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# Topic: Arithmetical functions

 Fixated Points of Arithmetical Functions The situation appears to be much rosier for "fixated points" of arithmetical functions. For simplicity, an arithmetical function in this discussion refers to a function having domain and range which are sets of integers. One such function is given by f(n) = Prime(n), the n-th prime, which has a rate of growth with respect to n that is approximately equal to ln(n) (the natural logarithm of n). www.geocities.com /windmill96/enlightened/fixated.html   (512 words)

 Learn more about Arithmetic in the online encyclopedia.   (Site not responding. Last check: 2007-10-24) Arithmetic is a branch of mathematics which records elementary properties of certain operations on numbers. The arithmetic of natural numbers, integers, rational numbers (in the form of fractions) and real numbers (in the form of decimal expansions) is typically studied by schoolchildren of the elementary grades. The term "arithmetic" is also used to refer to elementary number theory; it's in this context that one runs across the fundamental theorem of arithmetic and arithmetical functions. www.onlineencyclopedia.org /a/ar/arithmetic.html   (225 words)

 Arithmetic   (Site not responding. Last check: 2007-10-24) Arithmetic is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numeral s. The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. The arithmetic of natural number s, integer s, rational number s (in the form of fraction s), and real number s (using the decimal place-value system known as algorism) is typically studied by schoolchildren, who learn manual algorithms for arithmetic. www.nebulasearch.com /encyclopedia/article/Arithmetic.html   (655 words)

 Arithmetic function   (Site not responding. Last check: 2007-10-24) In number theory, an arithmetic function (or number-theoretic function) f(n) is a function defined for all positive integers and having values in the complex numbers. The most important arithmetic functions are the additive and the multiplicative ones. An important operation on arithmetic functions is the Dirichlet convolution. www.sciencedaily.com /encyclopedia/arithmetic_function   (262 words)

 Latte The parameters of Latte functions (see section 3.3 Groups) are variables whose scope is the body of the function for which they're defined. When the function is invoked, the first actual argument is assigned to the first positional parameter, the second actual argument is assigned to the second positional parameter, and so on. When the first subexpression of a group is a function (that is, when it is a variable reference whose value is a function, or when it is some other Latte expression yielding a function), then that function is called, passing the remaining subexpressions to it as arguments. www.latte.org /latte.html   (6362 words)

 [No title] An arithmetical function is a real- or complex-valued function defined on the integers. Properties of these and other functions are widely studied in elementary number theory and recreational mathematics. function has numerous applications and is fundamental to the study of congruences. www.math.columbia.edu /~rama/chapters/chap15.html   (194 words)

 Arithmetical Functions   (Site not responding. Last check: 2007-10-24) Functions f(n) of the positive integer n that express some arithmetical property of n are called arithmetical functions. An important example is Euler's function f(n), which counts the number of integers in a reduced residue system mod n. Some arithmetical functions are described by giving the first two function values f(1) and f(2) and then expressing f(n) for n ³ 3 in terms of f(n - 1) and f(n - 2).For example, the Fibonacci numbers are defined by: www.risberg.ws /Hypertextbooks/Mathematics/Numbers/arithmeticalfunctions.htm   (142 words)

 complex Package The complex package provides ways to perform arithmetical operations, such as initialization, assignment, input, and output, on complex values (that is, numbers with a real part and an imaginary part). Assigns the arithmetical sum of complex values to the complex object on the left side of an equation. Assigns the arithmetical quotient of two complex numbers to the complex object on the left side of an equation. www.ph.unimelb.edu.au /~jfm/dec_cxx_5_7/u_cxxlrm0003.html   (1089 words)

 [No title]   (Site not responding. Last check: 2007-10-24) This dependence is an expression that consists of arithmetical operators, functions, constants and symbols "x". This fact causes the function to break near the place 11000 (exp(11000) is an enormous number!). So everywhere you use function exp(x) the graph is limited at least by this place. www.physic.ut.ee /~nigul/tgraph_readme.txt   (536 words)

 Analytic Number Theory It is quite common that one studies the behavior of arithmetical functions f(n) for large values of n. This is called a Dirichlet series with coefficients f(n), and the function F(s) is called a generating function of the coefficients. This field is dealing with arithmetical functions related to additive properties of the the integers, as opposed to multiplicative number theory, which is concerned with properties arising from prime factorization. www.risberg.ws /Hypertextbooks/Mathematics/Numbers/analytic.htm   (724 words)

 Dynamical and spectral zeta functions   (Site not responding. Last check: 2007-10-24) It is shown as an example that the Riemann zeta function is the zeta functions of a quadratic polynomial, which is associated with the Laplacian on an interval. The spectral zeta function of the Sierpinski gasket is a product of the zeta function of a polynomial and a geometric part; the poles of the former are canceled by the zeros of the latter. V.Nesterenko and I. Pirozhenko, "Spectral zeta functions for a cylinder and a circle" www.maths.ex.ac.uk /~mwatkins/zeta/physics3.htm   (4444 words)

 Arithmetic   (Site not responding. Last check: 2007-10-24) Arithmetic is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals. The arithmetic of natural numbers, integers, rational numbers (in the form of fractions), and real numbers (using the decimal place-value system known as algorism) is typically studied by schoolchildren, who learn manual algorithms for arithmetic. However, in adult life, many people prefer to use tools such as calculators, computers, or the abacus to perform arithmetical computations. www.sciencedaily.com /encyclopedia/arithmetic   (237 words)

 Kids.net.au - Encyclopedia Arithmetic -   (Site not responding. Last check: 2007-10-24) Arithmetic is a branch of mathematics which records elementary properties of certain arithmetical operations on numbers. The term "arithmetic" is also sometimes used to refer to number theory; it's in this context that one runs across the fundamental theorem of arithmetic and arithmetical functions. Kids.net.au - Search engine for kids, children, educators and teachers - Searching sites designed for kids that are child safe and clean. www.kids.net.au /encyclopedia-wiki/ar/Arithmetic   (166 words)

 boats.com - Feature: Navigating With a Calculator   (Site not responding. Last check: 2007-10-24) Key sequences and functions vary with different calculators, and the manufacturer's handbook should always be studied first. Before using these functions consider the method by which they are derived as this will assist you in solving triangular problems. AB These functions are the same for angle BAC except that adjacent and opposite sides are different. www.boats.com /content/default_detail.jsp?contentid=2932   (1111 words)

 [No title] "Arithmetical properties of finite rings and algebras, and analytic number theory, III :Finite modules and algebras over Dedekind domains", J. f•r die reine und angew. Arithmetical properties of finite rings and algebras, and analytic number theory, IV : Relative asymptotic enumeration and L-series", J. f•r die reine und angew. "Arithmetical properties of finite rings and algebras, and analytic number theory, V : Categories and relative analytic number theory", J. f•r die reine und angew. www.wits.ac.za /science/number_theory/jplkpub.htm   (1305 words)

 Arithmetical Functions - Cambridge University Press The aim of this book is to characterize certain multiplicative and additive arithmetical functions by combining methods from number theory with some simple ideas from functional and harmonic analysis. The authors achieve this goal by considering convolutions of arithmetical functions, elementary mean-value theorems, and properties of related multiplicative functions. Mean-value theorems and multiplicative functions, II; Photographs; Appendix; Bibliography; Author index; Subject index; Photographs; Acknowledgements. www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521427258   (118 words)

 A+ Reference: Monadic Scalar Functions All monadic scalar functions produce scalars from scalars, and apply element by element to their arguments: they are applied to each element independently of the others. With only one exception, the error reports for monadic primitive scalar functions are common to all such functions. An inadvertent left argument results not in a valence error, but in the invocation of a dyadic function that shares the function symbol. www.aplusdev.org /APlusRefV2_6.html   (839 words)

 [No title]   (Site not responding. Last check: 2007-10-24) Dom::Integer which are represented by a subset of the arithmetical expressions. Multiplies an arbitrary number of arithmetical expressions and returns the (simplified) arithmetical expression f*g*... If one of the arguments cannot be evaluated to a number, then the function call with all non-numerical and the minimum of the numerical arguments is returned. www.sciface.com /STATIC/DOC30/eng/Dom_ArithmeticalExpression.html   (376 words)

 DEC C++   (Site not responding. Last check: 2007-10-24) Is a pointer to a generic error-handling function. Defines the noninline member functions of a class, specified by a macro with the name of the generic class. Specifies a function as the current error handler for a given instance of a parameterized class. nimbus.temple.edu /deccxx/cpcl001.htm   (1241 words)

 gf_poly_modulus   (Site not responding. Last check: 2007-10-24) If you need to do a lot of arithmetic modulo a fixed polynomial f, build a gf_poly_modulus F for f. Access Functions Let F be of type gf_poly_modulus. As described in the section polynomial, each polynomial carries a reference to an element of type galois_field representing the finite field over which the polynomial is defined. www.math.psu.edu /local_doc/LiDIA/node67.html   (384 words)

 Recursive Functions of the Integers   (Site not responding. Last check: 2007-10-24) In Reference 7 we develop the recursive functions of a class of symbolic expressions in terms of the conditional expression and recursive function formalism. As an example of the use of recursive function definitions, we shall give recursive definitions of a number of functions over the integers. We shall see how some of the properties of the arithmetical functions can conveniently be derived in this formalism in a later section. www-formal.stanford.edu /jmc/basis1/node4.html   (296 words)

 Temple University Advanced Technology Group   (Site not responding. Last check: 2007-10-24) In: The Theory of Arithmetic Functions (eds.: A.A. Gioia and D.L. Goldsmith), Lecture Notes in Mathematics, Springer Verlag, 251 (1971), 127-139. On the distribution of multiplicative arithmetical functions (J. Galambos and P. usz). The continuity of the limiting distribution of a function of two additive functions (J. Galambos and Imre Katai). www.math.temple.edu /~janos/PUBLICATIONS.html   (1966 words)

 1/f noise, signal processing and number theory   (Site not responding. Last check: 2007-10-24) It is first shown that Riemann zeta function is essentially the Mellin transform of the partition function $Z(x)$ of the non degenerate (one dimensional) perfect gas. Thermodynamical quantities carry a strong arithmetical structure: they are given by series with Fourier coefficients equal to summatory functions $\sigma_k(n)$ of the power of divisors, with k = -1 for the free energy, k = 0 for the number of particles and k = 1 for the internal energy. The common point of the three topics is found to be the Mangoldt function of prime number theory as the generator of low frequency noise in the coupling coefficient, the scattering coefficient and its quantum critical statistical states. www.maths.ex.ac.uk /~mwatkins/zeta/1fnoisesigprocNT.htm   (2570 words)

 Amazon.co.uk: Books: Non-Archimedean L-functions and Arithmetical Siegel Modular Forms (Lecture Notes in Mathematics)   (Site not responding. Last check: 2007-10-24) This book, now in its 2nd edition, is devoted to the arithmetical theory of Siegel modular forms and their L-functions. The given construction of these p-adic L-functions uses precise algebraic properties of the arithmetical Shimura differential operator.The book will be very useful for postgraduate students and for non-experts looking for a quick approach to a rapidly developing domain of algebraic number theory. This new edition is substantially revised to account for the new explanations that have emerged in the past 10 years of the main formulas for special L-values in terms of arithmetical theory of nearly holomorphic modular forms. www.amazon.co.uk /exec/obidos/ASIN/3540407294   (414 words)

 BOOLEAN ARITHMETIC and its Applications   (Site not responding. Last check: 2007-10-24) This Chapter focused on the theoretical background of the Boolean Arithmetic and its corresponding concepts to Boolean Algebra. In expression (4) we know the Boolean Arithmetical Function (BAF),, that is, the binary components of the abscissa of the Numerical Transform (NT). We believe that Boolean Arithmetic is a new area of research that may present relevant mathematical contributions to find the solutions for these problems. waneck.sites.uol.com.br   (8335 words)

 Recursion Theoretic Characterizations of Complexity Classes of Counting Functions - und, Wagner (ResearchIndex)   (Site not responding. Last check: 2007-10-24) Here, we characterize #P as the closure of a set of simple arithmetical functions under summation and weak product. Building on that result, the hierarchy of counting functions, which is the closure of #P under substitution, is characterized; remarkably without... We say that is closed under wea summation if for every 2 and every c 2 N we have that the function defined as (x)... citeseer.ist.psu.edu /vollmer94recursion.html   (655 words)

 Chapter 1 This result is very important in the theory of theta series, and has found broader application than simply the theory of partitions of numbers. Although it is beyond the scope of this thesis, the author aims to find Dirichlet series analogues for some of these results after aspects of chapters 5 to 9 of this thesis are more fully developed. FELD, J. The expansion of analytic functions in a generalised Lambert series, Annuls of Math., 33, 1932, 139-143. geocities.com /CapeCanaveral/Launchpad/9416/PhDthesis/introduction.html   (2623 words)

 Large Deviations of Combinatorial Distributions I: Central Limit Theorems - Hwang (ResearchIndex)   (Site not responding. Last check: 2007-10-24) This theorem has wide applications to combinatorial structures and to the distribution of additive arithmetical functions. The method of proof is an extension of Kubilius' version of Cramer's classical method based on analytic moment generating functions. where we derived a general central limit theorem for probabilities of large deviations applicable to many classes of combinatorial structures and arithmetic functions; we consider corresponding local limit... citeseer.ist.psu.edu /hwang95large.html   (796 words)

 Bibliography for On-Line Encyclopedia of Integer Sequences   (Site not responding. Last check: 2007-10-24) Abramowitz and I. Stegun, {\em Handbook of Mathematical Functions}, National Bureau of Standards, Washington DC, 1964. Odlyzko, {\em The $10^{20}$-th Zero of the Riemann Zeta Function and 175 Million of its Neighbors}, manuscript in preparation. Salvy and P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable,'' {\em ACM Trans. www.research.att.com /~njas/sequences/eisbib.html   (7328 words)

 [No title]   (Site not responding. Last check: 2007-10-24) Knopfmacher and R. Warlimont John Knopfmacher -A mathematical biography txt D.S.Lubinsky, 261 John Knopfmacher -Mathematical and other memories txt A.Knopfmacher, 263 John Knopfmacher and additive arithmetical semigroups ps R.Warlimont, 269 John Knopfmacher, [Abstract] Analytic number theory, and the theory of arithmetical functions ps W. Warlimont, 355 About the abscissa of convergence of the Zeta function of a multiplicative arithmetical semigroup ps R. Warlimont, 363 Arithmetically related ideal topologies and the infinitude of primes ps S. www.wits.ac.za /science/number_theory/qmissue2.htm   (112 words)

 Compaq C++ User Documentation These declarations are used by the iostream package but they are not members of any class. The following functions insert values into a stream, extract values from a stream, or specify the conversion base format. Keep in mind that when your application calls a member function for a predefined stream object, the member function will typically lock the object for the duration of the call. www.helsinki.fi /atk/unix/dec_manuals/cxx_6.3a/clu001.htm   (1629 words)

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