are rounded to the displayed precision. div div In mathematics , the factorial of a non negative ... 1988 Concrete Mathematics , Addison Wesley, Reading MA. ISBN 0 201 14236 8, p. 111 ref The factorial ... though. ref The definition of the factorial function can also be Extension of factorial to non integer ... Definition The factorial function is formally defined by math n prod k 1 n k math or Recursion recursively ... among a set of n is math tbinom nn tfrac n n 0 1 math . The factorial function can also be defined for non integer values using more advanced mathematics, detailed in the Extension of factorial to non ... the factorial function has its roots in combinatorics , formulas involving factorials occur in many ... n . The only factorial that is also a prime number is 2, but there are many primes of the form n 1, called factorial prime s. All factorials greater than 0 and 1 are Parity mathematics ... of growth Image Log factorial.svg 300px thumb right Plot of the natural logarithm of the factorial As n grows, the factorial n nowiki nowiki becomes larger than all polynomial s and exponential function ... if any will compute n nowiki nowiki , provided the result fits in the variable. Interestingly, the factorial ... 700 bits, so no reasonable specification of a factorial function using fixed size types can avoid .... The largest factorial that most calculators can handle is 69 , because 69 10 sup 100 ..., can handle factorials up to 170 , which is the largest factorial that can be represented as a IEEE ... will compute small factorials by direct multiplication or table lookup. Larger factorial values can ..., 376&ndash 380 1985 ref Peter Luschny presents source code and benchmarks for several efficient factorial ... math factorial FastFactorialFunctions.htm Fast Factorial Functions The Homepage of Factorial Algorithms . ref Extension of factorial to non integer values of argument The Gamma and Pi functions Main Gamma function Image Factorial plot.png thumb right 325px The factorial function, generalized to all ... more details
Unreferenced date December 2009 In probability theory , the n th factorial moment of a probability distribution , also called the n th factorial moment of any random variable X with that probability distribution, is math E X n math where math x n x x 1 x 2 cdots x n 1 math is the falling factorial confusingly, this same notation, the Pochhammer symbol x sub n sub , is used by some mathematicians, especially in the theory of special function s, to denote the rising factorial x x 1 x 2 ... x n &minus 1 the present notation is used by combinatorics combinatorialists . For example, if X has a Poisson distribution with expected value , then the n th factorial moment of X is math E X n lambda n. math One context in which factorial moments occur naturally is at an initial stage in the use of probability generating function s to derive the moments of discrete distributions. See also moment mathematics cumulant Factorial moment generating function DEFAULTSORT Factorial Moment Category Probability distributions Category Factorial and binomial topics hu Faktori lis momentum tr Fakt riyel moment ... more details
An exponential factorial is a positive integer n exponentiation raised to the power of n &minus 1, which in turn is raised to the power of n &minus 2, and so on and so forth, that is, math n n 1 n 2 cdots . , math The exponential factorial can also be defined with the recurrence relation math a 0 1, quad a n n a n 1 . , math The first few exponential factorials are 1 number 1 , 1 number 1 , 2 number 2 , 9 number 9 , 262144, etc. OEIS id A049384 . So, for example, 262144 is an exponential factorial since math 262144 4 3 2 1 . , math The exponential factorials grow much more quickly than regular factorial s or even hyperfactorial s. The exponential factorial of 5 is 5 sup 262144 sup which is approximately 6.206069878660874 × 10 sup 183230 sup . The sum of the reciprocals of the exponential factorials from 1 onwards is the irrational number 1.6111149258083767361111... OEIS2C id A080219 . Like tetration , there is currently no accepted method of extension of the exponential factorial function to real number real and complex number complex values of its argument, unlike the factorial function, for which such an extension is provided by the gamma function . Numtheory stub References Jonathan Sondow, http mathworld.wolfram.com ExponentialFactorial.html Exponential Factorial From Mathworld , a Wolfram Web resource Category Integer sequences Category Factorial and binomial topics Category Large integers es Factorial exponencial he uk ... more details
In statistics , a full factorial experiment is an experiment whose design consists of two or more factors ... combinations of these levels across all such factors. A full factorial design may also be called ... variable. For the vast majority of factorial experiments, each factor has only two levels. 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 ... Factorial designs were used in the 19th century by John Bennet Lawes and Joseph Henry Gilbert of the Rothamsted ... Fisher argued in 1926 that complex designs such as factorial designs were more efficient than studying .... Nature, he suggests, will best respond to a logical and carefully thought out questionnaire . A factorial ... of designs, by the Yates analysis . The term factorial may not have been used in print before ... Example The simplest factorial experiment contains two levels for each of two factors. Suppose an engineer ... of two different speeds, 2000 or 3000 RPM. The factorial experiment would consist of four experimental ... of a 2 sup 2 sup or 2x2 factorial experiment, so named because it considers two levels the base ... 2 sup 4 factorial points. File Factorial Design.svg thumb left Designs can involve many independent ... border 1 style float right margin 0 0 10px 1em text align center width 9em 2 2 factorial experiment A B 1 &minus &minus a &minus b &minus ab To save space, the points in a two level factorial experiment ... as math math , math math , math math , and math math . The factorial points can also be abbreviated ... or first values. Implementation For more than two factors, a 2 sup k sup factorial experiment can be usually recursively designed from a 2 sup k 1 sup factorial experiment by replicating the 2 ... three replicates for three level factors, etc . A factorial experiment allows for estimation ... more details
A factorial prime is a prime number that is one less or one more than a factorial all factorials above 1 are even . The first few factorial primes are 2 number 2 0 1 or 1 1 , 3 number 3 2 1 , 5 number 5 3 &minus 1 , 7 number 7 3 1 , 23 number 23 4 &minus 1 , 719 6 &minus 1 , 5039 7 &minus 1 , 39916801 11 1 , 479001599 12 &minus 1 , 87178291199 14 &minus 1 , ... OEIS id A088054 n &minus 1 is prime for OEIS id A002982 n 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, ... n 1 is prime for OEIS id A002981 n 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, ... No other factorial primes are known as of 2010. Absence of primes to both sides of a factorial n implies a relatively lengthy run of consecutive composite number s, since n k is divisible by k for 2 &le k &le n . For example, the next prime following 6227020777 13 &minus 23 is 6227020867 13 67 a run of 89 consecutive composites here the run is substantially longer than implied merely by the absence of factorial primes. Note that this is not the most efficient way to find large prime gap s. E.g., there are 95 consecutive composites between the primes 360653 and 360749. See also Primorial prime External links MathWorld urlname FactorialPrime title Factorial Prime http primes.utm.edu top20 page.php?id 30 List of largest known factorial primes from the Prime Pages Category Integer sequences Category Classes of prime numbers Category Factorial and binomial topics da Fakultetsprimtal de Fakult tsprimzahl fr Nombre premier factoriel it Primo fattoriale ja fi Kertoma alkuluku vi S nguy n t giai th a zh ... more details
In mathematics , an alternating factorial is the absolute value of the alternating sum of the first n factorial s. This is the same as their sum, with the odd indexed factorials multiplied by 1 number &minus 1 if n is even, and the even indexed factorials multiplied by &minus 1 if n is odd, resulting in an alternation of signs of the summands or alternation of addition and subtraction operators, if preferred . To put it algebraically, math mathrm af n sum i 1 n 1 n i i math or with the recurrence relation math mathrm af n n mathrm af n 1 math in which af 1 1. The first few alternating factorials are 1, 1 number 1 , 5 number 5 , 19 number 19 , 101 number 101 , 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019 OEIS id A005165 For example, the third alternating factorial is 1 &minus 2 3 . The fourth alternating factorial is &minus 1 2 3 4 19. Regardless of the parity of n , the last n sup th sup summand, n , is given a positive sign, the n 1 sup th sup summand is given a negative sign, and the signs of the lower indexed summands are alternated accordingly. This pattern of alternation ensures the resulting sums are all positive integers. Changing the rule so that either the odd or even indexed summands are given negative signs regardless of the parity of n changes the signs of the resulting sums but not their absolute values. Miodrag Zivkovi proved in 1999 that there are only a finite number of alternating factorials that are also prime number s, since 3612703 divides af 3612702 and therefore divides af n for all n &ge 3612702. As of 2006 , the known primes and probable prime s are af n for OEIS id A001272 n 3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164 Only the values up to n 661 have ... MathWorld urlname AlternatingFactorial title Alternating Factorial Yves Gallot, http perso.wanadoo.fr ... search for primes of form n n 1 n 2 n 3 ... 1 Category Integer sequences Category Factorial and binomial ... more details
This is a list of factorial and binomial topics in mathematics , by Wikipedia page. See also binomial disambiguation . Alternating factorial Antichain Beta function Binomial coefficient Binomial distribution Binomial proportion confidence interval Binomial QMF Daubechies wavelet filters Binomial series Binomial theorem Pascal s triangle Binomial transform Binomial type Carlson s theorem Catalan number Central binomial coefficient Combination De Polignac s formula Difference operator Difference polynomials Digamma function Erd s Ko Rado theorem Euler Mascheroni constant Fa di Bruno s formula FactorialFactorial moment Factorial prime Gamma distribution Gamma function Gaussian binomial coefficient Hyperfactorial Hypergeometric distribution Hypergeometric function identities Hypergeometric series Incomplete beta function Incomplete gamma function Lah number Lanczos approximation Lozani s triangle Mahler s theorem Multinomial distribution Multinomial coefficient , Multinomial formula , Multinomial theorem Multiplicities of entries in Pascal s triangle Multiset Multivariate gamma function Narayana numbers Negative binomial distribution N rlund Rice integral Pascal matrix Pascal s pyramid Pascal s simplex Pascal s triangle Permutation List of permutation topics Pochhammer symbol also falling, lower, rising, upper factorials Poisson distribution Polygamma function Primorial Proof of Bertrand s postulate Sierpinski triangle Star of David theorem Stirling number Stirling transform Stirling s approximation Subfactorial Table of Newtonian series Taylor series Trinomial expansion Vandermonde s identity Wilson prime Wilson s theorem Wolstenholme prime Category Mathematics related lists Factorial and binomial topics Category Combinatorics Category Factorial and binomial topics ... more details
Unreferenced date December 2009 In probability theory and statistics , the factorial moment generating function of the probability distribution of a real number real valued random variable X is defined as math M X t operatorname E bigl t X bigr math for all complex number s t for which this expected value exists. This is the case at least for all t on the unit circle math t 1 math , see characteristic function probability theory characteristic function . If X is a discrete random variable taking values only in the set 0,1, ... of non negative integer s, then math M X math is also called probability generating function of X and math M X t math is well defined at least for all t on the closed set closed unit disk math t le1 math . The factorial moment generating function generates the factorial moment s of the probability distribution . Provided math M X math exists in a neighbourhood mathematics neighbourhood of t 1, the n th factorial moment is given by math operatorname E X n M X n 1 left. frac mathrm d n mathrm d t n right t 1 M X t , math where the Pochhammer symbol x sub n sub is the falling factorial math x n x x 1 x 2 cdots x n 1 . , math Confusingly, some mathematicians, especially in the field of special function s, use the same notation to represent the rising factorial . Example Suppose X has a Poisson distribution with expected value , then its factorial moment generating function is math M X t sum k 0 infty t k underbrace operatorname P X k , lambda ke lambda k e lambda sum k 0 infty frac t lambda k k e lambda t 1 , qquad t in mathbb C , math use the Exponential function Formal definition definition of the exponential function and thus we have math operatorname E X n lambda n. math See also Moment mathematics Moment generating function Cumulant generating function DEFAULTSORT Factorial Moment Generating Function Category Factorial and binomial topics Category Theory of probability distributions Category Generating functions ... more details
In statistics , fractional factorial designs are experimental design s 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. Notation Fractional designs are expressed using the notation l sup k &minus p sup , 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 interaction statistics interaction s are confounded , i.e. , cannot be estimated independently of each other see below . A design with p such generators is a 1 l sup p sup fraction of the full factorial design. For example, a 2 sup 5 &minus 2 sup design is 1 4 of a two level, five factor factorial design. Rather than the 32 runs that would be required for the full 2 sup 5 sup 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 ... rely on statistical reference books to supply the standard fractional factorial designs, consisting ... factorial experiment is generated from a full factorial experiment by choosing an alias structure ... factor 2 sup 5 &minus 2 sup can be generated by using a full three factor factorial ... pri section3 pri334.htm Fractional Factorial Designs National Institute of Standards and Technology ... DEFAULTSORT Fractional Factorial Design Experimental design Statistics Category Design of experiments ... more details
numeral systems In combinatorics , the factorial number system , also called factoradic , is a mixed radix numeral system adapted to numbering permutation s. It is also called factorial base , although factorial s 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 ... Mathematik und Physik volume 14 year 1869 . ref The term factorial number system is used by Donald ... 16 176 0 volume 16 year 1888 . ref The term factoradic , which is a portmanteau of factorial and mixed ... us library aa302371.aspx year 2003 . ref Definition The factorial number system is a mixed radix numeral ..., and so on. The factorial number system is sometimes defined with the rightmost digit omitted, because it is always zero OEIS id A007623 . In this article a factorial number representation will be flagged ... systems apply to the factorial number system as well. For instance, one can convert a number into factorial ... Inversion Vector ref which are reflected factorial numbers below them. Another column shows the inversion sets. The digit sums of the inversion vectors or factorial numbers and the cardinalities of the inversion ... 4 permutohedron factorial number system.svg the same with factorial numbers br The arrows indicate the bitwise ... text align right height 662px height 34px decimal factorial 0 0 sub sub 1 10 sub sub 2 100 sub ... 1 1 0 0 . Clearly the next factorial number representation after 543210 sub sub is 1000000 sub sub which ... number, and its summed out expression above, is equal to 6 1. The factorial number system provides a unique ... is always the next factorial minus one math sum i 0 n i cdot i n 1 1. math This can be easily ... by subscripting each digit by its base, also given in decimal . In fact the factorial number system ... with n digits in factorial representation and permutation s of n elements in lexicographical order ... table . For example, with n 3, such a mapping is decimal factorial permutation ... more details
Demonstrating while loops These while loops will calculate the factorial of the number 5 Ada programming ... Factorial is Counter Integer 5 Factorial Integer 1 begin while Counter 0 loop FactorialFactorial Counter Counter Counter 1 end loop Ada.Integer Text IO.Put Factorial end Factorial source AS 3 programming ... i source Bash Unix shell Bash source lang bash counter 5 factorial 1 while counter gt 0 do factorialfactorial counter counter counter 1 done echo factorial source QBasic or Visual Basic source lang vb Initialize the variables Dim counter As Integer counter 5 Dim factorial As Long factorial 1 Do While counter 0 factorialfactorial counter Multiply counter counter 1 Decrement Loop Print factorial Prints ... long factorial 1 while counter 0 factorial counter Multiply and decrement printf lu , factorial source Fortran source lang fortran program FactorialProg integer counter 5 integer factorial 1 do while counter 0 factorialfactorial counter counter counter 1 end do print , factorial end program FactorialProg ... factorial 1 while counter 1 factorial counter source For Java the result is printed as follows source lang java System.out.println factorial source The same in C source lang csharp System.Console.WriteLine factorial source And finally in D source lang d writefln factorial source JavaScript source lang javascript var counter 5 var factorial 1 while counter 1 factorial counter document.body.appendChild document.createTextNode factorial source Lua programming language Lua source lang lua counter 5 factorial 1 while counter 0 do factorialfactorial counter counter counter 1 end print factorial source MATLAB source lang matlab counter 5 factorial 1 while counter 0 factorialfactorial counter Multiply counter counter 1 Decrement end factorial source Mathematica Block counter 5,factorial 1 , localize counter and factorial While counter 0, While loop factorial counter Multiply counter Decrement factorial Pascal programming language Pascal source lang pascal program Factorial1 var Counter, Factorial ... more details
Multifactorial can refer to The factorial Multifactorials multifactorial in mathematics. Multifactorial inheritance , a pattern of predisposition for a disease process. disambig ... more details
do while loops These example programs calculate the factorial of 5 using their respective ... Ada Programming Control source lang ada with Ada.Integer Text IO procedure Factorial is Counter Integer 10 Factorial Integer 1 begin loop FactorialFactorial Counter Counter Counter 1 exit when Counter 0 end loop Ada.Integer Text IO.Put Factorial end Factorial source C or C C demonstrates the canonical do while syntax. source lang c unsigned int counter 5 unsigned long factorial 1 do factorial counter Multiply, then decrement. while counter 0 printf lu n , factorial source A do while loop ... type, x, y do type tmp x x y y tmp while 0 source Java source lang java public class Factorial public static void main String args int counter 5 int factorial 1 do factorial counter Multiply, then decrement. while counter 0 System.out.println factorial source C source lang c int counter 5 int factorial 1 do factorial counter Multiply, then decrement. while counter 0 System.Console.WriteLine factorial source JavaScript source lang javascript var counter 5 var factorial 1 do factorial counter Multiply, then decrement. while counter 0 document.body.appendChild document.createTextNode factorial ... loop. source lang fortran program FactorialProg integer counter 5 integer factorial 1 do factorialfactorial counter counter counter 1 if counter 0 exit end do print , factorial end program FactorialProg source Pascal Pascal demonstrates the repeat until syntax. source lang pascal program Factorial var Counter, Factorial integer begin Counter 5 Factorial 1 repeat FactorialFactorial Counter Dec Counter until Counter 0 WriteLn Factorial end. source Perl source lang perl counter 5 factorial 1 do factorial counter while counter 0 print factorial source PHP source lang php ?php counter 5 factorial 1 do factorial counter while counter 0 echo factorial ? source Visual Basic.Net source lang vb Dim counter As Integer 5 Dim factorial As Integer 1 Do factorial counter counter 1 Loop While counter ... more details
Compile time function execution or compile time function evaluation , CTFE is the ability of a compiler , that would normally compile a function to machine code and execute it at Run time computing run time , to execute the function at compile time . This is possible if the arguments to the function are known at compile time, and the function does not make any reference to or attempt to modify any global state is a pure function . Even if the value of only some of the arguments are known, the compiler may still be able to perform some level of compile time function execution partial evaluation , possibly producing more optimized code than if no arguments were known. Example In C , template metaprogramming is often used to compute values at compile time, such as source lang CPP template int N struct Factorial enum value N Factorial N 1 value template struct Factorial 0 enum value 1 Factorial 4 value 24 Factorial 0 value 1 void foo int x Factorial 0 value 1 int y Factorial 4 value 24 source But with compile time function evaluation the code used to compute the factorial would be exactly the same as what one would write for run time evaluation this example code is in the D programming language ref http www.digitalmars.com d 1.0 function.html interpretation D 1.0 language specification Functions ref source lang D int factorial int n if n 0 return 1 return n factorial n 1 computed at compile time const int y factorial 0 1 const int x factorial 4 24 source The use of code const code tells the compiler that the initializer for the variables must be computed at compile time ref http www.digitalmars.com d 1.0 attribute.html const D 1.0 language specification Attributes ref . CTFE can be used to populate simple data structures at compile time in a simple way D version 2 this is a function template source lang D int N genFactorials int N int N result result 0 1 foreach i 1 .. N result i result i 1 i return result enum auto factorials genFactorials 13 Now factorials contains at ... more details
About a functional programming language used primarily by students the scripting language created by IBM REXX Orphan date February 2009 lowercase title rex rex is a functional programming language developed by Robert M. Keller for use in teaching functional programming to Harvey Mudd College students. The rex interpreter is written in Prolog . Example The Hello world program Hello World of functional languages is the factorial function. Expressed in rex factorial 0 1 factorial X X factorial X 1 The name rex derives from rewriting expressions , which is the basic principle underlying the implementation replace an instantation of an expression on the left hand side of a rule with the instantiated right hand side. External links http www.cs.hmc.edu keller rex Documentation for the rex Language and Interpreter Category Functional languages Category Educational programming languages it Rex linguaggio ... more details
wiktionarypar a double exclamation mark may refer to A Punctuation chess Brilliant move brilliant move in chess notation The Factorial Double factorial double factorial operator in mathematics A convert to Boolean data type Boolean pseudo operator in some computer languages This is through a computer type of double negative , where its Boolean value is negated, then negated again, converting to the value to either true or false. An operator for getting the n th element of a list in the programming language Haskell programming language Haskell . The double exclamation mark is present in Unicode as the single character U 203C DOUBLE EXCLAMATION MARK . See also disambiguation disambig de es fr ko it ru zh ... more details
final results. One example of a commonly encountered hylomorphism is the canonical factorial function. source lang haskell factorial Integer Integer factorial n n 0 1 n 0 n factorial n 1 source In the previous ... call tree isomorphism isomorphic to a list. For example, given n 5 it will produce the following factorial 5 5 factorial 4 120 factorial 4 4 factorial 3 24 factorial 3 3 factorial 2 6 factorial 2 2 factorial 1 2 factorial 1 1 factorial 0 1 factorial 0 1 In this example, the anamorphic part of the process ... elements of this list. Thus, in the notation given above, the factorial function may be written math text factorial 1, times , g, p math where math g n n, n 1 math and math p n n 0 math . Trees ... more details
In mathematics , two different function mathematics functions are known as the pi or Pi function math pi x , math pi function &ndash the prime counting function math Pi x , math Pi function &ndash the Gamma function when offset to coincide with the factorial disambig th ... more details
In the design of experiments and analysis of variance , a main effect is the effect of an independent variable on a dependent variable averaging across the levels of any other independent variables. The term is frequently used in the context of factorial design s and regression analysis regression models to distinguish main effects from interaction statistics interaction effects. For example, in factorial designs, the main effect is what the independent variables elicit when averaged out over each other. References McBurney, D.M., White, T.L. 2004 . Research Methods . CA Wadsworth Learning. Mook, Douglas G. 2001 . Psychological Research The Ideas Behind the Methods . NY W. W. Norton & Company. statistics stub Category Research Category Analysis of variance ... more details
The sparsity of effects principle states that a system is usually dominated by main effect s and low order interactions. Thus it is most likely that main single factor effects and two factor interactions are the most significant responses see factorial experiment . In other words, higher order interactions such as three factor interactions are very rare. Formally, C.F. Jeff Wu and Hamada 2000, page 112 refer to this as the hierarchical ordering principle . They state that the effect sparsity principle actually refers to the idea that only a few effects in a factorial experiment will be statistically significant. See also Occam s Razor References Wu, C. F. Jeff and Hamada, Michael 2000 Experiments Planning, analysis, and parameter design optimization, New York Wiley, ISBN 0 471 25511 4. Category Design of experiments Category Social sciences methodology Category Statistical principles ... more details
Mergeto Large numbers date March 2009 The Leviathan number in mathematics is defined as the factorial of the Number of the Beast 666 th power of ten 10 sup 666 sup . The baby Leviathan number is the factorial of Number of the Beast 666 , which has 1,594 digits. It s also 666 sup Legion s number sup or 911 sup 911 sup 911 sup sup In Satanism and numerology , the number of the Leviathan is 3. See also Number of the Beast Mathematics The Number of the Beast References http mathworld.wolfram.com LeviathanNumber.html LeviathanNumber at mathworld.wolfram.com cite book first Clifford A. last Pickover year 2000 title Wonders of Numbers Adventures in Math, Mind, and Meaning pages 196, pp. 350 1 publisher Oxford University Press isbn 0195133420 author link Clifford A. Pickover Category Large integers number stub pt N mero de Leviat ... more details
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