Here, a Christmas tree built out of the factorial sequence.
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It's similar to the old factorial formula, but with many terms removed.
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Factorial Digital is a growth marketing agency based in NYC that specializes in search and startups.
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Well, almost endless—according to Sancristoforo's website, the number of mixes Tableau can produce is 4x12 factorial.
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Yvonne Leow: How did you get into the game of growth, and why did you decide to start Factorial Digital?
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"So it's a multi-factorial issue that we view as a real critical issue in the cancer world right now."
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Some of the in-custody deaths were deemed "multi-factorial," with no distinct cause, and some were attributed to pre-existing health problems.
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Some of the in-custody deaths were deemed "multi-factorial," with no single cause, and some were attributed to pre-existing health problems.
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The factorial rule predicts that to watch six episodes in every possible order should require 873 episodes, but Houston found a way to do it in 872.
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The factorial rule for superpermutations became conventional wisdom (even though no one could prove it was true for every value of n), and mathematicians later confirmed it for n = 5.
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It's pretty well known that if you have n objects, the number of ways of putting them in a row is n factorial (the product n(n-1)(n-2)…1).
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At the time, mathematicians believed that the factorial formula for superpermutations was probably correct, and when you think you know the exact answer to a question, a lower bound isn't very interesting.
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"However it's going to be multiple factorial in terms of the solution set and it's going to require many sophisticated parties to make sure that we've got the right remedies in place for people who suffer from that."
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And since there is a simple way to convert a short superpermutation on n symbols into a short superpermutation on n + 993 symbols, Houston's example meant that the factorial rule fails for every value of n above 6 too.
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The study—titled "Situational and Dispositional Determinants of College Men's Perception of Women's Sexual Desire and Consent to Sex: A Factorial Vignette Analysis"—was comprised of 145 straight, male college students with a median age of 20 attending a large university in the south-east United States.
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And to actually get approval for those permits, your future home will have to satisfy a set of conditions that is a factorial of complex and conflicting federal, state and city building codes, separate sets of fire and energy requirements, and quasi-legal construction standards set by various independent agencies.
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For example, for a full factorial design with three factors, the design matrix is :\- - - :\+ - - :\- + - :\+ + - :\- - + :\+ - + :\- + + :\+ + + Determining the Yates order for fractional factorial designs requires knowledge of the confounding structure of the fractional factorial design.
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For example, with two factors each taking two levels, a factorial experiment would have four treatment combinations in total, and is usually called a 2×2 factorial design. If the number of combinations in a full factorial design is too high to be logistically feasible, a fractional factorial design may be done, in which some of the possible combinations (usually at least half) are omitted.
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ANOVA generalizes to the study of the effects of multiple factors. When the experiment includes observations at all combinations of levels of each factor, it is termed factorial. Factorial experiments are more efficient than a series of single factor experiments and the efficiency grows as the number of factors increases.Montgomery (2001, Section 5-2: Introduction to factorial designs; The advantages of factorials) Consequently, factorial designs are heavily used.
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In probability theory, the factorial moment is a mathematical quantity defined as the expectation or average of the falling factorial of a random variable. Factorial moments are useful for studying non-negative integer-valued random variables,D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Vol.
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For example, . The zero double factorial as an empty product. The sequence of double factorials for even = starts as : 1, 2, 8, 48, 384, 3840, 46080, 645120,... The sequence of double factorials for odd = starts as : 1, 3, 15, 105, 945, 10395, 135135,... The term odd factorial is sometimes used for the double factorial of an odd number.
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Z, would turn out to be the ordinary factorial of k.
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In this case, fractional factorial designs may be used. As with any statistical experiment, the experimental runs in a factorial experiment should be randomized to reduce the impact that bias could have on the experimental results. In practice, this can be a large operational challenge. Factorial experiments can be used when there are more than two levels of each factor.
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Consider the following pseudocode function to calculate the factorial of n: function factorial (n is a non-negative integer) if n is 0 then return 1 [by the convention that 0! = 1] else return factorial(n – 1) times n [recursively invoke factorial with the parameter 1 less than n] end if end function For every integer n such that `n≥0`, the final result of the function `factorial` is invariant; if invoked as `x = factorial(3)`, the result is such that x will always be assigned the value 6. The non-memoized implementation above, given the nature of the recursive algorithm involved, would require n + 1 invocations of `factorial` to arrive at a result, and each of these invocations, in turn, has an associated cost in the time it takes the function to return the value computed. Depending on the machine, this cost might be the sum of: # The cost to set up the functional call stack frame. # The cost to compare n to 0. # The cost to subtract 1 from n.
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The expectations of general functionals of simple point processes, provided some certain mathematical conditions, have (possibly infinite) expansions or series consisting of the corresponding factorial moment measures.B. Blaszczyszyn. Factorial-moment expansion for stochastic systems. Stoch. Proc. Appl., 56:321–335, 1995.
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However, the number of experimental runs required for three-level (or more) factorial designs will be considerably greater than for their two-level counterparts. Factorial designs are therefore less attractive if a researcher wishes to consider more than two levels.
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In statistics, fractional factorial designs are experimental designs consisting of a carefully chosen subset (fraction) of the experimental runs of a full factorial design . The subset is chosen so as to exploit the sparsity- of-effects principle to expose information about the most important features of the problem studied, while using a fraction of the effort of a full factorial design in terms of experimental runs and resources. In other words, it makes use of the fact that many experiments in full factorial design are often redundant, giving little or no new information about the system.
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Also, when , this definition is mathematically equivalent to the alternative extension of the double factorial.
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The gamma function interpolates the factorial function to non-integer values. The main clue is the recurrence relation generalized to a continuous domain. Besides nonnegative integers, the factorial can also be defined for non-integer values, but this requires more advanced tools from mathematical analysis. One function that fills in the values of the factorial (but with a shift of 1 in the argument), that is often used, is called the gamma function, denoted .
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Factorial moment measures are used when points are not allowed to repeat, hence points are distinct.
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For more than two factors, a 2k factorial experiment can usually be recursively designed from a 2k−1 factorial experiment by replicating the 2k−1 experiment, assigning the first replicate to the first (or low) level of the new factor, and the second replicate to the second (or high) level. This framework can be generalized to, e.g., designing three replicates for three level factors, etc. A factorial experiment allows for estimation of experimental error in two ways.
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It is also a sphenic number. 170 is the largest integer for which its factorial can be stored in double-precision floating-point format. This is probably why it is also the largest factorial that Google's built-in calculator will calculate, returning the answer as 170! = 7.25741562 × 10306.
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I. Probability and its Applications (New York). Springer, New York, second edition, 2003. The first factorial moment measure of a point process coincides with its first moment measure or intensity measure, which gives the expected or average number of points of the point process located in some region of space. In general, if the number of points in some region is considered as a random variable, then the moment factorial measure of this region is the factorial moment of this random variable.
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These example programs calculate the factorial of 5 using their respective languages' syntax for a do-while loop.
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The scientists at the Institute of Medicine said in their report that the factorial method tended to underestimate calorie expenditure.
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General mixed radix systems were studied by Georg Cantor.. The term "factorial number system" is used by Knuth,. while the French equivalent "numération factorielle" was first used in 1888.. The term "factoradic", which is a portmanteau of factorial and mixed radix, appears to be of more recent date.The term "factoradic" is apparently introduced in .
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Like tetration, there is currently no accepted method of extension of the exponential factorial function to real and complex values of its argument, unlike the factorial function, for which such an extension is provided by the gamma function. But it is possible to expand it if it is defined in a strip width of 1.
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For example, a 25 − 2 design is 1/4 of a two level, five factor factorial design. Rather than the 32 runs that would be required for the full 25 factorial experiment, this experiment requires only eight runs. In practice, one rarely encounters l > 2 levels in fractional factorial designs, since response surface methodology is a much more experimentally efficient way to determine the relationship between the experimental response and factors at multiple levels. In addition, the methodology to generate such designs for more than two levels is much more cumbersome.
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Note that the message passing approach works regardless of when `factorial(100000)` finishes computation and that no stinging/forcing is needed.
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The number of perfect matchings in a complete graph Kn (with n even) is given by the double factorial: (n-1)!!.
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1994 Have they been closed? If so, when? running experiment Is there grazing at your site? Yes, factorial simulated herbivory treatment.
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In statistics, a Yates analysis is an approach to analyzing data obtained from a designed experiment, where a factorial design has been used. Full- and fractional-factorial designs are common in designed experiments for engineering and scientific applications. In these designs, each factor is assigned two levels. These are typically called the low and high levels.
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This short collection of elementary superfunctions is illustrated in. D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.
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In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not function as base, but as place value of digits. By converting a number less than n! to factorial representation, one obtains a sequence of n digits that can be converted to a permutation of n in a straightforward way, either using them as Lehmer code or as inversion table representation; in the former case the resulting map from integers to permutations of n lists them in lexicographical order.
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313x313px In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all such factors. A full factorial design may also be called a fully crossed design. Such an experiment allows the investigator to study the effect of each factor on the response variable, as well as the effects of interactions between factors on the response variable. For the vast majority of factorial experiments, each factor has only two levels.
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Factorial designs were used in the 19th century by John Bennet Lawes and Joseph Henry Gilbert of the Rothamsted Experimental Station. Ronald Fisher argued in 1926 that "complex" designs (such as factorial designs) were more efficient than studying one factor at a time. Fisher wrote, Nature, he suggests, will best respond to "a logical and carefully thought out questionnaire". A factorial design allows the effect of several factors and even interactions between them to be determined with the same number of trials as are necessary to determine any one of the effects by itself with the same degree of accuracy.
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A factorial experiment can be analyzed using ANOVA or regression analysis. To compute the main effect of a factor "A", subtract the average response of all experimental runs for which A was at its low (or first) level from the average response of all experimental runs for which A was at its high (or second) level. Other useful exploratory analysis tools for factorial experiments include main effects plots, interaction plots, Pareto plots, and a normal probability plot of the estimated effects. When the factors are continuous, two-level factorial designs assume that the effects are linear.
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One-way designs are limited in that they allow researchers to look at only one independent variable at a time, whereas many phenomena of interest are dependent on multiple variables. Because of this, R.A Fisher popularized the use of factorial designs. Factorial designs contain two or more independent variables that are completely "crossed," which means that every level each independent variable appears in combination with every level of all other independent variables. Factorial designs carry labels that specify the number of independent variables and the number of levels of each independent variable there are in the design.
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In mathematics, and more particularly in number theory, primorial, denoted by “#”, is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors.
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This fact can be used to prove Euclid's theorem that the number of primes is infinite. Primes of the form are called factorial primes.
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However, this recursion does not permit us to compute the factorial of a negative integer; use of the formula to compute (−1)! would require a division of a nonzero value by zero, and thus blocks us from computing a factorial value for every negative integer. Similarly, the gamma function is not defined for zero or negative integers, though it is defined for all other complex numbers.
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Since the 18th century, Leonhard Euler and some other mathematicians had been trying to extend the domain of the factorial function to real numbers or even complex numbers, and eventually put forward the Gamma function. In 1997, Donald E. Knuth introduced the termial function in his The Art of Computer Programming, as an analog of factorial in addition, so as to illustrate the meaning of domain extension.
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If the function is a function of more than one point of the point process, the moment measures or factorial moment measures of the point process are needed, which can be compared to moments and factorial of random variables. The type of measure needed depends on whether the points of the point process in the random sum are need to be distinct or may repeat.
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For computational purposes, the factors are scaled so that the low level is assigned a value of -1 and the high level is assigned a value of +1. These are also commonly referred to as "-" and "+". A full factorial design contains all possible combinations of low/high levels for all the factors. A fractional factorial design contains a carefully chosen subset of these combinations.
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Preconditions are sometimes tested using guards or assertions within the code itself, and some languages have specific syntactic constructions for doing so. For example: the factorial is only defined for integers greater than or equal to zero. So a program that calculates the factorial of an input number would have preconditions that the number be an integer and that it be greater than or equal to zero.
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A factorial experiment is a statistically structured experiment in which several factors (watering levels, antibiotics, fertilizers, etc.) are applied to each experimental unit at varying (but integral) levels (high, low, or various intermediate levels). In a full factorial experiment all combinations of levels of the factors need to be tested, but to minimize confounding influences the levels should be varied within any experimental run. An orthogonal array of strength 2 can be used to design a factorial experiment. The columns represent the various factors and the entries are the levels that the factors can be applied at (assuming that all factors can be applied at the same number of levels).
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The design consists of three distinct sets of experimental runs: # A factorial (perhaps fractional) design in the factors studied, each having two levels; # A set of center points, experimental runs whose values of each factor are the medians of the values used in the factorial portion. This point is often replicated in order to improve the precision of the experiment; # A set of axial points, experimental runs identical to the centre points except for one factor, which will take on values both below and above the median of the two factorial levels, and typically both outside their range. All factors are varied in this way.
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Matrices are the input data for performing network analysis, factorial analysis or multidimensional scaling analysis; # Text mining of manuscripts (title, abstract, authors' keywords, etc.); # Co-word analysis.
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By Lagrange's theorem, the order of any finite permutation group of degree n must divide n! since n-factorial is the order of the symmetric group Sn.
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In 2003, Bingham and SitterBingham, D. and Sitter, R.R. (2003), "Fractional Factorial Split-Plot Designs for Robust Parameter Experiments", Technometrics, 45 80–89. defined maximum resolution and minimum aberration for two-level fractional factorial designs. Resolution determines the worst amount of aliasing present, and aberration determines how much of that worst-case aliasing is present in the design. Resolution III designs alias main effects with two-factor interactions.
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52-54, 1967. RIF uses the Principle of Moments or Varignon’s Theorem to calculate the first factorial moment of probability in order to define this center point of balance among all confidence weights (i.e., the point of Risk Equilibration). The formal derivation of the RIF is divided into three separate calculations: (1) calculation of the first factorial moment, (2) calculation of inclination, and (3) calculation of the risk inclination score.
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The Fibonorial numbers are used in the definition of Fibonomial coefficients (or Fibonacci-binomial coefficients) similarly as the factorial numbers are used in the definition of binomial coefficients.
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The function that does this value-for-function-object replacement can generically wrap any referentially transparent function. Consider the following pseudocode (where it is assumed that functions are first-class values): function memoized-call (F is a function object parameter) if F has no attached array values then allocate an associative array called values; attach values to F; end if; if F.values[arguments] is empty then F.values[arguments] = F(arguments); end if; return F.values[arguments]; end function In order to call an automatically memoized version of `factorial` using the above strategy, rather than calling `factorial` directly, code invokes `memoized- call(factorial(n))`. Each such call first checks to see if a holder array has been allocated to store results, and if not, attaches that array. If no entry exists at the position `values[arguments]` (where `arguments` are used as the key of the associative array), a real call is made to `factorial` with the supplied arguments.
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If a factorial moment measure is absolutely continuous, then with respect to the Lebesgue measure it is said to have a density (which is a generalized form of a derivative), and this density is known by a number of names such as factorial moment density and product density, as well as coincidence density, joint intensity , correlation function or multivariate frequency spectrumK. Handa. The two-parameter {Poisson-Dirichlet} point process. Bernoulli, 15(4):1082–1116, 2009. The first and second factorial moment densities of a point process are used in the definition of the pair correlation function, which gives a way to statistically quantify the strength of interaction or correlation between points of a point process.
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Zuckerman, M. (2002). Zuckerman-Kuhlman Personality Questionnaire (ZKPQ) An alternative five- factorial model. In B. DeRaad & M. Perugini (Eds.), Big Five Assessment (pp. 377-396). Seattle, WA: Hogrefe & Huber.
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For example, a 2x3 factorial design has two independent variables (because there are two numbers in the description), the first variable having two levels and the second having three.
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A factorial language L is one where if a word is in L, then all factors of that word are also in L. In combinatorics on words, a common problem is to determine the number A(n) of length-n words in a factorial language. Clearly A(m+n) \leq A(m)A(n), so \log A(n) is subadditive, and hence Fekete's lemma can be used to estimate the growth of A(n).
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Hadamard's gamma function plotted over part of the real axis. Unlike the classical gamma function, it is holomorphic; there are no poles. In mathematics, Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma function. This function, with its argument shifted down by 1, interpolates the factorial and extends it to real and complex numbers in a different way than Euler's gamma function.
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Five is the third prime number. Because it can be written as , five is classified as a Fermat prime; therefore, a regular polygon with 5 sides (a regular pentagon) is constructible with compass and an unmarked straightedge. Five is the third Sophie Germain prime, the first safe prime, the third Catalan number, and the third Mersenne prime exponent. Five is the first Wilson prime and the third factorial prime, also an alternating factorial.
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By convention, the value of 0! is defined as 1. This classical factorial function appears prominently in many theorems in number theory. The following are a few of these theorems.
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Measurement invariance is often tested in the framework of multiple-group confirmatory factor analysis (CFA). In the context of structural equation models, including CFA, measurement invariance is often termed factorial invariance.
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There are n! permutations of a set with n elements, and [n]q! maximal flags in F, where :[n]_q! := [1]_q [2]_q \dots [n]_q is the q-factorial.
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The function f is a unary operation on A. Common notations are prefix notation (e.g. +, −, ¬), postfix notation (e.g. factorial n!), functional notation (e.g. sin x or sin(x)), and superscripts (e.g.
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Programmers are encouraged to partake in the biannual programming competition "Pyweek". In 2014, a team from Denison called "101 Factorial" placed third for the entertainment value of their game "Hyperturret Defense".
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The risk inclination formula uses the principle of moments, or Varignon’s theorem, to calculate the first factorial moment of probability in order to define this center point of balance among all confidence weights (i.e., the point of risk equilibration). The formal derivation of the RIF is divided into three separate calculations: (1) calculation of 1st factorial moment, (2) calculation of inclination, and (3) calculation of the risk inclination score. The RIF , is a component of the risk inclination model.
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In the statistical analysis of the results from factorial experiments, the sparsity-of-effects principle states that a system is usually dominated by main effects and low-order interactions. Thus it is most likely that main (single factor) effects and two-factor interactions are the most significant responses in a factorial experiment. In other words, higher order interactions such as three-factor interactions are very rare. This is sometimes referred to as the hierarchical ordering principle.
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The SSD-12 consists of a three-factorial structure which reflects the three psychological criteria interpreted as cognitive, affective and behavioural aspects of the DSM-5 B criteria of somatic symptom disorder.
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Different DoE strategies are available, including random generator sequences, Factorial, Orthogonal and Iterative Techniques, as well as D-Optimal or Cross Validation. Monte Carlo and Latin hypercube are available for robustness analysis.
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The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5×4×3×2×1 = 120.
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The Second International Study of Infarct Survival (ISIS-2) was a 2×2 factorial placebo- controlled trial of aspirin and the thrombolytic drug streptokinase. It recruited 17,187 patients and was completed in 1988.
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Later supervised learning usually works much better when the raw input data is first translated into such a factorial code. For example, suppose the final goal is to classify images with highly redundant pixels. A naive Bayes classifier will assume the pixels are statistically independent random variables and therefore fail to produce good results. If the data are first encoded in a factorial way, however, then the naive Bayes classifier will achieve its optimal performance (compare Schmidhuber et al. 1996).
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The Friedman and Wise paper described only explicit futures, probably reflecting the difficulty of efficiently implementing implicit futures on stock hardware. The difficulty is that stock hardware does not deal with futures for primitive data types like integers. For example, an add instruction does not know how to deal with `3 + future factorial(100000)`. In pure actor or object languages this problem can be solved by sending `future factorial(100000)` the message `+[3]`, which asks the future to add `3` to itself and return the result.
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In probability and statistics, a factorial moment measure is a mathematical quantity, function or, more precisely, measure that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both. Moment measures generalize the idea of factorial moments, which are useful for studying non-negative integer-valued random variables.D. J. Daley and D. Vere- Jones. An introduction to the theory of point processes. Vol.
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Fractional designs are expressed using the notation lk − p, where l is the number of levels of each factor investigated, k is the number of factors investigated, and p describes the size of the fraction of the full factorial used. Formally, p is the number of generators, assignments as to which effects or interactions are confounded, i.e., cannot be estimated independently of each other (see below). A design with p such generators is a 1/(lp)=l-p fraction of the full factorial design.
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But it was hard to decide how representative was the crop sample chosen. Factorial design methodology showed how to estimate and correct for any random variation within the sample and also in the data collection procedures.
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Effects of training on Internet self-efficacy and computer user attitudes. Computers and Human Behavior, 18, 479-494.Torkzadeh, G., & Koufteros, X. (1994). Factorial validity of computer self-efficacy scale and the impact of computer training.
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Machin AJ. Why handaxes just aren't that sexy: a response to Kohn & Mithen (1999). Antiquity. 2008 Sep;82(317):761-6.Machin A. The role of the individual agent in Acheulean biface variability: a multi-factorial model.
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The factorial validity of the Maslach Burnout Inventory in a sample of California elementary and junior high school classroom teachers. Educational and Psychological Measurement, 44: 1009-1016. and general work populations.Leiter, M. P. & Schaufeli, W. B. (1996).
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BANKS, CHARLOTTE, and GERTRUDE KEIR. "A factorial analysis of items in the Bernreuter Personality Inventory." British Journal of Statistical Psychology 5.1 (1952): 19-29. The "Flanagan keys" eventually were incorporated into the published version of the test.
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A good solution to this is the gamma function. There are infinitely many continuous extensions of the factorial to non-integers: infinitely many curves can be drawn through any set of isolated points. The gamma function is the most useful solution in practice, being analytic (except at the non-positive integers), and it can be defined in several equivalent ways. However, it is not the only analytic function which extends the factorial, as adding to it any analytic function which is zero on the positive integers, such as , will give another function with that property.
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The sequence of these values, (0,0,2,1,1), gives the number : Consecutive permutations in the sequence generated by the Steinhaus–Johnson–Trotter algorithm have numbers of inversions that differ by one, forming a Gray code for the factorial number system.; .
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Frank Yates made significant contributions, particularly in the analysis of designs, by the Yates analysis. The term "factorial" may not have been used in print before 1935, when Fisher used it in his book The Design of Experiments.
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Version 9 incorporates split-plot factorial designs including two-level full and fractional factorials, general factorials and optimal factorials.“Design Expert Software Versión 9, herramienta de diseño de experimentos y simulación más avanzada ,” InfoWeek Online, December 4, 2014.
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The sparsity-of-effects principle actually refers to the idea that only a few effects in a factorial experiment will be statistically significant. This principle is only valid on the assumption of a factor space far from a stationary point.
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This rule can now be extended to any number of odd bells in changes, making Grandsire an easily extendable method. The hunt bell is changed many times during such ringing to enable the full factorial number of changes to be achieved.
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The Fourth International Study of Infarct Survival (ISIS-4) was a 2×2×2 factorial placebo-controlled trial of the angiotensin-converting enzyme inhibitor (ACE inhibitor) captopril, isosorbide mononitrate and magnesium sulphate. It recruited 58,050 patients and was completed in 1993.
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Minimum- Run Resolution IV and Minimum-Run Resolution V experimental designs were invented by Pat Whitcomb and Gary Oehlert in 2004.Pat Whitcomb, Mark Anderson, “Screening Process Factors in the Presence of Interactions,” American Society for Quality, 58th Annual Quality Congress, Session W202, 2004. Minimum-run resolution IV (“MR4”) factorial designs estimate all main effects, clear of two-factor or higher interactions, in a minimum of experimental runs. MR4 designs work well for factor screening. Minimum-run resolution V (“MR5”) factorial designs estimate all main effects and two-factor interactions in a minimum of experimental runs.
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The following sortable table shows the 24 permutations of four elements with different inversion related vectors. The left and right inversion counts l and r (the latter often called Lehmer code) are particularly eligible to be interpreted as factorial numbers. l gives the permutation's position in reverse colexicographic order (the default order of this table), and the latter the position in lexicographic order (both counted from 0). Sorting by a column that has the omissible 0 on the right makes the factorial numbers in that column correspond to the index numbers in the immovable column on the left.
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In the design of experiments and analysis of variance, a main effect is the effect of an independent variable on a dependent variable averaged across the levels of any other independent variables. The term is frequently used in the context of factorial designs and regression models to distinguish main effects from interaction effects. Relative to a factorial design, under an analysis of variance, a main effect test will test the hypotheses expected such as H0, the null hypothesis. Running a hypothesis for a main effect will test whether there is evidence of an effect of different treatments.
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The global optimum of this objective function corresponds to a factorial code represented in a distributed fashion across the outputs of the feature detectors. Painsky, Rosset and Feder (2016, 2017) further studied this problem in the context of independent component analysis over finite alphabet sizes. Through a series of theorems they show that the factorial coding problem can be accurately solved with a branch and bound search tree algorithm, or tightly approximated with a series of linear problems. In addition, they introduce a simple transformation (namely, order permutation) which provides a greedy yet very effective approximation of the optimal solution.
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Volatile anaesthetics may be the main cause of early but not delayed postoperative vomiting: a randomized controlled trial of factorial design. British Journal of Anaesthesia 2002; 88: 659–668Apfel CC, Korttila K, Abdalla M, Kerger H, Turan A, Vedder I, Zernak C, Danner K, Jokela R, Pocock SJ, Trenkler S, Kredel M, Biedler A, Sessler DI, Roewer N: A factorial trial of six interventions for the prevention of postoperative nausea and vomiting. NEJM. 2004 Jun 10;350(24):2441-51. Among these trials, many studies were aiming to get a label for the indication of PONV prevention (phase II and III trials).
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An easy way to estimate a first- degree polynomial model is to use a factorial experiment or a fractional factorial design. This is sufficient to determine which explanatory variables affect the response variable(s) of interest. Once it is suspected that only significant explanatory variables are left, then a more complicated design, such as a central composite design can be implemented to estimate a second- degree polynomial model, which is still only an approximation at best. However, the second-degree model can be used to optimize (maximize, minimize, or attain a specific target for) the response variable(s) of interest.
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D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. Stochastic geometry and its applications, volume 2. Wiley Chichester, 1995. Factorial moment measures completely characterize a wide class of point processes, which means they can be used to uniquely identify a point process.
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There are 52! (i.e., 52 factorial), or approximately 8, distinct deals. However, some games are effectively identical to others because suits assigned to cards are arbitrary or columns can be swapped. After taking these factors into account, there are approximately 1.75 distinct games.
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In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!. It is named after Adrien-Marie Legendre. It is also sometimes known as de Polignac's formula, after Alphonse de Polignac.
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His work on factorials is independent of that of James Stirling and Vandermonde. He was the first to use the notation n! (Elements d'arithmétique universelle, 1808). In fact, the more general concept of factorial was found at the same time by Arbogast.
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Townsend's major theoretical contributions include Systems Factorial Technology,Townsend, J.T. & Nozawa, G. (1995). Spatio-temporal properties of elementary perception: An investigation of parallel, serial and coactive theories. Journal of Mathematical Psychology, 39, 321-360. General Recognition TheoryAshby, F. G., & Townsend, J. T. (1986).
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A little-known feature of the calculator app included with the Android operating system is that it will accept fractional values as input to the factorial function and return the equivalent gamma function value. The same is true for Windows Calculator (in scientific mode).
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In a stable marriage instance chosen to maximize the number of different stable matchings, this number can be at least 2^{n-1}, and us also upper-bounded by an exponential function of (significantly smaller than the naive factorial bound on the number of matchings).
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In 2012 Ruscio and Roche introduced the comparative data (CD) procedure in an attempt improve upon the PA method. The authors state that "rather than generating random datasets, which only take into account sampling error, multiple datasets with known factorial structures are analyzed to determine which best reproduces the profile of eigenvalues for the actual data" (p. 258). The strength of the procedure is its ability to not only incorporate sampling error, but also the factorial structure and multivariate distribution of the items. Ruscio and Roche's (2012) simulation study determined that the CD procedure outperformed many other methods aimed at determining the correct number of factors to retain.
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At this point, gcd(110229-2,112729) = 139, so 139 is a non-trivial factor of 112729. Notice that p+1 = 140 = 22 × 5 × 7\. The number 7! is the lowest factorial which is multiple of 140, so the proper factor 139 is found in this step.
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Giuga's conjecture says that this equation is also a sufficient condition for p to be prime., The property of Giuga, pp. 21–22. Wilson's theorem says that an integer p>1 is prime if and only if the factorial (p-1)! is congruent to -1 mod p.
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In MCA, the relationship square provides a synthetic view of the connections between mixed variables, all the more valuable as there are many variables having many categories. This representation iscan be useful in any factorial analysis when there are numerous mixed variables, active and/or supplementary.
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Experimental designs sustain those basic principles of experimental statistics. There are three basic experimental designs to randomly allocate treatments in all plots of the experiment. They are completely randomized design, randomized block design, and factorial designs. Treatments can be arranged in many ways inside the experiment.
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Examples include sickle-cell anemia, Tay–Sachs disease, cystic fibrosis and xeroderma pigmentosa. A disease controlled by a single gene contrasts with a multi- factorial disease, like heart disease, which is affected by several loci (and the environment) as well as those diseases inherited in a non-Mendelian fashion.
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Most calculators use scientific notation with 2-digit decimal exponents, and the largest factorial that fits is then 69!, because . Other implementations (such as computer software such as spreadsheet programs) can often handle larger values. Most software applications will compute small factorials by direct multiplication or table lookup.
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ANOVA on ranks means that a standard analysis of variance is calculated on the rank-transformed data. Conducting factorial ANOVA on the ranks of original scores has also been suggested.Iman, R. L., & Conover, W. J. (1976). A comparison of several rank tests for the two-way layout (SAND76-0631).
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When looking at disruptive behaviors such as ODD, research has shown that the causes of behaviors are multi factorial. However, disruptive behaviors have been identified as being mostly due either to biological or environmental factors.Goldstein, S., & In DeVries, M. (2017). Handbook of DSM-5 Disorders in Children and Adolescents.
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720 (seven hundred [and] twenty) is the natural number following 719 and preceding 721. It is 6! (6 factorial), a composite number with thirty divisors, more than any number below, making it a highly composite number. It is a Harshad number in every base from binary to decimal.
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If there are two or more IVs, there may be a significant interaction, which means that the effect of one IV on the DV changes depending on the level of another factor. One can investigate the simple main effects using the same methods as in a factorial ANOVA.
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To fully understand the design criteria, an understanding of history and fractional factorial designs is necessary. FFDs seek to understand which factors have an effect on a response and seek to optimize the response by finding the appropriate factor settings. Unlike RPDs, FFDs do not distinguish between control and noise variables.
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I. Probability and its Applications (New York). Springer, New York, second edition, 2003 and arise in the use of probability- generating functions to derive the moments of discrete random variables. Factorial moments serve as analytic tools in the mathematical field of combinatorics, which is the study of discrete mathematical structures.
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Brocard's problem is a problem in mathematics that asks to find integer values of n and m for which :n!+1 = m^2, where n! is the factorial. It was posed by Henri Brocard in a pair of articles in 1876 and 1885, and independently in 1913 by Srinivasa Ramanujan.
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Other features include x-y exchange, percentage, register-current stack exchange, factorial, permutation/combination. Maximum open parenthesis level is 15, with 6 pending operation stacks. There are 2 operand registers (a, b) and 1 memory register. TI-35 PLUS uses Toshiba T7767 processor, while TI-36 SOLAR uses Toshiba T7768 processor.
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The factorial function, generalized to all real numbers except negative integers. For example, , , . In addition to this, the pi function satisfies the same recurrence as factorials do, but at every complex value where it is defined :\Pi(z) = z\Pi(z-1)\,. This is no longer a recurrence relation but a functional equation.
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The termial was coined by Donald E. Knuth in his The Art of Computer Programming. It is the additive analog of the factorial function, which is the product of integers from to . He used it to illustrate the extension of the domain from positive integers to the real numbers.Donald E. Knuth (1997).
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2010-02-10 Urinary retention in females is uncommon, occurring 1 in 100,000 every year, with a female-to-male incidence rate of 1:13. It is usually transient. The causes of UR in women can be multi- factorial, and can be postoperative and postpartum. Prompt urethral catheterization usually resolves the problem.
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The factorial function can be defined (using the above ## function) as follows. #>>>>>> Notice that the inner computation (#
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5040 is a factorial (7!) and one less than a square, making (7, 71) a Brown number pair, a highly composite number, superior highly composite number, abundant number, highly abundant number, superabundant number, colossally abundant number and the number of permutations of 4 items out of 10 choices (10 × 9 × 8 × 7 = 5040).
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Self and Identity 2(3): 223–250. The long scale consists of 26 items and the short scale consists of 12 items.Raes, F., Pommier, E., Neff, K. D. and Van Gucht, D. (2011), Construction and factorial validation of a short form of the Self-Compassion Scale. Clinical Psychology & Psychotherapy, 18: 250–255.
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Software that is used for designing factorial experiments plays an important role in scientific experiments and represents a route to the implementation of design of experiments procedures that derive from statistical and combinatorial theory. In principle, easy-to-use design of experiments (DOE) software should be available to all experimenters to foster use of DOE.
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A naive solution for the assignment problem is to check all the assignments and calculate the cost of each one. This may be very inefficient since, with n agents and n tasks, there are n! (factorial of n) different assignments. Fortunately, there are many algorithms for solving the problem in time polynomial in n.
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The subject codes so listed are used by the two major reviewing databases, Mathematical Reviews and Zentralblatt MATH. This list has some items that would not fit in such a classification, such as list of exponential topics and list of factorial and binomial topics, which may surprise the reader with the diversity of their coverage.
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Joseph Ser (1875-1954) was a French mathematician, of whom little was known till now. He published 45 papers between 1900 and 1954, among which four monographs, edited in Paris by Henry Gauthier-Villars. In the main, he worked on number theory and infinite series. He got important results in the domain of factorial series.
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However, the case was decided using the multi-factorial approach with reasons against finding a duty being: that Esanda, as a corporation, was not vulnerable as it could have made its own enquiries regarding the financial position of the borrower; and that allowing the appeal may have given rise to indeterminate liability to the auditor.
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The Third International Study of Infarct Survival (ISIS-3) was a 3×2 factorial trial that compared the three thrombolytic drugs streptokinase, tissue plasminogen activator (tPA) and anistreplase to each other, and also compared the anticoagulant heparin to no heparin. All patients were also given aspirin. It recruited 41,299 patients and was completed in 1991.
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Michael G. Akritas (born 1950) is a Greek American statistician and professor of Statistics at the Pennsylvania State University. His research has focused on nonparametric statistics, factorial designs, censored data, high- dimensional data modeling, astrostatistics, and social statistics. Akritas was elected Fellow of the American Statistical Association in 2001, and Fellow of the Institute of Mathematical Statistics in 2001.
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The number of different combinations possible with each additional button, however, increases according to a factorial progression.Fronczak, Tom. Monday Mind Teasers: GROW Games. Destructoid. 10 August 2009. Thus, a game with 5 buttons will have 120 possible combinations (of which only one leads to the good ending) and a game with 12 buttons will have 479 million possible combinations.
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Modifications also allow extending the test to examine more than two factors. The test strength of the Scheirer–Ray–Hare test, i.e. the probability of actually finding a statistically significant result, is significantly lower than that of the parametric multi-factorial ANOVA, so that it is considered more conservative in comparison of both methods.Scheirer–Ray–Hare test.
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If the sample has mean 0, standard deviation 1 then a line through 0 with slope 1 could be used. With more points, random deviations from a line will be less pronounced. Normal plots are often used with as few as 7 points, e.g., with plotting the effects in a saturated model from a 2-level fractional factorial experiment.
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Kramp could read and write in German and French. Kramp was appointed professor of mathematics at Strasbourg, the town of his birth, in 1809. He was elected to the geometry section of the French Academy of Sciences in 1817. As Bessel, Legendre and Gauss did, Kramp worked on the generalised factorial function which applied to non-integers.
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One dimension comprises nine industry attractiveness measures; the other comprises twelve internal business strength measures. The GE matrix helps a strategic business unit evaluate its overall strength. Each product, brand, service, or potential product is mapped in this industry attractiveness/business strength space. The GE multi factorial was first developed by McKinsey for General Electric in the 1970s.
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The development of birth defects associated with maternal hyperglycemia is multi-factorial. Environmental factors and genetic predisposition (paternal, maternal and offspring genome) are important in diabetic embryopathy. The diets of diabetic mothers impacts the rate at which malformations form in their offspring. Furthermore, there is evidence that resistance to certain malformations caused by diabetes is genetic.
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Unequal exchange is given if double factorial terms of trade of the respective country are < 1.0 (Raffer, 1987, Amin, 1975). The former ideological head of the Blekingegade Gang and political activist Torkil Lauesen argues in his book The Global Perspective that political theory and practice stemming from dependency theory are more relevant than ever.Lauesen, Torkil. The Global Perspective: Reflections on Imperialism and Resistance.
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It was also discovered that the Type I error rate problem was exacerbated in the context of Analysis of Covariance, particularly as the correlation between the covariate and the dependent variable increased.Headrick, T. C. (1997). Type I error and power of the rank transform analysis of covariance (ANCOVA) in a 3 x 4 factorial layout. Unpublished doctoral dissertation, University of South Florida.
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In statistics, a central composite design is an experimental design, useful in response surface methodology, for building a second order (quadratic) model for the response variable without needing to use a complete three-level factorial experiment. After the designed experiment is performed, linear regression is used, sometimes iteratively, to obtain results. Coded variables are often used when constructing this design.
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A Wilson prime, named after English mathematician John Wilson, is a prime number p such that p2 divides (p − 1)! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1\. The only known Wilson primes are 5, 13, and 563 ; if any others exist, they must be greater than 2.
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Two independent conditions that are both strictly stronger than the BFD condition are the half-factorial domain condition (HFD: any two factorizations of any given x have the same length), and the finite factorization domain condition (FFD: any x has but a finite number of non-associate divisors). Every unique factorization domain obviously satisfies these two conditions, but neither implies unique factorization.
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Clearly, a more convenient and flexible method of executing the calculator's instructions was needed. The HP-41C had a relatively small keyboard, and only one shift key, but provided hundreds of functions. Every function that was not assigned to a key could be invoked through the _XEQ_ key (pronounced EXEQTE -- "execute") and spelled out in full, e.g. _XEQ_ FACT for the factorial function.
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It is thus the integer after zero. Any number multiplied by one remains that number, as one is the identity for multiplication. As a result, 1 is its own factorial, its own square and square root, its own cube and cube root, and so on. One is also the result of the empty product, as any number multiplied by one is itself.
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An example of an algorithm that runs in factorial time is bogosort, a notoriously inefficient sorting algorithm based on trial and error. Bogosort sorts a list of n items by repeatedly shuffling the list until it is found to be sorted. In the average case, each pass through the bogosort algorithm will examine one of the n! orderings of the n items.
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The "plain Course" of Grandsire Doubles, still commonly rung today, is also shown. As more bells were added to towers, Stedman's methods were adapted to higher numbers. Since the number of possible changes varies with the factorial ( n! ) of the number of bells, it became impractical to ring extents on numbers above 7 (the extent on 10 bells would take around three months).
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Goodnow’s interest in culture and thought came after travelling to Hong Kong when she became interested in the thinking process of children from different cultures. She used Piaget’s conservation tasks and two combinational tasks. This study was conducted by giving 500 Chinese and European boys age ten to thirteen the Piagetian tasks of conservation of weight, volume, and space, along with Raven’s Progressive Matrices task and Piaget’s factorial problem. Children who were unschooled had difficulty in performing the tasks that were required for the factorial problem and the Progressive Matrices task. When she gave the tests to sample of “average” (IQ 101–120) and “dull” (IQ 64–88) boys in Montgomery County, Maryland she found that the US boys of average intelligence scores were similar to the schooled and semi-schooled Chinese boys on the conservation tasks.
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Since grandparents cannot know the length of their own life course, grandparents invest differentially in the children of those children who they deem most likely to support them in old age. Hechter’s engagement in value theory in general, and the measurement of values in particular, is shown in yet another paper, 'Prediction vs Explanation in the Measurement of Values' in the European Sociological Review, where rational action is explored from different measurement angles. In taking the role of values seriously as motivating factors in assessing social behavior, the authors construct a parallel factorial survey method to the more commonly used descriptive survey in value measurements. This method is based on an indirect form of measurement (factorial surveys) in that respondents have to make choices about a series of hypothetical vignettes, rather than responding to direct questions.
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Zuckerman retired in September 2002, aged 74 years and resided in Philadelphia where he wrote journal articles and invited book chapters for international handbooks until shortly before his death.Zuckerman, M. (2008). Zuckerman-Kuhlman Personality Questionnaire (ZKPQ): An operational definition of the alternative five factorial model of personality. In Boyle, G.J., Matthews, G., & Saklofske, D.H. (Eds.), The SAGE Handbook of Personality Theory and Assessment: Vol.
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This is often more efficient. The asymptotically best efficiency is obtained by computing from its prime factorization. As documented by Peter Borwein, prime factorization allows to be computed in time , provided that a fast multiplication algorithm is used (for example, the Schönhage–Strassen algorithm). Peter Luschny presents source code and benchmarks for several efficient factorial algorithms, with or without the use of a prime sieve.
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The formal algebraic manipulation of series investigated by Lagrange and Laplace in the 1770s has been put in the form of operator equalities by Arbogast in 1800. We owe him the general concept of factorial as a product of a finite number of terms in arithmetic progression. The original version of this article was taken from the public domain resource the Rouse History of Mathematics.
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Meta-modeling or Surrogate model is a process to win the mathematical relationship between design parameters and product characteristics. For each point in the parameter space, there is a corresponding point of the design space. Many model calculations should be performed to show the relationship between input and output systematically (Full Factorial Design). For a high computing effort of the product model, it is practically infeasible.
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Formann provided both theoretical and empirical evidence that the application of the parallel analysis for uncovering the factorial structure of binary variables is not appropriate. Results of a simulation study showed that sample size, item discrimination, and type of correlation coefficient considerably influence the performance of parallel analysis.Tran, U. S., & Formann, A. K. (2009). Performance of parallel analysis in retrieving unidimensionality in the presence of binary data.
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Microsoft Windows NT Calculator Version 3.1 A simple arithmetic calculator was first included with Windows 1.0.Windows 1.01 - Graphical User Interface Gallery In Windows 3.0, a scientific mode was added, which included exponents and roots, logarithms, factorial-based functions, trigonometry (supports radian, degree and gradians angles), base conversions (2, 8, 10, 16), logic operations, statistical functions such as single variable statistics and linear regression.
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Base systems corresponding to primorials (such as base 30, not to be confused with the primorial number system) have a lower proportion of repeating fractions than any smaller base. Every primorial is a sparsely totient number. The -compositorial of a composite number is the product of all composite numbers up to and including . The -compositorial is equal to the -factorial divided by the primorial .
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Cause of greying is incompletely understood. It is a complex multi-factorial process mainly considered to be an interplay of nutritional, genetic and environmental factors. Nutritional deficiencies like vitamin B12 deficiency, severe iron deficiency, chronic protein loss, copper deficiency are often found associated with premature greying of hairs. Other factors that have been incriminated are low serum ferritin, and low serum calcium and vitamin D3 levels.
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So a program that calculates the factorial of an input number would have postconditions that the result after the calculation be an integer and that it be greater than or equal to 1. Another example: a program that calculates the square root of an input number might have the postconditions that the result be a number and that its square be equal to the input.
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In languages such as Lua, more sophisticated techniques exist which allow a function to be replaced by a new function with the same name, which would permit: factorial = construct-memoized- functor(factorial) Essentially, such techniques involve attaching the original function object to the created functor and forwarding calls to the original function being memoized via an alias when a call to the actual function is required (to avoid endless recursion), as illustrated below: function construct-memoized-functor (F is a function object parameter) allocate a function object called memoized-version; let memoized-version(arguments) be if self has no attached array values then [self is a reference to this object] allocate an associative array called values; attach values to self; allocate a new function object called alias; attach alias to self; [for later ability to invoke F indirectly] self.alias = F; end if; if self.values[arguments] is empty then self.values[arguments] = self.
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Factorials were used to count permutations at least as early as the 12th century, by Indian scholars. In 1677, Fabian Stedman described factorials as applied to change ringing, a musical art involving the ringing of many tuned bells. After describing a recursive approach, Stedman gives a statement of a factorial (using the language of the original): The notation was introduced by the French mathematician Christian Kramp in 1808.
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For example, . The sequence of double factorials for starts as : 1, 3, 15, 105, 945, , ,... Double factorial notation may be used to simplify the expression of certain trigonometric integrals, to provide an expression for the values of the gamma function at half-integer arguments and the volume of hyperspheres,. and to solve many counting problems in combinatorics including counting binary trees with labeled leaves and perfect matchings in complete graphs..
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The symmetric group Sn (on a finite set of n symbols) is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself.Jacobson (2009), p. 31. Since there are n! (n factorial) possible permutations of a set of n symbols, it follows that the order (i.e.
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More significantly, the value of the coefficients of each term can be computed directly from the exponents. The formula is: , where x, y, z are the exponents of A, B, C, respectively, and "!" means factorial (e.g.: ). The exponent formulas for the 4th layer are: The exponents of each expansion term can be clearly seen and these formulae simplify to the expansion coefficients and the tetrahedron coefficients of layer 4.
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There are exactly 52 factorial (expressed in shorthand as 52!) possible orderings of the cards in a 52-card deck. In other words, there are 52 × 51 × 50 × 49 × ··· × 4 × 3 × 2 × 1 possible combinations of card sequence. This is approximately (80,658vigintillion) possible orderings, or specifically 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000. The magnitude of this number means that it is exceedingly improbable that two randomly selected, truly randomized decks will be the same.
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For some mathematical functions, a gold "f−1" prefix key would access the inverse of the gold-printed functions, e.g. "f−1" followed by "4" would calculate the inverse sine (sin^{-1}). Functions included square root, inverse, trigonometric (sine, cosine, tangent and their inverses), exponentiation, logarithms and factorial. The HP-65 was one of the first calculators to include a base conversion function, although it only supported octal (base 8) conversion.
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The parametric alternative to the Scheirer–Ray–Hare test is multi-factorial ANOVA, which requires a normal distribution of data within the samples. The Kruskal–Wallis test, from which the Scheirer–Ray–Hare test is derived, serves in contrast to this to investigate the influence of exactly one factor on the measured variable. A non-parametric test comparing exactly two unpaired samples is the Wilcoxon–Mann–Whitney test.
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There are other prime-related congruences that provide necessary and sufficient conditions on the primality of certain subsequences of the natural numbers. Many of these alternate statements characterizing primality are related to Wilson's theorem, or are restatements of this classical result given in terms of other special variants of generalized factorial functions. For instance, new variants of Wilson's theorem stated in terms of the hyperfactorials, subfactorials, and superfactorials are given in.
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GE multifactoral analysis is a technique used in brand marketing and product management to help a company decide what products to add to its portfolio and which opportunities in the market they should continue to invest in. It is conceptually similar to BCG analysis, but somewhat more complicated. Like in BCG analysis, a two-dimensional portfolio matrix is created. However, with the GE model the dimensions are multi factorial.
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Several variations on Euclid's proof exist, including the following: The factorial n! of a positive integer n is divisible by every integer from 2 to n, as it is the product of all of them. Hence, is not divisible by any of the integers from 2 to n, inclusive (it gives a remainder of 1 when divided by each). Hence is either prime or divisible by a prime larger than n.
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Uwe Storch Uwe Storch (born 12 July 1940, Leopoldshall- Lanzarote, 17 September 2017) was a German mathematician. His field of research was commutative algebra and analytic and algebraic geometry, in particular derivations, divisor class group, resultants. Storch studied mathematics, physics and mathematical logic in Münster and in Heidelberg. He got his PhD 1966 under the supervision of Heinrich Behnke with a thesis on almost (or Q) factorial rings.
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The book has 15 chapters. Its introductory chapter covers the history and applications of experimental designs, it has five chapters on balanced incomplete block designs and their existence, and three on Latin squares and mutually orthogonal Latin squares. Other chapters cover resolvable block designs, finite geometry, symmetric and asymmetric factorial designs, and partially balanced incomplete block designs. After this standard material, the remaining two chapters cover less-standard material.
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In this study two types of experiments were used to understand the way in which individuals interpret words in survey interviews, aptitude tests, and instructions accompanying an experiment. One experiment was a factorial experiment, and the other was a more naturalistic investigation. Participants were required to interpret ordinary survey concepts such as: household furniture. The results show that an investigators actions influence the participants’ answers in both experiments.
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The number 9! is the lowest factorial which is multiple of 810, so the proper factor 811 is found in this step. The factor 139 is not found this time because p−1 = 138 = 2 × 3 × 23 which is not a divisor of 9! As can be seen in these examples we do not know in advance whether the prime that will be found has a smooth p+1 or p−1.
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176, John Tukey, a proponent of simple statistical techniques, was another influence of Shainin's. As a result of Tukey's work, Shainin developed a simple confirmation test known as a "Six Pack Test." Six Pack Tests were known for being much simpler than t-tests, being non-parametric and having a basic rule set. Shainin further developed this work into an analysis of variance, or ANOVA, permitting non-parametric analysis of Fisher's full factorial experiments.
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FX-87 is a polymorphic typed functional language based on a system for static program analysis in which every expression has two static properties: a type and an effect. In a study done by MIT, FX-87 yields similar performance results as functional languages on programs that do not contain side effects (Fibonacci, Factorial). FX-87 did yield a great performance increase when matching DNA sequences. KFX is the kernel language of FX-87.
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Early childhood caries (ECC) is a multi-factorial disease, referring to various risk factors that inter-relate to increase risk of developing the disease. These risk factors include but not limited to, cariogenic bacteria, diet practices and socioeconomic factors. Normally after 6 months, deciduous teeth begin to erupt means, they are susceptible to tooth decay or dental caries. In some unfortunate cases, infants and young children have experienced severe tooth decay called ECC.
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Complete controllability implies state-to-state controllability. The computational task of finding a control field for a particular state-to-state transformation is difficult and becomes more difficult with the increase in the size of the system. This task is in the class of hard inversion problems of high computational complexity. The algorithmic task of finding the field that generates a unitary transformation scales factorial more difficult with the size of the system.
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In that study, the CD technique, making use of Pearson correlations accurately predicted the correct number of factors 87.14% of the time. However, the simulated study never involved more than five factors. Therefore, the applicability of the CD procedure to estimate factorial structures beyond five factors is yet to be tested. Courtney includes this procedure in his recommended list and gives guidelines showing how it can be easily carried out from within SPSS's user interface.
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As a result, an internationally reliable short-form, the I-PANAS-SF, has been developed and validated comprising two 5-item scales with internal reliability, cross-sample and cross-cultural factorial invariance, temporal stability, convergent and criterion-related validities. Mroczek and Kolarz have also developed another set of scales to measure positive and negative affect. Each of the scales has 6 items. The scales have shown evidence of acceptable validity and reliability across cultures.
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This emulator does provide a written symbolic instruction set, though this has been constructed by its authors rather than based on Babbage's original works. For example, a factorial program would be written as: N0 6 N1 1 N2 1 × L1 L0 S1 – L0 L2 S0 L2 L0 CB?11 where the CB is the conditional branch instruction or "combination card" used to make the control flow jump, in this case backwards by 11 cards.
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His work thus was central to the field of statistics of natural scenes that relates the statistics of images of real world scenes to the properties of the nervous system. Barlow also worked in the field of factorial codes. The goal was to encode images with statistically redundant components or pixels such that the code components are statistically independent. Such codes are hard to find but highly useful for purposes such as image classification.
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Statistical testing relies on design of experiments. Several methods in use for multivariate testing include: #Full factorial the most straightforward method whereby all possible combinations of content variants are served with equal probability. #Discrete choice and what has mutated to become choice modeling is the complex technique that won Daniel McFadden the Nobel Prize in Economics in 2000. Choice modeling models how people make tradeoffs in the context of a purchase decision.
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Thus the sequences considered in Fourier's theorem and in Budan's theorem have the same number of sign variations. This strong relationship between the two theorems may explain the priority controversy that occurred in 19th century, and the use of several names for the same theorem. In modern usage, for computer computation, Budan's theorem is generally preferred since the sequences have much larger coefficients in Fourier's theorem than in Budan's, because of the factorial factor.
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Two basic approaches to research design are within- subjects design and between-subjects design. In within-subjects or repeated measures designs, each participant serves in more than one or perhaps all of the conditions of a study. In between-subjects designs each participant serves in only one condition of an experiment. Within-subjects designs have significant advantages over between-subjects designs, especially when it comes to complex factorial designs that have many conditions.
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Dimitrios Trichopoulos was born in Volos, Greece, about 326 kilometres (203 miles) north of Athens. He studied Medicine at the University of Athens Medical School. He further studied Pathology, Microbiology, Public Health, and Epidemiology at universities of Athens, London, Oxford, and Harvard. Dr. Trichopoulos had also studied the multi-factorial etiology of hepatocellular carcinoma (liver cancer), with emphasis on the interactive effects of hepatitis B and hepatitis C viruses, tobacco smoking, and ethanol intake.
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In search of a subject that would bring her into more contact with other people, and discouraged by the job prospects for mathematicians and physicists in Australia, Barnard shifted her interests to biometrics. She began her doctoral studies in statistics with Ronald Fisher in the Galton Laboratory at University College London in 1934. She published three papers in this period, on craniometry, factorial experiments, and environmental statistics, and completed her Ph.D. in 1936.
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Tooth wear is majorly the result of three processes; attrition, abrasion and erosion. These forms of tooth wear can further lead to a condition known as abfraction, where by tooth tissue is 'fractured' due to stress lesions caused by extrinsic forces on the enamel. Tooth wear is a complex, multi-factorial problem and there is difficulty identifying a single causative factor. However, tooth wear is often a combination of the above processes.
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Brand was born in Preston, England on 1 June 1943. He went to Queen Elizabeth's Grammar School for Boys, and was a graduate of The Queen's College, Oxford, and a 1968–1970 Fellow of Nuffield College, Oxford. He was a Lecturer at University of Edinburgh, from 1970 to 1997, teaching in personality, psychopathology and philosophical problems and researching in factorial psychology. In the 1980s he served on the United Kingdom's Council for National Academic Awards.
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Her dissertation, Equivalence of symmetric factorial designs and characterization and ranking of two-level Split-lot designs, was supervised by Angela Dean. Katsaounis was president of the Caucus for Women in Statistics in 2007. She has also held leadership positions in the American Statistical Association's sections on Physical and Engineering Sciences and on Statistical Education, and organized the Section on Physical and Engineering Sciences Continuing Education component of the 2009 Joint Statistical Meetings.
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Albert Eagle was an English mathematician who wrote several books (some of them privately published) giving his forcefully expressed and somewhat eccentric views on science and mathematics. He was an assistant to J. J. Thomson, and was later a lecturer at the Victoria University of Manchester. His best-known book is on elliptic functions, where he uses his idiosyncratic mathematical notation, such as τ instead of π/2, and !n for n factorial.
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Maurice Auslander (August 3, 1926 – November 18, 1994) was an American mathematician who worked on commutative algebra and homological algebra. He proved the Auslander–Buchsbaum theorem that regular local rings are factorial, the Auslander–Buchsbaum formula, and introduced Auslander–Reiten theory and Auslander algebras. Born in Brooklyn, New York, Auslander received his bachelor's degree and his Ph.D. (1954) from Columbia University. He was a visiting scholar at the Institute for Advanced Study in 1956-57.
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In formal methods of computer science, a paramorphism (from Greek παρά, meaning "close together") is an extension of the concept of catamorphism first introduced by Lambert Meertens to deal with a form which “eats its argument and keeps it too”,Philip Wadler.Views: A way for pattern matching to cohabit with data abstraction. Technical Report 34, Programming Methodology Group, University of Göteburg and Chalmers University of Technology, March 1987. as exemplified by the factorial function.
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There are different perspectives that value the protection of the fetus, but opposing viewpoints ask "what/whom is the fetus being protected from?" Most things a mother does can harm her fetus, even things that may not seem harmful (e.g. eating feta cheese or owning a cat). It is difficult to prove causality as development is multi-factorial, making it difficult to define what or who the fetus needs to be protected from.
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In computer programming, a postcondition is a condition or predicate that must always be true just after the execution of some section of code or after an operation in a formal specification. Postconditions are sometimes tested using assertions within the code itself. Often, postconditions are simply included in the documentation of the affected section of code. For example: The result of a factorial is always an integer and greater than or equal to 1.
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Advanced planning and scheduling (APS, also known as advanced manufacturing) refers to a manufacturing management process by which raw materials and production capacity are optimally allocated to meet demand. APS is especially well-suited to environments where simpler planning methods cannot adequately address complex trade-offs between competing priorities. Production scheduling is intrinsically very difficult due to the (approximately) factorial dependence of the size of the solution space on the number of items/products to be manufactured.
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KCalc is the software calculator integrated with the KDE Software Compilation. In the default view it includes a number pad, buttons for adding, subtracting, multiplying, and dividing, brackets, memory keys, percent, reciprocal, factorial, square, and x to the power of y buttons. Additional buttons for scientific and engineering (trigonometric and logarithmic functions), statistics and logic functions can be enabled as needed. 6 additional buttons can be predefined with mathematical constants and physical constants or custom values.
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There are many ways to systematically generate all permutations of a given sequence. One classic, simple, and flexible algorithm is based upon finding the next permutation in lexicographic ordering, if it exists. It can handle repeated values, for which case it generates each distinct multiset permutation once. Even for ordinary permutations it is significantly more efficient than generating values for the Lehmer code in lexicographic order (possibly using the factorial number system) and converting those to permutations.
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The frequency of first permanent molar treatment for children with MIH is nearly 10 times greater compared to children without MIH. The available treatment modalities for MIH is extensive but the decision on which treatment should be used is complex and multi-factorial. Factors may include: condition severity, the patient’s dental age, the child/parent’s social background and expectations. There are treatment modalities available to manage children affected by MIH; however, the evidence supporting these modalities are still weak.
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In most cases of least squares fitting, the model coefficients for previously added terms change depending on what was successively added. For example, the X1 coefficient might change depending on whether or not an X2 term was included in the model. This is not the case when the design is orthogonal, as is a 23 full factorial design. For orthogonal designs, the estimates for the previously included terms do not change as additional terms are added.
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Paul Dirac first published the Dirac equation in 1928, and later (1936) extended it to particles of any half-integer spin before Fierz and Pauli subsequently found the same equations in 1939, and about a decade before Bargman, and Wigner. Eugene Wigner wrote a paper in 1937 about unitary representations of the inhomogeneous Lorentz group, or the Poincaré group. Wigner notes Ettore Majorana and Dirac used infinitesimal operators applied to functions. Wigner classifies representations as irreducible, factorial, and unitary.
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The noise in fluorescence measurements affects the precision of qPCR. It is typically a function of excitation source intensity variation, detector noise and mechanical noise. Multi factorial analysis has suggested that the contribution of mechanical noise is the most important factor, and that systems with no moving parts in their optical paths are likely to provide improved quantitative precision. In addition, when performing high resolution melting analyses, one factor that affects the sensitivity of heteroduplex detection is fluorimetric precision.
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In statistics, the relationship square is a graphical representation for use in the factorial analysis of a table individuals x variables. This representation completes classical representations provided by principal component analysis (PCA) or multiple correspondence analysis (MCA), namely those of individuals, of quantitative variables (correlation circle) and of the categories of qualitative variables (at the centroid of the individuals who possess them). It is especially important in factor analysis of mixed data (FAMD) and in multiple factor analysis (MFA).
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V. gilleni weighs, on average, .Pygmy Mulga monitor - Perth Zoo - Western Australia Males and females can be distinguished by the presence of spines on the sides of the male's vent. It is sometimes mistaken for the stripe tailed monitor, but the two species are allopatric and can be physically distinguished by the pygmy mulga monitor's slightly larger size. This species has very high endurance, owing to it having one of the highest factorial aerobic scopes of any vertebrate.
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The effects of independent variables in factorial studies, taken singly, are referred to as main effects. This refers to the overall effect of an independent variable, averaging across all levels of the other independent variables. A main effect is the only effect detectable in a one-way design. Often more important than main effects are "interactions", which occur when the effect of one independent variable on a dependent variable depends on the level of a second independent variable.
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Practically, they show that with a careful implementation, the favorable properties of the order permutation may be achieved in an asymptotically optimal computational complexity. Importantly, they provide theoretical guarantees, showing that while not every random vector can be efficiently decomposed into independent components, the majority of vectors do decompose very well (that is, with a small constant cost), as the dimension increases. In addition, they demonstrate the use of factorial codes to data compression in multiple setups (2017).
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Forty-eight is the double factorial of 6, a highly composite number. Like all other multiples of 6, it is a semiperfect number. 48 is the second 17-gonal number. 48 is the smallest number with exactly ten divisors. There are 11 solutions to the equation φ(x) = 48, namely 65, 104, 105, 112, 130, 140, 144, 156, 168, 180 and 210. This is more than any integer below 48, making 48 a highly totient number.
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Upper Saddle River, NJ: Prentice Hall. 2008, p. 300. replication—to reduce uncertainty, measurements should be repeated and experiments replicated to identify sources of variation; blocking—to arrange experimental units into groups of units that are similar to each other, and thus reducing irrelevant sources of variation; use of factorial experiments—efficient at evaluating the effects and possible interactions of several independent factors. The British Medical Research Council officially recognized the importance of clinical trials from the 1930s.
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De Moivre also managed to approximate the binomial coefficients and factorial, and found a closed form for the Fibonacci numbers by inventing generating functions. In the 18th century, Euler worked on problems of combinatorics, and several problems of probability which are linked to combinatorics. Problems Euler worked on include the Knights tour, Graeco-Latin square, Eulerian numbers, and others. To solve the Seven Bridges of Königsberg problem he invented graph theory, which also led to the formation of topology.
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Designed experiments with full factorial design (left), response surface with second-degree polynomial (right) In statistics, response surface methodology (RSM) explores the relationships between several explanatory variables and one or more response variables. The method was introduced by George E. P. Box and K. B. Wilson in 1951. The main idea of RSM is to use a sequence of designed experiments to obtain an optimal response. Box and Wilson suggest using a second-degree polynomial model to do this.
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Functional programming languages, such as Lisp, ML and Haskell, tend to substitute a factorial program for "Hello, World!", as functional programming emphasizes recursive techniques, whereas the original examples emphasize I/O, which violates the spirit of pure functional programming by producing side effects. Languages otherwise capable of printing "Hello, World!" (Assembly, C, VHDL) may also be used in embedded systems, where text output is either difficult (requiring additional components or communication with another computer) or nonexistent.
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The notion of minimum aberration can be understood from the definition provided in Leoppky, Bingham, and Sitter (2006): :For any two 2(m1+m2 )-(p1+p2) fractional factorial robust parameter designs, D1 and D2, we say that D1 has less aberration than D2 if there exists an r such that, Bi (D1) = Bi (D2) for all i < r – 1 and Br (D1) < Br (D2). If no other design has less aberration than D1, then D1 is the minimum aberration fractional factorial robust parameter design. Leoppky, Bingham, and Sitter (2006) also provide the RPD indicator function as: :For a given design, D, and a run, x∈D, define a contrast XL(x) = ∏l∈Lxl on D, where L ∈ P and P is the set of all subsets of {1, 2, …, m}. Further, define PC to be the set of all subsets of {1, 2,…, m} and PN to be the set of all subset of {1, 2, …, m}, where an element of P is of the form L ≡ J ∪ K where J ∈ PC and K ∈ PN.
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Previous genetics studies have shown that inter-individual genome variation plays a major role in differential normal development and disease processes. However, the details of how these relationships work are far from clear, even in the case of most Mendelian disorders where single genetic alterations are fully penetrant (essentially causative, rather than risk modifying). Background genetic effects (modifier genes), epistasis, somatic variation, and environmental factors all complicate the situation. This is particularly the case in complex, multi-factorial disorders (e.g.
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Having only 2 and 3 as its prime divisors, 162 is a 3-smooth number.. 162 is also an abundant number, since its sum of divisors 1+2+3+6+9+18+27+54+81 = 201 is greater than it. As the product 3\times 6\times 9=162 of numbers three units apart from each other, it is a triple factorial number. There are 162 ways of partitioning seven items into subsets of at least two items per subset.
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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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The simple formula for the factorial, x! = 1 \times 2 \times \cdots \times x, cannot be used directly for fractional values of x since it is only valid when is a natural number (or positive integer). There are, relatively speaking, no such simple solutions for factorials; no finite combination of sums, products, powers, exponential functions, or logarithms will suffice to express x!; but it is possible to find a general formula for factorials using tools such as integrals and limits from calculus.
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In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is factorial, usually written as , which means the product of all positive integers less than or equal to . Technically, a permutation of a set is defined as a bijection from to itself. That is, it is a function from to for which every element occurs exactly once as an image value.
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Wiley Chichester, 1995. also known as Campbell's formula, entails an integral equation for the aforementioned sum over a general point process, and not necessarily a Poisson point process. There also exist equations involving moment measures and factorial moment measures that are considered versions of Campbell's formula. All these results are employed in probability and statistics with a particular importance in the theory of point processes and queueing theory as well as the related fields stochastic geometry, continuum percolation theory,R.
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With Hua Luogeng (华罗庚, alternatively Hua Loo-Keng), he developed high-dimensional combinatorial designs for numerical integration on the unit cube. Their work came to the attention of the statistician Kai-Tai Fang, who realized that their results could be used in the design of experiments. In particular, their results could be used to investigate interaction, for example, in factorial experiments and response surface methodology. Collaborating with Fang led to uniform designs, which have been used also in computer simulations.
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In comparison to the Taylor series, which consists of a series of derivatives of some function, the nth factorial moment measure plays the roll as that of the n th derivative the Taylor series. In other words, given a general functional f of some simple point process, then this Taylor-like theorem for non-Poisson point processes means an expansion exists for the expectation of the function E, provided some mathematical condition is satisfied, which ensures convergence of the expansion.
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With this terminology, a coherent sheaf on an integral normal scheme is reflexive if and only if it is torsion-free and normal in the sense of Barth. A reflexive sheaf of rank one on an integral locally factorial scheme is invertible. A divisorial sheaf on a scheme X is a rank-one reflexive sheaf that is locally free at the generic points of the conductor DX of X. For example, a canonical sheaf (dualizing sheaf) on a normal projective variety is a divisorial sheaf.
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Sheep bears some resemblance to the video game Lemmings. The player can choose between 4 herders, the people Adam Halfpint and Bo Peep, and the dogs Motley and Shep. The player must then guide sheep of 4 types (Factorial, Longwool, NeoGenetic, and Pastoral), actually aliens from the planet Ovis Aries, through a series of obstacles to the finish line in the level. There are a series of different worlds, starting with Polygon Farm and on to others, such as Village Fete and Lost in Space.
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See E19 -- De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt, from The Euler Archive, which includes a scanned copy of the original article. Euler further discovered some of the gamma function's important functional properties, including the reflection formula. James Stirling, a contemporary of Euler, also attempted to find a continuous expression for the factorial and came up with what is now known as Stirling's formula. Although Stirling's formula gives a good estimate of , also for non-integers, it does not provide the exact value.
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The equivalent code in JOSS would be: I=1(2)5 Note that this code does not include a for. That is because in JOSS, ranges are first-class citizens of the language, not something that is part of a loop as in BASIC. Ranges can be defined and used in many contexts outside loops. For example, here is a conditional expression that evaluates the factorial of a parameter x: Let f(x)=[x=0:1 ; fp(x)=0:prod(i=1(1)x:i)].
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For this case, the probability of each microstate of the system is equal, so it was equivalent for Boltzmann to calculate the number of microstates associated with a macrostate. was historically misinterpreted as literally meaning the number of microstates, and that is what it usually means today. can be counted using the formula for permutations where ranges over all possible molecular conditions and "" denotes factorial. The "correction" in the denominator is due to the fact that identical particles in the same condition are indistinguishable.
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225 is the smallest number that is a polygonal number in five different ways. It is a square number , an octagonal number, and a squared triangular number . As the square of a double factorial, counts the number of permutations of six items in which all cycles have even length, or the number of permutations in which all cycles have odd length. And as one of the Stirling numbers of the first kind, it counts the number of permutations of six items with exactly three cycles.
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Many of the local Chinese dialects in Singapore, such as Hokkien, Teochew, or Cantonese, have also been largely acculturated and differ from what is spoken in China. ;Relation with current Northern Chinese Singaporeans and the Northern Chinese have had a testy relationship in recent years. While the reasons for such a contentious relationship are multi-factorial, one of those mentioned was the cultural differences between the vast majority of Singaporeans whose ancestors were Southern Chinese immigrants and the Northern Chinese immigrants who spoke Northern Mandarin dialect.
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The levels of a factor are commonly coded as +1 for the higher level, and −1 for the lower level. For a three-level factor, the intermediate value is coded as 0\. To save space, the points in a two-level factorial experiment are often abbreviated with strings of plus and minus signs. The strings have as many symbols as factors, and their values dictate the level of each factor: conventionally, - for the first (or low) level, and + for the second (or high) level.
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The Scheirer–Ray–Hare (SRH) test is a statistical test that can be used to examine whether a measure is affected by two or more factors. Since it does not require a normal distribution of the data, it is one of the non-parametric methods. It is an extension of the Kruskal–Wallis test, the non-parametric equivalent for one-way analysis of variance (ANOVA), to the application for more than one factor. It is thus a non-parameter alternative to multi- factorial ANOVA analyses.
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RSM assists in defining the effects and interactions of the physiological factors playing a role in biotechnological processes in the production of microbial metabolites including exopolysaccharides such as β-glucans. Statistical methodologies reduce the number of experiments to provide sufficient information for statistically acceptable results. The validation of the fermentation parameters by statistical factorial design improved botryosphaeran production by Botryosphaeria rhodina MAMB-05 over unoptimized conditions. Botryosphaeria rhodina MAMB-05 when grown on nutrient media containing different carbohydrate substrates produces a family of botryosphaerans.
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A Cayley graph of the symmetric group S4 The symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself. Since there are n! (n factorial) possible permutations of a set of n symbols, it follows that the order (the number of elements) of the symmetric group Sn is n!.
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To enable efficient estimation of driving and modulatory effects, a 2x2 factorial experimental design is often used - with one factor serving as the driving input and the other as the modulatory input. Resting state experiments have no experimental manipulations within the period of the neuroimaging recording. Instead, hypotheses are tested about the coupling of endogenous fluctuations in neuronal activity, or in the differences in connectivity between sessions or subjects. The DCM framework includes models and procedures for analysing resting state data, described in the next section.
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However, there exists a fully polynomial time randomized approximation scheme for counting the number of bipartite matchings. A remarkable theorem of Kasteleyn states that the number of perfect matchings in a planar graph can be computed exactly in polynomial time via the FKT algorithm. The number of perfect matchings in a complete graph Kn (with n even) is given by the double factorial (n − 1)!!.. The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are given by the telephone numbers..
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Most real world data sets consist of data vectors whose individual components are not statistically independent. In other words, knowing the value of an element will provide information about the value of elements in the data vector. When this occurs, it can be desirable to create a factorial code of the data, i. e., a new vector-valued representation of each data vector such that it gets uniquely encoded by the resulting code vector (loss-free coding), but the code components are statistically independent.
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One understated disorder that begins during pregnancy is pelvic girdle pain. It is complex, multi-factorial, and likely to be also represented by a series of sub-groups driven by pain varying from peripheral or central nervous system,Diagnosis and classification of pelvic girdle pain disorders— Part 1: A mechanism based approach within a bio psychosocial framework. Manual Therapy, Volume 12, Issue 2, May 2007, PB. O’Sullivan and DJ Beales. altered laxity/stiffness of muscles, laxity to injury of tendinous/ligamentous structures to maladaptive body mechanics.
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The book was the first monograph published within the series Palgrave Studies in Language History and Language Change. It provides new perspectives on etymology, word formation, language change, loanwords and contact linguistics. It establishes a principled classification of neologisms, their semantic fields, the roles of source languages, and the attitudes of purists and ordinary native speakers towards multi-factorial coinage. It analyses the tension between linguistic creativity and cultural flirting on the one hand, and the preservation of a distinct language identity on the other hand.
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Parthena (Tena) Ipsilantis Katsaounis is a Greek-American statistician interested in the factorial design of physical experiments. She is a lecturer in mathematics at The Ohio State University at Mansfield, and the former president of the Caucus for Women in Statistics. Katsaounis earned a bachelor's degree in mathematics from the Aristotle University of Thessaloniki in 1984. She went to Ohio State University for graduate study, earning a master's degree in mathematics in 1988, a master's degree in statistics in 1996, and a Ph.D. in statistics in 2006.
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We then analyze a blown-up difference x of the series representing e and its strictly smaller partial sum, which approximates the limiting value e. By choosing the magnifying factor to be the factorial of b, the fraction a⁄b and the partial sum are turned into integers, hence x must be a positive integer. However, the fast convergence of the series representation implies that the magnified approximation error x is still strictly smaller than 1\. From this contradiction we deduce that e is irrational.
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There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number of elements of the symmetric group is n!, and the number of elements of the general linear group over a finite field is the q-factorial [n]_q!; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the field with one element, which considers Coxeter groups to be simple algebraic groups over the field with one element.
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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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Core, Earl L. (1974), The Monongalia Story: A Bicentennial History, Vol. I: Prelude, Parsons, W. Va.: McClain Printing Co., pp 55-56. The origin of these grassy tracts has been the subject of wide-ranging speculation. Core favored a multi-factorial explanation involving a "bald-susceptible zone" present at mountain summits where various causes (extreme weather, fires, etc.) initially wiped out trees followed by succession by grasses that was so competitive with the new tree seedlings that the cleared areas persist for a much longer period than at lower elevations.
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The gamma function interpolates the factorial function to non-integer values. The gamma function can be seen as a solution to the following interpolation problem: : "Find a smooth curve that connects the points (x, y) given by y = (x - 1)! at the positive integer values for x." A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that precisely describes the curve, in which the number of operations does not depend on the size of x.
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Mobile games announced include: Exos Heroes (by OOZOO), Ravenix: The Card Master (also by OOZOO), Dark Summoners (by SkeinGlobe), Project PK (by Rock Square), Super String (by Factorial Games). Project NM by Space Dive was also announced for PC. Games to be released on mobile and PC include: Project NL (by MeerKat Games) and Uncharted Waters Origins (by LINE Games and Koei Tecmo). On 10 Jul 2019, Nintendo released Dr. Mario World co-developed by LINE Games. On July 18, 2019, First Summoner developed by SkeinGlobe was released.
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A vignette in psychological and sociological experiments presents a hypothetical situation, to which research participants respond thereby revealing their perceptions, values, social norms or impressions of events. Peter Rossi and colleaguesRossi, Peter H., and Steven L. Nock, Eds. Measuring Social Judgments: The Factorial Survey Approach (Sage, 1982); Rossi, P. H., and Richard A. Berk (1985). "Varieties of normative consensus" American Sociological Review 50: 333-347 developed a framework for creating vignettes by systematically combining predictor variables in order to dissect the effects of the variables on dependent variables.
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Recursion is the definition of a function using the function itself. Lambda calculus cannot express this as directly as some other notations: all functions are anonymous in lambda calculus, so we can't refer to a value which is yet to be defined, inside the lambda term defining that same value. However, recursion can still be achieved by arranging for a lambda expression to receive itself as its argument value, for example in `(λx.x x) E`. Consider the factorial function `F(n)` recursively defined by : `F(n) = 1, if n = 0; else n × F(n − 1)`.
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Plinabulin (provisional name BPI-2358, formerly NPI-2358) is a small molecule under development by BeyondSpring Pharmaceuticals, and is in a world-wide Phase 3 clinical trial for non-small cell lung cancer. Plinabulin is being investigated for the reduction of chemotherapy-induced neutropenia and for anti-cancer effects in combination with immune checkpoint inhibitors and in KRAS mutated tumors. Plinabulin blocks the polymerization of tubulin in a unique manner, resulting in multi-factorial effects including an enhanced immune-oncology response, activation of the JNK pathway and disruption of the tumor blood supply.
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Psychometric properties of the SSD-12 were examined in three different samples from Germany (psychosomatic outpatient clinic, n = 698 ; general population, n =2362 Toussaint A, Löwe B, Brähler E, Jordan P. The Somatic Symptom Disorder - B Criteria Scale (SSD-12): Factorial structure, validity and population-based norms. Journal of Psychosomatic Research. 2017;97:9-17.; primary care, n = 501 Toussaint A, Riedl B, Kehrer S, Schneider A, Löwe B, Linde K. Validity of the Somatic Symptom Disorder - B Criteria Scale (SSD-12) in Primary Care. BMC Family Practice. 2018;35:342-347).
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In general, ICA cannot identify the actual number of source signals, a uniquely correct ordering of the source signals, nor the proper scaling (including sign) of the source signals. ICA is important to blind signal separation and has many practical applications. It is closely related to (or even a special case of) the search for a factorial code of the data, i.e., a new vector-valued representation of each data vector such that it gets uniquely encoded by the resulting code vector (loss-free coding), but the code components are statistically independent.
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These conditions predispose to cholangiocarcinoma, possibly through an enhanced susceptibility of DNA to damage by carcinogens. Chronic hepatobiliary damage is reported to be multi-factorial and considered to arise from a continued mechanical irritation of the epithelium by the flukes present, particularly via their suckers, metabolites and excreted/secreted antigens as well as immunopathological processes. In silico analyses using techniques of genomics and bioinformatics is unraveling information on molecular mechanisms that may be relevant to the development of cholangiocarcinoma. In regions where Opisthorchis viverrini is highly endemic, the incidence of cholangiocarcinoma is unprecedented.
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In number theory, a Pillai prime is a prime number p for which there is an integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. To put it algebraically, n! \equiv -1 \mod p but p ot\equiv 1 \mod n. The first few Pillai primes are :23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, ... Pillai primes are named after the mathematician Subbayya Sivasankaranarayana Pillai, who studied these numbers.
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The relationship between sleep and women's psychological well‐being suggests that underlying causes of sleep disturbances across a woman's lifespan are often multi‐factorial. Initial variations of sleep in women begin with the menstrual cycle. However, most objective laboratory-based PSG measures of young healthy women do not confirm irregular sleep patterns, neither in sleep duration nor sleep quality across the menstrual cycle. One exception is the reduction of REM sleep and markedly more so the increase of Stage 2 sleep during the luteal phase of the menstrual cycle.
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Silica gel was also more effective than usual poisonous pesticides (particularly in cases with pesticide resistant bugs). When applied after being mixed with water and then sprayed, the outcome for silica gel was significantly lower, but still distinctly better than for the natural silica (used dry). Authors argued that the reason for the poor outcome for diatomaceous earth as a stand-alone treatment was multi-factorial. When tested in laboratory where the bed bugs had intensive, prolonged contact with diatomaceous earth and no access to a host, diatomaceous earth performed very well.
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Bressoud, David. 2002. "Was Calculus Invented in India?" The College Mathematics Journal (Mathematical Association of America). 33(1):2–13. The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as: where, for r = 1, the series reduce to the standard power series for these trigonometric functions, for example: (The Kerala school did not use the "factorial" symbolism.) The Kerala school made use of the rectification (computation of length) of the arc of a circle to give a proof of these results.
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To create factorial codes, Horace Barlow and co-workers suggested to minimize the sum of the bit entropies of the code components of binary codes (1989). Jürgen Schmidhuber (1992) re-formulated the problem in terms of predictors and binary feature detectors, each receiving the raw data as an input. For each detector there is a predictor that sees the other detectors and learns to predict the output of its own detector in response to the various input vectors or images. But each detector uses a machine learning algorithm to become as unpredictable as possible.
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The most common setting for Tukey's test of additivity is a two-way factorial analysis of variance (ANOVA) with one observation per cell. The response variable Yij is observed in a table of cells with the rows indexed by i = 1,..., m and the columns indexed by j = 1,..., n. The rows and columns typically correspond to various types and levels of treatment that are applied in combination. The additive model states that the expected response can be expressed EYij = μ + αi + βj, where the αi and βj are unknown constant values.
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Similar to a split-plot design, a strip- plot design can result when some type of restricted randomization has occurred during the experiment. A simple factorial design can result in a strip-plot design depending on how the experiment was conducted. Strip-plot designs often result from experiments that are conducted over two or more process steps in which each process step is a batch process, i.e., completing each treatment combination of the experiment requires more than one processing step with experimental units processed together at each process step.
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Her work was funded initially by a Bathurst studentship in 1904 and then Newnham College fellowship for 6 years, starting in 1909. In 1903, she joined William Bateson's genetics group at Cambridge where she began her study focusing on the inheritance of petal colour in Antirrhinum (snapdragons). William Bateson was the English biologist who was the first person to use the term genetics to describe the study of heredity, and the chief populariser of the ideas of Gregor Mendel following their rediscovery in 1900. Bateson and Onslow, alongside a research group mainly made up of Newnham College graduates, carried out a series of breeding experiments in various plant and animal species between 1903 and 1910. By 1906 she had enough data to formulate a rudimentary factorial analysis on snapdragon inheritance. In 1907, Wheldale published a full explanation what became termed epistasis, the phenomenon of dominant-like relationship between different pairs of nonallelomorphic factors. Wheldale's study of genetics on flower colouration ultimately gained her the most recognition, with the 1907 publication of a full factorial analysis of flower colour inheritance in snapdragons and the four subsequent papers she published from 1909 to 1910. Her interest was in the biochemistry underlying the petal colours, rather than understanding inheritance itself.
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ANOVA is a relatively robust procedure with respect to violations of the normality assumption. The one-way ANOVA can be generalized to the factorial and multivariate layouts, as well as to the analysis of covariance. It is often stated in popular literature that none of these F-tests are robust when there are severe violations of the assumption that each population follows the normal distribution, particularly for small alpha levels and unbalanced layouts. Furthermore, it is also claimed that if the underlying assumption of homoscedasticity is violated, the Type I error properties degenerate much more severely.
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Although Díaz and Pariguan restrict these symbols to k > 0, the Pochhammer k-symbol as they define it is well-defined for all real k, and for negative k gives the falling factorial, while for k = 0 it reduces to the power xn. The Díaz and Pariguan paper does not address the many analogies between the Pochhammer k-symbol and the power function, such as the fact that the binomial theorem can be extended to Pochhammer k-symbols. It is true, however, that many equations involving the power function xn continue to hold when xn is replaced by (x)n,k.
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Pregnancy related Pelvic Girdle Pain (PGP) can be either specific (trauma or injury to pelvic joints or genetical i.e. connective tissue disease) and non-specific. PGP disorder is complex and multi-factorial and likely to be also represented by a series of sub-groups driven by pain varying from peripheral or central nervous system,Diagnosis and classification of pelvic girdle pain disorders—Part 1: A mechanism based approach within a biopsychosocial framework Manual Therapy, Volume 12, Issue 2, May 2007, Peter B. O’Sullivan and Darren J. Beales. altered laxity/stiffness of muscles,European guidelines for the diagnosis and treatment of pelvic girdle pain.
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Modern research in genetics and molecular biology, boosted by information emerging from the Human Genome Project, offers new opportunities for the promotion of population health. Anticipated benefits include more effective population-based public health disease prevention and control activities; population screening and predictive genetic testing for common multi-factorial diseases; and targeted populations-based interventions. However, mining the genome-based information for potential benefits and ensuring that they are effectively implemented, with appropriate attention to avoiding harm, has proven to be an enormous task. A new field within public health, called public health genomics has evolved which addresses these issues.
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The exact cause of preterm birth is difficult to determine and it may be multi- factorial. Labor is a complex process involving many factors. Four different pathways have been identified that can result in preterm birth and have considerable evidence: precocious fetal endocrine activation, uterine overdistension (placental abruption), decidual bleeding, and intrauterine inflammation/infection. Identifying women at high risk of giving birth early would enable the health services to provide specialized care for these women to delay the birth or make sure they are in the best place to give birth (for example a hospital with a special care baby unit).
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Colour changes – now commonplace, a planned marketing tool – were rarer earlier, with most models being produced over the span of their inclusion in the series in just one or two major colour schemes. However, not only the colour of the model's body must be regarded, but rather the _entire_ model— including baseplate, interior, windows – and thus changes in different components can lead to a factorial increase in variation possibilities. Age also plays an important part in making a model rare. A model produced in standard quantities in the 1950s will likely be much rarer today than one produced in similar quantities the 1980s.
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A contemporary theory informed by awareness that an ever-expanding exposure to ideas made possible by the internet has changed both the act of creation, and the experience of perception, is known as Multi Factorial Apperception (MFA). This approach seeks to integrate a wide range of cultural variables to expand the contextual frame for viewing the creation, and perception, of aesthetic experience. Emphasis is on a dynamic mulit-layerd cultural framing of the act of creation at a particular moment in time, and admits that the meaning of a particular work will be in flux from that moment onward.
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116 is a noncototient, meaning that there is no solution to the equation , where stands for Euler's totient function.. 116! + 1 is a factorial prime.. There are 116 ternary Lyndon words of length six, and 116 irreducible polynomials of degree six over a three-element field, which form the basis of a free Lie algebra of dimension 116.. There are 116 different ways of partitioning the numbers from 1 through 5 into subsets in such a way that, for every k, the union of the first k subsets is a consecutive sequence of integers.. There are 116 different 6×6 Costas arrays..
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Most research on the CDI has been conducted with Caucasian participants of middle to lower class socioeconomic status throughout the world. The CDI can be given to children and youth across cultures, though its "internal consistency and factorial structure vary somewhat in different juvenile cohorts." Kovacs and other researchers have reported obtaining higher CDI scores for African- Americans (particularly boys),Steele, R.G., Little, T.D., Ilardi, S.S., Forehand, R., Brody, G.H., & Hunter, H.L. (2006). "A confirmatory comparison of the factor structure of the Children’s Depression Inventory between European American and African American youth." Journal of Child and Family Studies, 15 (6), 773-788. .
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Most medical organizations recognize obesity as a disease. According to the Obesity Medicine Association Obesity Algorithm: "Obesity is defined as a chronic, relapsing, multi-factorial, neurobehavioral disease, wherein an increase in body fat promotes adipose tissue dysfunction and abnormal fat mass physical forces, resulting in adverse metabolic, biomechanical, and psychosocial health consequences." Consistent with this functional definition, patients with the disease of obesity can generally be categorized into patients with "fat mass disease" and "sick fat disease." "Fat mass disease" represents pathologies related to abnormal physical forces, such as stress on weight-bearing joints, immobility, tissue compression, and tissue friction.
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Agricultural science advances served to meet the combination of larger city populations and fewer farms. But for crop scientists to take due account of widely differing geographical growing climates and needs, it was important to differentiate local growing conditions. To extrapolate experiments on local crops to a national scale, they had to extend crop sample testing economically to overall populations. As statistical methods advanced (primarily the efficacy of designed experiments instead of one-factor-at-a-time experimentation), representative factorial design of experiments began to enable the meaningful extension, by inference, of experimental sampling results to the population as a whole.
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In early 2005 SEB bought AGIO Bank () that later was transformed into SEB Bank (Ukraine) and in late 2007 SEB bought Factorial Bank () with offices in Kharkiv and eastern Ukraine. This bank is planned to be incorporated in SEB Bank (Ukraine). Today SEB has 85 offices in Ukraine, but the plan is to open additional 20–25 offices per year so it will reach about 300 offices in the coming years.SEB Bank's websiteThe newspaper Veckans affärer's website 2008-03-03 In mid-2007 Swedbank bought TAS-Komerzbank () that in December 2007 changed its name to Swedbank ().
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Every 4-regular undirected graph has an even number of Hamiltonian decompositions. More strongly, for every two edges e and f of a 4-regular graph, the number of Hamiltonian decompositions in which e and f belong to the same cycle is even. If a 2k-regular graph has a Hamiltonian decomposition, it has at least a triple factorial number of decompositions, :(3k-2)\cdot(3k-5)\cdots 7\cdot 4 \cdot 1. For instance, 4-regular graphs that have a Hamiltonian decomposition have at least four of them; 6-regular graphs that have a Hamiltonian decomposition have at least 28, etc.
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The points in this experiment can thus be represented as \--, +-, -+, and ++. The factorial points can also be abbreviated by (1), a, b, and ab, where the presence of a letter indicates that the specified factor is at its high (or second) level and the absence of a letter indicates that the specified factor is at its low (or first) level (for example, "a" indicates that factor A is on its high setting, while all other factors are at their low (or first) setting). (1) is used to indicate that all factors are at their lowest (or first) values.
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The Lite version features descriptive statistics, hypothesis testing (t-tests, chi-square, 1-way ANOVA with post-hocs, non-parametric tests), distribution testing, linear regression, correlation and a selection of basic graphing functions. It is limited to 254 rows and 12 columns of data, and does not offer import or export functions. The Pro version features advanced analyses (unlimited rows and columns, import and export, more means of analyses (2-way ANOVA, logistic and nonlinear regression, principal component analysis and multidimensional scaling, time series, power/sample-size, factorial design), more types of graph, word processing functions and options for formatting of results.
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A dodge means just that; two bells dodge round each other, thus changing their relationship to the treble, and giving rise to different changes. The plain bob pattern can be extended beyond the constraints of the plain course, to the full unique 720 changes possible ( this is factorial 6 on 6 bells, which is 1×2×3×4×5×6 = 720 changes). To do this, at set points in the sequences one of the ringers, called the "conductor" calls out commands such as "bob" or "single", which introduce further variations. The conductor follows a "composition" which they have to commit to memory.
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Intra-oral photograph showing full upper denture and natural lower teeth with mandibular central incisors missing Edentulism is the result of a mostly preventable oral disease process that is a worldwide public health concern. The loss of the permanent dentition is a multi-factorial process resulting from the impact of dental caries, periodontal disease and social factors. People who have lost teeth are referred to as (either partially or completely) edentulous (edentate), however those who have not lost teeth are referred to as dentate.Darby M, & Walsh, Margaret M. (2010), Dental hygiene: Theory and practice 3rd ed.
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Abrasion occurring on the cervical margins from the effects of friction from toothbrushing and abrasive toothpastes Cause of abrasion may arise from interaction of teeth with other objects such as toothbrushes, toothpicks, floss, and ill-fitting dental appliance like retainers and dentures. Apart from that, people with habits such as nail biting, chewing tobacco, lip or tongue piercing, and having occupation such as joiner, are subjected to higher risks of abrasion. Abrasion can also occur from the type of dentifrice being utilized as some have more abrasive qualities such as whitening toothpastes. The aetiology of dental abrasion can be due to a single stimuli or, as in most cases, multi-factorial.
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The gamma function, in blue, plotted along with in green. Notice the intersection at positive integers, both are valid analytic continuations of the factorials to the non-integers A more restrictive property than satisfying the above interpolation is to satisfy the recurrence relation defining a translated version of the factorial function, Extract of page 28 Expression G.2 on page 293 :f(1) = 1, :f(x+1) = x f(x), for any positive real number . But this would allow for multiplication by any periodic analytic function which evaluates to 1 on the positive integers, such as . One of several ways to finally resolve the ambiguity comes from the Bohr–Mollerup theorem.
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Although the symbol is an upper-case Greek gamma, there is no accepted standard for whether the function name should be written "gamma function" or "Gamma function" (some authors simply write "-function"). The alternative "pi function" notation due to Gauss is sometimes encountered in older literature, but Legendre's notation is dominant in modern works. It is justified to ask why we distinguish between the "ordinary factorial" and the gamma function by using distinct symbols, and particularly why the gamma function should be normalized to instead of simply using "". Consider that the notation for exponents, , has been generalized from integers to complex numbers without any change.
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In computing it may be required to generate permutations of a given sequence of values. The methods best adapted to do this depend on whether one wants some randomly chosen permutations, or all permutations, and in the latter case if a specific ordering is required. Another question is whether possible equality among entries in the given sequence is to be taken into account; if so, one should only generate distinct multiset permutations of the sequence. An obvious way to generate permutations of n is to generate values for the Lehmer code (possibly using the factorial number system representation of integers up to n!), and convert those into the corresponding permutations.
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The median polish is a simple and robust exploratory data analysis procedure proposed by the statistician John Tukey. The purpose of median polish is to find an additively-fit model for data in a two-way layout table (usually, results from a factorial experiment) of the form row effect + column effect + overall median. Median polish utilizes the medians obtained from the rows and the columns of a two-way table to iteratively calculate the row effect and column effect on the data. The results are not meant to be sensitive to the outliers, as the iterative procedure uses the medians rather than the means.
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Bronze Age sites in Kosovo The Bronze Age followed the Copper Age, stretching in the time period from 2500 up to 1100 BC. With the advancement of the metallurgy factorial industry, and with the mixing of copper with tin, the bronze was created as an alloy. The formula invention for creating this stronger metal, leads towards the preparation and production of weapons, tools and jewelry. The particularity of this period lies in the fact that matriarchate is replaced with patriarchate, the social differentiation was developed and the first elements of tribal aristocracy emerges. All new changes and developments are reflected in the material culture discovered in various sites across Kosovo.
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Multiple Factor Analysis by Example Using R. Chapman & Hall/CRC The R Series London 272 p It plays an important theoretical role because it opens the way to the simultaneous treatment of quantitative and qualitative variables. Two methods simultaneously analyze these two types of variables: factor analysis of mixed data and, when the active variables are partitioned in several groups: multiple factor analysis. This equivalence does not mean that MCA is a particular case of PCA as it is not a particular case of CA. Its only means that these methods are closely linked one another as they belong to the same family: the factorial methods.
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See also the factorial experimental design methods pioneered by Sir Ronald A. Fisher. Reasons for disfavoring OFAT include: #OFAT requires more runs for the same precision in effect estimation #OFAT cannot estimate interactions #OFAT can miss optimal settings of factors. Designed experiments remain nearly always preferred to OFAT with many types and methods available, in addition to fractional factorials which, though usually requiring more runs than OFAT, do address the three concerns above. One modern design over which OFAT has no advantage in number of runs is the Plackett-Burman which, by having all factors vary simultaneously (an important quality in experimental designs), gives generally greater precision in effect estimation.
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Interaction effect of education and ideology on concern about sea level riseIn statistics, an interaction may arise when considering the relationship among three or more variables, and describes a situation in which the effect of one causal variable on an outcome depends on the state of a second causal variable (that is, when effects of the two causes are not additive). Although commonly thought of in terms of causal relationships, the concept of an interaction can also describe non-causal associations. Interactions are often considered in the context of regression analyses or factorial experiments. The presence of interactions can have important implications for the interpretation of statistical models.
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In: Calvin Dytham: Choosing And Using Statistics: A Biologist's Guide. Wiley-Blackwell, 2003, , S. 145–150 For this reason, and because the method was described later than most other parametric and non-parametric variance analysis tests, it has found little use in textbooks and statistical analysis software. With computer programs that contain a function for parametric multi-factorial ANOVA, however, with additional manual effort and a calculation of the Scheirer Ray Hare test is possible. Since the Scheirer–Ray–Hare test only makes a statement about the diversity of all samples considered, it makes sense to perform a post-hoc test that compares the individual samples in pairs.
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Solubilization of botryosphaeran can be enhanced through chemical derivatization with various functional groups. The influence of the composition of the nutrient medium, including nitrogen, phosphate, minerals, supplements (soybean oil, Tween 80), and the carbon source (carbohydrates),, is important in enhancing the production of botryosphaeran and biomass during fermentation by Botryosphaeria rhodina MAMB-05. Catabolite repression, and the presence of β-glucan-hydrolyzing enzymes that attack botryosphaeran during the fermentation process are critical and limit the production of botryosphaeran. Statistical factorial design methods, such as the response surface methodology (RSM), are effective in investigating complex fermentation parameters and their interactions to optimize metabolite production by microorganisms.
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Each bells starts at a different place in this cyclical order. A dodge means just that: two bells dodge round each other, thus changing their relationship to the treble, and giving rise to different changes. The plain bob pattern can be extended beyond the constraints of the plain course of 60 changes, to the full unique 720 changes possible (this is 6 factorial on 6 bells, which is 1×2×3×4×5×6 = 720 changes). To do this, at set points in the sequences one of the ringers, called the "conductor" calls out commands such as "bob" or "single", which introduce further variations.
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The first recursion theorem (in particular, part 1) states that there is a set F such that Φ(F) = F. The set F will consist entirely of ordered pairs of natural numbers, and will be the graph of the factorial function f, as desired. The restriction to recursion equations that can be recast as recursive operators ensures that the recursion equations actually define a least fixed point. For example, consider the set of recursion equations: g(0) = 1 g(n + 1) = 1 g(2n) = 0 There is no function g satisfying these equations, because they imply g(2) = 1 and also imply g(2) = 0.
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Recursion that only contains a single self-reference is known as ', while recursion that contains multiple self- references is known as '. Standard examples of single recursion include list traversal, such as in a linear search, or computing the factorial function, while standard examples of multiple recursion include tree traversal, such as in a depth-first search. Single recursion is often much more efficient than multiple recursion, and can generally be replaced by an iterative computation, running in linear time and requiring constant space. Multiple recursion, by contrast, may require exponential time and space, and is more fundamentally recursive, not being able to be replaced by iteration without an explicit stack.
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Later analyses have shown that the factorial procedures Guilford presented as evidence for his theory did not provide support for it, and that the test data that he claimed provided evidence against g did in fact exhibit the usual pattern of intercorrelations after correction for statistical artifacts.Jensen 1998, 77–78, 115–117Mackintosh 2011, 52, 239 More recently, Howard Gardner has developed the theory of multiple intelligences. He posits the existence of nine different and independent domains of intelligence, such as mathematical, linguistic, spatial, musical, bodily- kinesthetic, meta-cognitive, and existential intelligences, and contends that individuals who fail in some of them may excel in others.
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Obesity medicine is a field of medicine dedicated to the comprehensive treatment of patients with obesity. Obesity medicine takes into account the multi-factorial etiology of obesity in which behavior, development, environment, epigenetic, genetic, nutrition, physiology, and psychosocial contributors all play a role. As time progresses, we become more knowledgeable about the complexity of obesity, and we have ascertained that there is a certain skill set and knowledge base that is required to treat this patient population. Clinicians in the field should understand how a myriad of factors contribute to obesity including: gut microbiota diversity, regulation of food intake and energy balance through enteroendocrine and neuroregulation, and adipokine physiology.
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The main area of expertise and emphasis of Kranke's scientific work was in the area of observational and interventional trials on risk factors for PONVC. C. Apfel, P. Kranke, L. H. J. Eberhart. Comparison of surgical site and patient's history with a simplified score for the prediction of postoperative nausea and vomiting. Anaesthesia 2004; 59: 1078–1082 as well as investigator initiated trials and industry-driven trials to investigate interventions aiming at the prevention or treatment of PONV.An International Multicenter Protocol to Assess the single and combined benefits of antiemetic interventions in a Controlled clinical Trial of a 2 ' 2 ' 2 ' 2 ' 2 ' 2 factorial design (IMPACT).
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Initially, the use of brute-force full factorial (gridded) designs meant that the simulations needed to run very quickly and the studies required high-performance computing. Even so, only a small number of factors (at a limited number of levels) could be investigated, due to the curse of dimensionality. The SEED Center for Data Farming at the Naval Postgraduate School also worked closely with Project Albert in model generation, output analysis, and the creation of new experimental designs to better leverage the computing capabilities at Maui and other facilities. Recent breakthroughs in designs specifically developed for data farming can be found in , among others.
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If efficiency is not a concern, computing factorials is trivial from an algorithmic point of view: successively multiplying a variable initialized to 1 by the integers up to (if any) will compute , provided the result fits in the variable. In functional languages, the recursive definition is often implemented directly to illustrate recursive functions. The main practical difficulty in computing factorials is the size of the result. To assure that the exact result will fit for all legal values of even the smallest commonly used integral type (8-bit signed integers) would require more than 700 bits, so no reasonable specification of a factorial function using fixed-size types can avoid questions of overflow.
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Hearing aids are used for a variety of pathologies including sensorineural hearing loss, conductive hearing loss, and single-sided deafness. Hearing aid candidacy is typically determined by a Doctor of Audiology, who will also fit the device based on the nature and degree of the hearing loss being treated. The amount of benefit experienced by the user of the hearing aid is multi-factorial, depending on the type, severity, and etiology of the hearing loss, the technology and fitting of the device, and on the motivation, personality, lifestyle, and overall health of the user. Hearing aids are incapable of truly correcting a hearing loss; they are an aid to make sounds more audible.
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It is sometimes convenient to consider positional numeral systems where the weights associated with the positions do not form a geometric sequence 1, b, b2, b3, etc., starting from the least significant position, as given in the polynomial form. In a mixed-radix system such as the factorial number system, the weights form a sequence where each weight is an integral multiple of the previous one, and the number of permitted digit values varies accordingly from position to position. For calendrical use, the Mayan numeral system was a mixed-radix system, since one of its positions represents a multiplication by 18 rather than 20, in order to fit a 360-day calendar.
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For the cake-baking example, the experimenter can fluctuate bake- time and oven-temperature to understand the effects of such fluctuation that may occur when control is no longer in his/her hands. Robust parameter designs are very similar to fractional factorial designs (FFDs) in that the optimal design can be found using Hadamard matrices, principles of effect hierarchy and factor sparsity are maintained, and aliasing is present when full RPDs are fractionated. Much like FFDs, RPDs are screening designs and can provide a linear model of the system at hand. What is meant by effect hierarchy for FFDs is that higher-order interactions tend to have a negligible effect on the response.
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Thus using letters A–Z to denote digits 10, 11, 12, ..., 35 as in other base-N make the largest representable number 36 × 36! − 1\. For arbitrarily greater numbers one has to choose a base for representing individual digits, say decimal, and provide a separating mark between them (for instance by subscripting each digit by its base, also given in decimal, like 24031201, this number also can be written as 2:0:1:0!). In fact the factorial number system itself is not truly a numeral system in the sense of providing a representation for all natural numbers using only a finite alphabet of symbols, as it requires an additional separation mark.
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Application of toxic units requires toxicity data for the individual components of the mixture as well as specialized mixture toxicity data. Evaluating the response of each individual chemical allows researchers to generate a new dosing metric, toxic units, which is standardized to the toxicity of each chemical. Since the toxicity of two compounds may vary widely, 1 toxic unit of two different compounds could correspond to two very different concentrations on a per mass basis. In addition to the toxicity of the individual components, use of toxic units requires a 2x2 factorial design concentration series where the response is measured to an increase of each contaminant with the other contaminant held constant.
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The study utilized various statistical methods and was retroactively performed on data from explorers of the 19th and 20th centuries. Exploratory factor analysis, structural equation modeling, and multiple correlation analysis were all run on this data and found that cultural variables such as political integration, sex morals, trends in autonomy, class stratification, and high gods, are correlated with intergroup conflict in the directions that regality theory would predicts. Studies that are considered related to this are studies that look that also attempt to isolate the relationship of cultural variables to collective dangers. Variations of factorial analyses appear to be most common statistical method that might be applied to regal and kungic society studies.
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Ringel's conjecture asks if the complete graph K2n+1 can be decomposed into copies of any tree with edges. This is known to be true for sufficiently large . The number of matchings of the complete graphs are given by the telephone numbers : 1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504, 2390480, 10349536, 46206736, ... . These numbers give the largest possible value of the Hosoya index for an n-vertex graph.. The number of perfect matchings of the complete graph Kn (with n even) is given by the double factorial (n − 1)!!.. The crossing numbers up to K27 are known, with K28 requiring either 7233 or 7234 crossings.
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The World Foundation for Medical Research and Prevention is a 501(c)(3) Houston-based grantmaking public charity established to provide leadership in developing resources to sponsor and fund the projects of Professor Luc Montagnier as well as other research aimed at the design of new types of AIDS vaccines, prolonging the active life of aging people and the diagnosis and treatment of microbial and viral factors associated with cancers, neurodegenerative, auto-immune and other multi-factorial disease processes such as Alzheimer's disease and Parkinson's disease. The World Foundation for Medical Research and Prevention supports its work through proceeds from special events, cause-related marketing projects, and voluntary contributions from individuals, corporations, and foundations.
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In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined before entering the loop). Primitive recursive functions form a strict subset of those general recursive functions that are also total functions. The importance of primitive recursive functions lies on the fact that most computable functions that are studied in number theory (and more generally in mathematics) are primitive recursive. For example, addition and division, the factorial and exponential function, and the function which returns the nth prime are all primitive recursive.
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The "plain course" of Grandsire Doubles; 30 changes Plain hunting is limited to a small number of possible different changes, which is numerically equal to twice the number of bells that are hunting. However, by introducing deviations from the plain hunt, by causing some of the bells to change their relationship to the others, change ringing "methods" were developed. These allow a large range of possible different changes to be rung; even to the extent of the full factorial sequence of changes. Grandsire, the oldest change ringing method, is based on a simple deviation to the plain hunt when the treble (bell No.1) is first in the sequence or it is said to "lead".
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Like the second recursion theorem, the first recursion theorem can be used to obtain functions satisfying systems of recursion equations. To apply the first recursion theorem, the recursion equations must first be recast as a recursive operator. Consider the recursion equations for the factorial function f: f(0)=1 f(n+1) = (n + 1) \cdot f(n) The corresponding recursive operator Φ will have information that tells how to get to the next value of f from the previous value. However, the recursive operator will actually define the graph of f. First, Φ will contain the pair ( \varnothing, (0, 1)). This indicates that f(0) is unequivocally 1, and thus the pair (0,1) is in the graph of f.
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Jones then completed a Master's degree at VSU under the guidance of Professor Glenn Harris while taking a systems biology approach to study Non- Insulin Dependent Diabetes Mellitus in mice in a project titled "Association between Genetic Variants and Inherited Multi-factorial Diseases". In addition, Jones also taught introductory biology courses to undergraduates. Towards the end of her master's program, Jones traveled to Ghana, where her mother's side of the family originated, and taught English, math, and science to young students. It was the excitement of the kids she was teaching that inspired her to pursue a Ph.D., so that she could continue to teach while creating opportunities for students to study-abroad.
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In a slightly more complex system, it is automated with a delivery pump, a timer and irrigation tubing to deliver nutrient solution with a delivery frequency that is governed by the key parameters of plant size, plant growing stage, climate, substrate, and substrate conductivity, pH, and water content. In a commercial setting, watering frequency is multi-factorial and governed by computers or PLCs. Commercial hydroponics production of large plants like tomatoes, cucumber, and peppers uses one form or another of run-to-waste hydroponics. In environmentally responsible uses, the nutrient-rich waste is collected and processed through an on-site filtration system to be used many times, making the system very productive.
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The device consisted of 25 aluminium discs attached to a four-and-a-half inch long rod, each disc containing the 26 letters of the Roman alphabet in scrambled order around its circumference (with the exception of the 17th disc, which began with the letters "ARMY OF THE US"). Each wheel had a different arrangement of the alphabet, and was stamped with an identifying number and letter; wheels were identified according to the letter following "A" on that wheel, from "B 1" to "Z 25". The wheels could be assembled on the rod in any order; the ordering used during encoding comprised the key. There were 25! (25 factorial) = 15,511,210,043,330,985,984,000,000 (more than 15 septillion) possible keys, which can be expressed as about an 84-bit key size.
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While memoization may be added to functions internally and explicitly by a computer programmer in much the same way the above memoized version of `factorial` is implemented, referentially transparent functions may also be automatically memoized externally. The techniques employed by Peter Norvig have application not only in Common Lisp (the language in which his paper demonstrated automatic memoization), but also in various other programming languages. Applications of automatic memoization have also been formally explored in the study of term rewriting and artificial intelligence. In programming languages where functions are first-class objects (such as Lua, Python, or Perl ), automatic memoization can be implemented by replacing (at run-time) a function with its calculated value once a value has been calculated for a given set of parameters.
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Turning a researcher into a policy entrepreneur, or a research institute or department into a policy-focused think tank involves a fundamental re-orientation towards policy engagement rather than academic achievement; engaging much more with the policy community; developing a research agenda focusing on policy issues rather than academic interests; acquiring new skills or building multidisciplinary teams; establishing new internal systems and incentives; spending much more on communications; producing a different range of outputs; and working more in partnerships and networks. These lessons show that the relationship between research, policy and practice is complex, multi-factorial, non-linear, and highly context specific. What works in one situation may not work in another. Developing effective strategies in complex environments is not straightforward.
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His approach moves from a revisitation of the traditional concepts of homology. According to Minelli, the homology relationships between two structures is necessarily limited to selected features of those structures, thus requiring the adoption a factorial, or combinatorial concept of homology. Minelli has introduced new concepts, such as axis paramorphism (useful for understanding the evolutionary relationships between the main axis of the body and its appendages) and those of eosegment and merosegment, through which he suggests a radical revisitation of the architecture of the body of segmented animals. Minelli has also explored the implication of evo-devo for biological systematics, speciation and the evolution of life cycles.< Minelli A., Brena Carlo, Deflorian Gianluca, Maruzzo Diego & Fusco G. (2006), “From embryo to adult.
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In the initial study describing the development and evaluation of the DOCS, the instrument's factorial validity was supported by exploratory and confirmatory factor analyses of 3 samples, including (a) individuals with OCD, (b) those with other anxiety disorders, and (c) non treatment-seeking individuals. Scores on the DOCS displayed excellent performance on indices of reliability (test-retest, internal consistency) and validity (convergent, divergent, construct), and the measure appears to be sensitive to treatment. The DOCS is also diagnostically sensitive and thus holds promise as a useful measure of OCD symptoms in clinical and research settings. The factor structure and psychometric properties of the DOCS have been examined in numerous studies in different cultures and languages, and via different methods of administration.
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In combinatorial mathematics, a Stirling permutation of order k is a permutation of the multiset 1, 1, 2, 2, ..., k, k (with two copies of each value from 1 to k) with the additional property that, for each value i appearing in the permutation, the values between the two copies of i are larger than i. For instance, the 15 Stirling permutations of order three are :1,1,2,2,3,3; 1,2,2,1,3,3; 2,2,1,1,3,3; :1,1,2,3,3,2; 1,2,2,3,3,1; 2,2,1,3,3,1; :1,1,3,3,2,2; 1,2,3,3,2,1; 2,2,3,3,1,1; :1,3,3,1,2,2; 1,3,3,2,2,1; 2,3,3,2,1,1; :3,3,1,1,2,2; 3,3,1,2,2,1; 3,3,2,2,1,1. The number of Stirling permutations of order k is given by the double factorial (2k − 1)!!. Stirling permutations were introduced by in order to show that certain numbers (the numbers of Stirling permutations with a fixed number of descents) are non-negative.
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105 is a triangular number, a dodecagonal number and the first Zeisel number. It is a sphenic number, and is the product of three consecutive prime numbers. 105 is the double factorial of 7. It is also the sum of the first five square pyramidal numbers. 105 comes in the middle of the prime quadruplet (101, 103, 107, 109). The only other such odd numbers less than a thousand are 9, 15, 195 and 825. 105 is also a pseudoprime to the prime bases 13, 29, 41, 43, 71, 83 and 97. The distinct prime factors of 105 add up to 15, and so do those of 104, hence the two numbers form a Ruth-Aaron pair under the first definition.
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The first interest of the relationship square is to represent the variables themselves, not their categories, which is all the more valuable as there are many variables. For this, we calculate for each qualitative variable j and each factor F_s ( F_s , rank s factor, is the vector of coordinates of the individuals along the axis of rank s ; in PCA, F_s is called principal component of rank s) , the square of the correlation ratio between the F_s and the variable j, usually denoted : \eta^2(j, F_s) Thus, to each factorial plane, we can associate a representation of qualitative variables themselves. Their coordinates being between 0 and 1 , the variables appear in the square having as vertices the points (0,0), ( 0,1), (1,0) and (1,1).
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A common computer programming tactic is to divide a problem into sub-problems of the same type as the original, solve those sub-problems, and combine the results. This is often referred to as the divide-and-conquer method; when combined with a lookup table that stores the results of solving sub-problems (to avoid solving them repeatedly and incurring extra computation time), it can be referred to as dynamic programming or memoization. A recursive function definition has one or more base cases, meaning input(s) for which the function produces a result trivially (without recurring), and one or more recursive cases, meaning input(s) for which the program recurs (calls itself). For example, the factorial function can be defined recursively by the equations and, for all , .
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In special cases when the keys are known in advance and the key set is static, a hash function can be found that achieves absolute (or collisionless) uniformity. Such a hash function is said to be perfect. There is no algorithmic way of constructing such a function - searching for one is a factorial function of the number of keys to be mapped versus the number of table slots they're mapped into. Finding a perfect hash function over more than a very small set of keys is usually computationally infeasible; the resulting function is likely to be more computationally complex than a standard hash function, and provides only a marginal advantage over a function with good statistical properties that yields a minimum number of collisions.
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While the model has greatly aided soil science, and spatially explains generalized soil- environmental relationships on landscapes, the device—supremely attractive in its parsimony—has constrained pedologic interpretation because its genetic- interpretive domain is constrained by its broad factorial (landscape context) tenets. The model allows for a wide-ranging generalized assessment of soilscapes, and is useful in soil chronosequence work (Birkeland 1974, 1984) but the biodynamic soil processes that are largely responsible in producing stonelayers are absent at a highest theoretical (five factors) level. Once mapped and classified, soils become static elements, which—while societally useful, scientifically misrepresents their true biodynamic nature. In a proposal to heal such Occam's razor wounds, process biodynamics supported with appropriate genetic language has been advanced to augment the traditional five factors genetic principles in pedology and soil-geomorphology.
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Pelvic girdle pain is complex and multi-factorial and likely to be represented by a series of sub- groups with different underlying pain drivers from peripheral or central nervous system, altered laxity/stiffness of muscles,European guidelines for the diagnosis and treatment of pelvic girdle pain.Eur Spine J. 2008 Feb 8 Vleeming A, Albert HB, Ostgaard HC, Sturesson B, Stuge B. laxity to injury of tendinous/ligamentous structuresPossible role of the long dorsal sacroiliac ligament in women with peripartum pelvic pain. Acta Obstetricia et Gynecologica Scandinavica Volume 81 Issue 5 Page 430-436, May 2002, Andry Vleeming, Haitze J. de Vries, Jan MA Mens, Jan-Paul van Wingerden to 'mal- adaptive' body mechanics. Musculo-skeletal mechanics involved in gait and weightbearing activities can be mild to grossly impaired.
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De Moivre's most notable achievement in probability was the discovery of the first instance of central limit theorem, by which he was able to approximate the binomial distribution with the normal distribution. To achieve this De Moivre developed an asymptotic sequence for the factorial function —- which we now refer to as Stirling's approximation —- and Bernoulli's formula for the sum of powers of numbers. Both Montmort and de Moivre adopted the term probability from Jacob Bernoulli, which had not been used in all the previous publications on gambling, and both their works were enormously popular. The refinement of Bernoulli's Golden Theorem, regarding the convergence of theoretical probability and empirical probability, was taken up by many notable latter day mathematicians like De Moivre, Laplace, Poisson, Chebyshev, Markov, Borel, Cantelli, Kolmogorov and Khinchin.
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The Scheirer–Ray–Hare test is analogous to the parametric multi-factorial ANOVA of investigating the influence of two different factors on a measure for which different samples are available for the factors. As with the parametric analysis of variance, the test can be used to investigate the null hypotheses that the two factors examined in each case have no influence on the positional parameter of the samples and thus on the measure, and that there are no interactions between the two factors. A p-value less than 0.05 for one or more of these three hypotheses leads to their rejection. As with many other non-parametric methods, the analysis in this method relies on the evaluation of the ranks of the samples in the samples rather than the actual observations.
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Anderson-Cook did her undergraduate studies at Western University and the University of Waterloo, earning a bachelor's degree in education from Western University and a bachelor's degree in mathematics from Waterloo in 1989. She moved to the University of Toronto for a master's degree in statistics in 1990, and returned to Waterloo for her Ph.D., which she completed in 1994. Her dissertation, Location and Dispersion Analysis for Factorial Experiments with Directional Data, was supervised by C. F. Jeff Wu. She was an assistant professor of statistics and actuarial science at Western University and then, beginning in 1996, an associate professor of statistics at Virginia Tech, before moving to Los Alamos in 2004. She chaired the American Statistical Association Section of Quality and Productivity in 2006, and the American Society for Quality Statistics Division in 2010.
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Split-plot designs result when a particular type of restricted randomization has occurred during the experiment. A simple factorial experiment can result in a split- plot type of design because of the way the experiment was actually executed. In many industrial experiments, three situations often occur: # some of the factors of interest may be 'hard to vary' while the remaining factors are easy to vary. As a result, the order in which the treatment combinations for the experiment are run is determined by the ordering of these 'hard-to-vary' factors # experimental units are processed together as a batch for one or more of the factors in a particular treatment combination # experimental units are processed individually, one right after the other, for the same treatment combination without resetting the factor settings for that treatment combination.
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Efficacy of vasopressin on systemic hemodynamics in catecholamine-resistant septic and postcardiotomy shock have been studied and published first in 2001 Later, the group concluded the ischemic skin lesions (ISL) developed in patients with catecholamine-resistant vasodilatory shock have multi-factorial cause and shall not necessarily been seen a side effect of AVP solely. The presence of septic shock and a history of peripheral arterial occlusive disease are independent risk factors for the development of ISL. In the last decade, in early hyperdynamic septic shock, the administration of high-dose AVP as a single agent proved to fail to increase mean arterial pressure in the first hour but maintains it above 70mmHg in two-thirds of patients at 48h. AVP decreases NE exposure, has no effect on the PrCO(2) - PaCO(2 )difference, and improves renal function and SOFA score.
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120 is the factorial of 5 and one less than a square, making (5, 11) a Brown number pair. 120 is the sum of a twin prime pair (59 + 61) and the sum of four consecutive prime numbers (23 + 29 + 31 + 37), four consecutive powers of 2 (8 + 16 + 32 + 64), and four consecutive powers of 3 (3 + 9 + 27 + 81). It is highly composite, superabundant, and colossally abundant number, with its 16 divisors being more than any number lower than it has, and it is also the smallest number to have exactly that many divisors. It is also a sparsely totient number. 120 is the smallest number to appear six times in Pascal's triangle. 120 is also the smallest multiple of 6 with no adjacent prime number, being adjacent to 119 = 7 × 17 and 121 = 112.
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The Lefschetz hyperplane theorem implies that for a smooth complex projective variety X of dimension at least 4 and a smooth ample divisor Y in X, the restriction Pic(X) → Pic(Y) is an isomorphism. For example, if Y is a smooth complete intersection variety of dimension at least 3 in complex projective space, then the Picard group of Y is isomorphic to Z, generated by the restriction of the line bundle O(1) on projective space. Grothendieck generalized Lefschetz's theorem in several directions, involving arbitrary base fields, singular varieties, and results on local rings rather than projective varieties. In particular, if R is a complete intersection local ring which is factorial in codimension at most 3 (for example, if the non-regular locus of R has codimension at least 4), then R is a unique factorization domain (and hence every Weil divisor on Spec(R) is Cartier).
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EIPH occurs when blood enters the air passages of a horse's lung, due to fractured capillaries. A variety of causes have been proposed, but EIPH is most likely a multi-factorial condition, involving airway, vascular, inflammatory, blood, cardiac, locomotory, and remodelling components. The primary mechanism is likely to be high pulmonary vascular pressures with concurrent negative airway pressures, causing extreme stress across the pulmonary capillary membrane (the fragile membrane separating blood in the pulmonary capillaries from the air-filled alveoli) and consequent hemorrhage into the air spaces of the lung. Other contributing factors may include upper airway obstruction, increased blood viscosity, abnormalities of cardiac origin (small cross-sectional area of atrioventricular valves, stiff valves, slow left ventricular relaxation time, right tricuspid valve regurgitation), preferential distribution of blood flow to the dorsocaudal lung regions, mechanical trauma, lower airway obstruction, inflammation, abnormalities of blood coagulation, inhomogeneity of ventilation and locomotory trauma.
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His data suggested that people think of intentionality of an action in terms of several psychological constituents: desire for outcome, belief about the expected outcome, intention to act (combination of desire and belief), skill to bring about the outcome, and awareness of action while performing that action. Consistent with this view as well as with our moral intuitions, studies found significant effects of the agent's intention, desire, and beliefs on various types of moral judgments, Using factorial designs to manipulate the content in the scenarios, Cushman showed that the agent's belief and desire regarding a harmful action significantly influenced judgments of wrongness, permissibility, punishment, and blame. However, whether the action actually brought about negative consequence or not only affected blame and punishment judgments, but not wrongness and permissibility judgments. Another study also provided neuroscientific evidence for the interplay between theory of mind and moral judgment.
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A Gray code for numbers in a given radix is a sequence that contains each number up to a given limit exactly once, in such a way that each pair of consecutive numbers differs by one in a single digit. The n! permutations of the n numbers from 1 to n may be placed in one-to-one correspondence with the n! numbers from 0 to n! − 1 by pairing each permutation with the sequence of numbers ci that count the number of positions in the permutation that are to the right of value i and that contain a value less than i (that is, the number of inversions for which i is the larger of the two inverted values), and then interpreting these sequences as numbers in the factorial number system, that is, the mixed radix system with radix sequence (1,2,3,4,...). For instance, the permutation (3,1,4,5,2) would give the values c1 = 0, c2 = 0, c3 = 2, c4 = 1, and c5 = 1\.
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In languages (such as C and Java) that favor iterative looping constructs, there is usually significant time and space cost associated with recursive programs, due to the overhead required to manage the stack and the relative slowness of function calls; in functional languages, a function call (particularly a tail call) is typically a very fast operation, and the difference is usually less noticeable. As a concrete example, the difference in performance between recursive and iterative implementations of the "factorial" example above depends highly on the compiler used. In languages where looping constructs are preferred, the iterative version may be as much as several orders of magnitude faster than the recursive one. In functional languages, the overall time difference of the two implementations may be negligible; in fact, the cost of multiplying the larger numbers first rather than the smaller numbers (which the iterative version given here happens to do) may overwhelm any time saved by choosing iteration.
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It also initiated much study of the contributions to sums of squares. Laplace knew how to estimate a variance from a residual (rather than a total) sum of squares.Stigler (1986, p 153) By 1827, Laplace was using least squares methods to address ANOVA problems regarding measurements of atmospheric tides.Stigler (1986, pp 154–155) Before 1800, astronomers had isolated observational errors resulting from reaction times (the "personal equation") and had developed methods of reducing the errors.Stigler (1986, pp 240–242) The experimental methods used in the study of the personal equation were later accepted by the emerging field of psychology Stigler (1986, Chapter 7 – Psychophysics as a Counterpoint) which developed strong (full factorial) experimental methods to which randomization and blinding were soon added.Stigler (1986, p 253) An eloquent non-mathematical explanation of the additive effects model was available in 1885.Stigler (1986, pp 314–315) Ronald Fisher introduced the term variance and proposed its formal analysis in a 1918 article The Correlation Between Relatives on the Supposition of Mendelian Inheritance.The Correlation Between Relatives on the Supposition of Mendelian Inheritance.
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Joseph Adolph Sonnabend (born 6 January 1933) is an Afrikaans physician, scientist and HIV/AIDS researcher, notable for pioneering community-based research, the propagation of safe sex to prevent infection, and an early multifactorial model of AIDS. As one of the first physicians to notice among his gay male patients the immune deficiency that would later be named AIDS, during the 1980s and 1990s he treated many hundreds of HIV-positive people. During the height of the AIDS crisis, Sonnabend helped create several AIDS organisations, including the AIDS Medical Foundation (now amfAR), the nonprofit Community Research Initiative (now ACRIA), which pioneered community-based research, and the PWA Health Group, the first and largest formally recognised buyers' club. Sonnabend became controversial for advocating that gay men change their sexual behaviors to avoid sexually- transmitted infections, rather than to just have fewer sexual partners, as advocated by Gay Men's Health Crisis and other gay community organizations and for hypothesizing a multi-factorial model of causation, including for a period of time after discovery of HIV .
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