for .
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In the context of photography, the dynamic range is often measured in "stops", which is the binary logarithm of the ratio of highest and lowest distinguishable exposures; in an engineering context, the dynamic range is usually given by its decadic logarithm expressed in decibels.
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This can be proved by taking the logarithm of the product and using limit comparison test.
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Then the natural logarithm could be recognized as the inverse function to the transcendental function ex.
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The (natural) exponential function is the unique function which is equal to its own derivative, with the initial value (and hence one may define as ). The natural logarithm, or logarithm to base , is the inverse function to the natural exponential function. The natural logarithm of a number can be defined directly as the area under the curve between and , in which case is the value of for which this area equals one (see image). There are various other characterizations.
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Lattice-based cryptographic constructions are the leading candidates for public-key post- quantum cryptography. Indeed, the main alternative forms of public-key cryptography are schemes based on the hardness of factoring and related problems and schemes based on the hardness of the discrete logarithm and related problems. However, both factoring and the discrete logarithm are known to be solvable in polynomial time on a quantum computer. Furthermore, algorithms for factorization tend to yield algorithms for discrete logarithm, and vice versa.
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The order of magnitude of a number is, intuitively speaking, the number of powers of 10 contained in the number. More precisely, the order of magnitude of a number can be defined in terms of the common logarithm, usually as the integer part of the logarithm, obtained by truncation. For example, the number has a logarithm (in base 10) of 6.602; its order of magnitude is 6. When truncating, a number of this order of magnitude is between 106 and 107.
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The Lambert function is named after Johann Heinrich Lambert. The principal branch is denoted in the Digital Library of Mathematical Functions, and the branch is denoted there. The notation convention chosen here (with and ) follows the canonical reference on the Lambert function by Corless, Gonnet, Hare, Jeffrey and Knuth. The name "product logarithm" can be understood as this: Since the inverse function of is called the logarithm, it makes sense to call the inverse function of the product as "product logarithm".
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However, finding the witness w such that g^w=x holds corresponds to solving the discrete logarithm problem.
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The computational complexity of computing the natural logarithm (using the arithmetic-geometric mean) is O(M(n) ln n). Here n is the number of digits of precision at which the natural logarithm is to be evaluated and M(n) is the computational complexity of multiplying two n-digit numbers.
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The common logarithm of a number is the index of that power of ten which equals the number.William Gardner (1742) Tables of Logarithms Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by Archimedes as the "order of a number".R.C. Pierce (1977) "A brief history of logarithm", Two-Year College Mathematics Journal 8(1):22–26. The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation.
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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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In mathematics, the binary logarithm of a number is often written as .For instance, this is the notation used in the Encyclopedia of Mathematics and The Princeton Companion to Mathematics. However, several other notations for this function have been used or proposed, especially in application areas. Some authors write the binary logarithm as ,.
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The invention of the telescope and the theodolite and the development of logarithm tables allowed exact triangulation and grade measurement.
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Archimedes had written The Quadrature of the Parabola in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647. The relation that the logarithm provides between a geometric progression in its argument and an arithmetic progression of values, prompted A. A. de Sarasa to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in prosthaphaeresis, leading to the term "hyperbolic logarithm", a synonym for natural logarithm. Soon the new function was appreciated by Christiaan Huygens, and James Gregory. The notation Log y was adopted by Leibniz in 1675,Florian Cajori (1913) "History of the exponential and logarithm concepts", American Mathematical Monthly 20: 5, 35, 75, 107, 148, 173, 205.
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Plots of logarithm functions, with three commonly used bases. The special points are indicated by dotted lines, and all curves intersect graph of the logarithm base 2 crosses the x-axis at and passes through the points , , and , depicting, e.g., and . The graph gets arbitrarily close to the -axis, but does not meet it.
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The complex logarithm is the complex number analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However, a multivalued function can be defined which satisfies most of the identities. It is usual to consider this as a function defined on a Riemann surface.
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This is because the formula used to calculate pH approximates the negative of the base 10 logarithm of the molar concentration of hydrogen ions in the solution. More precisely, pH is the negative of the base 10 logarithm of the activity of the H+ ion.Bates, Roger G. Determination of pH: theory and practice. Wiley, 1973.
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He noted that mapping x this way is not an algebraic function, but rather a transcendental function. For a > 1 these functions are monotonic increasing and form bijections of the real line with positive real numbers. Then each base a corresponds to an inverse function called the logarithm to base a, in chapter 6. In chapter 7, Euler introduces e as the number whose hyperbolic logarithm is 1. The reference here is to Gregoire de Saint-Vincent who performed a quadrature of the hyperbola y = 1/x through description of the hyperbolic logarithm.
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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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Furthermore, the efficiency is indifferent to choice of (positive) base , as indicated by the insensitivity within the final logarithm above thereto.
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It is shown by Stephen Pohlig and Martin Hellman in 1978 that if all the factors of p − 1 are less than log p, then the problem of solving discrete logarithm modulo p is in P. Therefore, for cryptosystems based on discrete logarithm, such as DSA, it is required that p − 1 have at least one large prime factor.
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Exponentiation has two inverse operations; roots and logarithms. Analogously, the inverses of tetration are often called the super-root, and the super-logarithm (In fact, all hyperoperations greater than or equal to 3 have analogous inverses); e.g., in the function {^3}y=x, the two inverses are the cube super-root of and the super logarithm base of .
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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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If the logarithm is taken as the forward function, the function taking the base to a given power is then called the antilogarithm.
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The D 37 of each protein was obtained from the curve of the Napierian logarithm of the remaining intact protein versus radiation dose.
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An attempted security proof for Dual_EC_DRBG states that it requires three problems to be mathematically hard in order for Dual_EC_DRBG to be secure: the decisional Diffie-Hellman problem, the x-logarithm problem, and the truncated point problem. The decisional Diffie-Hellman problem is widely accepted as hard. The x-logarithm problem is not widely accepted as hard, but some evidence is shown that this problem is hard but does not prove that the problem is hard. The security proof is therefore questionable and would be proven invalid if the x-logarithm problem is shown to be solvable instead of hard.
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His findings led to the natural logarithm function, once called the hyperbolic logarithm since it is obtained by integrating, or finding the area, under the hyperbola.Martin Flashman The History of Logarithms from Humboldt State University Before 1748 and the publication of Introduction to the Analysis of the Infinite, the natural logarithm was known in terms of the area of a hyperbolic sector. Leonhard Euler changed that when he introduced transcendental functions such as 10x. Euler identified e as the value of b producing a unit of area (under the hyperbola or in a hyperbolic sector in standard position).
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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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Plotting the logarithm of the number of trees per acre against the logarithm of the quadratic mean diameter (or the dbh of the tree of average basal area) of maximally stocked stands generally results in a straight-line relationship.Nyland, Ralph. 2002. Silvicultural Concepts and Applications 2nd edition. In most cases the line is used to define the limit of maximum stocking.
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An observer who can resolve details as small as 1 minute of visual angle scores LogMAR 0, since the base-10 logarithm of 1 is 0; an observer who can resolve details as small as 2 minutes of visual angle (i.e., reduced acuity) scores LogMAR 0.3, since the base-10 logarithm of 2 is near-approximately 0.3; and so on.
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Using this method, basic investigation of the properties of the super-logarithm and tetration can be performed with a small amount of computational overhead.
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Both analogies, with a unity mass particle and logarithm of number of photons, are useful in the analysis of behavior of the Toda oscillator.
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Some diffusions in random environment are even proportional to a power of the logarithm of the time, see for example Sinai's walk or Brox diffusion.
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Entropy is one of several ways to measure diversity. Specifically, Shannon entropy is the logarithm of , the true diversity index with parameter equal to 1.
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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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With base e the natural logarithm behaves like the common logarithm as ln(1e) = 0, ln(10e) = 1, ln(100e) = 2 and ln(1000e) = 3. The base e is the most economical choice of radix β > 1 , where the radix economy is measured as the product of the radix and the length of the string of symbols needed to express a given range of values.
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The element Σ Zntn is a group-like element of the Hopf algebra of formal power series over NSymm, so over the rationals its logarithm is primitive. The coefficients of its logarithm generate the free Lie algebra on a countable set of generators over the rationals. Over the rationals this identifies the Hopf algebra NSYmm with the universal enveloping algebra of the free Lie algebra.
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Sarason, Section IV.9. This construction is analogous to the real logarithm function , which is the inverse of the real exponential function , satisfying for positive real numbers . If a non-zero complex number is given in polar form as ( and real numbers, with ), then is one logarithm of . Since exactly for all integer , adding integer multiples to the argument gives all the numbers that are logarithms of : :.
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A plot of the multi-valued imaginary part of the complex logarithm function, which shows the branches. As a complex number z goes around the origin, the imaginary part of the logarithm goes up or down. This makes the origin a branch point of the function. For a function to have an inverse, it must map distinct values to distinct values, that is, it must be injective.
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The integral of 1/b thus yields the logarithm of the ratio of the upper and lower cut-offs. This number is known as the Coulomb logarithm and is designated by either \ln \Lambda or \lambda. It is the factor by which small-angle collisions are more effective than large-angle collisions. For many plasmas of interest it takes on values between 5 and 15.
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The diagram itself is a plot of the natural logarithm of the volume or yield against the natural logarithm of stems per acre. Just like a stocking diagram, the A-line, B-line, and C-line are plotted. In addition, the -3/2 rule maximum density line is plotted just above the A-line. The diagram works well for even aged, single cohort stands.
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Formally, LAT is 10 times the base 10 logarithm of the ratio of a root-mean-square A-weighted sound pressure during a stated time interval to the reference sound pressure and there is no time constant involved. To measure LAT an integrating-averaging meter is needed; this in concept takes the sound exposure, divides it by time and then takes the logarithm of the result.
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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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In computational number theory and computational algebra, Pollard's kangaroo algorithm (also Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm problem. The algorithm was introduced in 1978 by the number theorist J. M. Pollard, in the same paper J. Pollard, Monte Carlo methods for index computation (mod p), Mathematics of Computation, Volume 32, 1978 as his better-known Pollard's rho algorithm for solving the same problem. Although Pollard described the application of his algorithm to the discrete logarithm problem in the multiplicative group of units modulo a prime p, it is in fact a generic discrete logarithm algorithm—it will work in any finite cyclic group.
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This is not differentiable at t = 0, showing that the Cauchy distribution has no expectation. Also, the characteristic function of the sample mean of n independent observations has characteristic function , using the result from the previous section. This is the characteristic function of the standard Cauchy distribution: thus, the sample mean has the same distribution as the population itself. The logarithm of a characteristic function is a cumulant generating function, which is useful for finding cumulants; some instead define the cumulant generating function as the logarithm of the moment- generating function, and call the logarithm of the characteristic function the second cumulant generating function.
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By introducing the concept of thermodynamic probability as the number of microstates corresponding to the current macrostate, he showed that its logarithm is proportional to entropy.
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G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 4th Ed., Oxford 1975, footnote to paragraph 1.7: "log x is, of course, the 'Naperian' logarithm of x, to base e. 'Common' logarithms have no mathematical interest". Extract of page 9 Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
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If a calculation indicates a resistor of 165 ohms is required then log(150) = 2.176, log(165) = 2.217 and log(180) = 2.255. The logarithm of 165 is closer to the logarithm of 180 therefore a 180 ohm resistor would be the first choice if there are no other considerations. Whether a value rounds to or depends upon whether the squared value is greater than or less than the product .
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237, and another copy extended to negative powers appears on p. 249b. Earlier than Stifel, the 8th century Jain mathematician Virasena is credited with a precursor to the binary logarithm. Virasena's concept of ardhacheda has been defined as the number of times a given number can be divided evenly by two. This definition gives rise to a function that coincides with the binary logarithm on the powers of two,.
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The number of digits (bits) in the binary representation of a positive integer is the integral part of , i.e. : \lfloor \log_2 n\rfloor + 1. In information theory, the definition of the amount of self-information and information entropy is often expressed with the binary logarithm, corresponding to making the bit the fundamental unit of information. However, the natural logarithm and the nat are also used in alternative notations for these definitions..
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The discrete logarithm problem, the quadratic residuosity problem, the RSA inversion problem, and the problem of computing the permanent of a matrix are each random self-reducible problems.
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In optics, absorbance or decadic absorbance is the common logarithm of the ratio of incident to transmitted radiant power through a material, and spectral absorbance or spectral decadic absorbance is the common logarithm of the ratio of incident to transmitted spectral radiant power through a material. Absorbance is dimensionless, and in particular is not a length, though it is a monotonically increasing function of path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for absorbance is discouraged. In physics, a closely related quantity called "optical depth" is used instead of absorbance: the natural logarithm of the ratio of incident to transmitted radiant power through a material.
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Relative species abundance distributions are usually graphed as frequency histograms ("Preston plots"; Figure 2) or rank-abundance diagrams ("Whittaker Plots"; Figure 3).Whittaker, R. H. 1965. "Dominance and diversity in land plant communities", Science 147: 250–260 Frequency histogram (Preston plot): ::x-axis: logarithm of abundance bins (historically log2 as a rough approximation to the natural logarithm) ::y-axis: number of species at given abundance Rank-abundance diagram (Whittaker plot): ::x-axis: species list, ranked in order of descending abundance (i.e. from common to rare) ::y-axis: logarithm of % relative abundance When plotted in these ways, relative species abundances from wildly different data sets show similar patterns: frequency histograms tend to be right-skewed (e.g.
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The binary logarithm function is the inverse function of the power of two function. As well as , alternative notations for the binary logarithm include , , (the notation preferred by ISO 31-11 and ISO 80000-2), and (with a prior statement that the default base is 2) . Historically, the first application of binary logarithms was in music theory, by Leonhard Euler: the binary logarithm of a frequency ratio of two musical tones gives the number of octaves by which the tones differ. Binary logarithms can be used to calculate the length of the representation of a number in the binary numeral system, or the number of bits needed to encode a message in information theory.
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Y. Wang: "The law of the iterated logarithm for p-random sequences". In: Proc. 11th IEEE Conference on Computational Complexity (CCC), pages 180–189. IEEE Computer Society Press, 1996.
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This makes the Hausdorff dimension of this structure . Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question.
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Today's course may cover arithmetic and geometric sequences and series, but not the application by Saint-Vincent to gain his hyperbolic logarithm, which Euler used to finesse his precalculus.
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This is also the case for the expressions representing real numbers, which are built from the integers by using the arithmetical operations, the logarithm and the exponential (Richardson's theorem).
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Jones & Thron (1980) p. 202. Variations of this argument can be used to produce continued fraction expansions for the natural logarithm, the arcsin function, and the generalized binomial series.
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A common choice of branch cut is the negative real axis, although the choice is largely a matter of convenience. The logarithm has a jump discontinuity of 2i when crossing the branch cut. The logarithm can be made continuous by gluing together countably many copies, called sheets, of the complex plane along the branch cut. On each sheet, the value of the log differs from its principal value by a multiple of 2i.
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Suppose that we want to compare two models: one with a normal distribution of and one with a normal distribution of . We should not directly compare the AIC values of the two models. Instead, we should transform the normal cumulative distribution function to first take the logarithm of . To do that, we need to perform the relevant integration by substitution: thus, we need to multiply by the derivative of the (natural) logarithm function, which is .
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More precisely, the logarithm to any base is the only increasing function f from the positive reals to the reals satisfying and item (4.3.1) :f(xy)=f(x)+f(y).
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Thus these graphs are very useful for recognizing these relationships and estimating parameters. Any base can be used for the logarithm, though most commonly base 10 (common logs) are used.
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This technique was used to determine the surface gravity of Tau Ceti. The , or logarithm of the star's surface gravity, is about 4.4, very close to the for the Sun.
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By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is a transcendental number.
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The neper is the change in the level of a field quantity when the field quantity changes by a factor of e, that is , thereby relating all of the units as nondimensional natural log of field-quantity ratios, . Finally, the level of a quantity is the logarithm of the ratio of the value of that quantity to a reference value of the same kind of quantity. Therefore, the bel represents the logarithm of a ratio between two power quantities of 10:1, or the logarithm of a ratio between two field quantities of :1. Two signals whose levels differ by one decibel have a power ratio of 101/10, which is approximately 1.25893, and an amplitude (field quantity) ratio of 10 (1.12202).
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Mertens diplomatically describes his proof as more precise and rigorous. In reality none of the previous proofs are acceptable by modern standards: Euler's computations involve the infinity (and the hyperbolic logarithm of infinity, and the logarithm of the logarithm of infinity!); Legendre's argument is heuristic; and Chebyshev's proof, although perfectly sound, makes use of the Legendre- Gauss conjecture, which was not proved until 1896 and became better known as the prime number theorem. Mertens' proof does not appeal to any unproved hypothesis (in 1874), and only to elementary real analysis. It comes 22 years before the first proof of the prime number theorem which, by contrast, relies on a careful analysis of the behavior of the Riemann zeta function as a function of a complex variable.
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Returning to Göttingen, she completed her master's degree in 1987, with a thesis on the law of the iterated logarithm for self-similar processes. Czado continued to work with Taqqu, and joined the doctoral program in operations research and industrial engineering at Cornell. However, while she was there, Taqqu moved to Boston University. Frustrated with her progress on the law of the iterated logarithm, Czado took advantage of the opportunity to change topics under another advisor.
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In 1815, Peter Mark Roget invented the log log slide rule, which included a scale displaying the logarithm of the logarithm. This allowed the user to directly perform calculations involving roots and exponents. This was especially useful for fractional powers. In 1821, Nathaniel Bowditch, described in the American Practical Navigator a "sliding rule" that contained scales trigonometric functions on the fixed part and a line of log-sines and log-tans on the slider used to solve navigation problems.
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Mathematicians, on the other hand, wrote "log(x)" when they meant loge(x) for the natural logarithm. Today, both notations are found. Since hand-held electronic calculators are designed by engineers rather than mathematicians, it became customary that they follow engineers' notation. So the notation, according to which one writes "ln(x)" when the natural logarithm is intended, may have been further popularized by the very invention that made the use of "common logarithms" far less common, electronic calculators.
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A microarray for approximately 8700 genes. The expression rates of these genes are compared using binary logarithms. In bioinformatics, microarrays are used to measure how strongly different genes are expressed in a sample of biological material. Different rates of expression of a gene are often compared by using the binary logarithm of the ratio of expression rates: the log ratio of two expression rates is defined as the binary logarithm of the ratio of the two rates.
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The entropy is proportional to the logarithm of the number of microstates, the ways a system can be configured microscopically while leaving the macroscopic description unchanged. Black hole entropy is deeply puzzling — it says that the logarithm of the number of states of a black hole is proportional to the area of the horizon, not the volume in the interior. Later, Raphael Bousso came up with a covariant version of the bound based upon null sheets.
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New 2-variable statistic regression models include natural logarithm, exponent, power. Up to 42 sample points or pairs can be stored. In base calculation, binary base was removed. Complex function was restored.
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Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm.
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The human perception of the intensity of sound and light more nearly approximates the logarithm of intensity rather than a linear relationship (see Weber–Fechner law), making the dB scale a useful measure.
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The logarithms in this formula are usually taken (as shown in the graph) to the base 2. See binary logarithm. When p=\tfrac 1 2, the binary entropy function attains its maximum value.
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As the expansion gives the rate of change of the logarithm of the area density, this means the event horizon area can never go down, at least classically, assuming the null energy condition.
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The maximum-term method is a consequence of the large numbers encountered in statistical mechanics. It states that under appropriate conditions the logarithm of a summation is essentially equal to the logarithm of the maximum term in the summation. These conditions are (see also proof below) that (1) the number of terms in the sum is large and (2) the terms themselves scale exponentially with this number. A typical application is the calculation of a thermodynamic potential from a partition function.
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All these complex logarithms of are on a vertical line in the complex plane with real part . Since any nonzero complex number has infinitely many complex logarithms, the complex logarithm cannot be defined to be a single-valued function on the complex numbers, but only as a multivalued function. Settings for a formal treatment of this are, among others, the associated Riemann surface, branches, or partial inverses of the complex exponential function. Sometimes the instead of is used when addressing the complex logarithm.
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29, The International System of Units (SI), ed. Barry N. Taylor, NIST Special Publication 330, 2001. In astrophysics, the surface gravity may be expressed as log g, which is obtained by first expressing the gravity in cgs units, where the unit of acceleration is centimeters per second squared, and then taking the base-10 logarithm. Therefore, the surface gravity of Earth could be expressed in cgs units as 980.665 cm/s², with a base-10 logarithm (log g) of 2.992.
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In computer science, they count the number of steps needed for binary search and related algorithms. Other areas in which the binary logarithm is frequently used include combinatorics, bioinformatics, the design of sports tournaments, and photography. Binary logarithms are included in the standard C mathematical functions and other mathematical software packages. The integer part of a binary logarithm can be found using the find first set operation on an integer value, or by looking up the exponent of a floating point value.
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In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to transmitted radiant power through a material, and spectral optical depth or spectral optical thickness is the natural logarithm of the ratio of incident to transmitted spectral radiant power through a material. Optical depth is dimensionless, and in particular is not a length, though it is a monotonically increasing function of optical path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged. In chemistry, a closely related quantity called "absorbance" or "decadic absorbance" is used instead of optical depth: the common logarithm of the ratio of incident to transmitted radiant power through a material, that is the optical depth divided by ln 10.
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A single valued version, called the principal value of the logarithm, can be defined which is discontinuous on the negative x axis, and is equal to the multivalued version on a single branch cut.
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However, there is no mathematical reason to only use logarithm to base 2, and due to many discrepancies in describing the log2 fold changes in gene/protein expression, a new term "loget" has been proposed.
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Basin entropy is the logarithm of the attractors in one Boolean network. Employing approaches from statistical mechanics, the complexity, uncertainty, and randomness of networks can be described by network ensembles with different types of constraints.
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These tables listed the values of for any number in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of can be separated into an integer part and a fractional part, known as the characteristic and mantissa. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point.
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Another way to resolve the indeterminacy is to view the logarithm as a function whose domain is not a region in the complex plane, but a Riemann surface that covers the punctured complex plane in an infinite-to-1 way. Branches have the advantage that they can be evaluated at complex numbers. On the other hand, the function on the Riemann surface is elegant in that it packages together all branches of the logarithm and does not require an arbitrary choice as part of its definition.
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To finish the point, since the natural logarithm of a power law tail is exponential, one can take the logarithm of power law data and then test for outliers relative to an exponential tail. There are many test statistics and techniques for testing for outliers in an exponential sample. An inward test sequentially tests the largest point, then the second largest, and so on, until the first test that is not rejected (i.e., the null hypothesis that the point is not an outlier is not rejected).
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It is also possible to predict solubility from other physical constants such as the enthalpy of fusion. The octanol-water partition coefficient, usually expressed as its logarithm (Log P) is a measure of differential solubility of a compound in a hydrophobic solvent (1-octanol) and a hydrophilic solvent (water). The logarithm of these two values enables compounds to be ranked in terms of hydrophilicity (or hydrophobicity). The energy change associated with dissolving is usually given per mole of solute as the enthalpy of solution.
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This is the most common practice. When using the natural logarithm of base \displaystyle e, the unit will be the nat. For the base 10 logarithm, the unit of information is the hartley. As a quick illustration, the information content associated with an outcome of 4 heads (or any specific outcome) in 4 consecutive tosses of a coin would be 4 bits (probability 1/16), and the information content associated with getting a result other than the one specified would be ~0.09 bits (probability 15/16).
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In particular when we choose the rectangular hyperbola xy = 1, one recovers the natural logarithm. A student and co-worker of Saint-Vincent, A. A. de Sarasa noted that this area property of the hyperbola represented a logarithm, a means of reducing multiplication to addition. An approach to Vincent−Sarasa theorem may be seen with hyperbolic sectors and the area-invariance of squeeze mapping. In 1651 Christiaan Huygens published his Theoremata de Quadratura Hyperboles, Ellipsis, et Circuli which referred to the work of Saint-Vincent.
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The logarithm of a product is simply the sum of the logarithms of the factors. Therefore, when the logarithm of a product of random variables that take only positive values approaches a normal distribution, the product itself approaches a log-normal distribution. Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the products of different random factors, so they follow a log-normal distribution. This multiplicative version of the central limit theorem is sometimes called Gibrat's law.
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The cache maintenance algorithm ensures that each node maintains adequate knowledge of the "cloud". It is designed to ensure that the time to resolve a request varies as the logarithm of the size of the cloud.
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Strassen began his researches as a probabilist; his 1964 paper An Invariance Principle for the Law of the Iterated Logarithm defined a functional form of the law of the iterated logarithm, showing a form of scale invariance in random walks. This result, now known as Strassen's invariance principle or as Strassen's law of the iterated logarithm, has been highly cited and led to a 1966 presentation at the International Congress of Mathematicians. In 1969, Strassen shifted his research efforts towards the analysis of algorithms with a paper on Gaussian elimination, introducing Strassen's algorithm, the first algorithm for performing matrix multiplication faster than the O(n3) time bound that would result from a naive algorithm. In the same paper he also presented an asymptotically fast algorithm to perform matrix inversion, based on the fast matrix multiplication algorithm.
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Intermediate values are formed using the common logarithm. For example, a gas which is 99.97% pure would be described as N3.5, since log10(0.03%) = −3.523. Nines are used in a similar manner to describe computer system availability.
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The geometric mean applies only to positive numbers.The geometric mean only applies to numbers of the same sign in order to avoid taking the root of a negative product, which would result in imaginary numbers, and also to satisfy certain properties about means, which is explained later in the article. The definition is unambiguous if one allows 0 (which yields a geometric mean of 0), but may be excluded, as one frequently wishes to take the logarithm of geometric means (to convert between multiplication and addition), and one cannot take the logarithm of 0.
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Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors: : \log_b(xy) = \log_b x + \log_b y, \, provided that , and are all positive and .
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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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In the context of finite groups exponentiation is given by repeatedly multiplying one group element with itself. The discrete logarithm is the integer n solving the equation :b^n = x,\, where is an element of the group. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. This asymmetry has important applications in public key cryptography, such as for example in the Diffie–Hellman key exchange, a routine that allows secure exchanges of cryptographic keys over unsecured information channels.
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The Indian mathematician Virasena worked with the concept of ardhaccheda: the number of times a number of the form 2n could be halved. For exact powers of 2, this equals the binary logarithm, but it differs from the logarithm for other numbers. He described a product formula for this concept and also introduced analogous concepts for base 3 (trakacheda) and base 4 (caturthacheda). Michael Stifel published Arithmetica integra in Nuremberg in 1544, which contains a table of integers and powers of 2 that has been considered an early version of a table of binary logarithms.
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An order-of-magnitude estimate of a variable, whose precise value is unknown, is an estimate rounded to the nearest power of ten. For example, an order-of-magnitude estimate for a variable between about 3 billion and 30 billion (such as the human population of the Earth) is 10 billion. To round a number to its nearest order of magnitude, one rounds its logarithm to the nearest integer. Thus , which has a logarithm (in base 10) of 6.602, has 7 as its nearest order of magnitude, because "nearest" implies rounding rather than truncation.
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There is no formula to calculate prime numbers. However, the distribution of primes can be statistically modelled. The prime number theorem, which was proven at the end of the 19th century, says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits (logarithm). At the start of the 19th century, Adrien- Marie Legendre and Carl Friedrich Gauss suggested that as x goes very large, the number of primes up to x is asymptotic to x/\log x, where \log x is the natural logarithm of x.
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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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A measure is given by the area function on regions of the Cartesian plane. This measure becomes a charge in certain instances. For example, when the natural logarithm is defined by the area under the curve y = 1/x for x in the positive real numbers, the region with 0 < x< 1 is considered negative. The logarithm defined as an integral from University of California, Davis A region defined by a continuous function y = f(x), the x-axis, and lines x = a and x = b can be evaluated by Riemann integration.
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This timeline shows the whole history of the universe, the Earth, and mankind in one table. Each row is defined in years ago, that is, years before the present date, with the earliest times at the top of the chart. In each table cell on the right, references to events or notable people are given, more or less in chronological order within the cell. Each row corresponds to a change in log(time before present) (that is, the logarithm of the time before the present) of about 0.1 (using base 10 logarithm).
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386 The antiderivative of the natural logarithm is: : \int \ln(x) \,dx = x \ln(x) - x + C. Related formulas, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.
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The `fls` and `flsl` functions in the Linux kernelfls, Linux kernel API, kernel.org, retrieved 2010-10-17. and in some versions of the libc software library also compute the binary logarithm (rounded up to an integer, plus one).
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In statistics, the observed information, or observed Fisher information, is the negative of the second derivative (the Hessian matrix) of the "log- likelihood" (the logarithm of the likelihood function). It is a sample-based version of the Fisher information.
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In June 2012 the National Institute of Information and Communications Technology (NICT), Kyushu University, and Fujitsu Laboratories Limited improved the previous bound for successfully computing a discrete logarithm on a supersingular elliptic curve from 676 bits to 923 bits.
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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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This paper was communicated and signed by Edward Pickering, but the first sentence indicates that it was "prepared by Miss Leavitt". In the 1912 paper, Leavitt graphed the stellar magnitude versus the logarithm of the period and determined that, in her own words, Using the simplifying assumption that all of the Cepheids within the Small Magellanic Cloud were at approximately the same distance, the apparent magnitude of each star is equivalent to its absolute magnitude offset by a fixed quantity depending on that distance. 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). At the time, there was an unknown scale factor in this brightness since the distances to the Magellanic Clouds were unknown.
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The polar form requires 3/2 multiplications, 1/2 logarithm, 1/2 square root, and 1/2 division for each normal variate. The effect is to replace one multiplication and one trigonometric function with a single division and a conditional loop.
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In mathematics, particularly p-adic analysis, the p-adic exponential function is a p-adic analogue of the usual exponential function on the complex numbers. As in the complex case, it has an inverse function, named the p-adic logarithm.
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In the Toda oscillator model of self-pulsation, the logarithm of amplitude varies exponentially with time (for large amplitudes), thus the amplitude varies as doubly exponential function of time.. Dendritic macromolecules have been observed to grow in a doubly-exponential fashion.
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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 an acidic solution, the concentration of hydronium ions is greater than 10−7 moles per liter. Since pH is defined as the negative logarithm of the concentration of hydronium ions, acidic solutions thus have a pH of less than 7.
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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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Since the hyperbolic functions are rational functions of whose numerator and denominator are of degree at most two, these functions may be solved in terms of , by using the quadratic formula; then, taking the natural logarithm gives the following expressions for the inverse hyperbolic functions. For complex arguments, the inverse hyperbolic functions, the square root and the logarithm are multi-valued functions, and the equalities of the next subsections may be viewed as equalities of multi-valued functions. For all inverse hyperbolic functions (save the inverse hyperbolic cotangent and the inverse hyperbolic cosecant), the domain of the real function is connected.
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The definition of contrast ratio is therefore re-stated as follows : 'The ratio between the opacities of the darkest and lightest points in the film image', thus: contrast ratio = Omax. / Omin. As we have already seen, opacity is not easily measured with standard photographic equipment—but the logarithm of opacity is continually measured since, in fact, it is the unit of image saturation known as density. Since density is a logarithm we must take the ratio of the anti-logarithms of the maximum and minimum densities in the image in order to arrive at the contrast ratio.
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The "static" image is similar to the conventional radiographic image in most cases; however, there could be a significant difference if the studied object is moving. In this case, the "static" image is less affected by the motion noise than the ordinary calculated radiographic image. The reason is that the E(D) values are calculated by averaging the series of attenuation values computed from the corresponding series of photon numbers (from the underexposed images) and converted into a logarithm. On the other hand, the attenuation values of common radiographs are calculated from a single summed up photon number converted into a logarithm.
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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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Log–log plots are an alternative way of graphically examining the tail of a distribution using a random sample. Caution has to be exercised however as a log–log plot is necessary but insufficient evidence for a power law relationship, as many non power-law distributions will appear as straight lines on a log–log plot. This method consists of plotting the logarithm of an estimator of the probability that a particular number of the distribution occurs versus the logarithm of that particular number. Usually, this estimator is the proportion of times that the number occurs in the data set.
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Snellen chart The Snellen chart, which dates back to 1862, is also commonly used to estimate visual acuity. A Snellen score of 6/6 (20/20), indicating that an observer can resolve details as small as 1 minute of visual angle, corresponds to a LogMAR of 0 (since the base-10 logarithm of 1 is 0); a Snellen score of 6/12 (20/40), indicating an observer can resolve details as small as 2 minutes of visual angle, corresponds to a LogMAR of 0.3 (since the base-10 logarithm of 2 is near-approximately 0.3), and so on.
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Thus a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly, in contrast to essential singularities, such as 0 for the logarithm function, and branch points, such as 0 for the complex square root function.
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The identities of logarithms can be used to approximate large numbers. Note that , where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, . To get the base-10 logarithm, we would multiply 32,582,657 by , getting .
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Commentarii academiae scientiarum Petropolitanae 9, 1744, pp. 160-188. in order to denote "absolutus infinitus". Euler freely performed various operations on infinity, such as taking its logarithm. This symbol is not used anymore, and is not encoded as a separate character in Unicode.
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McCune, Bruce & Grace, James (2002) Analysis of Ecological Communities. Mjm Software Design; . Recently the Dice score (and its variations, e.g. logDice taking a logarithm of it) has become popular in computer lexicography for measuring the lexical association score of two given words.
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Because base–emitter voltage varies as the logarithm of the base–emitter and collector–emitter currents, a BJT can also be used to compute logarithms and anti-logarithms. A diode can also perform these nonlinear functions but the transistor provides more circuit flexibility.
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A contemporary example of using bilinear pairings is exemplified in the Boneh-Lynn-Shacham signature scheme. Pairing-based cryptography relies on hardness assumptions separate from e.g. the Elliptic Curve Discrete Logarithm Problem, which is older and has been studied for a longer time.
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Thorup's reduction is complicated and assumes the availability of either fast multiplication operations or table lookups, but he also provides an alternative priority queue using only addition and Boolean operations with time per operation, at most multiplying the time by an iterated logarithm.
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In mathematics, and particularly ordinary differential equations, a characteristic multiplier is an eigenvalue of a monodromy matrix. The logarithm of a characteristic multiplier is also known as characteristic exponent. They appear in Floquet theory of periodic differential operators and in the Frobenius method.
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If we consider the same time series, but increase the number of observations of it, the rescaled range will generally also increase. The increase of the rescaled range can be characterized by making a plot of the logarithm of R/S vs.
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In large deviations theory, the rate function is defined as the Legendre transformation of the logarithm of the moment generating function of a random variable. An important application of the rate function is in the calculation of tail probabilities of sums of i.i.d. random variables.
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For models with binary outcomes (Y = 1 or 0), the model can be scored with the logarithm of predictions : S = Y \log( p ) + ( 1 - Y ) ( \log( 1 - p ) ) where p is the probability in the model to be estimated and S is the score.
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In physics and chemistry, a plot of logarithm of pressure against temperature can be used to illustrate the various phases of a substance, as in the following for water: log-linear pressure–temperature phase diagram of water. The Roman numerals indicate various ice phases.
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By plotting Fnorm against the logarithm of the different concentrations of the dilution series, a sigmoidal binding curve is obtained. This binding curve can directly be fitted with the nonlinear solution of the law of mass action, with the dissociation constant KD as result.
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All quantities are in Gaussian (cgs) units except energy and temperature expressed in eV and ion mass expressed in units of the proton mass \mu = m_i/m_p; Z is charge state; k is Boltzmann's constant; K is wavenumber; \ln\Lambda is the Coulomb logarithm.
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He adds: > The observer and his powers of discrimination may have to be specified if > the variety is to be well defined. Variety can be stated as an integer, as above, or as the logarithm to the base 2 of the number i.e. in bits.
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The speed of an ADC varies by type. The Wilkinson ADC is limited by the clock rate which is processable by current digital circuits. For a successive-approximation ADC, the conversion time scales with the logarithm of the resolution, i.e. the number of bits.
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When the values are quadratic in the amplitude (e.g. power), they are first linearised by taking the square root before the logarithm is taken, or equivalently the result is halved. In the International System of Quantities, the neper is defined as .Thor, A. J. (1994).
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Post-quantum cryptography (sometimes referred to as quantum-proof, quantum- safe or quantum-resistant) refers to cryptographic algorithms (usually public- key algorithms) that are thought to be secure against an attack by a quantum computer. , this is not true for the most popular public-key algorithms, which can be efficiently broken by a sufficiently strong quantum computer. The problem with currently popular algorithms is that their security relies on one of three hard mathematical problems: the integer factorization problem, the discrete logarithm problem or the elliptic-curve discrete logarithm problem. All of these problems can be easily solved on a sufficiently powerful quantum computer running Shor's algorithm.
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The top image shows that leptokurtic densities in this family have a higher peak than the mesokurtic normal density, although this conclusion is only valid for this select family of distributions. The comparatively fatter tails of the leptokurtic densities are illustrated in the second image, which plots the natural logarithm of the Pearson type VII densities: the black curve is the logarithm of the standard normal density, which is a parabola. One can see that the normal density allocates little probability mass to the regions far from the mean ("has thin tails"), compared with the blue curve of the leptokurtic Pearson type VII density with excess kurtosis of 2.
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The logarithm of to base is denoted as , or without parentheses, , or even without the explicit base, , when no confusion is possible, or when the base does not matter such as in big O notation. More generally, exponentiation allows any positive real number as base to be raised to any real power, always producing a positive result, so for any two positive real numbers and , where is not equal to , is always a unique real number . More explicitly, the defining relation between exponentiation and logarithm is: : \log_b(x) = y \ exactly if \ b^y = x\ and \ x > 0 and \ b > 0 and \ b e 1. For example, , as .
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Public-key cryptography is based on the intractability of certain mathematical problems. Early public-key systems based their security on the assumption that it is difficult to factor a large integer composed of two or more large prime factors. For later elliptic-curve- based protocols, the base assumption is that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible: this is the "elliptic curve discrete logarithm problem" (ECDLP). The security of elliptic curve cryptography depends on the ability to compute a point multiplication and the inability to compute the multiplicand given the original and product points.
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Therefore, the secant method may occasionally be faster in practice. For instance, if we assume that evaluating f takes as much time as evaluating its derivative and we neglect all other costs, we can do two steps of the secant method (decreasing the logarithm of the error by a factor φ2 ≈ 2.6) for the same cost as one step of Newton's method (decreasing the logarithm of the error by a factor 2), so the secant method is faster. If, however, we consider parallel processing for the evaluation of the derivative, Newton's method proves its worth, being faster in time, though still spending more steps.
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In probability and statistics, the parabolic fractal distribution is a type of discrete probability distribution in which the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank (with the largest example having rank 1). This can markedly improve the fit over a simple power-law relationship (see references below). In the Laherrère/Deheuvels paper below, examples include galaxy sizes (ordered by luminosity), towns (in the USA, France, and world), spoken languages (by number of speakers) in the world, and oil fields in the world (by size). They also mention utility for this distribution in fitting seismic events (no example).
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The original idea for an anonymous credential system was derived from blind signatures, but relied on a trusted party for credential transfer—the translation from one pseudonym to another. The blind signature scheme introduced by Chaum was based on RSA signatures and based on the discrete logarithm problem can be used for constructing anonymous credential systems. Stefan Brands generalized digital credentials with secret-key certificate based credentials, improving on Chaum's basic blind-signature based system in both the discrete logarithm and strong RSA assumption settings. Brands credentials provide efficient algorithms and privacy in an unconditional commercial security setting, along with several other features such as a proof of non-membership blacklist.
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BLISS (short for Bimodal Lattice Signature Scheme) is a digital signature scheme proposed by Léo Ducas, Alain Durmus, Tancrède Lepoint and Vadim Lyubashevsky in their 2013 paper "Lattice Signature and Bimodal Gaussians". In cryptography, a digital signature ensures that a message is authentically from a specific person who has the private key to create such a signature, and can be verified using the corresponding public key. Current signature schemes rely either on integer factorization, discrete logarithm or elliptic curve discrete logarithm problem, all of which can be effectively attacked by a quantum computer. BLISS on the other hand, is a post-quantum algorithm, and is meant to resist quantum computer attacks.
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The logarithm of a Lomax(shape = 1.0, scale = λ)-distributed variable follows a logistic distribution with location log(λ) and scale 1.0. This implies that a Lomax(shape = 1.0, scale = λ)-distribution equals a log-logistic distribution with shape β = 1.0 and scale α = log(λ).
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In number theory, the integer complexity of an integer is the smallest number of ones that can be used to represent it using ones and any number of additions, multiplications, and parentheses. It is always within a constant factor of the logarithm of the given integer.
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Also in 1650 he proved that the sum of the alternating harmonic series is equal to the natural logarithm of 2. He also proved that the harmonic series does not converge, and provided a proof that Wallis' product for \pi is correct.Hofmann, Joseph Ehrenfried (1959). Classical Mathematics.
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This example will demonstrate standard quadratic sieve without logarithm optimizations or prime powers. Let the number to be factored N = 15347, therefore the ceiling of the square root of N is 124. Since N is small, the basic polynomial is enough: y(x) = (x + 124)2 − 15347\.
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Any differentiable function may be used for . The examples that follow use a variety of elementary functions; special functions may also be used. Note that multi-valued functions such as the natural logarithm may be used, but attention must be confined to a single Riemann surface.
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Mathematical tables containing common logarithms (base-10) were extensively used in computations prior to the advent of computers and calculators, not only because logarithms convert problems of multiplication and division into much easier addition and subtraction problems, but for an additional property that is unique to base-10 and proves useful: Any positive number can be expressed as the product of a number from the interval and an integer power of This can be envisioned as shifting the decimal separator of the given number to the left yielding a positive, and to the right yielding a negative exponent of Only the logarithms of these normalized numbers (approximated by a certain number of digits), which are called mantissas, need to be tabulated in lists to a similar precision (a similar number of digits). These mantissas are all positive and enclosed in the interval . The common logarithm of any given positive number is then obtained by adding its mantissa to the common logarithm of the second factor. This logarithm is called the characteristic of the given number.
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For the 25 years preceding the invention of the logarithm in 1614, prosthaphaeresis was the only known generally applicable way of approximating products quickly. It used the identities for the trigonometric functions of sums and differences of angles in terms of the products of trigonometric functions of those angles.
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In statistics, a support curve is the graph of the natural logarithm of the likelihood function. It has a relation to, but is distinct from, the support of a distribution. The term "support curve" was coined by A. W. F. Edwards. It refers to the hypotheses being tested, i.e.
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The "mass of the earth in gravitational measure" is stated as "9.81996×63709802" in The New Volumes of the Encyclopaedia Britannica (Vol. 25, 1902) with a "logarithm of earth's mass" given as "14.600522" []. This is the gravitational parameter in m3·s−2 (modern value ) and not the absolute mass.
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Graph of the equation . Here, is the unique number larger than 1 that makes the shaded area equal to 1. The number ', known as Euler's number, is a mathematical constant approximately equal to 2.71828, and can be characterized in many ways. It is the base of the natural logarithm.
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A LogMAR chart (Logarithm of the Minimum Angle of Resolution), also called a Bailey-Lovie chart or an ETDRS chart (Early Treatment Diabetic Retinopathy Study),Bailey IL, Lovie JE (2013). Visual acuity testing. From the laboratory to the clinic. Vision Research 90: 2-9. doi:10.1016/j.visres.2013.05.004.
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Am J Optom Physiol Opt. 53 (11): pp. 740–745. For this reason, the LogMAR chart is recommended, particularly in a research setting. When using a LogMAR chart, visual acuity is scored with reference to the logarithm of the minimum angle of resolution, as the chart's name suggests.
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In mathematics, a closed-form expression is a mathematical expression expressed using a finite number of standard operations. It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration.
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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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The logarithm and square root transformations are commonly used for positive data, and the multiplicative inverse (reciprocal) transformation can be used for non-zero data. The power transformation is a family of transformations parameterized by a non-negative value λ that includes the logarithm, square root, and multiplicative inverse as special cases. To approach data transformation systematically, it is possible to use statistical estimation techniques to estimate the parameter λ in the power transformation, thereby identifying the transformation that is approximately the most appropriate in a given setting. Since the power transformation family also includes the identity transformation, this approach can also indicate whether it would be best to analyze the data without a transformation.
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The integer factorization problem is believed to be intractable on any conventional computer if the primes are chosen at random and are sufficiently large. However, to factor the product of two n-bit primes, a quantum computer with roughly 6n bits of logical qubit memory and capable of executing a program known as Shor’s algorithm will easily accomplish the task. Shor's algorithm can also quickly break digital signatures based on what is known as the discrete logarithm problem and the more esoteric elliptic curve discrete logarithm problem. In effect, a relatively small quantum computer running Shor's algorithm could quickly break all of the digital signatures used to ensure the privacy and integrity of information on the internet today.
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Klein & Rosemann (1928), p. 163 Another important insight was the Laguerre formula by Edmond Laguerre (1853), who showed that the Euclidean angle between two lines can be expressed as the logarithm of a cross-ratio.Klein & Rosemann (1928), p. 138 Eventually, Cayley (1859) formulated relations to express distance in terms of a projective metric, and related them to general quadrics or conics serving as the absolute of the geometry.Klein & Rosemann (1928), p. 303Pierpont (1930), p. 67ff Klein (1871, 1873) removed the last remnants of metric concepts from von Staudt's work and combined it with Cayley's theory, in order to base Cayley's new metric on logarithm and the cross-ratio as a number generated by the geometric arrangement of four points.
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Discrete logarithm records are the best results achieved to date in solving the discrete logarithm problem, which is the problem of finding solutions x to the equation gx = h given elements g and h of a finite cyclic group G. The difficulty of this problem is the basis for the security of several cryptographic systems, including Diffie–Hellman key agreement, ElGamal encryption, the ElGamal signature scheme, the Digital Signature Algorithm, and the elliptic curve cryptography analogs of these. Common choices for G used in these algorithms include the multiplicative group of integers modulo p, the multiplicative group of a finite field, and the group of points on an elliptic curve over a finite field.
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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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TWINKLE, on the other hand, works one candidate smooth number (call it X) at a time. There is one LED corresponding to each prime smaller than B. At the time instant corresponding to X, the set of LEDs glowing corresponds to the set of primes that divide X. This can be accomplished by having the LED associated with the prime p glow once every p time instants. Further, the intensity of each LED is proportional to the logarithm of the corresponding prime. Thus, the total intensity equals the sum of the logarithms of all the prime factors of X smaller than B. This intensity is equal to the logarithm of X if and only if X is B-smooth.
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The additive persistence of a number, however, can become arbitrarily large (proof: For a given number n, the persistence of the number consisting of n repetitions of the digit 1 is 1 higher than that of n). The smallest numbers of additive persistence 0, 1, ... are: :0, 10, 19, 199, 19999999999999999999999, ... The next number in the sequence (the smallest number of additive persistence 5) is 2 × 102×(1022 − 1)/9 − 1 (that is, 1 followed by 2222222222222222222222 9's). For any fixed base, the sum of the digits of a number is proportional to its logarithm; therefore, the additive persistence is proportional to the iterated logarithm. More about the additive persistence of a number can be found here.
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The original statement of the law of the iterated logarithm is due to A. Ya. Khinchin (1924).A. Khinchine. "Über einen Satz der Wahrscheinlichkeitsrechnung", Fundamenta Mathematicae 6 (1924): pp. 9–20 (The author's name is shown here in an alternate transliteration.) Another statement was given by A. N. Kolmogorov in 1929.
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Y. Wang: Randomness and Complexity. PhD thesis, 1996. The Java-based software testing tool tests whether a pseudorandom generator outputs sequences that satisfy the LIL. A non-asymptotic version that holds over finite-time martingale sample paths has also been provedA. Balsubramani: "Sharp finite-time iterated- logarithm martingale concentration". arXiv:1405.2639.
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In statistics, the Champernowne distribution is a symmetric, continuous probability distribution, describing random variables that take both positive and negative values. It is a generalization of the logistic distribution that was introduced by D. G. Champernowne. Section 7.3 "Champernowne Distribution." Champernowne developed the distribution to describe the logarithm of income.
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A function is said to grow logarithmically if is (exactly or approximately) proportional to the logarithm of . (Biological descriptions of organism growth, however, use this term for an exponential function., chapter 19, p. 298) For example, any natural number N can be represented in binary form in no more than bits.
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The design of floating-point format allows various optimisations, resulting from the easy generation of a base-2 logarithm approximation from an integer view of the raw bit pattern. Integer arithmetic and bit-shifting can yield an approximation to reciprocal square root (fast inverse square root), commonly required in computer graphics.
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A high impedance voltmeter is used to measure the electromotive force or potential between the two electrodes when zero or no significant current flows between them. The potentiometric response is governed by the Nernst equation in that the potential is proportional to the logarithm of the concentration of the analyte.
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The inequality remains valid for n=\infty provided that a<\infty and b<\infty. The proof above holds for any function g such that f(x)=xg(x) is convex, such as all continuous non-decreasing functions. Generalizations to non-decreasing functions other than the logarithm is given in Csiszár, 2004.
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2 and 16 giving 4) does not depend on the base of the logarithm, just like in the case of log x (geometric mean, 2 and 8 giving 4), but unlike in the case of log log log x (4 and giving 16 if the base is 2, but not otherwise).
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This concept is closely related to half-value layer (HVL): a material with a thickness of one HVL will attenuate 50% of photons. A standard x-ray image is a transmission image, an image with negative logarithm of its intensities is sometimes called a number of mean free paths image.
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A quantum computer is a device that uses quantum mechanisms for computation. In this device the data are stored as qubits (quantum binary digits). That gives a quantum computer in comparison with a conventional computer the opportunity to solve complicated problems in a short time, e.g. discrete logarithm problem or factorization.
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MD5 and SHA-1 in particular both have published techniques more efficient than brute force for finding collisions. However, some hash functions have a proof that finding collisions is at least as difficult as some hard mathematical problem (such as integer factorization or discrete logarithm). Those functions are called provably secure.
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It is an open question whether any "natural" problem has the same property: Schaefer's dichotomy theorem provides conditions under which classes of constrained Boolean satisfiability problems cannot be in NPI. Some problems that are considered good candidates for being NP-intermediate are the graph isomorphism problem, factoring, and computing the discrete logarithm.
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The currently best known discrete logarithm attack is the generic Pollard's rho algorithm, requiring about 2^{122.5} group operations on average. Therefore, it typically belongs to the 128 bit security level. In order to prevent timing attacks, all group operations are done in constant time, i.e. without disclosing information about key material.
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In probability theory and statistics, the log-Laplace distribution is the probability distribution of a random variable whose logarithm has a Laplace distribution. If X has a Laplace distribution with parameters μ and b, then Y = eX has a log-Laplace distribution. The distributional properties can be derived from the Laplace distribution.
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Astronomers denote this value by the decimal logarithm of the gravitational force in cgs units, or log g. For IK Pegasi B, log g is 8.95. By comparison, log g for the Earth is 2.99. Thus the surface gravity on IK Pegasi is over 900,000 times the gravitational force on the Earth.
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Still other systems choose a middle ground, reducing the marginal value of additional points as the margin of victory increases. Sagarin chose to clamp the margin of victory to a predetermined amount. Other approaches include the use of a decay function, such as a logarithm or placement on a cumulative distribution function.
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100 is a Harshad number in base 10, and also in base 4, and in that base it is a self-descriptive number. There are exactly 100 prime numbers whose digits are in strictly ascending order (e.g. 239, 2357 etc.). 100 is the smallest number whose common logarithm is a prime number (i.e.
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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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This is followed by nautical devices such as compasses, chronometers and sextants. Although displaced by modern technology such as GPS, these devices are still mandatory on board. If they fail or do not work correctly, threaten penalties. Furthermore, nautical charts and logarithm tables are needed to determine the exact position of the ship.
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SEC Chromatogram of a biological sample. The elution volume (Ve) decreases roughly linear with the logarithm of the molecular hydrodynamic volume. Columns are often calibrated using 4-5 standard samples (e.g., folded proteins of known molecular weight), and a sample containing a very large molecule such as thyroglobulin to determine the void volume.
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The planimeter was a manual instrument to calculate the area of a closed figure by tracing over it with a mechanical linkage. 260x260px The slide rule was invented around 1620–1630, shortly after the publication of the concept of the logarithm. It is a hand- operated analog computer for doing multiplication and division.
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This paper was communicated and signed by Edward Pickering, but the first sentence indicates that it was "prepared by Miss Leavitt". Leavitt made a graph of magnitude versus logarithm of period and determined that, in her own words, magnitude. The lines drawn connect points corresponding to the stars' minimum and maximum brightness, respectively.
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The magnitude of an earthquakes can be roughly estimated by measuring the area affected by intensity level III or above in km2 and taking the logarithm. A more accurate estimate relies on the development of regional calibration functions derived using many isoseismal radii. Such approaches allow magnitudes to be estimated for historical earthquakes.
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James Thomson Bottomley (10 January 1845 – 18 May 1926) was an Irish-born British physicist. He is noted for his work on thermal radiation and on his creation of 4-figure logarithm tables, used to convert long multiplication and division calculations to simpler addition and subtraction before the introduction of fast calculators.
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The metallicity is given as a logarithm. 100.14 ≈ 1.4 In 2004, a gas giant planet was found in orbit around the star, but it was not until 2009 that this planet was confirmed. In 2017, five more planets were found. All have minimum masses significantly greater than that of the Earth, between and .
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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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The circular points at infinity are the points at infinity of the isotropic lines.C. E. Springer (1964) Geometry and Analysis of Projective Spaces, page 141, W. H. Freeman and Company They are invariant under translations and rotations of the plane. The concept of angle can be defined using the circular points, natural logarithm and cross-ratio:Duncan Sommerville (1914) Elements of Non-Euclidean Geometry, page 157, link from University of Michigan Historical Math Collection :The angle between two lines is a certain multiple of the logarithm of the cross-ratio of the pencil formed by the two lines and the lines joining their intersection to the circular points. Sommerville configures two lines on the origin as u : y = x \tan \theta, \quad u' : y = x \tan \theta ' .
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In 1924, Bell Telephone Laboratories received favorable response to a new unit definition among members of the International Advisory Committee on Long Distance Telephony in Europe and replaced the MSC with the Transmission Unit (TU). 1 TU was defined such that the number of TUs was ten times the base-10 logarithm of the ratio of measured power to a reference power. The definition was conveniently chosen such that 1 TU approximated 1 MSC; specifically, 1 MSC was 1.056 TU. In 1928, the Bell system renamed the TU into the decibel, being one tenth of a newly defined unit for the base-10 logarithm of the power ratio. It was named the bel, in honor of the telecommunications pioneer Alexander Graham Bell.
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Tables containing common logarithms (base-10) were extensively used in computations prior to the advent of electronic calculators and computers because logarithms convert problems of multiplication and division into much easier addition and subtraction problems. Base-10 logarithms have an additional property that is unique and useful: The common logarithm of numbers greater than one that differ only by a factor of a power of ten all have the same fractional part, known as the mantissa. Tables of common logarithms typically included only the mantissas; the integer part of the logarithm, known as the characteristic, could easily be determined by counting digits in the original number. A similar principle allows for the quick calculation of logarithms of positive numbers less than 1.
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Numerous topics relating to probability are named after him, including Feller processes, Feller's explosion test, Feller–Brown movement, and the Lindeberg–Feller theorem. Feller made fundamental contributions to renewal theory, Tauberian theorems, random walks, diffusion processes, and the law of the iterated logarithm. Feller was among those early editors who launched the journal Mathematical Reviews.
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By taking information per pulse N in bit/pulse to be the base-2-logarithm of the number of distinct messages M that could be sent, Hartley constructed a measure of the gross bitrate R as: :R = f_s \log_2(M) where fs is the baud rate in symbols/second or pulses/second. (See Hartley's law).
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The idea of logarithms was also used to construct the slide rule, which became ubiquitous in science and engineering until the 1970s. A breakthrough generating the natural logarithm was the result of a search for an expression of area against a rectangular hyperbola, and required the assimilation of a new function into standard mathematics.
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Among Napier's early followers were the instrument makers Edmund Gunter and John Speidell. The development of logarithms is given credit as the largest single factor in the general adoption of decimal arithmetic. The Trissotetras (1645) of Thomas Urquhart builds on Napier's work, in trigonometry. Henry Briggs (mathematician) was an early adopter of the Napierian logarithm.
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An alternative unit to the decibel used in electrical engineering, the neper, is named after Napier, as is Edinburgh Napier University in Edinburgh, Scotland. The crater Neper on the Moon is named after him.Neper Gazetteer of Planetary Nomenclature – USGS Astrogeology The natural logarithm is named after him in French (Logarithme Népérien) and Portuguese (Logaritmos Neperianos).
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The logarithm of the odds ratio, the difference of the logits of the probabilities, tempers this effect, and also makes the measure symmetric with respect to the ordering of groups. For example, using natural logarithms, an odds ratio of 27/1 maps to 3.296, and an odds ratio of 1/27 maps to −3.296.
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Springer, New York [Ch. 3, sec. 3.5] More precisely, this application is usually for the exact and near-exact distributions of the negative logarithm of such statistics. If necessary, it is then easy, through a simple transformation, to obtain the corresponding exact or near-exact distributions for the corresponding likelihood ratio test statistics themselves.
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The main difference is that where MuHASH applies a random oracle , ECOH applies a padding function. Assuming random oracles, finding a collision in MuHASH implies solving the discrete logarithm problem. MuHASH is thus a provably secure hash, i.e. we know that finding a collision is at least as hard as some hard known mathematical problem.
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CEILIDH is a public key cryptosystem based on the discrete logarithm problem in algebraic torus. This idea was first introduced by Alice Silverberg and Karl Rubin in 2003; Silverberg named CEILIDH after her cat. The main advantage of the system is the reduced size of the keys for the same security over basic schemes.
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Overview of the triple-alpha process. Logarithm of the relative energy output (ε) of proton–proton (PP), CNO and Triple-α fusion processes at different temperatures. The dashed line shows the combined energy generation of the PP and CNO processes within a star. At the Sun's core temperature, the PP process is more efficient.
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The planimeter was a manual instrument to calculate the area of a closed figure by tracing over it with a mechanical linkage. A slide rule. The slide rule was invented around 1620–1630, shortly after the publication of the concept of the logarithm. It is a hand-operated analog computer for doing multiplication and division.
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Product decompositions of matrix functions (which occur in coupled multi-modal systems such as elastic waves) are considerably more problematic since the logarithm is not well defined, and any decomposition might be expected to be non-commutative. A small subclass of commutative decompositions were obtained by Khrapkov, and various approximate methods have also been developed.
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The speed prior complexity of a program is its size in bits plus the logarithm of the maximum time we are willing to run it to get a prediction. When compared to traditional measures, use of the Speed Prior has the disadvantage of leading to less optimal predictions, and the advantage of providing computable predictions.
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Violations to this assumption result in a large reduction in power. Suggested solutions to this violation are: delete a variable, combine levels of one variable (e.g., put males and females together), or collect more data. 3\. The logarithm of the expected value of the response variable is a linear combination of the explanatory variables.
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A thermodynamic free entropy is an entropic thermodynamic potential analogous to the free energy. Also known as a Massieu, Planck, or Massieu–Planck potentials (or functions), or (rarely) free information. In statistical mechanics, free entropies frequently appear as the logarithm of a partition function. The Onsager reciprocal relations in particular, are developed in terms of entropic potentials.
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The third market model assumes that the logarithm of the return, or, log-return, of any risk factor typically follows a normal distribution. Collectively, the log-returns of the risk factors are multivariate normal. Monte Carlo algorithm simulation generates random market scenarios drawn from that multivariate normal distribution. For each scenario, the profit (loss) of the portfolio is computed.
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This gives: : b=\sqrt[y] It is less easy to make the subject of the expression. Logarithms allow us to do this: : y= This expression means that is equal to the power that you would raise to, to get . This operation undoes exponentiation because the logarithm of tells you the exponent that the base has been raised to.
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In order to provide the required high dynamic range with imperceptible luminance steps, LogLuv uses 16 bits to encode a fixed-point base 2 logarithm of the luminance, which allows an EV range of nearly 128 stops. The space occupied by one pixel is thus 32 bits (L16 + U8 + V8), marginally bigger than a standard 8 bit RGB-image.
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In this section it is explained how the concepts above using traces of elements can be applied to cryptography. In general, XTR can be used in any cryptosystem that relies on the (subgroup) Discrete Logarithm problem. Two important applications of XTR are the Diffie–Hellman key agreement and the ElGamal encryption. We will start first with Diffie–Hellman.
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John Todd proved that this sequence is neither finite nor cofinite. More precisely, the natural density of the Størmer numbers lies between 0.5324 and 0.905. It has been conjectured that their natural density is the natural logarithm of 2, approximately 0.693, but this remains unproven. Because the Størmer numbers have positive density, the Størmer numbers form a large set.
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Faulhaber made the first publication of Henry Briggs's Logarithm in Germany. He's also credited with the first printed solution of equal temperament.Date,name,ratio,cents: from equal temperament monochord tables p55-p78; J. Murray Barbour Tuning and Temperament, Michigan State University Press 1951 He died in Ulm. Faulhaber's major contribution was in calculating the sums of powers of integers.
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This can be an efficient implementation and can give essentially any filter response including excellent approximations to brickwall filters. There are some commonly used frequency domain transformations. For example, the cepstrum converts a signal to the frequency domain through Fourier transform, takes the logarithm, then applies another Fourier transform. This emphasizes the harmonic structure of the original spectrum.
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The planet has a minimal mass equivalent of 16 Earths. The star has the same stellar classification as the Sun: G2V. It has a similar temperature, at compared with for the Sun. It has a lower logarithm of metallicity ratio, at −0.06 compared with 0.00, and a slightly younger age, at 4.5 versus 4.6 billion years.
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As in the Scheiner system, speeds were expressed in 'degrees'. Originally the sensitivity was written as a fraction with 'tenths' (for example "18/10° DIN"), where the resultant value 1.8 represented the relative base 10 logarithm of the speed. 'Tenths' were later abandoned with DIN 4512:1957-11, and the example above would be written as "18° DIN".
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A set of networks that satisfies given structural characteristics can be treated as a network ensemble. Brought up by Ginestra Bianconi in 2007, the entropy of a network ensemble measures the level of the order or uncertainty of a network ensemble. The entropy is the logarithm of the number of graphs. Entropy can also be defined in one network.
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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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The min-entropy, in information theory, is the smallest of the Rényi family of entropies, corresponding to the most conservative way of measuring the unpredictability of a set of outcomes, as the negative logarithm of the probability of the most likely outcome. The various Rényi entropies are all equal for a uniform distribution, but measure the unpredictability of a nonuniform distribution in different ways. The min-entropy is never greater than the ordinary or Shannon entropy (which measures the average unpredictability of the outcomes) and that in turn is never greater than the Hartley or max-entropy, defined as the logarithm of the number of outcomes with nonzero probability. As with the classical Shannon entropy and its quantum generalization, the von Neumann entropy, one can define a conditional version of min-entropy.
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Logarithm of the relative energy output (ε) of proton–proton (PP), CNO and Triple-α fusion processes at different temperatures. The dashed line shows the combined energy generation of the PP and CNO processes within a star. At the Sun's core temperature, the PP process is more efficient. All main-sequence stars have a core region where energy is generated by nuclear fusion.
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Wittmann (1985) generalized Hartman–Wintner version of LIL to random walks satisfying milder conditions. Vovk (1987) derived a version of LIL valid for a single chaotic sequence (Kolmogorov random sequence). This is notable, as it is outside the realm of classical probability theory. Yongge Wang has shown that the law of the iterated logarithm holds for polynomial time pseudorandom sequences also.
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The first equation determines the magnitude of the deviatoric stress \ q needed to keep the soil flowing continuously as the product of a frictional constant \ M (capital \ \mu) and the mean effective stress \ p'. The second equation states that the specific volume \ u occupied by unit volume of flowing particles will decrease as the logarithm of the mean effective stress increases.
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The interpreter supported a full set of scientific functions (trigonometric functions, logarithm etc.) at this accuracy. The language supported two-dimensional arrays, and a ROM extension made high-level functions such as matrix multiplication and inversion available. For the larger HP-86 and HP-87 series, HP also offered a plug-in CP/M processor card with a separate Zilog Z-80 processor.
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It became very popular and was republished 14 times (last edition in 1938). In 1919–1921, he published other textbooks on plane trigonometry, history of math, algebra, logarithm. He also wrote a textbook on physics (1922) and two textbooks on learning to write (1907 and 1921). The textbook on plane trigonometry was reworked and republished by his son in 1938.
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The logarithm transformation may help to overcome cases where the Kolmogorov test data does not seem to fit the assumption that it came from the normal distribution. Using estimated parameters, the questions arises which estimation method should be used. Usually this would be the maximum likelihood method, but e.g. for the normal distribution MLE has a large bias error on sigma.
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There is one type of isometry in one dimension that may leave the probability distribution unchanged, that is reflection in a point, for example zero. A possible symmetry for randomness with positive outcomes is that the former applies for the logarithm, i.e., the outcome and its reciprocal have the same distribution. However this symmetry does not single out any particular distribution uniquely.
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There is some uncertainty in the scientific press concerning the higher ratio of elements heavier than hydrogen compared to those found in the Sun; what astronomers term the metallicity. Santos et al. (2004) report the logarithm of the abundance ratio of iron to hydrogen, [Fe/H], to be 0.12 dex, whereas Cenarro et al. (2007) published a value of –0.15 dex.
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This process of increasing and decreasing inhibition can be described as a random oscillator. The IQ concept can now be defined as the natural logarithm of the ratio a0 / a1. This is the first time in the history of intelligence testing that the IQ concept has been formally defined. The test measures two factors of intelligence: mental speed and distractibility.
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Consequently, it is always possible to take a logarithm of probabilities. Because template comparisons with itself lower ApEn values, the signals are interpreted to be more regular than they actually are. These self-matches are not included in SampEn. However, since SampEn makes direct use of the correlation integrals, it is not a real measure of information but an approximation.
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In cryptography, a Schnorr signature is a digital signature produced by the Schnorr signature algorithm that was described by Claus Schnorr. It is a digital signature scheme known for its simplicity, among the first whose security is based on the intractability of certain discrete logarithm problems. It is efficient and generates short signatures. It was covered by which expired in February 2008.
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The natural logarithm f(x) = \ln x scales additively and so is not homogeneous. This can be demonstrated with the following examples: f(5x) = \ln 5x = \ln 5 + f(x), f(10x) = \ln 10 + f(x), and f(15x) = \ln 15 + f(x). This is because there is no k such that f(\alpha \cdot x) = \alpha^k \cdot f(x).
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Modular exponentiation similar to the one described above is considered easy to compute, even when the integers involved are enormous. On the other hand, computing the modular discrete logarithm – that is, the task of finding the exponent when given , , and – is believed to be difficult. This one-way function behavior makes modular exponentiation a candidate for use in cryptographic algorithms. .
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In 1998 Gerhard Frey firstly proposed using trace zero varieties for cryptographic purpose. These varieties are subgroups of the divisor class group on a low genus hyperelliptic curve defined over a finite field. These groups can be used to establish asymmetric cryptography using the discrete logarithm problem as cryptographic primitive. Trace zero varieties feature a better scalar multiplication performance than elliptic curves.
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Tsien claimed that the security-stamped documents were mostly written by himself and had outdated classifications, adding that, "There were some drawings and logarithm tables, etc., which someone might have mistaken for codes."Chang (1995), p. 157. Included in the material was a scrapbook with news clippings about the trials of those charged with atomic espionage, such as Klaus Fuchs.Chang (1995), p. 160.
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Thus the set of rational numbers q for which is dense in the rational numbers, as is the set of q for which . This means that the function (−1)q is not continuous at any rational number q where it is defined. On the other hand, arbitrary complex powers of negative numbers b can be defined by choosing a complex logarithm of b.
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The Sinclair Scientific Programmable was introduced in 1975, with the same case as the Sinclair Oxford. It was larger than the Scientific, at , and used a larger PP3 battery, but could also be powered by mains electricity. It had 24-step programming abilities, which meant it was highly limited for many purposes. It also lacked functions for the natural logarithm and exponential function.
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Alternatively, if the exponential function, denoted or , has been defined first, say by using an infinite series, then the natural logarithm may be defined as its inverse function. In other words, is that function such that . Since the range of the exponential function is all positive real numbers, and since the exponential function is strictly increasing, this is well-defined for all positive .
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Henry Briggs' 1617 Logarithmorum Chilias Prima showing the base-10 (common) logarithm of the integers 0 to 67 to fourteen decimal places. Part of a 20th-century table of common logarithms in the reference book Abramowitz and Stegun. A page from a table of logarithms of trigonometric functions from the 2002 American Practical Navigator. Columns of differences are included to aid interpolation.
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Because the costs are additive, they behave like the logarithm of the probability (since log-likelihoods are additive), or equivalently, somewhat like the entropy (since entropies are additive). This makes Link Grammar compatible with machine learning techniques such as hidden Markov models and the Viterbi algorithm, because the link costs correspond to the link weights in Markov networks or Bayesian networks.
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Logarithm of the relative energy output (ε) of proton–proton (PP), CNO and Triple-α fusion processes at different temperatures. The dashed line shows the combined energy generation of the PP and CNO processes within a star. At the Sun's core temperature, the PP process is more efficient. Stellar nucleosynthesis is the creation (nucleosynthesis) of chemical elements by nuclear fusion reactions within stars.
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He also calculated a 7-digit logarithm table and extended a table of integer factorizations from 6,000,000 to 9,000,000. Dase had very little knowledge of mathematical theory. The mathematician Julius Petersen tried to teach him some of Euclid's theorems, but gave up the task once he realized that their comprehension was beyond Dase's capabilities. Preston, Richard, 2008, Panic in Level 4, p. 32.
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The logarithm of fitness as a function of the number of deleterious mutations. Synergistic epistasis is represented by the red line - each subsequent deleterious mutation has a larger proportionate effect on the organism's fitness. Antagonistic epistasis is in blue. The black line shows the non-epistatic case, where fitness is the product of the contributions from each of its loci.
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Aqua ions are subject to hydrolysis. The logarithm of the first hydrolysis constant is proportional to z2/r for most aqua ions. The aqua ion is associated, through hydrogen bonding with other water molecules in a secondary solvation shell. Water molecules in the first hydration shell exchange with molecules in the second solvation shell and molecules in the bulk liquid.
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In probability theory, a log-Cauchy distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Cauchy distribution. If X is a random variable with a Cauchy distribution, then Y = exp(X) has a log-Cauchy distribution; likewise, if Y has a log-Cauchy distribution, then X = log(Y) has a Cauchy distribution.
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The traditional negative binomial regression model, commonly known as NB2, is based on the Poisson-gamma mixture distribution. This model is popular because it models the Poisson heterogeneity with a gamma distribution. Poisson regression models are generalized linear models with the logarithm as the (canonical) link function, and the Poisson distribution function as the assumed probability distribution of the response.
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Beer's law stated that the transmittance of a solution remains constant if the product of concentration and path length stays constant. The modern derivation of the Beer–Lambert law combines the two laws and correlates the absorbance, which is the negative decadic logarithm of the transmittance, to both the concentrations of the attenuating species and the thickness of the material sample.
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Since the attenuation is defined as proportional to the logarithm of the ratio between P(x) and P(y), where P is the power at point x and y respectively. Using the cutback technique, the power transmitted through a fiber of known length is measured and compared with the same measurement for the same fiber cut to a length of 2m approximately.
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"Tabula logarithmorum vulgarium", 1797 Vega published a series of books of logarithm tables. The first one appeared in 1783. Much later, in 1797 it was followed by a second volume that contained a collection of integrals and other useful formulae. His Handbook, which was originally published in 1793, was later translated into several languages and appeared in over 100 issues.
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The number field sieve algorithm, which is generally the most effective in solving the discrete logarithm problem, consists of four computational steps. The first three steps only depend on the order of the group G, not on the specific number whose finite log is desired.Whitfield Diffie, Paul C. Van Oorschot, and Michael J. Wiener "Authentication and Authenticated Key Exchanges", in Designs, Codes and Cryptography, 2, 107–125 (1992), Section 5.2, available as Appendix B to It turns out that much Internet traffic uses one of a handful of groups that are of order 1024 bits or less. By precomputing the first three steps of the number field sieve for the most common groups, an attacker need only carry out the last step, which is much less computationally expensive than the first three steps, to obtain a specific logarithm.
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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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A major mathematical difficulty in symbolic integration is that in many cases, a closed formula for the antiderivative of a rather simple-looking function does not exist. For instance, it is known that the antiderivatives of the functions and cannot be expressed in the closed form involving only rational and exponential functions, logarithm, trigonometric functions and inverse trigonometric functions, and the operations of multiplication and composition; in other words, none of the three given functions is integrable in elementary functions, which are the functions which may be built from rational functions, roots of a polynomial, logarithm, and exponential functions. The Risch algorithm provides a general criterion to determine whether the antiderivative of an elementary function is elementary, and, if it is, to compute it. Unfortunately, it turns out that functions with closed expressions of antiderivatives are the exception rather than the rule.
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The law of the iterated logarithm (LIL) for a sum of independent and identically distributed (i.i.d.) random variables with zero mean and bounded increment dates back to Khinchin and Kolmogorov in the 1920s. Since then, there has been a tremendous amount of work on the LIL for various kinds of dependent structures and for stochastic processes. Following is a small sample of notable developments.
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516 on the other hand a new method for studying partial waves using the Laplace transform.A.Martin, « Analyticity of partial waves obtained from the Schrödinger equation », Nuovo cimento, 14, (1959), p. 516 After the proof, due to Froissart, that the total effective cross section cannot grow faster than the logarithm squared of the energy, using the Mandelstam representation,M.Froissart, « Asymptotic Behavior and Subtractions in the Mandelstam Representation », Phys.
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Some of these methods used tables derived from trigonometric identities.Enrique Gonzales-Velasco (2011) Journey through Mathematics – Creative Episodes in its History, §2.4 Hyperbolic logarithms, p. 117, Springer Such methods are called prosthaphaeresis. Invention of the function now known as the natural logarithm began as an attempt to perform a quadrature of a rectangular hyperbola by Grégoire de Saint-Vincent, a Belgian Jesuit residing in Prague.
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The brighter star has an effective temperature of , a logarithm of surface gravity of 7.75, and a mass 0.6 times the Sun. Its radius is 0.0156 that of the Sun. The dimmer star is cooler, with a temperature of under , and has a mass 0.21 that of the Sun. It is physically larger than the brighter star at 0.0314 the radius of the Sun.
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A page from Henry Briggs' 1617 Logarithmorum Chilias Prima showing the base-10 (common) logarithm of the integers 0 to 67 to fourteen decimal places. Part of a 20th-century table of common logarithms in the reference book Abramowitz and Stegun. A page from a table of logarithms of trigonometric functions from the 2002 American Practical Navigator. Columns of differences are included to aid interpolation.
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In GWAS Manhattan plots, genomic coordinates are displayed along the X-axis, with the negative logarithm of the association p-value for each single nucleotide polymorphism (SNP) displayed on the Y-axis, meaning that each dot on the Manhattan plot signifies a SNP. Because the strongest associations have the smallest p-values (e.g., 10−15), their negative logarithms will be the greatest (e.g., 15).
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15–18, (1996) Jaynes' analysis of handing the infinities of the Lamb shift calculation. This model leads to the same type of Bethe logarithm (an essential part of the Lamb shift calculation), vindicating Jaynes' claim that two different physical models can be mathematically isomorphic to each other and therefore yield the same results, a point also apparently made by Scott and Moore on the issue of causality.
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The study of functions with arguments from a Clifford algebra is called Clifford analysis. A matrix may be considered a hypercomplex number. For example, study of functions of 2 × 2 real matrices shows that the topology of the space of hypercomplex numbers determines the function theory. Functions such as square root of a matrix, matrix exponential, and logarithm of a matrix are basic examples of hypercomplex analysis.
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Generally, the pzc in electrochemistry is the value of the negative decimal logarithm of the activity of the potential-determining ion in the bulk fluid.IUPAC Gold Book The pzc is of fundamental importance in surface science. For example, in the field of environmental science, it determines how easily a substrate is able to adsorb potentially harmful ions. It also has countless applications in technology of colloids, e.g.
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The value of a partial function is undefined when its argument is out of its domain of definition. This include numerous arithmetical cases such as division by zero, square root or logarithm of a negative number etc.; see NaN. Even some mathematically well-defined expressions like exp(100000) may be undefined in floating point arithmetic because the result is so large that it cannot be represented.
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For instance, the first prime gap of size larger than 14 occurs between the primes 523 and 541, while 15! is the vastly larger number 1307674368000. The average gap between primes increases as the natural logarithm of the integer, and therefore the ratio of the prime gap to the integers involved decreases (and is asymptotically zero). This is a consequence of the prime number theorem.
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One of the simplest and frequently used proofs of knowledge, the proof of knowledge of a discrete logarithm, is due to Schnorr.C P Schnorr, Efficient identification and signatures for smart cards, in G Brassard, ed. Advances in Cryptology – Crypto '89, 239–252, Springer-Verlag, 1990. Lecture Notes in Computer Science, nr 435 The protocol is defined for a cyclic group G_q of order q with generator g.
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Torus-based cryptography involves using algebraic tori to construct a group for use in ciphers based on the discrete logarithm problem. This idea was first introduced by Alice Silverberg and Karl Rubin in 2003 in the form of a public key algorithm by the name of CEILIDH. It improves on conventional cryptosystems by representing some elements of large finite fields compactly and therefore transmitting fewer bits.
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Its expression is :(1 + 1) × (1 + 1) × (1 + 1 + 1) × (1 + 1 + 1). Thus, the complexity of an integer is at least . The complexity of is at most (approximately ): an expression of this length for can be found by applying Horner's method to the binary representation of .. Almost all integers have a representation whose length is bounded by a logarithm with a smaller constant factor, ..
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The problem of finding collisions in ECOH is similar to the subset sum problem. Solving a subset sum problem is almost as hard as the discrete logarithm problem. It is generally assumed that this is not doable in polynomial time. However a significantly loose heuristic must be assumed, more specifically, one of the involved parameters in the computation is not necessarily random but has a particular structure.
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By comparison, stars like the Sun enter the main sequence after 30 million years. The average proportion of elements with higher atomic numbers than helium is termed the metallicity by astronomers. This is expressed by the logarithm of the ratio of iron to hydrogen, compared to the same proportion in the Sun. For M34, the metallicity has a value of [Fe/H] = +0.07 ± 0.04.
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This quantity is normally listed as the base–10 logarithm of the proportion relative to the Sun; for NGC 6809 the metallicity is given by: [Fe/H] = –1.94 dex. Taking this exponent to the powers of 10 yields an abundance equal to 1.1% of the proportion of such elements in the Sun. Only about 55 variable stars have been discovered in the central part of M55.
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For example, the notation "[O/Fe]" represents the difference in the logarithm of the star's oxygen abundance versus its iron content compared to that of the Sun. In general, a given stellar nucleosynthetic process alters the proportions of only a few elements or isotopes, so a star or gas sample with nonzero [X/Fe] values may be showing the signature of particular nuclear processes.
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The first volume of Introduction to the Analysis of the Infinite had no diagrams, allowing teachers and students to draw their own illustrations. There is a gap in Euler's text where Lorentz transformations arise. A feature of natural logarithm is its interpretation as area in hyperbolic sectors. In relativity the classical concept of velocity is replaced with rapidity, a hyperbolic angle concept built on hyperbolic sectors.
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Predominance diagram for chromate In aqueous solution, chromate and dichromate anions exist in a chemical equilibrium. :2 \+ 2 H+ \+ H2O The predominance diagram shows that the position of the equilibrium depends on both pH and the analytical concentration of chromium.pCr is equal to minus the decimal logarithm of the analytical concentration of chromium. Thus, when pCr = 2, the chromium concentration is 10−2 mol/L.
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His mathematical work included works on the determinant, hyperbolic functions, and parabolic logarithms and trigonometry.This is about connecting the rectified length of line segments along a parabola, giving logarithms for appropriate coordinates, and trigonometric values for suitable angles, in a similar way as the area under a hyperbola defines the natural logarithm, and a hyperbolic angle is defined via the area of a hyperbolically truncated triangle.
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Shor's algorithm solves the discrete logarithm problem and the integer factorization problem in polynomial time, whereas the best known classical algorithms take super-polynomial time. These problems are not known to be in P or NP-complete. It is also one of the few quantum algorithms that solves a non-black-box problem in polynomial time where the best known classical algorithms run in super-polynomial time.
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A Gauss sum is a type of exponential sum. The best known classical algorithm for estimating these sums takes exponential time. Since the discrete logarithm problem reduces to Gauss sum estimation, an efficient classical algorithm for estimating Gauss sums would imply an efficient classical algorithm for computing discrete logarithms, which is considered unlikely. However, quantum computers can estimate Gauss sums to polynomial precision in polynomial time.
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Several modern authors still consider non-Euclidean geometry and hyperbolic geometry synonyms. Arthur Cayley noted that distance between points inside a conic could be defined in terms of logarithm and the projective cross-ratio function. The method has become called the Cayley–Klein metric because Felix Klein exploited it to describe the non-Euclidean geometries in articlesF. Klein, Über die sogenannte nichteuklidische Geometrie, Mathematische Annalen, 4(1871).
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Dixon proposed a widely used way of plotting enzyme inhibition data, commonly known as the Dixon plot, in which reciprocal rate is plotted against the inhibitor concentration. Rather confusingly, the same name is sometimes given to a quite different plot proposed by Dixon in the same year for analysing pH dependences, in which the logarithm of a Michaelis–Menten parameter is plotted against pH.
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His work and that of Black–Scholes changed the nature of the finance literature. Influential mathematical textbook treatments were by Fleming and Rishel, and by Fleming and Soner. These techniques were applied by Stein to the financial crisis of 2007–08. The maximization, say of the expected logarithm of net worth at a terminal date T, is subject to stochastic processes on the components of wealth.
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This then determines the rest of the B_k. In some cases the constant must be zero. For example, consider the following differential equation (Kummer's equation with and ): :zu+(2-z)u'-u=0 The roots of the indicial equation are −1 and 0. Two independent solutions are 1/z and (e^z)/z, so we see that the logarithm does not appear in any solution.
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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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Thus, the matrix exponential of a Hamiltonian matrix is symplectic. However the logarithm of a symplectic matrix is not necessarily Hamiltonian because the exponential map from the Lie algebra to the group is not surjective.. The characteristic polynomial of a real Hamiltonian matrix is even. Thus, if a Hamiltonian matrix has as an eigenvalue, then , and are also eigenvalues. It follows that the trace of a Hamiltonian matrix is zero.
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The sum of three points P, Q, and R on an elliptic curve E (red) is zero if there is a line (blue) passing through these points. A widely applied cryptographic routine uses the fact that discrete exponentiation, i.e., computing : ( factors, for an integer ) in a (large) finite field can be performed much more efficiently than the discrete logarithm, which is the inverse operation, i.e., determining the solution to an equation :.
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Together with the linearity of the integral, this formula allows one to compute the integrals of all polynomials. The term "quadrature" is a traditional term for area; the integral is geometrically interpreted as the area under the curve y = xn. Traditionally important cases are y = x2, the quadrature of the parabola, known in antiquity, and y = 1/x, the quadrature of the hyperbola, whose value is a logarithm.
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This is about as far as we can go using thermodynamics alone. Note that the above equation is flawed – as the temperature approaches zero, the entropy approaches negative infinity, in contradiction to the third law of thermodynamics. In the above "ideal" development, there is a critical point, not at absolute zero, at which the argument of the logarithm becomes unity, and the entropy becomes zero. This is unphysical.
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Black = no data. Soil pH is a measure of the acidity or basicity (alkalinity) of a soil. pH is defined as the negative logarithm (base 10) of the activity of hydronium ions ( or, more precisely, ) in a solution. In soils, it is measured in a slurry of soil mixed with water (or a salt solution, such as 0.01 M ), and normally falls between 3 and 10, with 7 being neutral.
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An important property of base-10 logarithms, which makes them so useful in calculations, is that the logarithm of numbers greater than 1 that differ by a factor of a power of 10 all have the same fractional part. The fractional part is known as the mantissa.This use of the word mantissa stems from an older, non-numerical, meaning: a minor addition or supplement, e.g., to a text.
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If problem A is hard, there exists a formal security reduction from a problem which is widely considered unsolvable in polynomial time, such as integer factorization problem or discrete logarithm problem. However, non-existence of a polynomial time algorithm does not automatically ensure that the system is secure. The difficulty of a problem also depends on its size. For example, RSA public key cryptography relies on the difficulty of integer factorization.
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A closely related notion is the probability that a universal computer outputs some string x when fed with a program chosen at random. This Algorithmic "Solomonoff" Probability (AP) is key in addressing the old philosophical problem of induction in a formal way. The major drawback of AC and AP are their incomputability. Time-bounded "Levin" complexity penalizes a slow program by adding the logarithm of its running time to its length.
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This means that, if the Jacobian has n elements, that the running time is exponential in \log(n). This makes it possible to use Jacobians of a fairly small order, thus making the system more efficient. But if the hyperelliptic curve is chosen poorly, the DLP will become quite easy to solve. In this case there are known attacks which are more efficient than generic discrete logarithm solvers or even subexponential.
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In an adjacency list in which the neighbors of each vertex are unsorted, testing for the existence of an edge may be performed in time proportional to the minimum degree of the two given vertices, by using a sequential search through the neighbors of this vertex. If the neighbors are represented as a sorted array, binary search may be used instead, taking time proportional to the logarithm of the degree.
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A polylogarithmic function in n is a polynomial in the logarithm of n, : a_k (\log n)^k + \cdots + a_1(\log n) + a_0. The notation \log^k n is often used as a shorthand for (\log n)^k, analogous to \sin^2 \theta for (\sin \theta)^2. In computer science, polylogarithmic functions occur as the order of time or memory used by some algorithms (e.g., "it has polylogarithmic order").
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In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory).
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Science advances, 5(5): eaau6253. . 50px Material was copied from this source, which is available under a Creative Commons Attribution 4.0 International License. (A) The natural logarithm of the annual mean of monthly phytoplankton richness is shown as a function of sea temperature (k, Boltzmann’s constant; T, temperature in kelvin). Filled and open circles indicate areas where the model results cover 12 or less than 12 months, respectively.
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In the complete form: : F = Se^{(r+y-q-u)T} Where: : F, S represent the cost of the good on the futures market and the spot market, respectively. : e is the mathematical constant for the base of the natural logarithm. : r is the applicable interest rate (for arbitrage, the cost of borrowing), stated at the continuous compounding rate. : y is the storage cost over the life of the contract.
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His mathematical work concerned in particular the calculations of the lengths of the parabola and cycloid, and the quadrature of the hyperbola,W. Brouncker (1667) The Squaring of the Hyperbola, Philosophical Transactions of the Royal Society of London, abridged edition 1809, v. i, pp 233–6, link form Biodiversity Heritage Library which requires approximation of the natural logarithm function by infinite series.Julian Coolidge Mathematics of Great Amateurs, chapter 11, pp.
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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 initial mass function is typically graphed on a logarithm scale of log(N) vs log(m). Such plots give approximately straight lines with a slope Γ equal to 1-α. Hence Γ is often called the slope of the initial mass function. The present-day mass function, for coeval formation, has the same slope except that it rolls off at higher masses which have evolved away from the main sequence.
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In 1728, Gabriel Cramer had produced fundamentally the same theory in a private letter.Cramer, Garbriel; letter of 21 May 1728 to Nicolaus Bernoulli (excerpted in PDF ). Each had sought to resolve the St. Petersburg paradox, and had concluded that the marginal desirability of money decreased as it was accumulated, more specifically such that the desirability of a sum were the natural logarithm (Bernoulli) or square root (Cramer) thereof.
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The fact that finding a square root of a number modulo a large composite n is equivalent to factoring (which is widely believed to be a hard problem) has been used for constructing cryptographic schemes such as the Rabin cryptosystem and the oblivious transfer. The quadratic residuosity problem is the basis for the Goldwasser-Micali cryptosystem. The discrete logarithm is a similar problem that is also used in cryptography.
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400x400px A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model.
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Plot from Leavitt's 1912 paper. The horizontal axis is the logarithm of the period of the corresponding Cepheid, and the vertical axis is its apparent magnitude. The lines drawn correspond to the stars' minimum and maximum brightness, respectively. Leavitt, a graduate of Radcliffe College, worked at the Harvard College Observatory as a "computer", tasked with examining photographic plates in order to measure and catalog the brightness of stars.
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The generation number can be calculated as the logarithm to base 2 of the ahnentafel number, and rounding down to a full integer by truncating decimal digits. For example, the number 38 is between 25=32 and 26=64, so log2(38) is between 5 and 6. This means that ancestor no.38 belongs to generation five, and was a great-great-great- grandparent of the reference person who is no.
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However, other cryptographic algorithms do not appear to be broken by those algorithms.See also pqcrypto.org, a bibliography maintained by Daniel J. Bernstein and Tanja Lange on cryptography not known to be broken by quantum computing. Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the McEliece cryptosystem based on a problem in coding theory.
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In October 1938 Hitler specially requested that Rommel be seconded to command the Führerbegleitbatallion (his escort battalion). This unit accompanied Hitler whenever he travelled outside of Germany. During this period Rommel indulged his interest in engineering and mechanics by learning about the inner workings and maintenance of internal combustion engines and heavy machine guns. He memorized logarithm tables in his spare time and enjoyed skiing and other outdoor sports.
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Orthovanadate V is used in protein crystallography to study the biochemistry of phosphate. The tetrathiovanadate [VS4]3− is analogous to the orthovanadate ion. At lower pH values, the monomer [HVO4]2− and dimer [V2O7]4− are formed, with the monomer predominant at vanadium concentration of less than c. 10−2M (pV > 2, where pV is equal to the minus value of the logarithm of the total vanadium concentration/M).
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It has been found that the F number linearly correlates with the log k' value (logarithm of the retention factor) in aqueous reversed- phase liquid chromatography. This relationship can be used to understand the significance of different aspects of molecular architecture on their separation using different stationary phases. This size analysis is complementary to the length-to-breadth (L/B) ratio, which classifies molecules according to their "rodlike" or "squarelike" shape.
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The scaling factor may be proportional to , for any ; it may also be multiplied by a slowly varying function of . The law of the iterated logarithm specifies what is happening "in between" the law of large numbers and the central limit theorem. Specifically it says that the normalizing function , intermediate in size between of the law of large numbers and of the central limit theorem, provides a non-trivial limiting behavior.
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According to Henk Bos, :The Introduction is meant as a survey of concepts and methods in analysis and analytic geometry preliminary to the study of the differential and integral calculus. [Euler] made of this survey a masterly exercise in introducing as much as possible of analysis without using differentiation or integration. In particular, he introduced the elementary transcendental functions, the logarithm, the exponential function, the trigonometric functions and their inverses without recourse to integral calculus — which was no mean feat, as the logarithm was traditionally linked to the quadrature of the hyperbola and the trigonometric functions to the arc-length of the circle.H. J. M. Bos (1980) "Newton, Leibnitz and the Leibnizian tradition", chapter 2, pages 49–93, quote page 76, in From the Calculus to Set Theory, 1630 – 1910: An Introductory History, edited by Ivor Grattan-Guinness, Duckworth Euler accomplished this feat by introducing exponentiation ax for arbitrary constant a in the positive real numbers.
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Three subfields of the complex numbers C have been suggested as encoding the notion of a "closed-form number"; in increasing order of generality, these are the Liouville numbers (not to be confused with Liouville numbers in the sense of rational approximation), EL numbers and elementary numbers. The Liouville numbers, denoted L, form the smallest algebraically closed subfield of C closed under exponentiation and logarithm (formally, intersection of all such subfields)—that is, numbers which involve explicit exponentiation and logarithms, but allow explicit and implicit polynomials (roots of polynomials); this is defined in . L was originally referred to as elementary numbers, but this term is now used more broadly to refer to numbers defined explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms. A narrower definition proposed in , denoted E, and referred to as EL numbers, is the smallest subfield of C closed under exponentiation and logarithm—this need not be algebraically closed, and correspond to explicit algebraic, exponential, and logarithmic operations.
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The scale of everything. It begins with spacetime (quantum foam) and moves through small elementary particles, intermediate elementary particles, large elementary particles, components of composite particles, the components of atoms, electromagnetic waves, simple atoms, complex atoms, molecules, small viruses, large viruses, chromosomes, cells, hairs, body parts, species, groups of species, small areas such as craters, large areas such as land masses, planets, orbits, stars, small planetary systems, intermediate planetary systems, large planetary systems, collections of stars, star clusters, galaxies, galaxy groups, galaxy clusters, galaxy superclusters, the cosmic web, Hubble volumes, and ends with the Universe. An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually ten, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic distributions are common in nature and considering the order of magnitude of values sampled from such a distribution can be more intuitive.
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In his twentieth year (1826) Graves engaged in researches on the exponential function and the complex logarithm; they were printed in the Philosophical Transactions for 1829 under the title An Attempt to Rectify the Inaccuracy of some Logarithmic Formulæ. M. Vincent of Lille claimed to have arrived in 1825 at similar results, which, however, were not published by him till 1832. The conclusions announced by Graves were not at first accepted by George Peacock, who referred to them in his Report on Algebra, nor by Sir John Herschel. Graves communicated to the British Association in 1834 (Report for that year) on his discovery, and in the same report is a supporting paper by Hamilton, On Conjugate Functions or Algebraic Couples, as tending to illustrate generally the Doctrine of Imaginary Quantities, and as confirming the Results of Mr. Graves respecting the existence of Two independent Integers in the complete expression of an Imaginary Logarithm.
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The decadic (base-10) logarithm of the reciprocal of the transmittance is called the absorbance or density. DMax and DMin refer to the maximum and minimum density that can be produced by the material. The difference between the two is the density range. The density range is related to the exposure range (dynamic range), which is the range of light intensity that is represented by the recording, via the Hurter–Driffield curve.
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Double-precision floating-point implementations of the gamma function and its logarithm are now available in most scientific computing software and special functions libraries, for example TK Solver, Matlab, GNU Octave, and the GNU Scientific Library. The gamma function was also added to the C standard library (math.h). Arbitrary-precision implementations are available in most computer algebra systems, such as Mathematica and Maple. PARI/GP, MPFR and MPFUN contain free arbitrary-precision implementations.
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The hidden subgroup problem (HSP) is a topic of research in mathematics and theoretical computer science. The framework captures problems like factoring, discrete logarithm, graph isomorphism, and the shortest vector problem. This makes it especially important in the theory of quantum computing because Shor's quantum algorithm for factoring is essentially equivalent to the hidden subgroup problem for finite Abelian groups, while the other problems correspond to finite groups that are not Abelian.
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There are molecular weight size markers available that contain a mixture of molecules of known sizes. If such a marker was run on one lane in the gel parallel to the unknown samples, the bands observed can be compared to those of the unknown to determine their size. The distance a band travels is approximately inversely proportional to the logarithm of the size of the molecule. There are limits to electrophoretic techniques.
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The Mincer earnings function is a single-equation model that explains wage income as a function of schooling and experience, named after Jacob Mincer. The equation has been examined on many datasets and Thomas Lemieux argues it is "one of the most widely used models in empirical economics". Typically the logarithm of earnings is modelled as the sum of years of education and a quadratic function of "years of potential experience".Lemieux, Thomas.
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In 1975, Richard E. Ladner showed that if P ≠ NP then there exist problems in NP that are neither in P nor NP- complete. Such problems are called NP-intermediate problems. The graph isomorphism problem, the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP- complete.
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409–26 Similarly, the merge sort algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time approximately proportional to . The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor is usually disregarded in the analysis of algorithms under the standard uniform cost model.
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In the US, the base of the natural logarithm is often denoted e (italicized), while it is usually denoted e (roman type) in the UK and Continental Europe. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called the elementary positive charge. From the 2019 redefinition of SI base units, which took effect on 20 May 2019, its value is exactly , by definition of the coulomb.
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Conversely, if the pH is above the pzc value, the surface charge would be negative so that the cations can be adsorbed. For example, the charge on the surface of silver iodide crystals may be determined by the concentration of iodide ions in the solution above the crystals. Then, the pzc value of the AgI surface will be described by the concentration of I− in the solution (or negative decimal logarithm of this concentration, pI−).
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One method of calculating outcome factorisation is as follow:Sun Ging-Yun, Empirical methodology in the tautological study of qualitative and quantitative social interaction data, Albany: State University of New York Press 1982 # Prepare a qualitative questionnaire asking the respondents to give a score of between −10 and +10. # Average out the scores. A weighted average can be used. Calculate the average of all the questionnaires from respondents # Calculate the inverse natural logarithm.
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The polynomial basis is frequently used in cryptographic applications that are based on the discrete logarithm problem such as elliptic curve cryptography. The advantage of the polynomial basis is that multiplication is relatively easy. For contrast, the normal basis is an alternative to the polynomial basis and it has more complex multiplication but squaring is very simple. Hardware implementations of polynomial basis arithmetic usually consume more power than their normal basis counterparts.
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Now an alphabet of 32 characters can carry 5 bits of information per character (as 32 = 25). In general the number of bits of information per character is , where N is the number of characters in the alphabet and is the binary logarithm. So for English each character can convey bits of information. However the average amount of actual information carried per character in meaningful English text is only about 1.5 bits per character.
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An ion-selective electrode (ISE), also known as a specific ion electrode (SIE), is a transducer (or sensor) that converts the activity of a specific ion dissolved in a solution into an electrical potential. The voltage is theoretically dependent on the logarithm of the ionic activity, according to the Nernst equation. Ion-selective electrodes are used in analytical chemistry and biochemical/biophysical research, where measurements of ionic concentration in an aqueous solution are required.
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If a particular problem involves performing the same operation on a group of numbers (such as taking the sine or logarithm of each in turn), a language that provides implicit parallelism might allow the programmer to write the instruction thus: numbers = [0 1 2 3 4 5 6 7]; result = sin(numbers); The compiler or interpreter can calculate the sine of each element independently, spreading the effort across multiple processors if available.
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The Digital Signature Algorithm (DSA) is a Federal Information Processing Standard for digital signatures, based on the mathematical concept of modular exponentiation and the discrete logarithm problem. DSA is a variant of the Schnorr and ElGamal signature schemes. The National Institute of Standards and Technology (NIST) proposed DSA for use in their Digital Signature Standard (DSS) in 1991, and adopted it as FIPS 186 in 1994. Four revisions to the initial specification have been released.
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This was the original definition of Sørensen in 1909, which was superseded in favor of pH in 1924. [H] is the concentration of hydrogen ions, denoted [H+] in modern chemistry, which appears to have units of concentration. More correctly, the thermodynamic activity of H+ in dilute solution should be replaced by [H+]/c0, where the standard state concentration c0 = 1 mol/L. This ratio is a pure number whose logarithm can be defined.
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It was shown by Ladner that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete. Such problems are called NP-intermediate problems. The graph isomorphism problem, the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP- complete.
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John M. Pollard (born 1941) is a British mathematician who has invented algorithms for the factorization of large numbers and for the calculation of discrete logarithms. His factorization algorithms include the rho, p − 1, and the first version of the special number field sieve, which has since been improved by others. His discrete logarithm algorithms include the rho algorithm for logarithms and the kangaroo algorithm. He received the RSA Award for Excellence in Mathematics.
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The gel sieves the DNA by the size of the DNA molecule whereby smaller molecules travel faster. Double-stranded DNA moves at a rate that is approximately inversely proportional to the logarithm of the number of base pairs. This relationship however breaks down with very large DNA fragments and it is not possible to separate them using standard agarose gel electrophoresis. The limit of resolution depends on gel composition and field strength.
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On the other hand, the half-pitch is equal to 2(3/π2)1/3aN2/3χ1/6. The fluctuations of the pattern widths are actually only weakly (square root) dependent on the logarithm of the half-pitch, so they become more significant relative to smaller half-pitches. DSA has not yet been implemented in manufacturing, due to defect concerns, where a feature does not appear as expected by the guided self-assembly.A. Gharbi et al.
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In contrast, also shown is a picture of the natural logarithm function and some of its Taylor polynomials around . These approximations converge to the function only in the region ; outside of this region the higher-degree Taylor polynomials are worse approximations for the function. This is similar to Runge's phenomenon. The error incurred in approximating a function by its th-degree Taylor polynomial is called the remainder or residual and is denoted by the function .
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De Moivre's formula does not hold for non-integer powers. The derivation of de Moivre's formula above involves a complex number raised to the integer power . If a complex number is raised to a non-integer power, the result is multiple-valued (see failure of power and logarithm identities). For example, when , de Moivre's formula gives the following results: :for the formula gives 1 = 1, and :for the formula gives 1 = −1.
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In particular, most of the popular public key ciphers are based on the difficulty of factoring integers or the discrete logarithm problem, both of which can be solved by Shor's algorithm. In particular, the RSA, Diffie–Hellman, and elliptic curve Diffie–Hellman algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security.
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This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of calculus, such as computation of antiderivatives, limits, asymptotic expansion, and numerical evaluation to any precision, with a certified error bound.
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The photon structure function, in quantum field theory, describes the quark content of the photon. While the photon is a massless boson, through certain processes its energy can be converted into the mass of massive fermions. The function is defined by the process hadrons. It is uniquely characterized by the linear increase in the logarithm of the electronic momentum transfer log and by the approximately linear rise in , the fraction of the quark momenta within the photon.
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Similarly to the preceding section, the binary number b of the form b_0\dots b_n with m 1s equals the skew binary number b_1\dots b_n plus m. Note that since addition is not defined, adding m corresponds to incrementing the number m times. However, m is bounded by the logarithm of b and incrementation takes constant time. Hence transforming a binary number into a skew binary number runs in time linear in the length of the number.
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One of the main uses of the generic group model is to analyse computational hardness assumptions. An analysis in the generic group model can answer the question: "What is the fastest generic algorithm for breaking a cryptographic hardness assumption". A generic algorithm is an algorithm that only makes use of the group operation, and does not consider the encoding of the group. This question was answered for the discrete logarithm problem by Victor Shoup using the generic group model.
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Neutral density (ND) filters have a constant attenuation across the range of visible wavelengths, and are used to reduce the intensity of light by reflecting or absorbing a portion of it. They are specified by the optical density (OD) of the filter, which is the negative of the common logarithm of the transmission coefficient. They are useful for making photographic exposures longer. A practical example is making a waterfall look blurry when it is photographed in bright light.
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Alice chooses a ring of prime order p, with multiplicative generator g. Alice randomly picks a secret value x from 0 to p − 1 to commit to and calculates c = gx and publishes c. The discrete logarithm problem dictates that from c, it is computationally infeasible to compute x, so under this assumption, Bob cannot compute x. On the other hand, Alice cannot compute a x' <> x, such that gx' = c, so the scheme is binding.
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An example of an information-theoretically hiding commitment scheme is the Pedersen commitment scheme, which is binding under the discrete logarithm assumption. Additionally to the scheme above, it uses another generator h of the prime group and a random number r. The commitment is set c=g^x h^r. These constructions are tightly related to and based on the algebraic properties of the underlying groups, and the notion originally seemed to be very much related to the algebra.
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The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm. Every Riemann surface is a two- dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of holomorphic functions.
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With the restriction to only this exponential, as shown by Galois theory, only compositions of Abelian extensions may be constructed, which suffices only for equations of the fourth degree and below. Something more general is required for equations of higher degree, so to solve the quintic, Hermite, et al. replaced the exponential by an elliptic modular function and the integral (logarithm) by an elliptic integral. Kronecker believed that this was a special case of a still more general method.
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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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Most operators encountered in programming and mathematics are of the binary form. For both programming and mathematics, these can be the multiplication operator, the radix operator, the often omitted exponentiation operator, the logarithm operator, the addition operator, the division operator. Logical predicates such as OR, XOR, AND, IMP are typically used as binary operators with two distinct operands. In CISC architectures, it is common to have two source operands (and store result in one of them).
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In mathematics, logarithm function is defined for positive argument only. A log-antilog circuit built with npn transistors will only accept positive input voltage VX, or only negative VX in case of pnp transistors. This is unacceptable in audio applications, which have to handle alternating signals. Adding DC offset to audio signal, as was proposed by Embley in 1970, will work at a fixed gain setting, but any changes in gain will modulate output DC offset.
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Using the finite form of Jensen's inequality for the natural logarithm, we can prove the inequality between the weighted arithmetic mean and the weighted geometric mean stated above. Since an with weight has no influence on the inequality, we may assume in the following that all weights are positive. If all are equal, then equality holds. Therefore, it remains to prove strict inequality if they are not all equal, which we will assume in the following, too.
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XNUMBERS is a multi-precision floating point computing and numerical methods library for Microsoft Excel. Xnumbers claims to be an open source Excel addin (xla), the license however is not an open source license. XNUMBERS performs multi-precision floating point arithmetic from 15 up to 200 significant digits. The version 5.6 as of 2008 is compatible with Excel 2003/XP and consists of a set of more than 300 functions for arithmetic, complex, trigonometric, logarithm, exponential calculus.
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Resistivity logging measures the subsurface electrical resistivity, which is the ability to impede the flow of electric current. This helps to differentiate between formations filled with salty waters (good conductors of electricity) and those filled with hydrocarbons (poor conductors of electricity). Resistivity and porosity measurements are used to calculate water saturation. Resistivity is expressed in ohms or ohms/meter, and is frequently charted on a logarithm scale versus depth because of the large range of resistivity.
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Then, in 1866, he moved to Chillán, where in 1869 he married Clorinda Pardo, the daughter of a Chilean army colonel. It is not known if the couple had any children. He made the news again in 1883 when he published "Large logarithm tables to twelve decimal points" (Grandes Tablas de Logaritmos a doce decimales) in Chile and France, financed by the Chilean government. He then travelled back to France, after which there is no further record of him.
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Eric Weisstein Rectangular hyperbola from Wolfram Mathworld The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of spacetime as a pseudo-Euclidean space. There the asymptotes of the unit hyperbola form a light cone. Further, the attention to areas of hyperbolic sectors by Gregoire de Saint-Vincent led to the logarithm function and the modern parametrization of the hyperbola by sector areas.
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Prof. Smart is best known for his work in elliptic curve cryptography, especially work on the ECDLP.S. D. Galbraith and N. P. Smart, A cryptographic application of the Weil descent, Cryptography and Coding, 1999.P. Gaudry, F. Hess, and N. P. Smart, Constructive and destructive facets of Weil descent on elliptic curves, Hewlett Packard Laboratories Technical Report, 2000.N. Smart, The discrete logarithm problem on elliptic curves of trace one, Journal of Cryptology, Volume 12, 1999.
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After odds ratios and P-values have been calculated for all SNPs, a common approach is to create a Manhattan plot. In the context of GWA studies, this plot shows the negative logarithm of the P-value as a function of genomic location. Thus the SNPs with the most significant association stand out on the plot, usually as stacks of points because of haploblock structure. Importantly, the P-value threshold for significance is corrected for multiple testing issues.
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In pursuing the history in years before Lorentz enunciated his expressions, one looks to the essence of the concept. In mathematical terms, Lorentz transformations are squeeze mappings, the linear transformations that turn a square into a rectangles of the same area. Before Euler, the squeezing was studied as quadrature of the hyperbola and led to the hyperbolic logarithm. In 1748 Euler issued his precalculus textbook where the number e is exploited for trigonometry in the unit circle.
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Polynomials can be used to approximate complicated curves, for example, the shapes of letters in typography, given a few points. A relevant application is the evaluation of the natural logarithm and trigonometric functions: pick a few known data points, create a lookup table, and interpolate between those data points. This results in significantly faster computations. Polynomial interpolation also forms the basis for algorithms in numerical quadrature and numerical ordinary differential equations and Secure Multi Party Computation, Secret Sharing schemes.
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The natural logarithm of 10, which has the decimal expansion 2.30258509..., plays a role for example in the computation of natural logarithms of numbers represented in scientific notation, as a mantissa multiplied by a power of 10: : \ln(a\cdot 10^n) = \ln a + n \ln 10. This means that one can effectively calculate the logarithms of numbers with very large or very small magnitude using the logarithms of a relatively small set of decimals in the range [1,10).
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Hash trees are a generalization of hash lists and hash chains. Demonstrating that a leaf node is a part of a given binary hash tree requires computing a number of hashes proportional to the logarithm of the number of leaf nodes of the tree; this contrasts with hash lists, where the number is proportional to the number of leaf nodes itself. The concept of hash trees is named after Ralph Merkle who patented it in 1979.
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The first two are the easiest because they each only require two tables. Using the second formula, however, has the unique advantage that if only a cosine table is available, it can be used to estimate inverse cosines by searching for the angle with the nearest cosine value. Notice how similar the above algorithm is to the process for multiplying using logarithms, which follows these steps: scale down, take logarithms, add, take inverse logarithm, scale up.
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Analytic continuation of natural logarithm (imaginary part) Analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent. The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value).
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The unfolding limb is generated in a similar fashion by mixing denaturant-free protein with a concentrated denaturant solution in buffer. When the logarithm of these relaxation rates are plotted as a function of the final denaturant concentration, a chevron plot results. The mixing of the solutions determines the dead time of the instrument, which is about a millisecond. Therefore, a stopped-flow apparatus can be employed only for proteins with a relaxation time of a few milliseconds.
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Leonhard Euler was the first to apply binary logarithms to music theory, in 1739. The powers of two have been known since antiquity; for instance, they appear in Euclid's Elements, Props. IX.32 (on the factorization of powers of two) and IX.36 (half of the Euclid–Euler theorem, on the structure of even perfect numbers). And the binary logarithm of a power of two is just its position in the ordered sequence of powers of two.
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The logarithm of the amplitude squared is usually quoted in dB, so a null amplitude corresponds to −∞ dB. Loudness is related to amplitude and intensity and is one of the most salient qualities of a sound, although in general sounds it can be recognized independently of amplitude. The square of the amplitude is proportional to the intensity of the wave. For electromagnetic radiation, the amplitude of a photon corresponds to the changes in the electric field of the wave.
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In telecommunications, net gain is the overall gain of a transmission circuit. Net gain is measured by applying a test signal at an appropriate power level at the input port of a circuit and measuring the power delivered at the output port. The net gain in dB is calculated by taking 10 times the common logarithm of the ratio of the output power to the input power. The net gain expressed in dB may be positive or negative.
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Very large groups of prime order constructed in elliptic curve cryptography serve for public-key cryptography. Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the discrete logarithm very hard to calculate. One of the earliest encryption protocols, Caesar's cipher, may also be interpreted as a (very easy) group operation. Most cryptographic schemes use groups in some way.
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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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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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These include 133 RR Lyrae variables, of which about a third display the Blazhko effect of long-period modulation. The overall abundance of elements other than hydrogen and helium, what astronomers term the metallicity, is in the range of –1.34 to –1.50 dex. This value gives the logarithm of the abundance relative to the Sun; the actual proportion is 3.2–4.6% of the solar abundance. Messier 3 is the prototype for the Oosterhoff type I cluster, which is considered "metal- rich".
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The Poisson assumption means that :\Pr(0) = \exp(-\mu), where μ is a positive number denoting the expected number of events. If p represents the proportion of observations with at least one event, its complement :(1-p) = \Pr(0) = \exp(-\mu), and then :(-\log(1-p)) = \mu. A linear model requires the response variable to take values over the entire real line. Since μ must be positive, we can enforce that by taking the logarithm, and letting log(μ) be a linear model.
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This result is known as Hölder's theorem. A definite and generally applicable characterization of the gamma function was not given until 1922. Harald Bohr and Johannes Mollerup then proved what is known as the Bohr–Mollerup theorem: that the gamma function is the unique solution to the factorial recurrence relation that is positive and logarithmically convex for positive and whose value at 1 is 1 (a function is logarithmically convex if its logarithm is convex). Another characterisation is given by the Wielandt theorem.
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Any problem of numeric calculation can be viewed as the evaluation of some function ƒ for some input . The condition number of the problem is the ratio of the relative error in the function's output to the relative error in the input, and varies with both the function and the input. The condition number describes how error grows during the calculation. Its base-10 logarithm tells how many fewer digits of accuracy exist in the result than existed in the input.
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The A20, or addressing line 20, is one of the electrical lines that make up the system bus of an x86-based computer system. The A20 line in particular is used to transmit the 21st bit on the address bus. A microprocessor typically has a number of addressing lines equal to the base-two logarithm of its physical addressing space. For example, a processor with 4 GB of byte- addressable physical space requires 32 lines, which are named A0 through A31.
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A dose–response curve is a coordinate graph relating the magnitude of a stimulus to the response of the receptor. A number of effects (or endpoints) can be studied. The measured dose is generally plotted on the X axis and the response is plotted on the Y axis. In some cases, it is the logarithm of the dose that is plotted on the X axis, and in such cases the curve is typically sigmoidal, with the steepest portion in the middle.
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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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Holdsworth S.D, Aseptic Processing and Packaging of Food Products, 1992. , 9781851667758 The z-value of an organism in a particular medium is the temperature change required for the D-value to change by a factor of ten, or put another way, the temperature required for the thermal destruction curve to move one log cycle. It is the reciprocal of the slope resulting from the plot of the logarithm of the D-value versus the temperature at which the D-value was obtained.
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The ratio is then converted to a logarithm and expressed as a log odds score, as for PAM. BLOSUM matrices are usually scaled in half-bit units. A score of zero indicates that the frequency with which a given two amino acids were found aligned in the database was as expected by chance, while a positive score indicates that the alignment was found more often than by chance, and negative score indicates that the alignment was found less often than by chance.
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Their scheme allows these trees to be encoded in a number of bits that is close to the information-theoretic lower bound (the base-2 logarithm of the Wedderburn–Etherington number) while still allowing constant-time navigation operations within the tree.. use unordered binary trees, and the fact that the Wedderburn–Etherington numbers are significantly smaller than the numbers that count ordered binary trees, to significantly reduce the number of terms in a series representation of the solution to certain differential equations..
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The cepstrum can be seen as information about the rate of change in the different spectrum bands. It was originally invented for characterizing the seismic echoes resulting from earthquakes and bomb explosions. It has also been used to determine the fundamental frequency of human speech and to analyze radar signal returns. Cepstrum pitch determination is particularly effective because the effects of the vocal excitation (pitch) and vocal tract (formants) are additive in the logarithm of the power spectrum and thus clearly separate.
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In mathematics, a metric space with metric is said to be doubling if there is some doubling constant such that for any and , it is possible to cover the ball with the union of at most balls of radius . The base-2 logarithm of is often referred to as the doubling dimension of . Euclidean spaces equipped with the usual Euclidean metric are examples of doubling spaces where the doubling constant depends on the dimension . For example, in one dimension, ; and in two dimensions, .
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Here, the AVHRR to MODIS ratio of reflectances is modeled as a third-order polynomial using the natural logarithm of TWP from the NCEP reanalysis. Using these two methods, monthly calibration slopes are generated with a linear fit forced through the origin of the adjusted MODIS reflectances versus AVHRR counts. To extend the MODIS reference back for AVHRRs prior to the MODIS era (pre-2000), Heidinger et al. [2010] use the stable Earth targets of Dome C in Antarctica and the Libyan Desert.
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It gets its efficiency by eschewing binary arithmetic for an "optical" adder which can add hundreds of thousands of quantities in a single clock cycle. The key idea used is "time-space inversion". Conventional NFS sieving is carried out one prime at a time. For each prime, all the numbers to be tested for smoothness in the range under consideration which are divisible by that prime have their counter incremented by the logarithm of the prime (similar to the sieve of Eratosthenes).
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In Hick's experiment, the RT is found to be a function of the binary logarithm of the number of available choices (n). This phenomenon is called "Hick's law" and is said to be a measure of the "rate of gain of information". The law is usually expressed by the formula RT = a + b\log_2(n + 1), where a and b are constants representing the intercept and slope of the function, and n is the number of alternatives.Hick's Law at Encyclopedia.
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Strong acids and bases are compounds that, for practical purposes, are completely dissociated in water. Under normal circumstances this means that the concentration of hydrogen ions in acidic solution can be taken to be equal to the concentration of the acid. The pH is then equal to minus the logarithm of the concentration value. Hydrochloric acid (HCl) is an example of a strong acid. The pH of a 0.01M solution of HCl is equal to −log10(0.01), that is, pH = 2.
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In mathematics, the Liouvillian functions comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be recursively defined as integrals of other Liouvillian functions. More explicitly, it is a function of one variable which is the composition of a finite number of arithmetic operations , exponentials, constants, solutions of algebraic equations (a generalization of nth roots), and antiderivatives. The logarithm function does not need to be explicitly included since it is the integral of 1/x.
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Exponentiation in finite fields has applications in public key cryptography. For example, the Diffie–Hellman key exchange uses the fact that exponentiation is computationally inexpensive in finite fields, whereas the discrete logarithm (the inverse of exponentiation) is computationally expensive. Any finite field F has the property that there is a unique prime number p such that px=0 for all x in F; that is, x added to itself p times is zero. For example, in F_2, the prime number has this property.
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If the uppermost bit was set to one, this was an Extracode and was implemented as a special kind of subroutine jump to a location in the fixed store (ROM), its address being determined by the other nine bits. About 250 extracodes were implemented, of the 512 possible. Extracodes were what would be called software interrupt or trap today. They were used to call mathematical procedures which would have been too inefficient to implement in hardware, for example sine, logarithm, and square root.
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The table below lists the language editions of Wikipedia roughly sorted by the number of active users (registered users who have made at least one edit in the last thirty days); in particular, the "power of ten" of the count of active users (i.e., the common logarithm rounded down to a whole number) is used: so "5" means at least 105 (or 100,000), "4" means at least 104 (10,000), and so on. Script names are listed as their ISO codes.
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In 2013, Ma embarked on the light design for the Water Cube. The project Nature and Man in Rhapsody of Light at the Water Cube incorporates traditional Eastern philosophy and modern technology. The building's celluloid body shined in different colors, rhythms, movements, and compositions in patterns calculated daily by a computer logarithm based on daily I Ching readings, and societal conditions as reflected in social media statistics. Ma's competence to implement large installations and projects stands out among her contemporaries.
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If a 2 × 2 real matrix has a negative determinant, it has no real logarithm. Note first that any 2 × 2 real matrix can be considered one of the three types of the complex number z = x + y ε, where ε² ∈ { −1, 0, +1 }. This z is a point on a complex subplane of the ring of matrices. The case where the determinant is negative only arises in a plane with ε² =+1, that is a split-complex number plane.
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The simplest interesting case is an n-cycle. Richard Cole and Uzi Vishkin, see also show that there is a distributed algorithm that reduces the number of colors from n to O(log n) in one synchronous communication step. By iterating the same procedure, it is possible to obtain a 3-coloring of an n-cycle in O( n) communication steps (assuming that we have unique node identifiers). The function , iterated logarithm, is an extremely slowly growing function, "almost constant".
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Let p be the number of pairs of vertices that are not connected by an edge in the given graph , and let be the unique integer for which . Then the intersection number of is at most .. As cited by . Graphs that are the complement of a sparse graph have small intersection numbers: the intersection number of any -vertex graph is at most , where is the base of the natural logarithm and d is the maximum degree of the complement graph of ..
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A coloring of a given graph is distinguishing for that graph if and only if it is distinguishing for the complement graph. Therefore, every graph has the same distinguishing number as its complement. For every graph , the distinguishing number of is at most proportional to the logarithm of the number of automorphisms of . If the automorphisms form a nontrivial abelian group, the distinguishing number is two, and if it forms a dihedral group then the distinguishing number is at most three.
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However, the term bipolar coordinates is reserved for the coordinates described here, and never used for systems associated with those other curves, such as elliptic coordinates. Geometric interpretation of the bipolar coordinates. The angle σ is formed by the two foci and the point P, whereas τ is the logarithm of the ratio of distances to the foci. The corresponding circles of constant σ and τ are shown in red and blue, respectively, and meet at right angles (magenta box); they are orthogonal.
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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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The Spectronic 20 measures the absorbance of light at a pre-determined concentration, and the concentration is calculated from the Beer-Lambert relationship. The absorbance of the light is the base 10 logarithm of the ratio of the Transmittance of the pure solvent to the transmittance of the sample, and so the two absorbance and transmittance can be interconverted. Either transmittance or absorbance can therefore be plotted versus concentration using measurements from the Spectronic 20. Plotting a curve using percent transmittance of light yields an exponential curve.
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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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Such a model is termed an exponential-response model (or log- linear model, since the logarithm of the response is predicted to vary linearly). Similarly, a model that predicts a probability of making a yes/no choice (a Bernoulli variable) is even less suitable as a linear-response model, since probabilities are bounded on both ends (they must be between 0 and 1). Imagine, for example, a model that predicts the likelihood of a given person going to the beach as a function of temperature.
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Shor's algorithm can also efficiently solve the discrete logarithm problem, which is the basis for the security of Diffie–Hellman, elliptic curve Diffie–Hellman, elliptic curve DSA, Curve25519, ed25519, and ElGamal. Although quantum computers are currently in their infancy, the ongoing development of quantum computers and their theoretical ability to compromise modern cryptographic protocols (such as TLS/SSL) has prompted the development of post-quantum cryptography. SIDH was created in 2011 by De Feo, Jao, and Plut. It uses conventional elliptic curve operations and is not patented.
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If an improved algorithm can be found to solve the problem, then the system is weakened. For example, the security of the Diffie–Hellman key exchange scheme depends on the difficulty of calculating the discrete logarithm. In 1983, Don Coppersmith found a faster way to find discrete logarithms (in certain groups), and thereby requiring cryptographers to use larger groups (or different types of groups). RSA's security depends (in part) upon the difficulty of integer factorization — a breakthrough in factoring would impact the security of RSA.
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In the field of genomics (and more generally in bioinformatics), the modern usage is to define fold change in terms of ratios, and not by the alternative definition. However, log-ratios are often used for analysis and visualization of fold changes. The logarithm to base 2 is most commonly used, as it is easy to interpret, e.g. a doubling in the original scaling is equal to a log2 fold change of 1, a quadrupling is equal to a log2 fold change of 2 and so on.
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From a heuristic view, we expect the probability that the ratio of the length of the gap to the natural logarithm is greater than or equal to a fixed positive number k to be ; consequently the ratio can be arbitrarily large. Indeed, the ratio of the gap to the number of digits of the integers involved does increase without bound. This is a consequence of a result by Westzynthius.. In the opposite direction, the twin prime conjecture posits that for infinitely many integers n.
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The following protocol was suggested by David Chaum. A group, G, is chosen in which the discrete logarithm problem is intractable, and all operation in the scheme take place in this group. Commonly, this will be the finite cyclic group of order p contained in Z/nZ, with p being a large prime number; this group is equipped with the group operation of integer multiplication modulo n. An arbitrary primitive element (or generator), g, of G is chosen; computed powers of g then combine obeying fixed axioms.
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The most famous of the name was John Napier the seventeenth Laird of Merchiston who developed the system of Logarithm. In 1617 he was succeeded by his son, Archibald Napier, 1st Lord Napier who accompanied James VI and I to claim his new throne in England. Napier married the daughter of the fourth Earl of Montrose and sister of James Graham, 1st Marquess of Montrose. As the brother-in-law of the king's captain the Napiers supported the king throughout the Scottish Civil War.
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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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Safe primes are also important in cryptography because of their use in discrete logarithm-based techniques like Diffie–Hellman key exchange. If is a safe prime, the multiplicative group of numbers modulo has a subgroup of large prime order. It is usually this prime- order subgroup that is desirable, and the reason for using safe primes is so that the modulus is as small as possible relative to p. A prime number p = 2q + 1 is called a safe prime if q is prime.
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ECOH does not use random oracles and its security is not strictly directly related to the discrete logarithm problem, yet it is still based on mathematical functions. ECOH is related to the Semaev's problem of finding low degree solutions to the summation polynomial equations over binary field, called the Summation Polynomial Problem. An efficient algorithm to solve this problem has not been given so far. Although the problem was not proven to be NP-hard, it is assumed that such an algorithm does not exist.
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Theon was a great philosopher of harmony and he discusses semitones in his treatise. There are several semitones used in Greek music, but of this variety, there are two that are very common. The “diatonic semitone” with a value of 16/15 and the “chromatic semitone” with a value of 25/24 are the two more commonly used semitones (Papadopoulos, 2002). In these times, Pythagoreans did not rely on irrational numbers for understanding of harmonies and the logarithm for these semitones did not match with their philosophy.
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The AIC values of the candidate models must all be computed with the same data set. Sometimes, though, we might want to compare a model of the response variable, , with a model of the logarithm of the response variable, . More generally, we might want to compare a model of the data with a model of transformed data. Following is an illustration of how to deal with data transforms (adapted from : "Investigators should be sure that all hypotheses are modeled using the same response variable").
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Neither the logarithm method nor the rational exponent method can be used to define br as a real number for a negative real number b and an arbitrary real number r. Indeed, er is positive for every real number r, so ln(b) is not defined as a real number for . The rational exponent method cannot be used for negative values of b because it relies on continuity. The function has a unique continuous extension from the rational numbers to the real numbers for each .
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The most important characteristics of a DAC are: ;Resolution: The number of possible output levels the DAC is designed to reproduce. This is usually stated as the number of bits it uses, which is the binary logarithm of the number of levels. For instance a 1-bit DAC is designed to reproduce 2 (21) levels while an 8-bit DAC is designed for 256 (28) levels. Resolution is related to the effective number of bits which is a measurement of the actual resolution attained by the DAC.
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Small Proth primes (less than 10200) have been used in constructing prime ladders, sequences of prime numbers such that each term is "close" (within about 1011) to the previous one. Such ladders have been used to empirically verify prime-related conjectures. For example, Goldbach's weak conjecture was verified in 2008 up to 8.875·1030 using prime ladders constructed from Proth primes. (The conjecture was later proved by Harald Helfgott.) Also, Proth primes can optimize den Boer reduction between the Diffie-Hellman problem and the Discrete logarithm problem.
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In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the matrix exponential or matrix logarithm. In other areas, such as formal analysis, it is more convenient to work directly with the power series themselves. Thus one may define a solution of a differential equation as a power series which, one hopes to prove, is the Taylor series of the desired solution.
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Bernoulli, Nicolas; letter of 5 April 1732, acknowledging receipt of "Specimen theoriae novae metiendi sortem pecuniariam" (excerpted in PDF ). In 1728, Gabriel Cramer produced fundamentally the same theory in a private letter.Cramer, Garbriel; letter of 21 May 1728 to Nicolaus Bernoulli (excerpted in PDF ). Each had sought to resolve the St. Petersburg paradox, and had concluded that the marginal desirability of money decreased as it was accumulated, more specifically such that the desirability of a sum were the natural logarithm (Bernoulli) or square root (Cramer) thereof.
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Irreducible polynomials over finite fields are also useful for Pseudorandom number generators using feedback shift registers and discrete logarithm over F2n. The number of irreducible monic polynomials of degree n over Fq is the number of aperiodic necklaces, given by Moreau's necklace- counting function Mq(n). The closely related necklace function Nq(n) counts monic polynomials of degree n which are primary (a power of an irreducible); or alternatively irreducible polynomials of all degrees d which divide n.Christophe Reutenauer, Mots circulaires et polynomes irreductibles, Ann. Sci.
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A related description of CTC physics was given in 2001 by Michael Devin, and applied to thermodynamics. The same model with the introduction of a noise term allowing for inexact periodicity, allows the grandfather paradox to be resolved, and clarifies the computational power of a time machine assisted computer. Each time traveling qubit has an associated negentropy, given approximately by the logarithm of the noise of the communication channel. Each use of the time machine can be used to extract as much work from a thermal bath.
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If the log is taken base 2, the unit of information is the binary digit or bit (so named by John Tukey); if we use a natural logarithm instead, we might call the resulting unit the "nat." In magnitude, a nat is apparently identical to Boltzmann's constant k or the ideal gas constant R, although these particular quantities are usually reserved to measure physical information that happens to be entropy, and that are expressed in physical units such as joules per kelvin, or kilocalories per mole-kelvin.
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The candidates are issued identification passes in advance, which are presented to the staff at the examination site. The site itself must not be the same school where a candidate is from; to ensure impartiality, the candidate must travel to a different school to take the examination. For the same reason, the candidate may not identify himself/herself on the answer sheet except with an identity-masking number. Use of calculation aids other than logarithm tables, which are provided by the examination center, is prohibited.
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In frequentist statistics, the likelihood function is itself a statistic that summarizes a single sample from a population, whose calculated value depends on a choice of several parameters θ1 ... θp, where p is the count of parameters in some already-selected statistical model. The value of the likelihood serves as a figure of merit for the choice used for the parameters, and the parameter set with maximum likelihood is the best choice, given the data available. The specific calculation of the likelihood is the probability that the observed sample would be assigned, assuming that the model chosen and the values of the several parameters θ give an accurate approximation of the frequency distribution of the population that the observed sample was drawn from. Heuristically, it makes sense that a good choice of parameters is those which render the sample actually observed the maximum possible post-hoc probability of having happened. Wilks' theorem quantifies the heuristic rule by showing that the difference in the logarithm of the likelihood generated by the estimate’s parameter values and the logarithm of the likelihood generated by population’s "true" (but unknown) parameter values is χ² distributed.
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In its use as a television antenna, it was common to combine a log-periodic design for VHF with a Yagi for UHF, with both halves being roughly equal in size. This resulted in much higher gain for UHF, typically on the order of 10 to 14 dB on the Yagi side and 6.5 dB for the log-periodic. But this extra gain was needed anyway in order to make up for a number of problems with UHF signals. It should be strictly noted that the log- periodic shape, according to the IEEE definition,“Log-periodic antenna Any one of a class of antennas having a structural geometry such that its impedance and radiation characteristics repeat periodically as the logarithm of frequency.” (see The new IEEE Standard Dictionary of Electrical and Electronics Terms, 1993 ⓒ IEEE.) “Log-periodic antenna Any one of a class of antennas having a structural geometry such that its impedance and radiation characteristics repeat periodically as the logarithm of frequency.” (see Acknowledgments, and footnote in page 1), Self-Complementary Antennas―Principle of Self-Complementarity for Constant Impedance―, by Y. Mushiake, Springer-Verlag London Ltd.
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Multiple or multivariate regression is an approach to look at the relationship between several independent or predictor variables and a dependent or influential variable. It is best used in geometric morphometrics when analyzing shape variables based on an external influence. For example, it can be used in studies with attached functional or environmental variables like age or the development over time in certain environments. The multivariate regression of shape based on the logarithm of centroid size (square root of the sum of squared distances of landmarks) is ideal for allometric studies.
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WASP-13 is a sunlike, G-type star that is situated approximately 230 parsecs (750 light years) in the Lynx constellation. With an apparent magnitude of 10.42, the star cannot be seen with the unaided eye from the perspective of someone on Earth. The star's effective temperature, at , is slightly hotter than that of the Sun, and the radius of is also larger, leading to a bolometric luminosity of . However, its metallicity is similar; this can be seen in how the logarithm of the concentration of iron, or [Fe/H], is approximately 0.
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WASP-13 has a mass of and the logarithm of its surface gravity is measured at , while the rate at which it rotates is at most . The evolutionary status of WASP-13, as shown from its position in the Hertsprung-Russel diagram is near the main sequence turnoff, and it is considered very close to exhausting its core hydrogen and becoming a subgiant. Comparison with theoretical isochrones and stars with accurately-determined ages gives an age for WASP-13 of around . Earlier estimates had given an older age, but with a very large uncertainty.
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Larger factorial values can be approximated using Stirling's formula. Wolfram Alpha can calculate exact results for the ceiling function and floor function applied to the binary, natural and common logarithm of for values of up to , and up to for the integers. If the exact values of large factorials are needed, they can be computed using arbitrary-precision arithmetic. Instead of doing the sequential multiplications , a program can partition the sequence into two parts, whose products are roughly the same size, and multiply them using a divide-and-conquer method.
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Quoting Fisher: The concept of likelihood should not be confused with probability as mentioned by Sir Ronald Fisher Fisher's invention of statistical likelihood was in reaction against an earlier form of reasoning called inverse probability. His use of the term "likelihood" fixed the meaning of the term within mathematical statistics. A. W. F. Edwards (1972) established the axiomatic basis for use of the log-likelihood ratio as a measure of relative support for one hypothesis against another. The support function is then the natural logarithm of the likelihood function.
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Avoiding the use of expensive trigonometric functions improves speed over the basic form. It discards of the total input uniformly distributed random number pairs generated, i.e. discards uniformly distributed random number pairs per Gaussian random number pair generated, requiring input random numbers per output random number. The basic form requires two multiplications, 1/2 logarithm, 1/2 square root, and one trigonometric function for each normal variate.Note that the evaluation of 2U1 is counted as one multiplication because the value of 2 can be computed in advance and used repeatedly.
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In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after imposing some constraint. If the constraint (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero.
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Logarithm of odds (LOD) is a statistical technique used to estimate the probability of gene linkage between traits. LOD is often used in conjunction with pedigrees, maps of a family's genetic make- up, in order to yield more accurate estimations. A key benefit of this technique is its ability to give reliable results in both large and small sample sizes, which is a marked advantage in laboratory research. Quantitative trait loci (QTL) mapping is another statistical method used to determine the chromosomal positions of a set of genes responsible for a given trait.
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A scatterplot in which the areas of the sovereign states and dependent territories in the world are plotted on the vertical axis against their populations on the horizontal axis. The upper plot uses raw data. In the lower plot, both the area and population data have been transformed using the logarithm function. In statistics, data transformation is the application of a deterministic mathematical function to each point in a data set—that is, each data point zi is replaced with the transformed value yi = f(zi), where f is a function.
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Transforms are usually applied so that the data appear to more closely meet the assumptions of a statistical inference procedure that is to be applied, or to improve the interpretability or appearance of graphs. Nearly always, the function that is used to transform the data is invertible, and generally is continuous. The transformation is usually applied to a collection of comparable measurements. For example, if we are working with data on peoples' incomes in some currency unit, it would be common to transform each person's income value by the logarithm function.
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The Finite Field Diffie-Hellman algorithm has roughly the same key strength as RSA for the same key sizes. The work factor for breaking Diffie-Hellman is based on the discrete logarithm problem, which is related to the integer factorization problem on which RSA's strength is based. Thus, a 2048-bit Diffie-Hellman key has about the same strength as a 2048-bit RSA key. Elliptic-curve cryptography (ECC) is an alternative set of asymmetric algorithms that is equivalently secure with shorter keys, requiring only approximately twice the bits as the equivalent symmetric algorithm.
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This explains why the normal distribution can be used successfully for certain data sets of ratios. Another related distribution is the log-harmonic law, which is the probability distribution of a random variable whose logarithm follows an harmonic law. This family has an interesting property, the Pitman estimator of the location parameter does not depend on the choice of the loss function. Only two statistical models satisfy this property: One is the normal family of distributions and the other one is a three-parameter statistical model which contains the log-harmonic law.
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Roughness length (z_0) is a parameter of some vertical wind profile equations that model the horizontal mean wind speed near the ground; in the log wind profile, it is equivalent to the height at which the wind speed theoretically becomes zero. In reality the wind at this height no longer follows a mathematical logarithm. It is so named because it is typically related to the height of terrain roughness elements. Whilst it is not a physical length, it can be considered as a length-scale representation of the roughness of the surface.
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By taking the logarithm of the above equation, one obtains: and if one assumes that the atmospheric disturbance \tau does not change during the observations (which last for a morning or an afternoon), the plot of ln I versus m is a straight line with a slope equal to \tau. Then, by linear extrapolation to m = 0, one obtains I0, i.e. the Sun's radiance that would be observed by an instrument placed above the atmosphere. Points are Langley extrapolation to top of atmosphere of direct solar radiation measured at Niamey, Niger 24 December 2006.
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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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Details for the required modifications to the test statistic and for the critical values for the normal distribution and the exponential distribution have been published by Pearson & Hartley (1972, Table 54). Details for these distributions, with the addition of the Gumbel distribution, are also given by Shorak & Wellner (1986, p239). Details for the logistic distribution are given by Stephens (1979). A test for the (two parameter) Weibull distribution can be obtained by making use of the fact that the logarithm of a Weibull variate has a Gumbel distribution.
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Like approximate entropy (ApEn), Sample entropy (SampEn) is a measure of complexity. But it does not include self-similar patterns as ApEn does. For a given embedding dimension m , tolerance r and number of data points N , SampEn is the negative natural logarithm of the probability that if two sets of simultaneous data points of length m have distance < r then two sets of simultaneous data points of length m+1 also have distance < r . And we represent it by SampEn(m,r,N) (or by SampEn(m,r,\tau,N) including sampling time \tau).
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There are some logic systems that do not have any notion of equality. This reflects the undecidability of the equality of two real numbers, defined by formulas involving the integers, the basic arithmetic operations, the logarithm and the exponential function. In other words, there cannot exist any algorithm for deciding such an equality. The binary relation "is approximately equal" (denoted by the symbol \approx) between real numbers or other things, even if more precisely defined, is not transitive (since many small differences can add up to something big).
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In information theory an entropy encoding is a lossless data compression scheme that is independent of the specific characteristics of the medium. One of the main types of entropy coding creates and assigns a unique prefix-free code to each unique symbol that occurs in the input. These entropy encoders then compress data by replacing each fixed-length input symbol with the corresponding variable-length prefix-free output codeword. The length of each codeword is approximately proportional to the negative logarithm of the probability of occurrence of that codeword.
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When working with continued fractions, the number of terms is limited by the available accuracy and by the size of each term. An approximate formula is 2 decimal fraction digit accuracy for each (term times the base ten logarithm of the term). The only way to do such work safely is to do it twice, in parallel, with the initial input to one dithered in the final several digits (at least 1 word). Then when the two calculations do not give identical terms, stop at the previous term.
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Robert Lees obtained a value for the "glottochronological constant" (r) of words by considering the known changes in 13 pairs of languages using the 200 word list. He obtained a value of 0.805 ± 0.0176 with 90% confidence. For his 100-word list Swadesh obtained a value of 0.86, the higher value reflecting the elimination of semantically unstable words. The constant is related to the retention rate of words by the following formula: :L = 2\ln(r) L is the rate of replacement, ln represents the natural logarithm and r is the glottochronological constant.
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Briggs helped to develop the common logarithm, described as "one of the most useful systems for mathematics". The third professor, John Wallis, introduced the use of for infinity, and was regarded as "one of the leading mathematicians of his time". Both Edmond Halley, who successfully predicted the return of the comet named in his honour, and his successor Nathaniel Bliss held the post of Astronomer Royal in addition to the professorship. Stephen Rigaud (professor 1810–27) has been called "the foremost historian of astronomy and mathematics in his generation".
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Within the chapter on business, a section entitled ' (Details of calculation and recording) describes the accounting methods then in use among northern-Italian merchants, including double-entry bookkeeping, trial balances, balance sheets and various other tools still employed by professional accountants. The business chapter also introduces the rule of 72 for predicting an investment's future value, anticipating the development of the logarithm by more than century. These techniques did not originate with Pacioli, who merely recorded and explained the established best practices of contemporary businesspeople in his region.
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Traditionally the pattern has been explored by plotting the number of species on the y-axis and the logarithm to the base two of the species body mass (g) on the x-axis. This yields a highly right skewed body size distribution with a mode centered near species with a mass ranging from 50-100 grams. Although this relationship is very distinct at large spatial scales, the pattern breaks down when the sampling area is small (Hutchinson and MacArthur, 1959; Brown and Maurer 1989; Brown and Nicoletto 1991; Bakker and Kelt 2000).
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This is done such that the input sequence can be precisely reconstructed from the representation at the highest level. The system effectively minimises the description length or the negative logarithm of the probability of the data. Given a lot of learnable predictability in the incoming data sequence, the highest level RNN can use supervised learning to easily classify even deep sequences with long intervals between important events. It is possible to distill the RNN hierarchy into two RNNs: the "conscious" chunker (higher level) and the "subconscious" automatizer (lower level).
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For example, any two notes an octave apart have a frequency ratio of 2:1. This means that successive increments of pitch by the same interval result in an exponential increase of frequency, even though the human ear perceives this as a linear increase in pitch. For this reason, intervals are often measured in cents, a unit derived from the logarithm of the frequency ratio. In Western music theory, the most common naming scheme for intervals describes two properties of the interval: the quality (perfect, major, minor, augmented, diminished) and number (unison, second, third, etc.).
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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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In algebraic number theory, Leopoldt's conjecture, introduced by , states that the p-adic regulator of a number field does not vanish. The p-adic regulator is an analogue of the usual regulator defined using p-adic logarithms instead of the usual logarithms, introduced by . Leopoldt proposed a definition of a p-adic regulator Rp attached to K and a prime number p. The definition of Rp uses an appropriate determinant with entries the p-adic logarithm of a generating set of units of K (up to torsion), in the manner of the usual regulator.
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In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is , where is the prime- counting function and is the natural logarithm of .
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When the pressure is below the triple point, solid nitrogen directly sublimes to gas. The triple point is at 63.14±0.06 K and 0.1255±0.0005 bar. The vapour pressure has been measured from 20 K up to the triple point. For α-nitrogen (below 35 K) the logarithm of the pressure is given by 12.40 −807.4 × T−1 −3926 T−2 +6.297×10\+ 4T−3 −4.633× 10 +5T−4 1.325× 10\+ 6T−5. For β-nitrogen it is given by 8.514 −458.4T−1 −19870 T−2 4.800 × 10\+ 5T−3 −4.524 × 10+6T−4.
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Logarithm of the relative energy output (ε) of proton–proton (PP), CNO and Triple-α fusion processes at different temperatures. The dashed line shows the combined energy generation of the PP and CNO processes within a star. At the Sun's core temperature, the PP process is more efficient. Scheme of the proton–proton branch I chain reaction The proton–proton chain reaction, also commonly referred to as the p-p chain, is one of two known sets of nuclear fusion reactions by which stars convert hydrogen to helium.
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Third, he specified the Wood–Anderson seismograph as the standard instrument for producing seismograms. Magnitude was then defined as "the logarithm of the maximum trace amplitude, expressed in microns", measured at a distance of . The scale was calibrated by defining a magnitude 0 shock as one that produces (at a distance of ) a maximum amplitude of 1 micron (1 µm, or 0.001 millimeters) on a seismogram recorded by a Wood–Anderson torsion seismograph.. See also ; . Finally, Richter calculated a table of distance corrections,, Table I. in that for distances less than 200 kilometers.
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An efficient algorithm to solve the discrete logarithm problem would make it easy to compute a or b and solve the Diffie–Hellman problem, making this and many other public key cryptosystems insecure. Fields of small characteristic may be less secure. The order of G should have a large prime factor to prevent use of the Pohlig–Hellman algorithm to obtain a or b. For this reason, a Sophie Germain prime q is sometimes used to calculate , called a safe prime, since the order of G is then only divisible by 2 and q.
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The potential between electric charges obeys the usual Colomb law, it is inversely proportional to the distance between the charges. When M is greater than 3N, the theory is free in the infrared, and so the force between two charges is inversely proportional to the product of the distance times the logarithm of the distance between the charges. However the theory is ill-defined in the ultraviolet, unless one includes additional heavy degrees of freedom which lead, for example, to a Seiberg dual theory of the type described above at N+1
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The implication of genetics in psychiatric illnesses is not unique to schizophrenia, though the heritability of schizophrenia has been calculated as high as 80%. The continued research of the family following the discovery of the translocation yielded statistical analysis of the probability of observing the simultaneous occurrence, or co-inheritance, of psychological afflictions and the translocation. This concept was measured quantitatively using the LOD, or logarithm of the odds value. The higher the LOD value, the stronger the correlation between the presence of the translocation and given disease(s) is thought to be.
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A plot of chain length vs. the logarithm of the lipid bilayer/buffer partition coefficient K is linear, with the addition of each methylene group causing a change in the Gibbs free energy of -3.63 kJ/mol. The cutoff effect was first interpreted as evidence that anaesthetics exert their effect not by acting globally on membrane lipids but rather by binding directly to hydrophobic pockets of well-defined volumes in proteins. As the alkyl chain grows, the anaesthetic fills more of the hydrophobic pocket and binds with greater affinity.
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The term significand was introduced by George Forsythe and Cleve Moler in 1967 and is the word used in the IEEE standard. However, in 1946 Arthur Burks used the terms mantissa and characteristic to describe the two parts of a floating-point number (Burks et al.) and that usage remains common among computer scientists today. Mantissa and characteristic have long described the two parts of the logarithm found on tables of common logarithms. While the two meanings of exponent are analogous, the two meanings of mantissa are not equivalent.
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The plot is similar to that of the multivalued complex logarithm function except that the spacing between sheets is not constant and the connection of the principal sheet is different There are countably many branches of the function, denoted by , for integer ; being the main (or principal) branch. is defined for all complex numbers z while with is defined for all non-zero z. We have and for all . The branch point for the principal branch is at , with a branch cut that extends to along the negative real axis.
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This computation appears independent of the kind of black hole, since the given Immirzi parameter is always the same. However, Krzysztof Meissner and Marcin Domagala with Jerzy Lewandowski have corrected the assumption that only the minimal values of the spin contribute. Their result involves the logarithm of a transcendental number instead of the logarithms of integers mentioned above. The Immirzi parameter appears in the denominator because the entropy counts the number of edges puncturing the event horizon and the Immirzi parameter is proportional to the area contributed by each puncture.
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The basic idea of the algorithm is due to Western and Miller (1968),Western and Miller (1968) Tables of indices and primitive roots, Royal Society Mathematical Tables, vol 9, Cambridge University Press. which ultimately relies on ideas from Kraitchik (1922).M. Kraitchik, Théorie des nombres, Gauthier--Villards, 1922 The first practical implementations followed the 1976 introduction of the Diffie-Hellman cryptosystem which relies on the discrete logarithm. Merkle's Stanford University dissertation (1979) was credited by Pohlig (1977) and Hellman and Reyneri (1983), who also made improvements to the implementation.
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The number of radioactive decays per second in a given mass of 40K is the number of atoms in that mass, divided by the average lifetime of a 40K atom in seconds. The number of atoms in one gram of 40K is Avogadro's number 6.022 (the number of atoms per mole) divided by the atomic weight of potassium-40 (39.96 grams per mole), which is about 0.1507 per gram. As in any exponential decay, the average lifetime is the half-life divided by the natural logarithm of 2, or about 56.82 seconds.
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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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Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not converge if is far from . That is, the Taylor series diverges at if the distance between and is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point.
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A holonomic function, also called a D-finite function, is a function that is a solution of a homogeneous linear differential equation with polynomial coefficients. Most functions that are commonly considered in mathematics are holonomic or quotients of holonomic functions. In fact, holonomic functions include polynomials, algebraic functions, logarithm, exponential function, sine, cosine, hyperbolic sine, hyperbolic cosine, inverse trigonometric and inverse hyperbolic functions, and many special functions such as Bessel functions and hypergeometric functions. Holonomic functions have several closure properties; in particular, sums, products, derivative and integrals of holonomic functions are holonomic.
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1986 edition of TI-35 PLUS TI-36 SOLAR, first edition It can display 10 digits mantissa with 2 digits exponent, and calculates with 12-digit precision internally. TI-36 SOLAR was based on 1985 version of TI-35 PLUS, but incorporates solar cells. It addition to standard features such as trigonometric functions, exponents, logarithm, and intelligent order of operations found in TI-30 and TI-34 series of calculators, it also include base (decimal, hexadecimal, octal, binary) calculations, complex values, statistics. Conversions include polar- rectangular coordinates (P←→R), angles.
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A typical chevron plot observed in protein folding experiments. A chevron plot is a way of representing protein folding kinetic data in the presence of varying concentrations of denaturant that disrupts the protein's native tertiary structure. The plot is known as "chevron" plot because of the canonical v, or chevron shape observed when the logarithm of the observed relaxation rate is plotted as a function of the denaturant concentration. In a two-state system, folding and unfolding rates dominate the observed relaxation rates below and above the denaturation midpoint (Cm).
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This gives rise to the terminology of folding and unfolding arms for the limbs of the chevron. A priori information on the Cm of a protein can be obtained from equilibrium experiments. In fitting to a two-state model, the logarithm of the folding and unfolding rates is assumed to depend linearly on the denaturant concentration, thus resulting in the slopes mf and mu, called the folding and unfolding m-values, respectively (also called the kinetic m-values). The sum of the two rates is the observed relaxation rate.
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This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function f. This limit is the derivative of the function f at x = a, denoted f ′(a). Using derivatives, the equation of the tangent line can be stated as follows: : y=f(a)+f'(a)(x-a).\, Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the power function, trigonometric functions, exponential function, logarithm, and their various combinations.
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Shannon entropy or information content measured as the surprise value of a particular event, is essentially inversely proportional to the logarithm of the event's probability, i = log(1/p). Claude Shannon's information theory arose from research at Bell labs, building upon George Boole's digital logic. As information theory predicts common and easily predicted words tend to become shorter for optimal communication channel efficiency while less common words tend to be longer for redundancy and error correction. Vedral compares the process of life to John von Neumann's self replicating automata.
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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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The quadrature of the hyperbola xy = 1 by Gregoire de Saint-Vincent established the natural logarithm as the area of a hyperbolic sector, or an equivalent area against an asymptote. In spacetime theory, the connection of events by light divides the universe into Past, Future, or Elsewhere based on a Here and Now . On any line in space, a light beam may be directed left or right. Take the x-axis as the events passed by the right beam and the y-axis as the events of the left beam.
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Miessler G.L. and Tarr D.A. Inorganic Chemistry (2nd ed., Prentice-Hall 1998, p.170) (To prevent ambiguity, in the rest of this article, "strong acid" will, unless otherwise stated, refer to an acid that is strong as measured by its pKa value (pKa < –1.74). This usage is consistent with the common parlance of most practicing chemists.) When the acidic medium in question is a dilute aqueous solution, the H0 is approximately equal to the pH value, which is a negative logarithm of the concentration of aqueous H+ in solution.
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The hydroxide ion is a natural part of water because of the self-ionization reaction in which its complement, hydronium, is passed hydrogen: :H3O+ \+ OH− 2H2O The equilibrium constant for this reaction, defined as :Kw = [H+][OH−][H+] denotes the concentration of hydrogen cations and [OH−] the concentration of hydroxide ions has a value close to 10−14 at 25 °C, so the concentration of hydroxide ions in pure water is close to 10−7 mol∙dm−3, in order to satisfy the equal charge constraint. The pH of a solution is equal to the decimal cologarithm of the hydrogen cation concentration;Strictly speaking pH is the cologarithm of the hydrogen cation activity the pH of pure water is close to 7 at ambient temperatures. The concentration of hydroxide ions can be expressed in terms of pOH, which is close to (14 − pH),pOH signifies the minus the logarithm to base 10 of [OH−], alternatively the logarithm of so the pOH of pure water is also close to 7. Addition of a base to water will reduce the hydrogen cation concentration and therefore increase the hydroxide ion concentration (increase pH, decrease pOH) even if the base does not itself contain hydroxide.
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In order for this generator to be secure, the prime number p needs to be large enough so that computing discrete logarithms modulo p is infeasible. To be more precise, any method that predicts the numbers generated will lead to an algorithm that solves the discrete logarithm problem for that prime. There is a paper discussing possible examples of the quantum permanent compromise attack to the Blum–Micali construction. This attacks illustrate how a previous attack to the Blum–Micali generator can be extended to the whole Blum–Micali construction, including the Blum Blum Shub and Kaliski generators.
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By measuring a series of standards and creating the standard curve, it is possible to quantify the amount or concentration of a substance within a sample by determining the absorbance on the Spec 20 and finding the corresponding concentration on the calibration curve. Alternatively, the logarithm of percent transmittance can be plotted versus concentration to create a standard curve using the same procedure. The absorbance measured by the Spectronic 20 is the sum of the absorbance of each of the constituents of the solution. Therefore, the Spectronic 20 can be used to analyze more complex solutions.
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His algorithms include: Baby-step giant-step algorithm for computing the discrete logarithm, which is useful in public-key cryptography; Shanks's square forms factorization, an integer factorization method that generalizes Fermat's factorization method; and the Tonelli–Shanks algorithm that finds square roots modulo a prime, which is useful for the quadratic sieve method of integer factorization. In 1974, Shanks and John Wrench did some of the first computer work on estimating the value of Brun's constant, the sum of the reciprocals of the twin primes, calculating it over the twin primes among the first two million primes.
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In applied mathematics, stretching fields provide the local deformation of an infinitesimal circular fluid element over a finite time interval ∆t. The logarithm of the stretching (after first dividing by ∆t) gives the finite-time Lyapunov exponent λ for separation of nearby fluid elements at each point in a flow. For periodic two-dimensional flows, stretching fields have been shown to be closely related to the mixing of a passive scalar concentration field. Until recently, however, the extension of these ideas to systems that are non- periodic or weakly turbulent has been possible only in numerical simulations.
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In probability theory and intertemporal portfolio choice, the Kelly criterion (or Kelly strategy, Kelly bet, ...), also known as the scientific gambling method, is a formula for bet sizing that leads almost surely to higher wealth compared to any other strategy in the long run (i.e. approaching the limit as the number of bets goes to infinity). The Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric growth rate. The Kelly Criterion is to bet a predetermined fraction of assets, and it can seem counterintuitive.
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Pickover stalks are certain kinds of details that are empirically found in the Mandelbrot set in the study of fractal geometry. In the 1980s, Pickover proposed that experimental mathematicians and computer artists examine the behavior of orbit trajectories for the Mandelbrot in order to study how closely the orbits of interior points come to the x and y axes in the complex plane. In some renditions of this behavior, the closer that the point approaches, the higher up the color scale, with red denoting the closest approach. The logarithm of the distance is taken to accentuate the details.
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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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An ivory set of Napier's Bones from around 1650 A set of Napier's calculating tables from around 1680 His work, Mirifici Logarithmorum Canonis Descriptio (1614) contained fifty-seven pages of explanatory matter and ninety pages of tables of numbers related to natural logarithms (see Napierian logarithm). The book also has an excellent discussion of theorems in spherical trigonometry, usually known as Napier's Rules of Circular Parts. See also Pentagramma mirificum. Modern English translations of both Napier's books on logarithms and their description can be found on the web, as well as a discussion of Napier's bones and Promptuary (another early calculating device).
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The distribution of X2 is a chi-squared distribution for the following reason; under the null hypothesis for test i, the p-value pi follows a uniform distribution on the interval [0,1]. The negative natural logarithm of a uniformly distributed value follows an exponential distribution. Scaling a value that follows an exponential distribution by a factor of two yields a quantity that follows a chi-squared distribution with two degrees of freedom. Finally, the sum of k independent chi-squared values, each with two degrees of freedom, follows a chi-squared distribution with 2k degrees of freedom.
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In quantum computing, the quantum Fourier transform (for short: QFT) is a linear transformation on quantum bits, and is the quantum analogue of the inverse discrete Fourier transform. The quantum Fourier transform is a part of many quantum algorithms, notably Shor's algorithm for factoring and computing the discrete logarithm, the quantum phase estimation algorithm for estimating the eigenvalues of a unitary operator, and algorithms for the hidden subgroup problem. The quantum Fourier transform was invented by Don Coppersmith. The quantum Fourier transform can be performed efficiently on a quantum computer, with a particular decomposition into a product of simpler unitary matrices.
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The forking lemma was later generalized by Mihir Bellare and Gregory Neven.Mihir Bellare and Gregory Neven, "Multi-Signatures in the Plain Public-Key Model and a General Forking Lemma", Proceedings of the 13th Association for Computing Machinery (ACM) Conference on Computer and Communications Security (CCS), Alexandria, Virginia, 2006, pp. 390-399. The forking lemma has been used and further generalized to prove the security of a variety of digital signature schemes and other random-oracle based cryptographic constructions. Ali Bagherzandi, Jung Hee Cheon, Stanislaw Jarecki: Multisignatures secure under the discrete logarithm assumption and a generalized forking lemma.
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Commonly used substitution matrices include the blocks substitution (BLOSUM) and point accepted mutation (PAM) matrices. Both are based on taking sets of high-confidence alignments of many homologous proteins and assessing the frequencies of all substitutions, but they are computed using different methods. Scores within a BLOSUM are log-odds scores that measure, in an alignment, the logarithm for the ratio of the likelihood of two amino acids appearing with a biological sense and the likelihood of the same amino acids appearing by chance. The matrices are based on the minimum percentage identity of the aligned protein sequence used in calculating them.
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Michael Thoreau Lacey (born September 26, 1959)The Library of Congress "Lacey, Michael T." is an American mathematician. Lacey received his Ph.D. from the University of Illinois at Urbana-Champaign in 1987, under the direction of Walter Philipp.. His thesis was in the area of probability in Banach spaces, and solved a problem related to the law of the iterated logarithm for empirical characteristic functions. In the intervening years, his work has touched on the areas of probability, ergodic theory, and harmonic analysis. His first postdoctoral positions were at the Louisiana State University, and the University of North Carolina at Chapel Hill.
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The Jacobian on a hyperelliptic curve is an Abelian group and as such it can serve as group for the discrete logarithm problem (DLP). In short, suppose we have an Abelian group G and g an element of G, the DLP on G entails finding the integer a given two elements of G, namely g and g^a. The first type of group used was the multiplicative group of a finite field, later also Jacobians of (hyper)elliptic curves were used. If the hyperelliptic curve is chosen with care, then Pollard's rho method is the most efficient way to solve DLP.
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Historically, weak coloring served as the first non-trivial example of a graph problem that can be solved with a local algorithm (a distributed algorithm that runs in a constant number of synchronous communication rounds). More precisely, if the degree of each node is odd and bounded by a constant, then there is a constant-time distributed algorithm for weak 2-coloring.. This is different from (non-weak) vertex coloring: there is no constant-time distributed algorithm for vertex coloring; the best possible algorithms (for finding a minimal but not necessarily minimum coloring) require communication rounds... Here is the iterated logarithm of .
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The aforementioned process achieves a t-bit security level with 4t-bit signatures. For example, a 128-bit security level would require 512-bit (64-byte) signatures. The security is limited by discrete logarithm attacks on the group, which have a complexity of the square-root of the group size. In Schnorr's original 1991 paper, it was suggested that since collision resistance in the hash is not required, then therefore shorter hash functions may be just as secure, and indeed recent developments suggest that a t-bit security level can be achieved with 3t-bit signatures.
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It seems that Zipf's law holds for frequency lists drawn from longer texts of any natural language. Frequency lists are a useful tool when building an electronic dictionary, which is a prerequisite for a wide range of applications in computational linguistics. German linguists define the Häufigkeitsklasse (frequency class) N of an item in the list using the base 2 logarithm of the ratio between its frequency and the frequency of the most frequent item. The most common item belongs to frequency class 0 (zero) and any item that is approximately half as frequent belongs in class 1.
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In 1791, Thomas Wright Hill courageously tried to save the apparatus of Dr Joseph Priestley from a mob in the Birmingham 'Church and King' riots of 1791—the offer was declined. He was interested in astronomy, being a Fellow of the Royal Astronomical Society, and in computers, as is shown by a letter of his to Charles Babbage, dated 23 March 1836, among the Babbage manuscripts at the British Library, returning some logarithm tables that he had borrowed and adding "How happy I shall be when I can see such a work verified and enlarged by your divine machine".
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"Why is infrared, or IR for short, bad?" Eyewear is rated for optical density (OD), which is the base-10 logarithm of the attenuation factor by which the optical filter reduces beam power. For example, eyewear with OD 3 will reduce the beam power in the specified wavelength range by a factor of 1000. In addition to an optical density sufficient to reduce beam power to below the maximum permissible exposure (see above), laser eyewear used where direct beam exposure is possible should be able to withstand a direct hit from the laser beam without breaking.
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Usually, both parameters ~u~ and ~v~ are supposed to be positive; then this speed-proportional friction coefficient grows exponentially at large positive values of coordinate ~x~. The potential ~\Phi(x)=e^x-x-1~ is a fixed function, which also shows exponential growth at large positive values of coordinate ~x~. In the application in laser physics, ~x~ may have a sense of logarithm of number of photons in the laser cavity, related to its steady-state value. Then, the output power of such a laser is proportional to ~\exp(x)~ and may show pulsation at oscillation of ~x~.
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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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Later, scientists such as Ludwig Boltzmann, Josiah Willard Gibbs, and James Clerk Maxwell gave entropy a statistical basis. In 1877 Boltzmann visualized a probabilistic way to measure the entropy of an ensemble of ideal gas particles, in which he defined entropy as proportional to the natural logarithm of the number of microstates such a gas could occupy. Henceforth, the essential problem in statistical thermodynamics has been to determine the distribution of a given amount of energy E over N identical systems. Carathéodory linked entropy with a mathematical definition of irreversibility, in terms of trajectories and integrability.
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Since modular arithmetic has such a wide range of applications, it is important to know how hard it is to solve a system of congruences. A linear system of congruences can be solved in polynomial time with a form of Gaussian elimination, for details see linear congruence theorem. Algorithms, such as Montgomery reduction, also exist to allow simple arithmetic operations, such as multiplication and exponentiation modulo , to be performed efficiently on large numbers. Some operations, like finding a discrete logarithm or a quadratic congruence appear to be as hard as integer factorization and thus are a starting point for cryptographic algorithms and encryption.
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Kozen states that Cobham and Edmonds are "generally credited with the invention of the notion of polynomial time." Cobham invented the class as a robust way of characterizing efficient algorithms, leading to Cobham's thesis. However, H. C. Pocklington, in a 1910 paper, analyzed two algorithms for solving quadratic congruences, and observed that one took time "proportional to a power of the logarithm of the modulus" and contrasted this with one that took time proportional "to the modulus itself or its square root", thus explicitly drawing a distinction between an algorithm that ran in polynomial time versus one that did not.
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On the other hand, the global symmetry group is an observable so it is essential that it is the same, SU(M), in both descriptions. The dual magnetic theory is free in the infrared, the coupling constant shrinks logarithmically, and so by the Dirac quantization condition the electric coupling constant grows logarithmically in the infrared. This implies that the potential between two electric charges, at long distances, scales as the logarithm of their distance divided by the distance. When M is between 3N/2 and 3N, in the theory has an infrared fixed point where it becomes a nontrivial conformal field theory.
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Although there are infinitely many possible values for a general complex logarithm, there are only a finite number of values for the power in the important special case where and is a positive integer. These are the th roots of ; they are solutions of the equation . As with real roots, a second root is also called a square root and a third root is also called a cube root. It is usual in mathematics to define as the principal value of the root, which is, conventionally, the th root whose argument has the smallest absolute value.
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When subjected to an electric field in PAGE, the negatively charged polypeptide chains travel toward the anode with different mobility. Their mobility, or the distance traveled by molecules, is inversely proportional to the logarithm of their molecular weight. By comparing the relative ratio of the distance traveled by each protein to the length of the gel (Rf) one can make conclusions about the relative molecular weight of the proteins, where the length of the gel is determined by the distance traveled by a small molecule like a tracking dye. For nucleic acids, urea is the most commonly used denaturant.
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Al-Khwarizmi: The Inventor of Algebra, by Corona Brezina (2006) Al-Khwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through his other book, the Algebra.Foremost mathematical texts in history, according to Carl B. Boyer. In late medieval Latin, algorismus, the corruption of his name, simply meant the "decimal number system" that is still the meaning of modern English algorism. During the 17th century, the French form for the word – but not its meaning – was changed to algorithm, following the model of the word logarithm, this form alluding to the ancient Greek .
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A plot of the logarithm of the deposition rate against the reciprocal of the absolute temperature in the surface-reaction-limited region results in a straight line whose slope is equal to –qEa/k. At reduced pressure levels for VLSI manufacturing, polysilicon deposition rate below 575 °C is too slow to be practical. Above 650 °C, poor deposition uniformity and excessive roughness will be encountered due to unwanted gas-phase reactions and silane depletion. Pressure can be varied inside a low-pressure reactor either by changing the pumping speed or changing the inlet gas flow into the reactor.
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The graph of for real and . The upper branch (blue) with is the graph of the function (principal branch), the lower branch (magenta) with is the graph of the function . The minimum value of x is at {−1/e,−1} In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the inverse relation of the function , where is any complex number and is the exponential function. For each integer there is one branch, denoted by , which is a complex-valued function of one complex argument.
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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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Years later he declined to join Mensa International, saying that his IQ was too low. Physicist Steve Hsu stated of the test: When Feynman was 15, he taught himself trigonometry, advanced algebra, infinite series, analytic geometry, and both differential and integral calculus. Before entering college, he was experimenting with and deriving mathematical topics such as the half-derivative using his own notation. He created special symbols for logarithm, sine, cosine and tangent functions so they did not look like three variables multiplied together, and for the derivative, to remove the temptation of canceling out the d's.
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In cryptography, for certain groups, it is assumed that the DHP is hard, and this is often called the Diffie–Hellman assumption. The problem has survived scrutiny for a few decades and no "easy" solution has yet been publicized. As of 2006, the most efficient means known to solve the DHP is to solve the discrete logarithm problem (DLP), which is to find x given g and gx. In fact, significant progress (by den Boer, Maurer, Wolf, Boneh and Lipton) has been made towards showing that over many groups the DHP is almost as hard as the DLP.
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Integrated Encryption Scheme (IES) is a hybrid encryption scheme which provides semantic security against an adversary who is allowed to use chosen- plaintext and chosen-ciphertext attacks. The security of the scheme is based on the computational Diffie–Hellman problem. Two incarnations of the IES are standardized: Discrete Logarithm Integrated Encryption Scheme (DLIES) and Elliptic Curve Integrated Encryption Scheme (ECIES), which is also known as the Elliptic Curve Augmented Encryption Scheme or simply the Elliptic Curve Encryption Scheme. These two incarnations are identical up to the change of an underlying group and so to be concrete we concentrate on the latter.
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In Book 6, part 4, page 586, Proposition CIX, he proves that if the abscissas of points are in geometric proportion, then the areas between a hyperbola and the abscissas are in arithmetic proportion. This finding allowed Saint-Vincent's former student, Alphonse Antonio de Sarasa, to prove that the area between a hyperbola and the abscissa of a point is proportional to the abscissa's logarithm, thus uniting the algebra of logarithms with the geometry of hyperbolas. See also: Enrique A. González-Velasco, Journey through Mathematics: Creative Episodes in Its History (New York, New York: Springer, 2011), page 118.
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Proofs of knowledge relying on the discrete logarithm problem for groups of known order and on the special RSA problem for groups of hidden order form the basis for most of today's group signature and anonymous credential systems. Moreover, direct anonymous attestation a protocol for authenticating trusted platform modules is based on the same techniques. Direct anonymous attestation can be seen as the first commercial application of multi show anonymous digital credentials, even though in this case credentials are not attached to persons, but to chips and consequently computer platforms. From an applications' point of view, the main advantage of Camenisch et al.
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Pollard's rho algorithm for logarithms is an algorithm introduced by John Pollard in 1978 to solve the discrete logarithm problem, analogous to Pollard's rho algorithm to solve the integer factorization problem. The goal is to compute \gamma such that \alpha ^ \gamma = \beta, where \beta belongs to a cyclic group G generated by \alpha. The algorithm computes integers a, b, A, and B such that \alpha^a \beta^b = \alpha^A \beta^B. If the underlying group is cyclic of order n, \gamma is one of the solutions of the equation (B-b) \gamma = (a-A) \pmod n.
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Consequently, the interaction may be characterised by a dimensionful parameter , namely the value of the RG scale at which the coupling constant diverges. In the case of quantum chromodynamics, this energy scale is called the QCD scale, and its value 220 MeV supplants the role of the original dimensionless coupling constant in the form of the logarithm (at one- loop) of the ratio and . Perturbation theory, which produced this type of running formula, is only valid for a (dimensionless) coupling ≪ 1. In the case of QCD, the energy scale is an infrared cutoff, such that implies , with the RG scale.
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A number of fields such as stellar photometry, Gaussian beam characterization, and emission/absorption line spectroscopy work with sampled Gaussian functions and need to accurately estimate the height, position, and width parameters of the function. There are three unknown parameters for a 1D Gaussian function (a, b, c) and five for a 2D Gaussian function (A; x_0,y_0; \sigma_X,\sigma_Y). The most common method for estimating the Gaussian parameters is to take the logarithm of the data and fit a parabola to the resulting data set.Hongwei Guo, "A simple algorithm for fitting a Gaussian function," IEEE Sign. Proc. Mag.
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An amount of (classical) physical information may be quantified, as in information theory, as follows.Claude E. Shannon and Warren Weaver, Mathematical Theory of Communication, University of Illinois Press, 1963. For a system S, defined abstractly in such a way that it has N distinguishable states (orthogonal quantum states) that are consistent with its description, the amount of information I(S) contained in the system's state can be said to be log(N). The logarithm is selected for this definition since it has the advantage that this measure of information content is additive when concatenating independent, unrelated subsystems; e.g.
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In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters. A Poisson regression model is sometimes known as a log-linear model, especially when used to model contingency tables. Negative binomial regression is a popular generalization of Poisson regression because it loosens the highly restrictive assumption that the variance is equal to the mean made by the Poisson model.
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Binary search trees allow binary search for fast lookup, addition and removal of data items, and can be used to implement dynamic sets and lookup tables. The order of nodes in a BST means that each comparison skips about half of the remaining tree, so the whole lookup takes time proportional to the binary logarithm of the number of items stored in the tree. This is much better than the linear time required to find items by key in an (unsorted) array, but slower than the corresponding operations on hash tables. Several variants of the binary search tree have been studied.
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In algebraic geometry, the Witten conjecture is a conjecture about intersection numbers of stable classes on the moduli space of curves, introduced by Edward Witten in the paper , and generalized in . Witten's original conjecture was proved by Maxim Kontsevich in the paper . Witten's motivation for the conjecture was that two different models of 2-dimensional quantum gravity should have the same partition function. The partition function for one of these models can be described in terms of intersection numbers on the moduli stack of algebraic curves, and the partition function for the other is the logarithm of the τ-function of the KdV hierarchy.
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Both are written as exponentiation modulo a composite number, and both are related to the problem of prime factorization. Functions related to the hardness of the discrete logarithm problem (either modulo a prime or in a group defined over an elliptic curve) are not known to be trapdoor functions, because there is no known "trapdoor" information about the group that enables the efficient computation of discrete logarithms. A trapdoor in cryptography has the very specific aforementioned meaning and is not to be confused with a backdoor (these are frequently used interchangeably, which is incorrect). A backdoor is a deliberate mechanism that is added to a cryptographic algorithm (e.g.
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The first, that applies where only heat changes cause a change in entropy, is the entropy change (\Delta S) to a system containing a sub-system which undergoes heat transfer to its surroundings (inside the system of interest). It is based on the macroscopic relationship between heat flow into the sub-system and the temperature at which it occurs summed over the boundary of that sub-system. The second calculates the absolute entropy (S) of a system based on the microscopic behaviour of its individual particles. This is based on the natural logarithm of the number of microstates possible in a particular macrostate (\Omega) called the thermodynamic probability.
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Similar to how the concentration of hydrogen ion determines the acidity or pH of an aqueous solution, the tendency of electron transfer between a chemical species and an electrode determines the redox potential of an electrode couple. Like pH, redox potential represents how easily electrons are transferred to or from species in solution. Redox potential characterises the ability under the specific condition of a chemical species to lose or gain electrons instead of the amount of electrons available for oxidation or reduction. In fact, it is possible to define pe, the negative logarithm of electron concentration (−log[e−]) in a solution, which will be directly proportional to the redox potential.
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Many of the techniques he used and conclusions he drew would drive the field forward. Early analysis relied on statistical interpretation through processes such as LOD (logarithm of odds) scores of pedigrees and other observational methods such as affected sib-pairs, which looks at phenotype and IBD (identity by descent) configuration. Many of the disorders studied early on including Alzheimer's, Huntington's and amyotrophic lateral sclerosis (ALS) are still at the center of much research to this day. By the late 1980s new advances in genetics such as recombinant DNA technology and reverse genetics allowed for the broader use of DNA polymorphisms to test for linkage between DNA and gene defects.
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A unique feature of the Mark II is that it had built- in hardware for several functions such as the reciprocal, square root, logarithm, exponential, and some trigonometric functions. These took between five and twelve seconds to execute. The Mark I and Mark II were not a stored- program computer – it read an instruction of the program one at a time from a tape and executed it (like the Mark I). This separation of data and instructions is known as the Harvard architecture. The Mark II had a peculiar programming method that was devised to ensure that the contents of a register were available when needed.
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See, e.g., Marlow Anderson, Victor J. Katz, Robin J. Wilson, Sherlock Holmes in Babylon and Other Tales of Mathematical History, Mathematical Association of America, 2004, p. 114\. The first full proof of the fundamental theorem of calculus was given by Isaac Barrow. Translator: J. M. Child (1916)Review of J.M. Child's translation (1916) The geometrical lectures of Isaac Barrow reviewer: Arnold Dresden (Jun 1918) p.454 Barrow has the fundamental theorem of calculus Shaded area of one unit square measure when x = 2.71828... The discovery of Euler’s number e, and its exploitation with functions ex and natural logarithm, completed integration theory for calculus of rational functions.
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A Cox point process, Cox process or doubly stochastic Poisson process is a generalization of the Poisson point process by letting its intensity measure \textstyle \Lambda to be also random and independent of the underlying Poisson process. The process is named after David Cox who introduced it in 1955, though other Poisson processes with random intensities had been independently introduced earlier by Lucien Le Cam and Maurice Quenouille. The intensity measure may be a realization of random variable or a random field. For example, if the logarithm of the intensity measure is a Gaussian random field, then the resulting process is known as a log Gaussian Cox process.
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The National Cyclopaedia of Useful Knowledge, Vol III, (1847), London, Charles Knight, p.808 At this time, Briggs obtained a copy of Mirifici Logarithmorum Canonis Descriptio, in which Napier introduced the idea of logarithms. It has also been suggested that he knew of the method outlined in Fundamentum Astronomiae published by the Swiss clockmaker Jost Bürgi, through John Dee. Napier's formulation was awkward to work with, but the book fired Briggs' imagination – in his lectures at Gresham College he proposed the idea of base 10 logarithms in which the logarithm of 10 would be 1; and soon afterwards he wrote to the inventor on the subject.
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A page from Henry Briggs' 1617 Logarithmorum Chilias Prima showing the base-10 (common) logarithm of the integers 0 to 67 to fourteen decimal places. In 1616 Briggs visited Napier at Edinburgh in order to discuss the suggested change to Napier's logarithms. The following year he again visited for a similar purpose. During these conferences the alteration proposed by Briggs was agreed upon; and on his return from his second visit to Edinburgh, in 1617, he published the first chiliad of his logarithms. In 1619 he was appointed Savilian Professor of Geometry at the University of Oxford, and resigned his professorship of Gresham College in July 1620.
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The rank-size rule (or law) describes the remarkable regularity in many phenomena, including the distribution of city sizes, the sizes of businesses, the sizes of particles (such as sand), the lengths of rivers, the frequencies of word usage, and wealth among individuals. All are real-world observations that follow power laws, such as Zipf's law, the Yule distribution, or the Pareto distribution. If one ranks the population size of cities in a given country or in the entire world and calculates the natural logarithm of the rank and of the city population, the resulting graph will show a log-linear pattern. This is the rank-size distribution.
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Logarithm of the relative energy output (ε) of proton–proton (p-p), CNO, and triple-α fusion processes at different temperatures. The dashed line shows the combined energy generation of the p-p and CNO processes within a star. The CNO cycle (for carbon–nitrogen–oxygen) is one of the two known sets of fusion reactions by which stars convert hydrogen to helium, the other being the proton–proton chain reaction (p-p cycle), which is more efficient at the Sun's core temperature. The CNO cycle is hypothesized to be dominant in stars that are more than 1.3 times as massive as the Sun.
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Modelling the recent common ancestry of all living humans. Nature. 2004: Vol 43: 562-565. They modeled the human population as a set of randomly mating subpopulations that are connected by occasional migrants. If the size of the population is n, then the time to the most recent common ancestor is a small multiple of the base-2 logarithm of n, even if the levels of migration among the populations are very low. Using a model of the world’s landmasses and populations with moderate levels of migration, the authors calculated that the most recent common ancestor could have lived as recently as AD 55.
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Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter for the base of the natural logarithm (now also known as Euler's number), the Greek letter Σ for summations and the letter to denote the imaginary unit. The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it originated with Welsh mathematician William Jones.
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The history of logarithms in seventeenth-century Europe is the discovery of a new function that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms). Prior to Napier's invention, there had been other techniques of similar scopes, such as the prosthaphaeresis or the use of tables of progressions, extensively developed by Jost Bürgi around 1600. Napier coined the term for logarithm in Middle Latin, "logarithmorum," derived from the Greek, literally meaning, "ratio-number," from logos "proportion, ratio, word" + arithmos "number".
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The above equation is a good approximation only when the argument of the logarithm is much larger than unity – the concept of an ideal gas breaks down at low values of . Nevertheless, there will be a "best" value of the constant in the sense that the predicted entropy is as close as possible to the actual entropy, given the flawed assumption of ideality. A quantum- mechanical derivation of this constant is developed in the derivation of the Sackur–Tetrode equation which expresses the entropy of a monatomic ( = ) ideal gas. In the Sackur–Tetrode theory the constant depends only upon the mass of the gas particle.
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As n, the number of compounding periods per year, increases without limit, the case is known as continuous compounding, in which case the effective annual rate approaches an upper limit of , where is a mathematical constant that is the base of the natural logarithm. Continuous compounding can be thought of as making the compounding period infinitesimally small, achieved by taking the limit as n goes to infinity. See definitions of the exponential function for the mathematical proof of this limit. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as :P(t)=P_0 e ^ {rt}.
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The point on the finite length slowed down as it reached the end of the line, so never actually reaching it. He used the distance between P & Q to define the logarithm. By repeated subtractions Napier calculated for L ranging from 1 to 100. The result for L=100 is approximately 0.99999 = 1 − 10−5. Napier then calculated the products of these numbers with for L from 1 to 50, and did similarly with and . These computations, which occupied 20 years, allowed him to give, for any number N from 5 to 10 million, the number L that solves the equation :N=10^7 (1-10^{-7})^L.
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The Blackmer RMS detector is an electronic true RMS converter invented by David E. Blackmer in 1971. The Blackmer detector, coupled with the Blackmer gain cell, forms the core of the dbx noise reduction system and various professional audio signal processors developed by dbx, Inc. Unlike earlier RMS detectors that time-averaged algebraic square of input signal, the Blackmer detector performs time-averaging on the logarithm of the input, being the first successful, commercialized instance of log-domain filter. The circuit, created by trial and error, computes root mean squared of various waveforms with high precision, although exact nature of its operation was not known to the inventor.
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Above, it was assumed that the variable X_i was being tested for normal distribution. Any other family of distributions can be tested but the test for each family is implemented by using a different modification of the basic test statistic and this is referred to critical values specific to that family of distributions. The modifications of the statistic and tables of critical values are given by Stephens (1986) for the exponential, extreme-value, Weibull, gamma, logistic, Cauchy, and von Mises distributions. Tests for the (two-parameter) log-normal distribution can be implemented by transforming the data using a logarithm and using the above test for normality.
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In photography reciprocity is the inverse relationship between the intensity and duration of light that determines the reaction of light-sensitive material. Within a normal exposure range for film stock, for example, the reciprocity law states that the film response will be determined by the total exposure, defined as intensity × time. Therefore, the same response (for example, the optical density of the developed film) can result from reducing duration and increasing light intensity, and vice versa. The reciprocal relationship is assumed in most sensitometry, for example when measuring a Hurter and Driffield curve (optical density versus logarithm of total exposure) for a photographic emulsion.
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In group theory, a branch of mathematics, the baby-step giant-step is a meet- in-the-middle algorithm for computing the discrete logarithm or order of an element in a finite abelian group due to Daniel Shanks. The discrete log problem is of fundamental importance to the area of public key cryptography. Many of the most commonly used cryptography systems are based on the assumption that the discrete log is extremely difficult to compute; the more difficult it is, the more security it provides a data transfer. One way to increase the difficulty of the discrete log problem is to base the cryptosystem on a larger group.
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He translated and improved the most commonly used logarithm tables in Chile, and invented a new way of calculating divisions by creating a table that allowed mathematicians to divide any number up to 10,000 with a simple sum. He also improved the Lalande algorithm tables which were widely used by engineers, architects, surveyors, merchants, or anyone needing to solve complex mathematical problems. Picarte asked other Chilean mathematicians examine his work but did not receive an enthusiastic response. He tried to sell the copyrights for it at a very low price so that it could be published and distributed, but was unable to find a buyer.
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Their model demonstrated that with the addition of only a small number of long-range links, a regular graph, in which the diameter is proportional to the size of the network, can be transformed into a "small world" in which the average number of edges between any two vertices is very small (mathematically, it should grow as the logarithm of the size of the network), while the clustering coefficient stays large. It is known that a wide variety of abstract graphs exhibit the small-world property, e.g., random graphs and scale-free networks. Further, real world networks such as the World Wide Web and the metabolic network also exhibit this property.
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The decibannage represented the reduction in (the logarithm of) the total number of possibilities (similar to the change in the Hartley information); and also the log-likelihood ratio (or change in the weight of evidence) that could be inferred for one hypothesis over another from a set of observations. The expected change in the weight of evidence is equivalent to what was later called the Kullback discrimination information. But underlying this notion was still the idea of equal a-priori probabilities, rather than the information content of events of unequal probability; nor yet any underlying picture of questions regarding the communication of such varied outcomes.
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Forward secrecy is designed to prevent the compromise of a long- term secret key from affecting the confidentiality of past conversations. However, forward secrecy cannot defend against a successful cryptanalysis of the underlying ciphers being used, since a cryptanalysis consists of finding a way to decrypt an encrypted message without the key, and forward secrecy only protects keys, not the ciphers themselves. A patient attacker can capture a conversation whose confidentiality is protected through the use of public-key cryptography and wait until the underlying cipher is broken (e.g. large quantum computers could be created which allow the discrete logarithm problem to be computed quickly).
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Diffie–Hellman key exchange depends for its security on the presumed difficulty of solving the discrete logarithm problem. The authors took advantage of the fact that the number field sieve algorithm, which is generally the most effective method for finding discrete logarithms, consists of four large computational steps, of which the first three depend only on the order of the group G, not on the specific number whose finite log is desired. If the results of the first three steps are precomputed and saved, they can be used to solve any discrete log problem for that prime group in relatively short time. This vulnerability was known as early as 1992.
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One way the safe-life approach is planning and envisaging the toughness of the mechanisms in the automotive industry. When the repetitive loading on mechanical structures intensified with the advent of the steam engine, back in the mid-1800s, this approach was established (Oja 2013). According to Michael Oja, “Engineers and academics began to understand the effect that cyclic stress (or strain) has on the life of a component; a curve was developed relating the magnitude of the cyclic stress (S) to the logarithm of the number of cycles to failure (N)” (Oja 2013). The S-N curve because the fundamental relation is in safe life designs.
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Thorne has investigated the quantum statistical mechanical origin of the entropy of a black hole. With his postdoc Wojciech Zurek, he showed that the entropy of a black hole is the logarithm of the number of ways that the hole could have been made. With Igor Novikov and Don Page, he developed the general relativistic theory of thin accretion disks around black holes, and using this theory he deduced that with a doubling of its mass by such accretion a black hole will be spun up to 0.998 of the maximum spin allowed by general relativity, but not any farther. This is probably the maximum black- hole spin allowed in nature.
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The Science of Conjecture: Evidence and Probability before Pascal, James Franklin, JHU Press, 2015, Caramuel's Mathesis biceps presents some original contributions to the field of mathematics: he proposed a new method of approximation for trisecting an angle and proposed a form of logarithm that prefigure cologarithms, although he was not understood by his contemporaries.Juan Vernet, Dictionary of Scientific Biography [1971], cited in Jens Høyrup, Barocco e scienza secentesca: un legame inesistente?, published in Analecta Romana Instituti Danici, 25 (1997), 141-172. Caramuel was also the first mathematician who made a reasoned study on non-decimal counts, thus making a significant contribution to the development of the binary numeral system.
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The effective population size is most commonly measured with respect to the coalescence time. In an idealised diploid population with no selection at any locus, the expectation of the coalescence time in generations is equal to twice the census population size. The effective population size is measured as within-species genetic diversity divided by four times the mutation rate \mu, because in such an idealised population, the heterozygosity is equal to 4N\mu. In a population with selection at many loci and abundant linkage disequilibrium, the coalescent effective population size may not reflect the census population size at all, or may reflect its logarithm.
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Film speed is used in the exposure equations to find the appropriate exposure parameters. Four variables are available to the photographer to obtain the desired effect: lighting, film speed, f-number (aperture size), and shutter speed (exposure time). The equation may be expressed as ratios, or, by taking the logarithm (base 2) of both sides, by addition, using the APEX system, in which every increment of 1 is a doubling of exposure; this increment is commonly known as a "stop". The effective f-number is proportional to the ratio between the lens focal length and aperture diameter, the diameter itself being proportional to the square root of the aperture area.
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It then replaces by the subset that it has determined to contain the bottleneck weight, and starts the next iteration with this new set . The number of subsets into which can be split increases exponentially with each step, so the number of iterations is proportional to the iterated logarithm function, , and the total time is . In a model of computation where each edge weight is a machine integer, the use of repeated bisection in this algorithm can be replaced by a list-splitting technique of , allowing to be split into smaller sets in a single step and leading to a linear overall time bound..
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He notes that multiplication and division could be done with logarithm tables, but to keep the tables small enough, interpolation would be needed and this in turn requires multiplication, though perhaps with less precision. Numbers are to be represented in binary notation. He estimates 27 binary digits (he did not use the term "bit," which was coined by Claude Shannon in 1948) would be sufficient (yielding 8 decimal place accuracy) but rounds up to 30-bit numbers with a sign bit and a bit to distinguish numbers from orders, resulting in 32-bit word he calls a minor cycle. Two’s complement arithmetic is to be used, simplifying subtraction.
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This section should aid in resolving any uncertainties. (see Criticism section for more on the variety of terms) compensation effect/rule : umbrella term for the observed linear relationship between: (i) the logarithm of the preexponential factors and the activation energies, (ii) enthalpies and entropies of activation, or (iii) between the enthalpy and entropy changes of a series of similar reactions. enthalpy- entropy compensation : the linear relationship between either the enthalpies and entropies of activation or the enthalpy and entropy changes of a series of similar reactions. isoequilibrium relation (IER), isoequilibrium effect : On a Van 't Hoff plot, there exists a common intersection point describing the thermodynamics of the reactions.
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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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The Helmholtz free energy has a special theoretical importance since it is proportional to the logarithm of the partition function for the canonical ensemble in statistical mechanics. (Hence its utility to physicists; and to gas-phase chemists and engineers, who do not want to ignore work.) Historically, the term 'free energy' has been used for either quantity. In physics, free energy most often refers to the Helmholtz free energy, denoted by A (or F), while in chemistry, free energy most often refers to the Gibbs free energy. The values of the two free energies are usually quite similar and the intended free energy function is often implicit in manuscripts and presentations.
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Consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition, and to find the symbolic answer whenever it exists. The Risch algorithm, implemented in Mathematica, Maple and other computer algebra systems, does just that for functions and antiderivatives built from rational functions, radicals, logarithm, and exponential functions. Some special integrands occur often enough to warrant special study.
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The LOD score (logarithm (base 10) of odds), developed by Newton Morton, is a statistical test often used for linkage analysis in human, animal, and plant populations. The LOD score compares the likelihood of obtaining the test data if the two loci are indeed linked, to the likelihood of observing the same data purely by chance. Positive LOD scores favour the presence of linkage, whereas negative LOD scores indicate that linkage is less likely. Computerised LOD score analysis is a simple way to analyse complex family pedigrees in order to determine the linkage between Mendelian traits (or between a trait and a marker, or two markers).
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The Tsiolkovsky rocket equation shows that the delta-v of a rocket (stage) is proportional to the logarithm of the fuelled-to-empty mass ratio of the vehicle, and to the specific impulse of the rocket engine. A key goal in designing space-mission trajectories is to minimize the required delta-v to reduce the size and expense of the rocket that would be needed to successfully deliver any particular payload to its destination. The simplest delta-v budget can be calculated with Hohmann transfer, which moves from one circular orbit to another coplanar circular orbit via an elliptical transfer orbit. In some cases a bi-elliptic transfer can give a lower delta-v.
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The distribution of the product of two random variables which have lognormal distributions is again lognormal. This is itself a special case of a more general set of results where the logarithm of the product can be written as the sum of the logarithms. Thus, in cases where a simple result can be found in the list of convolutions of probability distributions, where the distributions to be convolved are those of the logarithms of the components of the product, the result might be transformed to provide the distribution of the product. However this approach is only useful where the logarithms of the components of the product are in some standard families of distributions.
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The function on the larger domain is said to be analytically continued from its values on the smaller domain. This allows the extension of the definition of functions, such as the Riemann zeta function, which are initially defined in terms of infinite sums that converge only on limited domains to almost the entire complex plane. Sometimes, as in the case of the natural logarithm, it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface. All this refers to complex analysis in one variable.
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Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying \mu^\lambda = \kappa. However, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy u^\lambda = \kappa. The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of positive real numbers possess.
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See, for example, Euclid's algorithm for finding the greatest common divisor of two numbers. By the High Middle Ages, the positional Hindu–Arabic numeral system had reached Europe, which allowed for systematic computation of numbers. During this period, the representation of a calculation on paper actually allowed calculation of mathematical expressions, and the tabulation of mathematical functions such as the square root and the common logarithm (for use in multiplication and division) and the trigonometric functions. By the time of Isaac Newton's research, paper or vellum was an important computing resource, and even in our present time, researchers like Enrico Fermi would cover random scraps of paper with calculation, to satisfy their curiosity about an equation.
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These values are typical of the received ranging signals of the GPS, where the navigation message is sent at 50 bit/s (below the channel capacity for the given S/N), and whose bandwidth is spread to around 1 MHz by a pseudo-noise multiplication before transmission. # As stated above, channel capacity is proportional to the bandwidth of the channel and to the logarithm of SNR. This means channel capacity can be increased linearly either by increasing the channel's bandwidth given a fixed SNR requirement or, with fixed bandwidth, by using higher-order modulations that need a very high SNR to operate. As the modulation rate increases, the spectral efficiency improves, but at the cost of the SNR requirement.
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Log reduction is a measure of how thoroughly a decontamination process reduces the concentration of a contaminant. It is defined as the common logarithm of the ratio of the levels of contamination before and after the process, so an increment of 1 corresponds to a reduction in concentration by a factor of 10. In general, an n-log reduction means that the concentration of remaining contaminants is only 10−n times that of the original. So for example, a 0-log reduction is no reduction at all, while a 1-log reduction corresponds to a reduction of 90 percent from the original concentration, and a 2-log reduction corresponds to a reduction of 99 percent from the original concentration.
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Cayley–Klein model of Lobachevsky geometry Before 1900, there was known the Cayley–Klein model of Lobachevsky geometry in the unit disk, according to which geodesic lines are chords of the disk and the distance between points is defined as a logarithm of the cross-ratio of a quadruple. For two-dimensional Riemannian metrics, Eugenio Beltrami (1835–1900) proved that flat metrics are the metrics of constant curvature.E. Beltrami, Risoluzione del Problema: Riportare i punti di una superficie sobra un piano in modo che le linee geodetiche Vengano rappresentate da linee rette, Annali di Matematica Pura ed Applicata, № 7 (1865), 185—204. For multidimensional Riemannian metrics this statement was proved by E. Cartan in 1930.
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This is a measure of interval strength or stability and finality. Notice that it is similar to the more common measure of interval strength, which is determined by its approximation to a lower, stronger, or higher, weaker, position in the harmonic series. The reason for the effect of finality of such interval ratios may be seen as follows. If F = h_2/2^n is the interval ratio in consideration, where n is a positive integer and h_2 is the higher harmonic number of the ratio, then its interval can be determined by taking the base-2 logarithm I=12log_2(h_2/2^n)=12log_2(h_2) - 12n (3/2=7.02 and 4/3=4.98).
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Example: An ion-selective electrode might be calibrated using dilute solutions of the analyte in distilled water. If this calibration is used to calculate the concentration of the analyte in sea water (high ionic strength), significant error is introduced by the difference between the activity of the analyte in the dilute solutions and the concentrated sample. This can be avoided by adding a small amount of ionic-strength buffer to the standards, so that the activity coefficients match more closely. Adding a TISAB buffer to increase the ionic strength of the solution helps to "fix" the ionic strength at a stable level, making a linear correlation between the logarithm of the concentration of analyte and the measured voltage.
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The scorecard tries to predict the probability that the customer, if given the product, would become "bad" within a given timeframe, incurring losses for the lender. The exact definition of what constitutes "bad" varies across different lenders, product types and target markets, however, examples may be "missing three payments within the next 18 months" or "default within the next 12 months". The score given to a customer is usually a three or four digit integer, and in most cases is proportional to the natural logarithm of the odds (or logit) of the customer becoming "bad". In general, a low score indicates a low quality (a high chance of going "bad") and a high score indicates the opposite.
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This amount of information he quantified as :H = \log S^n \, where S was the number of possible symbols, and n the number of symbols in a transmission. The natural unit of information was therefore the decimal digit, much later renamed the hartley in his honour as a unit or scale or measure of information. The Hartley information, H0, is still used as a quantity for the logarithm of the total number of possibilities. A similar unit of log10 probability, the ban, and its derived unit the deciban (one tenth of a ban), were introduced by Alan Turing in 1940 as part of the statistical analysis of the breaking of the German second world war Enigma cyphers.
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After being ingested, clonidine is absorbed into the blood stream rapidly and nearly completely, with peak concentrations in human plasma occurring within 60–90 minutes. Clonidine is fairly lipid soluble with the logarithm of its partition coefficient (log P) equal to 1.6; to compare, the optimal log P to allow a drug that is active in the human central nervous system to penetrate the blood brain barrier is 2.0. Less than half of the absorbed portion of an orally administered dose will be metabolized by the liver into inactive metabolites, with roughly the other half being excreted unchanged by the kidneys. About one-fifth of an oral dose will not be absorbed, and is thus excreted in the feces.
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The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes having just one even number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers.
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For example, to evaluate the form 00: The right-hand side is of the form \infty/\infty, so L'Hôpital's rule applies to it. Note that this equation is valid (as long as the right-hand side is defined) because the natural logarithm (ln) is a continuous function; it is irrelevant how well-behaved f and g may (or may not) be as long as f is asymptotically positive. Although L'Hôpital's rule applies to both 0/0 and \infty/\infty, one of these forms may be more useful than the other in a particular case (because of the possibility of algebraic simplification afterwards). One can change between these forms, if necessary, by transforming f/g to (1/g)/(1/f).
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Smaller molecules travel faster than larger molecules in gel, and double-stranded DNA moves at a rate that is inversely proportional to the logarithm of the number of base pairs. This relationship however breaks down with very large DNA fragments, and separation of very large DNA fragments requires the use of pulsed field gel electrophoresis (PFGE), which applies alternating current from two different directions and the large DNA fragments are separated as they reorient themselves with the changing current. For standard agarose gel electrophoresis, larger molecules are resolved better using a low concentration gel while smaller molecules separate better at high concentration gel. High concentrations gel however requires longer run times (sometimes days).
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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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If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, one can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction. A person can calculate division with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset. A person can use logarithm tables to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result.
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Indeed, several classic results in random graph theory show that even networks with no real topological structure exhibit the small-world phenomenon, which mathematically is expressed as the diameter of the network growing with the logarithm of the number of nodes (rather than proportional to the number of nodes, as in the case for a lattice). This result similarly maps onto networks with a power-law degree distribution, such as scale-free networks. In computer science, the small-world phenomenon (although it is not typically called that) is used in the development of secure peer-to-peer protocols, novel routing algorithms for the Internet and ad hoc wireless networks, and search algorithms for communication networks of all kinds.
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It was developed by George Marsaglia and others in the 1960s. A typical value produced by the algorithm only requires the generation of one random floating-point value and one random table index, followed by one table lookup, one multiply operation and one comparison. Sometimes (2.5% of the time, in the case of a normal or exponential distribution when using typical table sizes) more computations are required. Nevertheless, the algorithm is computationally much faster than the two most commonly used methods of generating normally distributed random numbers, the Marsaglia polar method and the Box–Muller transform, which require at least one logarithm and one square root calculation for each pair of generated values.
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The lack of the notion of prime elements in the group of points on elliptic curves makes it impossible to find an efficient factor base to run index calculus method as presented here in these groups. Therefore this algorithm is incapable of solving discrete logarithms efficiently in elliptic curve groups. However: For special kinds of curves (so called supersingular elliptic curves) there are specialized algorithms for solving the problem faster than with generic methods. While the use of these special curves can easily be avoided, in 2009 it has been proven that for certain fields the discrete logarithm problem in the group of points on general elliptic curves over these fields can be solved faster than with generic methods.
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The discovery of Benford's law goes back to 1881, when the Canadian-American astronomer Simon Newcomb noticed that in logarithm tables the earlier pages (that started with 1) were much more worn than the other pages. (subscription required) Newcomb's published result is the first known instance of this observation and includes a distribution on the second digit, as well. Newcomb proposed a law that the probability of a single number N being the first digit of a number was equal to log(N + 1) − log(N). The phenomenon was again noted in 1938 by the physicist Frank Benford, who tested it on data from 20 different domains and was credited for it.
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As an important special case, which is used as a subroutine in the general algorithm (see below), the Pohlig–Hellman algorithm applies to groups whose order is a prime power. The basic idea of this algorithm is to iteratively compute the p-adic digits of the logarithm by repeatedly "shifting out" all but one unknown digit in the exponent, and computing that digit by elementary methods. (Note that for readability, the algorithm is stated for cyclic groups — in general, G must be replaced by the subgroup \langle g\rangle generated by g, which is always cyclic.) :Input. A cyclic group G of order n=p^e with generator g and an element h\in G. :Output.
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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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In competitive games and sports involving two players or teams in each game or match, the binary logarithm indicates the number of rounds necessary in a single-elimination tournament required to determine a winner. For example, a tournament of players requires rounds to determine the winner, a tournament of teams requires rounds, etc. In this case, for players/teams where is not a power of 2, is rounded up since it is necessary to have at least one round in which not all remaining competitors play. For example, is approximately , which rounds up to , indicating that a tournament of teams requires rounds (either two teams sit out the first round, or one team sits out the second round).
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Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been conceived, we can express his conclusion anachronistically by writing: V = log (F/R).Clagett, Marshall (1961) The Science of Mechanics in the Middle Ages, (Madison: University of Wisconsin Press), pp. 421–40. Bradwardine's analysis is an example of transferring a mathematical technique used by al-Kindi and Arnald of Villanova to quantify the nature of compound medicines to a different physical problem.Murdoch, John E. (1969) "Mathesis in Philosophiam Scholasticam Introducta: The Rise and Development of the Application of Mathematics in Fourteenth Century Philosophy and Theology", in Arts libéraux et philosophie au Moyen Âge (Montréal: Institut d'Études Médiévales), at pp. 224–27.
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Lalande 21185 is a typical type-M main-sequence star (red dwarf) with about 46% of the mass of the Sun and is much cooler than the Sun at 3,828 K. It is intrinsically dim with an absolute magnitude of 10.48, emitting most of its energy in the infrared. Lalande 21185 is a high-proper- motion star moving at about 5 arc seconds a year in an orbit perpendicular to the plane of the Milky Way. The proportion of elements other than hydrogen and helium is estimated based on the ratio of iron to hydrogen in the star when compared to the Sun. The logarithm of this ratio is −0.20, indicating that the proportion of iron is about 10−0.20, or 63% of the Sun.
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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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Potentiometric titrimetry has been the predominant automated titrimetric technique since the 1970s, so it is worthwhile considering the basic differences between it and thermometric titrimetry. Potentiometrically-sensed titrations rely on a free energy change in the reaction system. Measurement of a free energy dependent term is necessary. : ΔG0 = -RT lnK (1) Where: : ΔG0 = change on free energy : R = universal gas constant : T = temperature in kelvins (K) or degrees Rankine (°R) : K = equilibrium constant at temperature T : ln is the natural logarithm function In order for a reaction to be amenable to potentiometric titrimetry, the free energy change must be sufficient for an appropriate sensor to respond with a significant inflection (or "kink") in the titration curve where sensor response is plotted against the amount of titrant delivered.
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Another interpretation of this is that the "inverse" of the complex exponential function is a multivalued function taking each nonzero complex number z to the set of all logarithms of z. There are two solutions to this problem. One is to restrict the domain of the exponential function to a region that does not contain any two numbers differing by an integer multiple of 2πi: this leads naturally to the definition of branches of , which are certain functions that single out one logarithm of each number in their domains. This is analogous to the definition of on as the inverse of the restriction of to the interval : there are infinitely many real numbers θ with , but one arbitrarily chooses the one in .
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Saltwater freezing point Freezing-point depression is the decrease of the freezing point of a solvent on the addition of a non-volatile solute. Examples include salt in water, alcohol in water, or the mixing of two solids such as impurities into a finely powdered drug. In all cases, the substance added/present in smaller amounts is considered the solute, while the original substance present in larger quantity is thought of as the solvent. The resulting liquid solution or solid-solid mixture has a lower freezing point than the pure solvent or solid because the chemical potential of the solvent in the mixture is lower than that of the pure solvent, the difference between the two being proportional to the natural logarithm of the mole fraction.
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Tang's classical algorithm, inspired by the fast quantum algorithm of Kerenidis and Prakash, is able to perform the same calculations but on a normal computer without the need for quantum machine learning. Both approaches run in polylogarithmic time which means the total computation time scales with the logarithm of the problem variables such as the total number of products and users, except Tang utilises a classical replication of the quantum sampling techniques. Prior to Tang's results, it was widely assumed that no fast classical algorithm existed; Kerenidis and Prakash did not attempt to study the classical solution, and the task assigned to Tang by Aaronson was to prove its nonexistence. As he said, "that seemed to me like an important 't' to cross to complete this story".
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They noted that graphs could be classified according to two independent structural features, namely the clustering coefficient, and average node-to-node distance (also known as average shortest path length). Purely random graphs, built according to the Erdős–Rényi (ER) model, exhibit a small average shortest path length (varying typically as the logarithm of the number of nodes) along with a small clustering coefficient. Watts and Strogatz measured that in fact many real-world networks have a small average shortest path length, but also a clustering coefficient significantly higher than expected by random chance. Watts and Strogatz then proposed a novel graph model, currently named the Watts and Strogatz model, with (i) a small average shortest path length, and (ii) a large clustering coefficient.
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Later he applied methods from the metric theory of functions to problems in probability theory and number theory. He became one of the founders of modern probability theory, discovering the law of the iterated logarithm in 1924, achieving important results in the field of limit theorems, giving a definition of a stationary process and laying a foundation for the theory of such processes. Khinchin made significant contributions to the metric theory of Diophantine approximations and established an important result for simple real continued fractions, discovering a property of such numbers that leads to what is now known as Khinchin's constant. He also published several important works on statistical physics, where he used the methods of probability theory, and on information theory, queuing theory and mathematical analysis.
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Even in PC-based implementations, it's a common optimization to speed up sieving by adding approximate logarithms of small primes together. Similarly, TWINKLE has much room for error in its light measurements; as long as the intensity is at about the right level, the number is very likely to be smooth enough for the purposes of known factoring algorithms. The existence of even one large factor would imply that the logarithm of a large number is missing, resulting in a very low intensity; because most numbers have this property, the device's output would tend to consist of stretches of low intensity output with brief bursts of high intensity output. In the above it is assumed that X is square-free, i.e.
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The DSA algorithm works in the framework of public-key cryptosystems and is based on the algebraic properties of modular exponentiation, together with the discrete logarithm problem, which is considered to be computationally intractable. The algorithm uses a key pair consisting of a public key and a private key. The private key is used to generate a digital signature for a message, and such a signature can be verified by using the signer's corresponding public key. The digital signature provides message authentication (the receiver can verify the origin of the message), integrity (the receiver can verify that the message has not been modified since it was signed) and non-repudiation (the sender cannot falsely claim that they have not signed the message).
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Bit-length or bit width is the number of binary digits, called bits, necessary to represent an integer as a binary number. Formally, the number of bits of zero is 1 and any other natural number n>0 is a function, bitLength(n), of the binary logarithm of n: :bitLength(n)= \lfloor log_2(n) + 1 \rfloor = \lceil log_2(n+1) \rceil At their most fundamental level, digital computers and telecommunications devices (as opposed to analog devices) can process only data that has been expressed in binary format. The binary format expresses data as an arbitrary length series of values with one of two choices: Yes/No, 1/0, True/False, etc., all of which can be expressed electronically as On/Off.
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An explanation of this is that although the logarithm of the lognormal density function is quadratic in , yielding a "bowed" shape in a log–log plot, if the quadratic term is small relative to the linear term then the result can appear almost linear, and the lognormal behavior is only visible when the quadratic term dominates, which may require significantly more data. Therefore, a log–log plot that is slightly "bowed" downwards can reflect a log-normal distribution – not a power law. In general, many alternative functional forms can appear to follow a power-law form for some extent. Stumpf proposed plotting the empirical cumulative distribution function in the log-log domain and claimed that a candidate power-law should cover at least two orders of magnitude.
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NL consists of the decision problems that can be solved by a nondeterministic Turing machine with a read-only input tape and a separate read-write tape whose size is limited to be proportional to the logarithm of the input length. Similarly, L consists of the languages that can be solved by a deterministic Turing machine with the same assumptions about tape length. Because there are only a polynomial number of distinct configurations of these machines, both L and NL are subsets of the class P of deterministic polynomial-time decision problems. Formally, a decision problem is NL-complete when it belongs to NL, and has the additional property that every other decision problem in NL can be reduced to it.
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Choosing one out of many solutions as the principal value leaves us with functions that are not continuous, and the usual rules for manipulating powers can lead us astray. Any nonrational power of a complex number has an infinite number of possible values because of the multi-valued nature of the complex logarithm. The principal value is a single value chosen from these by a rule which, amongst its other properties, ensures powers of complex numbers with a positive real part and zero imaginary part give the same value as does the rule defined above for the corresponding real base. Exponentiating a real number to a complex power is formally a different operation from that for the corresponding complex number.
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It is expected that a necessary condition for f(T) to make sense is f be defined on the spectrum of T. For example, the spectral theorem for normal matrices states every normal matrix is unitarily diagonalizable. This leads to a definition of f(T) when T is normal. One encounters difficulties if f(λ) is not defined for some eigenvalue λ of T. Other indications also reinforce the idea that f(T) can be defined only if f is defined on the spectrum of T. If T is not invertible, then (recalling that T is an n x n matrix) 0 is an eigenvalue. Since the natural logarithm is undefined at 0, one would expect that ln(T) can not be defined naturally.
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The idea of the probit function was published by Chester Ittner Bliss in a 1934 article in Science on how to treat data such as the percentage of a pest killed by a pesticide. Bliss proposed transforming the percentage killed into a "probability unit" (or "probit") which was linearly related to the modern definition (he defined it arbitrarily as equal to 0 for 0.0001 and 1 for 0.9999). He included a table to aid other researchers to convert their kill percentages to his probit, which they could then plot against the logarithm of the dose and thereby, it was hoped, obtain a more or less straight line. Such a so-called probit model is still important in toxicology, as well as other fields.
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The doubling time is time it takes for a population to double in size/value. It is applied to population growth, inflation, resource extraction, consumption of goods, compound interest, the volume of malignant tumours, and many other things that tend to grow over time. When the relative growth rate (not the absolute growth rate) is constant, the quantity undergoes exponential growth and has a constant doubling time or period, which can be calculated directly from the growth rate. This time can be calculated by dividing the natural logarithm of 2 by the exponent of growth, or approximated by dividing 70 by the percentage growth rateDonella Meadows, Thinking in Systems: A Primer, Chelsea Green Publishing, 2008, page 33 (box "Hint on reinforcing feedback loops and doubling time").
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Genetic studies using common fruit flies as experimental models reveal a link between night sleep and brain development mediated by evolutionary conserved transcription factors such as AP-2 Sleepwalking may be inherited as an autosomal dominant disorder with reduced penetrance. Genome-wide multipoint parametric linkage analysis for sleepwalking revealed a maximum logarithm of the odds score of 3.14 at chromosome 20q12-q13.12 between 55.6 and 61.4 cM."Neurology" Journal, January 4, 2011 76:12-13 published by the American Academy of Neurology www.Neurology.org Sleepwalking has been hypothesized to be linked to the neurotransmitter serotonin, which also appears to be metabolized differently in migraine patients and people with Tourette syndrome, both populations being four to nine times more likely to experience an episode of sleepwalking.
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The root node of the tree is the middle element of the array. The middle element of the lower half is the left child node of the root, and the middle element of the upper half is the right child node of the root. The rest of the tree is built in a similar fashion. Starting from the root node, the left or right subtrees are traversed depending on whether the target value is less or more than the node under consideration. In the worst case, binary search makes \lfloor \log_2 (n) + 1 \rfloor iterations of the comparison loop, where the \lfloor \rfloor notation denotes the floor function that yields the greatest integer less than or equal to the argument, and \log_2 is the binary logarithm.
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Information content is defined as the logarithm of the reciprocal of the probability that a system is in a specific microstate, and the information entropy of a system is the expected value of the system's information content. This definition of entropy is equivalent to the standard Gibbs entropy used in classical physics. Applying this definition to a physical system leads to the conclusion that, for a given energy in a given volume, there is an upper limit to the density of information (the Bekenstein bound) about the whereabouts of all the particles which compose matter in that volume. In particular, a given volume has an upper limit of information it can contain, at which it will collapse into a black hole.
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However, knowledge that a particular number will win a lottery has high value because it communicates the outcome of a very low probability event. The information content (also called the surprisal) of an event E is a function which decreases as the probability p(E) of an event increases, defined by I(E) = -\log_2(p(E)) or equivalently I(E) = \log_2(1/p(E)), where \log is the logarithm. Entropy measures the expected (i.e., average) amount of information conveyed by identifying the outcome of a random trial. This implies that casting a die has higher entropy than tossing a coin because each outcome of a die toss has smaller probability (about p=1/6) than each outcome of a coin toss (p=1/2).
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Daniel Bernoulli proposed that a nonlinear function of utility of an outcome should be used instead of the expected value of an outcome, accounting for risk aversion, where the risk premium is higher for low-probability events than the difference between the payout level of a particular outcome and its expected value. Bernoulli further proposed that it was not the goal of the gambler to maximize his expected gain but to instead maximize the logarithm of his gain. Bernoulli's paper was the first formalization of marginal utility, which has broad application in economics in addition to expected utility theory. He used this concept to formalize the idea that the same amount of additional money was less useful to an already- wealthy person than it would be to a poor person.
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RLWE is more properly called Learning with Errors over Rings and is simply the larger learning with errors (LWE) problem specialized to polynomial rings over finite fields. Because of the presumed difficulty of solving the RLWE problem even on a quantum computer, RLWE based cryptography may form the fundamental base for public-key cryptography in the future just as the integer factorization and discrete logarithm problem have served as the base for public key cryptography since the early 1980s. An important feature of basing cryptography on the ring learning with errors problem is the fact that the solution to the RLWE problem can be used to solve the NP-hard shortest vector problem (SVP) in a lattice (a polynomial-time reduction from the SVP problem to the RLWE problem has been presented).
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In statistics, the likelihood function (often simply called the likelihood) measures the goodness of fit of a statistical model to a sample of data for given values of the unknown parameters. It is formed from the joint probability distribution of the sample, but viewed and used as a function of the parameters only, thus treating the random variables as fixed at the observed values. The likelihood function describes a hypersurface whose peak, if it exists, represents the combination of model parameter values that maximize the probability of drawing the sample obtained. The procedure for obtaining these arguments of the maximum of the likelihood function is known as maximum likelihood estimation, which for computational convenience is usually done using the natural logarithm of the likelihood, known as the log- likelihood function.
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The application of rare-earth elements to geology is important to understanding the petrological processes of igneous, sedimentary and metamorphic rock formation. In geochemistry, rare-earth elements can be used to infer the petrological mechanisms that have affected a rock due to the subtle atomic size differences between the elements, which causes preferential fractionation of some rare earths relative to others depending on the processes at work. In geochemistry, rare-earth elements are typically presented in normalized "spider" diagrams, in which concentration of rare- earth elements are normalized to a reference standard and are then expressed as the logarithm to the base 10 of the value. Commonly, the rare-earth elements are normalized to chondritic meteorites, as these are believed to be the closest representation of unfractionated solar system material.
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This scheme isn't perfectly concealing as someone could find the commitment if he manages to solve the discrete logarithm problem. In fact, this scheme isn't hiding at all with respect to the standard hiding game, where an adversary should be unable to guess which of two messages he chose were committed to - similar to the IND-CPA game. One consequence of this is that if the space of possible values of x is small, then an attacker could simply try them all and the commitment would not be hiding. A better example of a perfectly binding commitment scheme is one where the commitment is the encryption of x under a semantically secure, public-key encryption scheme with perfect completeness, and the decommitment is the string of random bits used to encrypt x.
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Ronald Fisher in 1913 Early users of maximum likelihood were Carl Friedrich Gauss, Pierre-Simon Laplace, Thorvald N. Thiele, and Francis Ysidro Edgeworth. However, its widespread use rose between 1912 and 1922 when Ronald Fisher recommended, widely popularized, and carefully analyzed maximum- likelihood estimation (with fruitless attempts at proofs). Maximum-likelihood estimation finally transcended heuristic justification in a proof published by Samuel S. Wilks in 1938, now called Wilks' theorem. The theorem shows that the error in the logarithm of likelihood values for estimates from multiple independent observations is asymptotically χ 2-distributed, which enables convenient determination of a confidence region around any estimate of the parameters. The only difficult part of Wilks’ proof depends on the expected value of the Fisher information matrix, which is provided by a theorem proven by Fisher.
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The forking lemma is any of a number of related lemmas in cryptography research. The lemma states that if an adversary (typically a probabilistic Turing machine), on inputs drawn from some distribution, produces an output that has some property with non-negligible probability, then with non- negligible probability, if the adversary is re-run on new inputs but with the same random tape, its second output will also have the property. This concept was first used by David Pointcheval and Jacques Stern in "Security proofs for signature schemes," published in the proceedings of Eurocrypt 1996.Ernest Brickell, David Pointcheval, Serge Vaudenay, and Moti Yung, "Design Validations for Discrete Logarithm Based Signature Schemes", Third International Workshop on Practice and Theory in Public Key Cryptosystems, PKC 2000, Melbourne, Australia, January 18-20, 2000, pp. 276-292.
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In computer software and hardware, find first set (ffs) or find first one is a bit operation that, given an unsigned machine word, designates the index or position of the least significant bit set to one in the word counting from the least significant bit position. A nearly equivalent operation is count trailing zeros (ctz) or number of trailing zeros (ntz), which counts the number of zero bits following the least significant one bit. The complementary operation that finds the index or position of the most significant set bit is log base 2, so called because it computes the binary logarithm . This is closely related to count leading zeros (clz) or number of leading zeros (nlz), which counts the number of zero bits preceding the most significant one bit.
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In some cases, binding between the two partners will occur, which will become visible in the force curve, as the use of a flexible linker gives rise to a characteristic curve shape (see Worm-like chain model) distinct from adhesion. The collected rupture forces can then be analysed as a function of the bond loading rate. The resulting graph of the average rupture force as a function of the loading rate is called the force spectrum and forms the basic dataset for dynamic force spectroscopy. In the ideal case of a single sharp energy barrier for the tip-sample interactions the dynamic force spectrum will show a linear increase of the rupture force as function of a logarithm of the loading rate, as described by a model proposed by Bell et al.
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In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function f(z) has a root at w, then f(z)/(z−w), taking the limit value at w, is an entire function. On the other hand, neither the natural logarithm nor the square root is an entire function, nor can they be continued analytically to an entire function.
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A tabular data card proposed for Babbage's Analytical Engine showing a key–value pair, in this instance a number and its base-ten logarithm A key–value database, or key–value store, is a data storage paradigm designed for storing, retrieving, and managing associative arrays, and a data structure more commonly known today as a dictionary or hash table. Dictionaries contain a collection of objects, or records, which in turn have many different fields within them, each containing data. These records are stored and retrieved using a key that uniquely identifies the record, and is used to find the data within the database. A table showing different formatted data values associated with different keys Key–value databases work in a very different fashion from the better known relational databases (RDB).
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If the Mark–Houwink–Sakurada constants K and α are known (see Mark–Houwink equation), a plot of log [η]M versus elution volume (or elution time) for a particular solvent, column and instrument provides a universal calibration curve which can be used for any polymer in that solvent. By determining the retention volumes (or times) of monodisperse polymer standards (e.g. solutions of monodispersed polystyrene in THF), a calibration curve can be obtained by plotting the logarithm of the molecular weight versus the retention time or volume. Once the calibration curve is obtained, the gel permeation chromatogram of any other polymer can be obtained in the same solvent and the molecular weights (usually Mn and Mw) and the complete molecular weight distribution for the polymer can be determined.
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The inequalities are equivalent to the inequalities of Goluzin, discovered in 1947. Roughly speaking, the Grunsky inequalities give information on the coefficients of the logarithm of a univalent function; later generalizations by Milin, starting from the Lebedev–Milin inequality, succeeded in exponentiating the inequalities to obtain inequalities for the coefficients of the univalent function itself. The Grunsky matrix and its associated inequalities were originally formulated in a more general setting of univalent functions between a region bounded by finitely many sufficiently smooth Jordan curves and its complement: the results of Grunsky, Goluzin and Milin generalize to that case. Historically the inequalities for the disk were used in proving special cases of the Bieberbach conjecture up to the sixth coefficient; the exponentiated inequalities of Milin were used by de Branges in the final solution.
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It is usual in the computer industry to specify password strength in terms of information entropy which is measured in bits and is a concept from information theory. Instead of the number of guesses needed to find the password with certainty, the base-2 logarithm of that number is given, which is commonly referred to as the number of "entropy bits" in a password, though this is not exactly the same quantity as information entropy. A password with an entropy of 42 bits calculated in this way would be as strong as a string of 42 bits chosen randomly, for example by a fair coin toss. Put another way, a password with an entropy of 42 bits would require 242 (4,398,046,511,104) attempts to exhaust all possibilities during a brute force search.
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Hergé started collecting these types of words for use in Haddock's outbursts, and on occasion even searched dictionaries to come up with inspiration. As a result, Captain Haddock's colourful insults began to include "bashi-bazouk", "visigoths", "kleptomaniac", "sea gherkin", "anacoluthon", "pockmark", "nincompoop", "abominable snowman", "nitwits", "scoundrels", "steam rollers", "parasites", "vegetarians", "floundering oath", "carpet seller", "blundering Bazookas", "Popinjay", "bragger", "pinheads", "miserable slugs", "ectomorph", "maniacs", "pickled herring"; "freshwater swabs", "miserable molecule of mildew","Logarithm", "bandits", "orang-outangs", "cercopithecuses", "Polynesians", "iconoclasts", "ruffians", "fancy-dress freebooter", "ignoramus", "sycophant", "dizzard", "black-beetle", "pyrographer", "slave- trader" and "Fuzzy Wuzzy", but again, nothing actually considered a swear word. On one occasion, this scheme appeared to backfire. In one particularly angry state, Hergé had the captain yell the word "pneumothorax" (a medical emergency caused by the collapse of the lung within the chest).
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However, its designers, Adi Shamir and Eran Tromer, estimate that if TWIRL were built, it would be able to factor 1024-bit numbers in one year at the cost of "a few dozen million US dollars". TWIRL could therefore have enormous repercussions in cryptography and computer security — many high-security systems still use 1024-bit RSA keys, which TWIRL would be able to break in a reasonable amount of time and for reasonable costs. The security of some important cryptographic algorithms, notably RSA and the Blum Blum Shub pseudorandom number generator, rests in the difficulty of factorizing large integers. If factorizing large integers becomes easier, users of these algorithms will have to resort to using larger keys (computationally more expensive) or to using different algorithms, whose security rests on some other computationally hard problem (like the discrete logarithm problem).
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Assuming no drawn games, determining a clear winner (and, incidentally, a clear loser) would require the same number of rounds as that of a knockout tournament, which is the binary logarithm of the number of players rounded up. Thus three rounds can handle eight players, four rounds can handle sixteen players and so on. If fewer than this minimum number of rounds are played, two or more players could finish the tournament with a perfect score, having won all their games but never having faced each other. Due to the fact that players should meet each other at most once and pairings are chosen dependent on the results, there is a natural upper bound on the number of rounds of a Swiss-system tournament, which is equal to half of the number of players rounded up.
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Since illumination and reflectance combine multiplicatively, the components are made additive by taking the logarithm of the image intensity, so that these multiplicative components of the image can be separated linearly in the frequency domain. Illumination variations can be thought of as a multiplicative noise, and can be reduced by filtering in the log domain. To make the illumination of an image more even, the high-frequency components are increased and low-frequency components are decreased, because the high- frequency components are assumed to represent mostly the reflectance in the scene (the amount of light reflected off the object in the scene), whereas the low-frequency components are assumed to represent mostly the illumination in the scene. That is, high-pass filtering is used to suppress low frequencies and amplify high frequencies, in the log-intensity domain.
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In statistics a quasi-maximum likelihood estimate (QMLE), also known as a pseudo-likelihood estimate or a composite likelihood estimate, is an estimate of a parameter θ in a statistical model that is formed by maximizing a function that is related to the logarithm of the likelihood function, but in discussing the consistency and (asymptotic) variance-covariance matrix, we assume some parts of the distribution may be mis-specified. In contrast, the maximum likelihood estimate maximizes the actual log likelihood function for the data and model. The function that is maximized to form a QMLE is often a simplified form of the actual log likelihood function. A common way to form such a simplified function is to use the log-likelihood function of a misspecified model that treats certain data values as being independent, even when in actuality they may not be.
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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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In 1977, Schwartz pointed out that the hypercolumn model of Hubel and Weisel implied the existence of a periodic vortex like pattern of orientation singularities across the surface of visual cortex. Specifically, the angular part of the complex logarithm function, viewed as a spatial map provided a possible explanation of the hypercolumn structure, which in current language is termed the "pinwheel" structure of visual cortex . In 1990, together with Alan Rojer, Schwartz showed that such "vortex" or "pinwheel" structures, together with the associated ocular dominance column pattern in cortex, could be caused by spatial filtering of random vector or scalar spatial noise, respectively. Prior to this work, most modeling of cortical columns was in terms of somewhat opaque and clumsy "neural network" models—bandpass-filtered noise quickly became a standard modeling technique for cortical columnar structure.
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For any fixed choice of a value in a given set of numbers, if one randomly permutes the numbers and forms a binary tree from them as described above, the expected value of the length of the path from the root of the tree to is at most , where "" denotes the natural logarithm function and the introduces big O notation. For, the expected number of ancestors of is by linearity of expectation equal to the sum, over all other values in the set, of the probability that is an ancestor of . And a value is an ancestor of exactly when is the first element to be inserted from the elements in the interval . Thus, the values that are adjacent to in the sorted sequence of values have probability of being an ancestor of , the values one step away have probability , etc.
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The research activities into Speech coding started even before the ones on speech recognition and synthesis, aiming to build equipment such as CODEC and echo canceler to be able to increase as much as possible the number of telephone conversations that can flow through a single cable (or satellite connection) without losing voice intelligibility. In the late seventies, studies and experiments led to the creation of algorithms to encode the telephonic speech signal and set-up the European regulation CCITT known as encoding A-law (8-bit logarithm encoding law "A" for audio signal 8 kHz band limited). This standard was then used in the CODEC for 64 kbit/s ISDN telephone lines. In subsequent years they built stronger codecs (used telephone exchanges) and, within the PAN-Europe consortium GSM, the codec to use in second-generation mobile phones.
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However, this is potentially misleading. Using a unary input is slower for any given number, not faster; the distinction is that a binary (or larger base) input is proportional to the base 2 (or larger base) logarithm of the number while unary input is proportional to the number itself. Therefore, while the run- time and space requirement in unary looks better as function of the input size, it does not represent a more efficient solution.. In computational complexity theory, unary numbering is used to distinguish strongly NP-complete problems from problems that are NP-complete but not strongly NP-complete. A problem in which the input includes some numerical parameters is strongly NP- complete if it remains NP-complete even when the size of the input is made artificially larger by representing the parameters in unary.
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The full title of Rivas's 11-folio booklet is translated in English as Syzygies and Lunar Quadratures Aligned to the Meridian of Mérida of the Yucatán by an Anctitone or Inhabitant of the Moon, and Addressed to the Scholar Don Ambrosio de Echevarria, Reciter of Funeral Kyries in the Parish of Jesus of Said City, and Presently Teacher of Logarithm in the Town of Mama of the Yucatán Peninsula, in the Year of the Lord 1775. No actual date of composition is given in the manuscript. The manuscript was rediscovered in 1958 by Pablo González Casanova, "hidden among the dusty volumes of the National Archives in Mexico City"—in fact, among the documents compiled by the Inquisition pertaining to Rivas's trial. It was referenced in a 1977 study of Mexican literature, but was not commented on or published until 1994.
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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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The Meissel–Mertens constant is analogous to the Euler–Mascheroni constant, but the harmonic series sum in its definition is only over the primes rather than over all integers and the logarithm is taken twice, not just once. Mertens's theorems are three 1874 results related to the density of prime numbers. Erwin Schrödinger was taught calculus and algebra by Mertens. His memory is honoured by the Franciszek Mertens Scholarship granted to those outstanding pupils of foreign secondary schools who wish to study at the Faculty of Mathematics and Computer Science of the Jagiellonian University in Kraków and were finalists of the national-level mathematics, or computer science olympiads, or they have participated in one of the following international olympiads: in mathematics (IMO), computer science (IOI), astronomy (IAO), physics (IPhO), linguistics (IOL), or they were participants of the European Girls' Mathematical Olympiad (EGMO).
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Lexington: Lexington Books by Johansen, Ledoit and Sornette.A. Johansen, D. Sornette and O. Ledoit, Predicting Financial Crashes using discrete scale invariance, Journal of Risk 1 (4), 5–32 (1999)A. Johansen, O. Ledoit and D. Sornette, Crashes as critical points, International Journal of Theoretical and Applied Finance 3 (2), 219–255 (2000) This approach is now referred to in the literature as the JLS model. Recently, Sornette has added the S to the LPPL acronym of "log-periodic power law" to make clear that the "power law" part should not be confused with power law distributions: indeed, the "power law" refers to the hyperbolic singularity of the form \ln[p(t)] = \ln[p(t_c)] - B (t_c -t)^m, where \ln[p(t)] is the logarithm of the price at time t, 00 and t_c is the critical time of the end of the bubble.
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According to the concept of glass developed by M. Shultz, in analogy with pH for aqueous solutions he proposed an innovative idea to establish for glasses and melts—the degree of acidity pO (negative logarithm of the activity of oxygen ions O2−) and standards for methods of measurement: pO is inversely proportional to the degree of basicity and concentration of the oxide. Under the guidance of M. Shultz developed are heat resistant inorganic coatings for the protection of structural materials of space technique (including military rockets, and for the spacecraft Buran) and lamellar coatings on semiconductor silicon for industrial electronics, organo- silicate corrosion-resistant, anti-icing, dielectric, thermal insulation, radiation proof covers for construction, electrical engineering and shipbuilding. Large enough the contribution of the scientist is in the sphere of developing new construction materials. M. Shultz is a founder one of Russian scientific schools.
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Logluv TIFF's design solves two specific problems: storing high dynamic image data and doing so within a reasonable amount of space. Traditional image format generally stores pixel data in RGB- space occupying 24 bits, with 8 bits for each color component. This limits the representable colors to a subset of all visible and distinguishable colors, introducing quantization and clamping artifacts clearly visible to human observers. Using a triplet of floats to represent RGB would be a viable solution, but it would quadruple the size of the file (occupying 32 bits for each color-component, as opposed to 8 bits). Instead of using RGB, LogLuv uses the logarithm of the luminance and the CIELUV (u’, v’) chromaticity coordinates in order to provide a perceptually uniform color space. LogLuv allocates 8 bits for each of the u’ and v’ coordinates, which allows encoding the full visible gamut with imperceptible step sizes.
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The metric distance between two points inside the absolute is the logarithm of the cross ratio formed by these two points and the two intersections of their line with the absolute In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance"Cayley (1859), p 82, §§209 to 229 where he calls the quadric the absolute. The construction was developed in further detail by Felix Klein in papers in 1871 and 1873, and subsequent books and papers.Klein (1871, 1873), Klein (1893ab), Fricke/Klein (1897), Klein (1910), Klein/Ackerman (1926/1979), Klein/Rosemann (1928) The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in hyperbolic geometry, elliptic geometry, and Euclidean geometry.
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By taking information per pulse in bit/pulse to be the base-2-logarithm of the number of distinct messages M that could be sent, Hartley constructed a measure of the line rate R as: : R = f_p \log_2(M), where f_p is the pulse rate, also known as the symbol rate, in symbols/second or baud. Hartley then combined the above quantification with Nyquist's observation that the number of independent pulses that could be put through a channel of bandwidth B hertz was 2B pulses per second, to arrive at his quantitative measure for achievable line rate. Hartley's law is sometimes quoted as just a proportionality between the analog bandwidth, B, in Hertz and what today is called the digital bandwidth, R, in bit/s. Other times it is quoted in this more quantitative form, as an achievable line rate of R bits per second: : R \le 2B \log_2(M).
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Before the advent of computers, printed lookup tables of values were used by people to speed up hand calculations of complex functions, such as in trigonometric tables, logarithm tables, and tables of statistical density functions School children are often taught to memorize "times tables" to avoid calculations of the most commonly used numbers (up to 9 x 9 or 12 x 12). Even as early as 493 A.D., Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144" Maher, David. W. J. and John F. Makowski. "Literary Evidence for Roman Arithmetic With Fractions", 'Classical Philology' (2001) Vol.
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Given a curve, E, defined along some equation in a finite field (such as E: ), point multiplication is defined as the repeated addition of a point along that curve. Denote as for some scalar (integer) n and a point that lies on the curve, E. This type of curve is known as a Weierstrass curve. The security of modern ECC depends on the intractability of determining n from given known values of Q and P if n is large (known as the elliptic curve discrete logarithm problem by analogy to other cryptographic systems). This is because the addition of two points on an elliptic curve (or the addition of one point to itself) yields a third point on the elliptic curve whose location has no immediately obvious relationship to the locations of the first two, and repeating this many times over yields a point nP that may be essentially anywhere.
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The poetry of O Fortuna was actually the work of itinerant goliards, found in the German Benedictine monastery of Benediktbeuern Abbey. The hoax was lent an air of credibility because often medieval monks did discover scientific and mathematical theories, only to have them hidden or shelved due to persecution or simply ignored because publication prior to the invention of the printing press was difficult at best. Mr. Girvan adds to this suggestion by associating Udo with several other more legitimate discoveries where an author was considered ahead of his time in terms of a scientific theory of some sort that is now established as a mainstream theory but was considered fringe science at the time. Another aspect of the deception was that it was very common for pre-20th century mathematicians to spend incredible amounts of time on hand calculations such as a logarithm table or trigonometric functions.
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In other words, the signal has equal power in any band of a given bandwidth (power spectral density) when the bandwidth is measured in Hz. For example, with a white noise audio signal, the range of frequencies between 40 Hz and 60 Hz contains the same amount of sound power as the range between 400 Hz and 420 Hz, since both intervals are 20 Hz wide. Note that spectra are often plotted with a logarithmic frequency axis rather than a linear one, in which case equal physical widths on the printed or displayed plot do not all have the same bandwidth, with the same physical width covering more Hz at higher frequencies than at lower frequencies. In this case a white noise spectrum that is equally sampled in the logarithm of frequency (i.e., equally sampled on the X axis) will slope upwards at higher frequencies rather than being flat.
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See Rickey reference for discussion and further references. However, as in the rest of his work, Fermat's techniques were more ad hoc tricks than systematic treatments, and he is not considered to have played a significant part in the subsequent development of calculus. Of note is that Cavalieri only compared areas to areas and volumes to volumes – these always having dimensions, while the notion of considering an area as consisting of units of area (relative to a standard unit), hence being unitless, appears to have originated with Wallis;Ball, 281Britannica, 171 Wallis studied fractional and negative powers, and the alternative to treating the computed values as unitless numbers was to interpret fractional and negative dimensions. The exceptional case of −1 (the standard hyperbola) was first successfully treated by Grégoire de Saint-Vincent in his Opus geometricum quadrature circuli et sectionum coni (1647), though a formal treatment had to wait for the development of the natural logarithm, which was accomplished by Nicholas Mercator in his Logarithmotechnia (1668).
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It implies that the optimal win probability is always at least 1/e (where e is the base of the natural logarithm), and that the latter holds even in a much greater generality (2003). The optimal stopping rule prescribes always rejecting the first \sim n/e applicants that are interviewed and then stopping at the first applicant who is better than every applicant interviewed so far (or continuing to the last applicant if this never occurs). Sometimes this strategy is called the 1/e stopping rule, because the probability of stopping at the best applicant with this strategy is about 1/e already for moderate values of n. One reason why the secretary problem has received so much attention is that the optimal policy for the problem (the stopping rule) is simple and selects the single best candidate about 37% of the time, irrespective of whether there are 100 or 100 million applicants.
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Separators have been used as part of data compression algorithms for representing planar graphs and other separable graphs using a small number of bits. The basic principle of these algorithms is to choose a number k and repeatedly subdivide the given planar graph using separators into O(n/k) subgraphs of size at most k, with O(n/√k) vertices in the separators. With an appropriate choice of k (at most proportional to the logarithm of n) the number of non-isomorphic k-vertex planar subgraphs is significantly less than the number of subgraphs in the decomposition, so the graph can be compressed by constructing a table of all the possible non-isomorphic subgraphs and representing each subgraph in the separator decomposition by its index into the table. The remainder of the graph, formed by the separator vertices, may be represented explicitly or by using a recursive version of the same data structure.
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Analytic continuation of natural logarithm (imaginary part) Suppose f is an analytic function defined on a non-empty open subset U of the complex plane \Complex. If V is a larger open subset of \Complex, containing U, and F is an analytic function defined on V such that :F(z) = f(z) \qquad \forall z \in U, then F is called an analytic continuation of f. In other words, the restriction of F to U is the function f we started with. Analytic continuations are unique in the following sense: if V is the connected domain of two analytic functions F1 and F2 such that U is contained in V and for all z in U :F_1(z) = F_2(z) = f(z), then :F_1=F_2 on all of V. This is because F1 − F2 is an analytic function which vanishes on the open, connected domain U of f and hence must vanish on its entire domain.
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Horowitz investigates gravitational phenomena, such as black holes, in string theory. In the 1990s, he worked with, among others, Andrew StromingerCounting states of near extremal black holes, Phys. Rev. Lett., Bd.77, 1996, S.2368 and Joseph Polchinski showing that string theory provides a description of the quantum microstates of certain black holes (following earlier work of Strominger and Cumrun Vafa ).For general black holes, one can only show the proportionality of the logarithm of the number of string states to the surface area (which corresponds to the entropy), Horowitz, Polchinski A correspondence principle for black holes and strings, Physical Review D, Bd.55, 1997, S.6189 In 1985 Horowitz published an influential paper with Philip Candelas, Andrew Strominger and Edward Witten on the compactification of superstrings in Calabi-Yau spaces.Vacuum configuration of superstrings, Nuclear Physics B, Bd.258, 1985, S.46-76 In the early 1990s, Horowitz and Strominger found black brane solutions in string theory.
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In photography, the differences between an "objective" and "subjective" tone reproduction, and between "accurate" and "preferred" tone reproduction, have long been recognized. Many steps in the process of photography are recognized as having their own nonlinear curves, which in combination form the overall tone reproduction curve; the Jones diagram was developed as a way to illustrate and combine curves, to study and explain the photographic process. The luminance range of a scene maps to the focal-plane illuminance and exposure in a camera, not necessarily directly proportionally, as when a graduated neutral density filter is used to reduce the exposure range to less than the scene luminance range. The film responds nonlinearly to the exposure, as characterized by the film's characteristic curve, or Hurter–Driffield curve; this plot of optical density of the developed negative versus the logarithm of the exposure (also called a D–logE curve) has central straight section whose slope is called the gamma of the film.
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However, when the coefficients are integers, rational numbers or polynomials, these arithmetic operations imply a number of GCD computations of coefficients which is of the same order and make the algorithm inefficient. The subresultant pseudo- remainder sequences were introduced to solve this problem and avoid any fraction and any GCD computation of coefficients. A more efficient algorithm is obtained by using the good behavior of the resultant under a ring homomorphism on the coefficients: to compute a resultant of two polynomials with integer coefficients, one computes their resultants modulo sufficiently many prime numbers and then reconstructs the result with the Chinese remainder theorem. The use of fast multiplication of integers and polynomials allows algorithms for resultants and greatest common divisors that have a better time complexity, which is of the order of the complexity of the multiplication, multiplied by the logarithm of the size of the input (\log(s(d+e)), where is an upper bound of the number of digits of the input polynomials).
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For reasons of conceptual clarification, the various puzzles that remain with regard to genome size variation instead have been suggested by one author to more accurately comprise a puzzle or an enigma (the so-called "C-value enigma"). Genome size correlates with a range of measurable characteristics at the cell and organism levels, including cell size, cell division rate, and, depending on the taxon, body size, metabolic rate, developmental rate, organ complexity, geographical distribution, or extinction risk. Based on currently available completely sequenced genome data (as of April 2009), log-transformed gene number forms a linear correlation with log- transformed genome size in bacteria, archaea, viruses, and organelles combined, whereas a nonlinear (semi-natural logarithm) correlation is seen for eukaryotes. Although the latter contrasts with the previous view that no correlation exists for the eukaryotes, the observed nonlinear correlation for eukaryotes may reflect disproportionately fast-increasing non-coding DNA in increasingly large eukaryotic genomes.
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The area- preserving property of squeeze mapping has an application in setting the foundation of the transcendental functions natural logarithm and its inverse the exponential function: Definition: Sector(a,b) is the hyperbolic sector obtained with central rays to (a, 1/a) and (b, 1/b). Lemma: If bc = ad, then there is a squeeze mapping that moves the sector(a,b) to sector(c,d). Proof: Take parameter r = c/a so that (u,v) = (rx, y/r) takes (a, 1/a) to (c, 1/c) and (b, 1/b) to (d, 1/d). Theorem (Gregoire de Saint-Vincent 1647) If bc = ad, then the quadrature of the hyperbola xy = 1 against the asymptote has equal areas between a and b compared to between c and d. Proof: An argument adding and subtracting triangles of area ½, one triangle being {(0,0), (0,1), (1,1)}, shows the hyperbolic sector area is equal to the area along the asymptote.
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It was described, in the late 18th century, as "one of the most entertaining narratives in our language", in particular for the historical portrayal it leaves of men like John Dee, Simon Forman, John Booker, Edward Kelley, including a whimsical first meeting of John Napier and Henry Briggs, respective co-inventors of the logarithm and Briggsian logarithms,David Stuart & John Minto in "Account of the Life of John Napier of Merchiston," The Edinburgh Magazine, or Literary Miscellany (1787) Vol.6 and for its curious tales about the effects of crystals and the appearance of Queen Mab. In it, Lilly describes the friendly support of Oliver Cromwell during a period in which he faced prosecution for issuing political astrological predictions. He also writes about the 1666 Great Fire of London, and how he was brought before the committee investigating the cause of the fire, being suspected of involvement because of his publication of images, 15 years earlier, which depicted a city in flames surrounded by coffins.
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Many real-world examples of Benford's law arise from multiplicative fluctuations. For example, if a stock price starts at $100, and then each day it gets multiplied by a randomly chosen factor between 0.99 and 1.01, then over an extended period the probability distribution of its price satisfies Benford's law with higher and higher accuracy. The reason is that the logarithm of the stock price is undergoing a random walk, so over time its probability distribution will get more and more broad and smooth (see above). (More technically, the central limit theorem says that multiplying more and more random variables will create a log-normal distribution with larger and larger variance, so eventually it covers many orders of magnitude almost uniformly.) To be sure of approximate agreement with Benford's law, the distribution has to be approximately invariant when scaled up by any factor up to 10; a lognormally distributed data set with wide dispersion would have this approximate property.
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Above the paneled walls, six portraits depict prominent Yugoslavs. On the front wall are portraits of Vuk Stefanović Karadžić (1787–1864) who compiled the Serbian dictionary and collected, edited, and published Serbian national ballads and folk songs; and Croatian statesman Bishop Josip Juraj Strossmayer (1815–1905) who was known for his efforts to achieve understanding between the Roman Catholic and Greek Orthodox churches, founder of the Yugoslav Academy of Sciences and Arts (now the Croatian Academy of Sciences and Arts). On the corridor wall are likenesses of Baron George von Vega (1754–1802), a Slovenian officer in the Austrian army and mathematician recognized for various works including a book of logarithm tables; and Petar Petrović Njegoš (1813–1851), the last prince-bishop of Montenegro, who was celebrated for his poetry. Represented on the rear wall are Rugjer Bošković (1711–1787), a Croatian scientist distinguished for his achievements in the fields of mathematics, optics, and astronomy; and France Ksaver Prešeren (1800–1849) who is considered one of the greatest native-language Slovenian poets.
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349-350 Pierre Ageron, studying superficially8 a copy of Ibn Hamza's manuscript in Ottoman Turkish, kept at the Süleymaniye Kütüphanesi library, and dated to the Hegirian year 1013, highlights an example linking geometric progression and arithmetic progression: the first written in oriental Arabic numerals (۱ ۲ ٤ ۸ ۱٦ ۳۲ ٦٤ ۱۲۸), and the second in alphabetical numbers (ا ب ج د ه و ز ح). In the margin is a figure which gives two graduations of the same segment: a regular one above, and a "logarithmic" graduation below. But, for the latter, the use of alphabetic and therefore whole numbers suggests that Ibn Hamza did not think of inserting non-integers and no approximate logarithm calculation is recorded in the manuscript9. Nevertheless we can note that, in the text in Ottoman Turkish, where Pierre Agero identifies the Arabic words us (exponent), dil'ayn (two sides) and a series of powers of 2 in oriental Arabic numerals and that of the corresponding exponents in numerals alphabetic, he was unable to read the actual text of the book because he did not master Ottoman Turkish.
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One cent compared to a semitone on a truncated monochord. The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each. Typically, cents are used to express small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is too small to be perceived between successive notes. Cents, as described by Alexander J. Ellis, follow a tradition of measuring intervals by logarithms that began with Juan Caramuel y Lobkowitz in the 17th century.Caramuel mentioned the possible use of binary logarithms for music in a letter to Athanasius Kircher in 1647; this usage often is attributed to Leonhard Euler in 1739 (see Binary logarithm). Isaac Newton described musical logarithms using the semitone () as base in 1665; Gaspard de Prony did the same in 1832. Joseph Sauveur in 1701, and Felix Savart in the first half of the 19th century, divided the octave in 301 or 301,03 units.
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The Sylow subgroups of general linear groups are another fundamental family of examples. Let V be a vector space of dimension n with basis { e1, e2, …, en } and define Vi to be the vector space generated by { ei, ei+1, …, en } for 1 ≤ i ≤ n, and define Vi = 0 when i > n. For each 1 ≤ m ≤ n, the set of invertible linear transformations of V which take each Vi to Vi+m form a subgroup of Aut(V) denoted Um. If V is a vector space over Z/pZ, then U1 is a Sylow p-subgroup of Aut(V) = GL(n, p), and the terms of its lower central series are just the Um. In terms of matrices, Um are those upper triangular matrices with 1s one the diagonal and 0s on the first m−1 superdiagonals. The group U1 has order pn·(n−1)/2, nilpotency class n, and exponent pk where k is the least integer at least as large as the base p logarithm of n.
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The so-called Landauer limit implied by the laws of physics sets a lower limit on the energy required to perform a computation of per bit erased in a computation, where T is the temperature of the computing device in kelvins, k is the Boltzmann constant, and the natural logarithm of 2 is about 0.693. No irreversible computing device can use less energy than this, even in principle. Thus, in order to simply flip through the possible values for a 128-bit symmetric key (ignoring doing the actual computing to check it) would, theoretically, require 2128 − 1 bit flips on a conventional processor. If it is assumed that the calculation occurs near room temperature (~300 K), the Von Neumann-Landauer Limit can be applied to estimate the energy required as ~1018 joules, which is equivalent to consuming 30 gigawatts of power for one year. This is equal to 30×109 W×365×24×3600 s = 9.46×1017 J or 262.7 TWh (about 0.1% of the yearly world energy production).
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In computational complexity theory, a decision problem is PSPACE-complete if it can be solved using an amount of memory that is polynomial in the input length (polynomial space) and if every other problem that can be solved in polynomial space can be transformed to it in polynomial time. The problems that are PSPACE-complete can be thought of as the hardest problems in PSPACE, because a solution to any one such problem could easily be used to solve any other problem in PSPACE. The PSPACE-complete problems are widely suspected to be outside the more famous complexity classes P and NP, but that is not known. It is known that they lie outside of the class NC (a class of problems with highly efficient parallel algorithms), because problems in NC can be solved in an amount of space polynomial in the logarithm of the input size, and the class of problems solvable in such a small amount of space is strictly contained in PSPACE by the space hierarchy theorem.
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John Farey, Sr. (1766–1826) was a polymath, well known today for his work as a geologist and for his investigations of mathematics. He was greatly interested in the mathematics of sound, and the schemes of temperament used in tuning musical instruments then. He was a prolific contributor to contemporary journals, such as the Philosophical Magazine, and the Monthly Magazine as well as the Edinburgh Encyclopædia on this topic.See the bibliography appended to Trever D. Ford and Hugh S. Torrens introduction ('John Farey (1766–1826) an unrecognised polymath') to the 1989 reprinting of the first volume of John Farey's General View of the Agriculture and Minerals of Derbyshire. His articles for Rees abound in extensive mathematical calculations sometimes extending to many places of decimals, as this brief example from Vol 18 shows: ::INCOMPOSIT Ditone of the Enharmonic Genus is the excess of a fourth tone above half a tone major, or 3² ÷ 8 root 2, which is 202 Σ + 4f + 17½m, or 202.00393 Σ + 4f + 17½m, whose common logarithm is .
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But the exact formulation in the standard was written such that use of the alleged backdoored P and Q was required for FIPS 140-2 validation, so the OpenSSL project chose to implement the backdoored P and Q, even though they were aware of the potential backdoor and would have preferred generating their own secure P and Q. New York Times would later write that NSA had worked during the standardization process to eventually become the sole editor of the standard. A security proof was later published for Dual_EC_DRBG by Daniel R.L. Brown and Kristian Gjøsteen, showing that the generated elliptic curve points would be indistinguishable from uniformly random elliptic curve points, and that if fewer bits were output in the final output truncation, and if the two elliptic curve points P and Q were independent, then Dual_EC_DRBG is secure. The proof relied on the assumption that three problems were hard: the decisional Diffie–Hellman assumption (which is generally accepted to be hard), and two newer less-known problems which are not generally accepted to be hard: the truncated point problem, and the x-logarithm problem.Kristian Gjøsteen.
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At the isoequilibrium temperature β, all the reactions in the series should have the same equilibrium constant (Ki) ::: ΔGi(β) = α isokinetic relation (IKR), isokinetic effect : On an Arrhenius plot, there exists a common intersection point describing the kinetics of the reactions. At the isokinetic temperature β, all the reactions in the series should have the same rate constant (ki) ::: ki(β) = exp(α) isoequilibrium temperature : used for thermodynamic LFERs; refers to β in the equations where it possesses dimensions of temperature isokinetic temperature : used for kinetic LFERs; refers to β in the equations where it possesses dimensions of temperature kinetic compensation : an increase in the preexponential factors tends to compensate for the increase in activation energy: ::: lnA = lnA0 \+ αΔE0 Meyer-Neldel rule (MNR) : primarily used in materials science and condensed matter physics; the MNR is often stated as the plot of the logarithm of the preexponential factor against activation energy is linear: ::: σ(T) = σ0exp(-Ea/kBT) where lnσ0 is the preexponential factor, Ea is the activation energy, σ is the conductivity, and kB is Boltzmann's constant, and T is temperature.Abtew, T. A.; Zhang, M.; Pan, Y.; Drabold, D. A. Electrical conductivity and Meyer-Neldel rule: The role of localized states in hydrogenated amorphous silicon. J. Non Cryst.
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