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Topic: Divisor function

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	Divisor function - Wikipedia, the free encyclopedia
		In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer.
		Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities.
		The behaviour of the sigma function is irregular.
	en.wikipedia.org /wiki/Divisor_function (606 words)

	Euler's totient function - Wikipedia, the free encyclopedia
		In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n.
		The totient function is important mainly because it gives the size of the multiplicative group of integers modulo n.
		The growth of φ(n) as a function of n is an interesting question, since the first impression from small n that φ(n) might be noticeably smaller than n is somewhat misleading.
	en.wikipedia.org /wiki/Euler's_totient_function (648 words)

	Divisors
		Divisors are created by naming the divisor group in which they should lie (if that is not otherwise clear) and then giving enough data to identify precisely the divisor in question.
		The principal divisor corresponding to f, that is, the divisor of D of zeros and poles of function field element, ambient polynomial or ambient rational polynomial f.
		The divisor in the divisor group D corresponding to the differential a.
	www.umich.edu /~gpcc/scs/magma/text1032.htm (2782 words)

	XEmacs Lisp Reference Manual - Numbers
		To understand how various functions work on integers, especially the bitwise operators (see section Bitwise Operations on Integers), it is often helpful to view the numbers in their binary form.
		This function tests whether its first argument is less than or equal to its second argument.
		This function tests whether its first argument is greater than or equal to its second argument.
	www.tau.ac.il /cc/pages/docs/xemacso/lispref_4.html (2730 words)

	Using the Symbolic Math Toolbox (Symbolic Math Toolbox)
		For example, the greatest common divisor of 14 and 21 is 7.
		The first thing we need to know is how to call the greatest common divisor function in Maple.
		gcd - greatest common divisor of polynomials lcm - least common multiple of polynomials Calling Sequence: gcd(a,b,'cofa','cofb') lcm(a,b,...) Parameters: a, b - multivariate polynomials over an algebraic number field or an algebraic function field cofa,cofb - (optional) unevaluated names Description: - The gcd function computes the greatest common divisor of two polynomials a and b.
	www-rohan.sdsu.edu /doc/matlab/toolbox/symbolic/ch226.html (524 words)

	[No title]
		The function fails if the biggest elementary divisor cannot be completely factored and the non-factored part is not a divisor of the biggest elementary divisor only.
		If it is not and this function returns an integer, then this is the rank of A for all prime divisors of p.
		The idea to make this function useful for non primes was to use it with large factors of the biggest elementary divisor of A whose prime factorization cannot be found easily.
	www-history.mcs.st-and.ac.uk /~gap/Manuals/pkg/edim/xmldoc/main.xml (2288 words)

	Class Field Theory
		The ray class group modulo D is a quotient of the group of divisors that are coprime to D modulo certain principal divisors.
		Since this function uses the class group of the function field in an essential way, it may be necessary for large examples to precompute the class group.
		Given an effective divisor D and a subgroup U of the ray class group, (see RayClassGroup), Cl_D of D such that the quotient Cl_D/U is finite, compute defining equations for the corresponding (ray) class field.
	magma.maths.usyd.edu.au /magma/htmlhelp/text716.htm (2760 words)

	GNU Emacs Lisp Reference Manual - Numbers
		People have pointed out that this function is misnamed, because the term "natural number" is usually understood as excluding zero.
		As the examples illustrate, shifting the pattern of bits one place to the left produces a number that is twice the value of the previous number.
		Note, however that functions do not check for overflow, and a returned value may be negative (and in any case, no more than a 24 bit value) when an integer is sufficiently left shifted.
	www.cs.utah.edu /dept/old/texinfo/elisp-manual-19-2.02.2/elisp_4.html (2579 words)

	Divisors
		Given a function field F/k and a divisor D belonging to F/k, return a vector space V and a k-linear mapping h: V longrightarrow F such that V is isomorphic to the Riemann-Roch space (L)(D) subset F under h.
		Return the complementary divisor D^# of D. The function field F/k of D must be a finite extension of a rational function field k(x).
		The divisor class group of F/k as an Abelian group, a map of representatives from the class group to the divisor group and the homomorphism from the divisor group onto the divisor class group.
	www.umich.edu /~gpcc/scs/magma/text696.htm (2708 words)

	Spectra of Rings - Page 3 (Site not responding. Last check: 2007-11-05)
		Then the theory of (doubly periodic) P functions shows that X is isomorphic to the quotient of C^2 by a lattice, hence to a torus.
		Second a rational function defines a map from a smooth projective curve to P^1, which is a branched covering, and again every point of p^1 has the same number of preimages counted properly.
		Anyway, from the fact that the quotient of two differentials is a rational function and the fact that the divisor of a rational function is zero, it follws that all differentials on a curve have divisors of the same degree.
	www.physicsforums.com /showthread.php?p=316894#post316894 (2219 words)

	Arithmetic Operations
		The functions to add or subtract 1 are provided because they are traditional in Lisp and commonly used.
		This function is not analogous to the C operator
		For negative arguments, the remainder is in principle machine-dependent since the quotient is; but in practice, all known machines behave alike.
	www.cse.unsw.edu.au /help/doc/info/elisp/Arithmetic_Operations.html (394 words)

	CLHS: Function FLOOR, FFLOOR, CEILING, FCEILING...
		ffloor, fceiling, ftruncate, and fround handle arguments of different types in the following way: If number is a float, and divisor is not a float of longer format, then the first result is a float of the same type as number.
		When only number is given, the two results are exact; the mathematical sum of the two results is always equal to the mathematical value of number.
		(where function is any of one of floor, ceiling, ffloor, fceiling, truncate, round, ftruncate, and fround) return the same first value, but they return different remainders as the second value.
	www.lispworks.com /documentation/HyperSpec/Body/f_floorc.htm (401 words)

	[No title]
		The first element in the list is the name of the function to be called, all other elements are the arguments to apply to it.
		Functions (note Functions::.) and macros (note Macros::.) have more complex descriptions; as well as the name of the object being described, they also have a list of parameters which the object will accept.
		Special forms (note Special Forms::.) are described similarly to functions* except that the argument list is formatted differently, since special forms are, by definition, more flexible in how they treat their arguments.
	web.mit.edu /afs/sipb/project/sawfish/info/librep.info-1 (5244 words)

	Totient Function -- from MathWorld
		Since a number less than or equal to and relatively prime to a given number is called a totative, the totient function
		The totient function is given by the Möbius transform of 1, 2, 3, 4,...
		Function." §2.27 in Solved and Unsolved Problems in Number Theory, 4th ed.
	users.skynet.be /fa956617/math/topics/TotientFunction.html (677 words)

	divisor - Hutchinson encyclopedia article about divisor
		In mathematics, any number that is to be divided into another number.
		For example in the computation 100 ÷ 25 = 4, 25 is the divisor.
		I am afraid so--thirteen is a dreadful divisor for a small family estate.
	encyclopedia.farlex.com /divisor (109 words)

	GNU Emacs Lisp Reference Manual: Numbers
		To understand how various functions work on integers, especially the bitwise operators (see section 3.8 Bitwise Operations on Integers), it is often helpful to view the numbers in their binary form.
		Many of the functions described in this chapter accept markers for arguments in place of numbers.
		Markers.) Since the actual arguments to such functions may be either numbers or markers, we often give these arguments the name
	www.cs.huji.ac.il /~osigor/emacs/elisp-help/elisp-manref/elisp_3.html (2968 words)

	[No title]
		To understand how various functions work on integers, especially the bitwise operators (*note Bitwise Operations::.), it is often helpful to view the numbers in their binary form.
		The functions `integerp' and `floatp' can take any type of Lisp object as argument (the predicates would not be of much use otherwise); but the `zerop' predicate requires a number as its argument.
		For example, (setq foo 4) => 4 (1+ foo) => 5 This function is not analogous to the C operator `++'--it does not increment a variable.
	www.math.psu.edu /Doc/emacs/info/elisp-4 (5200 words)

	XEmacs Lisp Reference Manual: Numbers
		To understand how various functions work on integers, especially the bitwise operators (see section 9.8 Bitwise Operations on Integers), it is often helpful to view the numbers in their binary form.
		Markers.) More precisely, the actual arguments to such functions may be either integers or markers, which is why we often give these arguments the name
		In addition to numbers, all of the following functions also accept characters and markers as arguments, and treat them as their number equivalents.
	theoryx5.uwinnipeg.ca /gnu/xemacs/lispref_10.html (2888 words)

	Arithmetic Operations (Site not responding. Last check: 2007-11-05)
		A remainder function supplements the (integer) division function.
		* Function: 1- MARKER-OR-INT This function subtracts one from MARKER-OR-INT.
		This is because the Lisp function `/' is implemented using the C division operator, which has the same possibility for machine-dependent rounding.
	www.cis.ksu.edu /VirtualHelp/Info/gnu/elisp.Arithmetic_Operations.html (376 words)

	Algebraic Function Fields [HB 57]
		An algorithm for computing the divisor class group of a global function field has been implemented.
		The following functions relating to the class group may be applied to maximal finite orders and their ideals in a global function field:
		The intersection, sum etc. of such modules is possible as well as coercion between the field or space and the module.
	www.math.niu.edu /help/math/magmahelp/rel/node36.html (175 words)

	G415-Project 7 (Site not responding. Last check: 2007-11-05)
		The function should take 1 input (the vector of taxable incomes) and send back to the calling process one output (the vector of taxes).
		You are to write a function that calculates the greatest common divisor (gcd).
		The point is that the function, which MUST be called mygcd, have 2 input arguments (the 2 integers) and 1 output argument (the gcd of the 2 input integers).
	userwww.sfsu.edu /~garfield/proj_07.html (219 words)

	Overview of Shor's Algorithm
		is a periodic function, where x is an integer coprime to n.
		When two numbers are coprime it means that their greatest common divisor is 1.
		This Fourier transform maps functions in the time domain to functions in the frequency domain.
	www.dhushara.com /book/quantcos/qcompu/shor/s.htm (1298 words)

	Only Python: Python as pseudo-code and Optional Static Typing
		Yet, one of the strength of Python is its syntax, and many objections (NIMPY as GvR described it) to the proposed addition(s) to the language came from an unease with the suggested syntax.
		I will use GvR's version of a Greatest Common Divisor finding function as an example and build from it; the complexity of the examples will increase as we go in an attempt to fully illustrate the suggested syntax.
		However, given that the optional static typing proposal is new, it should be looked at objectively, without rejecting off-hand proposals that were rejected in a different context.
	aroberge.blogspot.com /2005/01/python-as-pseudo-code-and-optional.html (978 words)

	Quadrillion - Psychology Central
		The divisor function calculated on a quadrillion is equal to:
		The divisor function of a quadrillion (and indeed, of all high powers of ten) may be calculated with the formula:
		which may be seen by applying the formula for the partial sum of a geometric series twice to the sum of the divisors.
	psychcentral.com /psypsych/Quadrillion (260 words)

	Divisor - divisor - definition of divisor by the Free Online Dictionary (Site not responding. Last check: 2007-11-05)
		The initial divisor of each Dow Jones Average was the number of stocks in the By changing the divisor in this way, Dow Jones preserves the continuity of
		>Divisor: number by which dividend is to be divided; number that divides definition, but it is clear that a factor is a divisor.
		The divisor function can also be generalized to Gaussian integers.
	bewiner.com /?q=divisor (238 words)

	Divisor - LKML: "Stuart MacDonald": RE: serial port, custom divisor
		The greatest common divisor of $ a$ and $ b$, denoted by $ \gcd(a,b)$, is the More intuitively, the greatest common divisor is the largest integer
		The divisor class group of a normal affine monoid M (or the monoid algebra K[M] The conic divisor classes appear in the decomposition of (1/n)M, n Î N, deviser, devisor, divisor.
		The Greatest Common Divisor, GCD for short, of two positive integers can be computed with If the remainder c is zero, b is the greatest common divisor.
	infofeedtech.com /ifft/divisor.htm (471 words)

	CS 51: Introduction to Computer Science II
		If THING is a symbol that names a not-yet-compiled function, the function definition is disassembled.
		The result is a sequence such that the i-th element is the result of applying FUNCTION to the i-th elements of the SEQUENCEs.
		----------------------------------------------------------------------------- EQL [Function] EQL Function in LISP package: Args: (x y) Returns T if X and Y are EQ, or if they are numbers of the same type with the same value, or if they are character objects that represent the same character.
	www.deas.harvard.edu /courses/cs51/clisp.html (8605 words)

	[No title]
		It's more important in header files where variables and functions are declared.
		Hence, when compiling it is usually not necessary to tell the compiler were such files are loacted on the file-system.
		The initialization routine is common to a number * of functions, so we simply abstracted it out.
	www.cs.umass.edu /~heeringa/classes/winter2001/cmpsci197c/assignment0/Rational.cc (380 words)

	Integer and Number-Theoretical Functions
		This gives a list of all the divisors of 24.
		gives the total number of distinct divisors of 24.
		The plot of the digit count function is self-similar.
	documents.wolfram.com /v4/MainBook/3.2.4.html (900 words)

	[No title]
		// // The following public member functions are declared to operate on the // class Fraction: // default and initializer constructors; // algebraic operators +, -, * and /; // conversion to decimal equivalent; // simplification; // output to write fractions in form a/b; // equality test.
		Fraction Fraction::operator/ (Fraction frac) { int num_result = numer * frac.denom; int den_result = denom * frac.numer; Fraction result(num_result, den_result); result.simplify(); return(result); } // Utilities // The function find_gcd determines the greatest common divisor of two // variables a and b using Euclid's algorithm.
		The loop // repeats until b is zero, at which time a is the greatest common divisor.
	www.ce.washington.edu /~kkantner/cae/hw5/vector/fraction.h (538 words)

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