Related papers: Various Arithmetic Functions and their Application…
Standard probability theory has been extremely successful but there are some conceptually possible scenarios, such as fair infinite lotteries, that it does not model well. For this reason alternative probability theories have been…
For enumerative problems, i.e. computable functions f from N to Z, we define the notion of an effective (or closed) formula. It is an algorithm computing f(n) in the number of steps that is polynomial in the combined size of the input n and…
In this expository article we collect the integer sequences that count several different types of matrices over finite fields and provide references to the Online Encyclopedia of Integer Sequences (OEIS). Section 1 contains the sequences,…
This is the second volume of a textbook for a two-semester course in mathematical analysis. This second volume is about analysis of multi-variable functions. The topics covered include Euclidean spaces, convergence of sequences, open sets…
Two approximations of the integral of a class of sinusoidal composite functions, for which an explicit form does not exist, are derived. Numerical experiments show that the proposed approximations yield an error that does not depend on the…
We study inverse factorial series and their relation to Stirling numbers of the first kind. We prove a special representation of the polylogarithm function in terms of series with such numbers. Using various identities for Stirling numbers…
We establish that the sequences formed by logarithms and by "fractional" powers of integers, as well as the sequence of prime numbers, are non-holonomic, thereby answering three open problems of Gerhold [Electronic Journal of Combinatorics…
We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists…
This paper presents a brief survey of the most important and the most remarkable inequalities involving the basic arithmetic functions.
Windowed recurrences are sliding window calculations where a function is applied iteratively across the window of data, and are ubiquitous throughout the natural, social, and computational sciences. In this monograph we explore the…
Work in progress concerning alternative formalizations of arithmetic.
We study enumeration functions for unimodal sequences of positive integers, where the size of a sequence is the sum of its terms. We survey known results for a number of natural variants of unimodal sequences, including Auluck's generalized…
A way to construct and classify the three dimensional polynomially deformed algebras is given and the irreducible representations is presented. for the quadratic algebras 4 different algebras are obtained and for cubic algebras 12 different…
Algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials, including the solution of linear and nonlinear periodic ordinary…
By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.
An arithmetic formula is an expression involving only the constant $1$, and the binary operations of addition and multiplication, with multiplication by $1$ not allowed. We obtain an asymptotic formula for the number of arithmetic formulas…
These notes are designed to offer some (perhaps new) codicils to related work, a list of problems and conjectures seeking (preferably) combinatorial proofs. The main items are Eulerian polynomials and hook/contents of Young diagram, mostly…
We study the ring of arithmetical functions with unitary convolution, giving an isomorphism to a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett between the ring of arithmetical…
This is a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are…
Special functions have always played a central role in physics and in mathematics, arising as solutions of particular differential equations, or integrals, during the study of particular important physical models and theories in Quantum…