Related papers: Additive transform of an arithmetic function : Par…
We prove an isomorphism between the finite domain from 1 up to the product of the first n primes and the new defined set of prime modular numbers. This definition provides some insights about relative prime numbers. We provide an inverse…
The arithmetic partial derivative (with respect to a prime $p$) is a function from the set of integers that sends $p$ to 1 and satisfies the Leibniz rule. In this paper, we prove that the $p$-adic valuation of the sequence of higher order…
The purpose of this paper is to introduce the notion of a generalized derivation which derivates a prescribed family of smooth vector-valued functions of several variables. The basic calculus rules are established and then a result derived…
We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make…
In this paper we will give a scheme-theoretic discussion on the unramified extensions of an arithmetic function field in several variables. The notion of unramified discussed here is parallel to that in algebraic number theory and for the…
Vector-valued discrete Fourier transforms (DFTs) and ambiguity functions are defined. The motivation for the definitions is to provide realistic modeling of multi-sensor environments in which a useful time-frequency analysis is essential.…
Image reconstruction is likely the most predominant auxiliary task for image classification, but we would like to think twice about this convention. In this paper, we investigated "approximating the Fourier Transform of the input image" as…
Fixed-order perturbative calculations for differential cross sections can suffer from non-physical artifacts: they can be non-positive, non-normalizable, and non-finite, none of which occur in experimental measurements. We propose a…
Expanding upon recent work, a new class of $A$-functions is introduced that can be viewed as an appropriate generalization of the class of regular $A$-functions, the class of structured $A$-functions, and the class of perfect $A$-functions.…
A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of…
The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature.…
The logarithmic model offers new tools for image processing. An efficient method for image enhancement is to use an affine transformation with the logarithmic operations: addition and scalar multiplication. We define some criteria for…
Formula for the nth prime using elementary arithmetical functions based in a previous formula changing the characteristic function of prime numbers.
In this paper we introduce a family of rational approximations of the reciprocal of a $\phi$-function involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The…
In the context of the stream calculus, we present an Implicit Function Theorem (IFT) for polynomial systems, and discuss its relations with the classical IFT from calculus. In particular, we demonstrate the advantages of the stream IFT from…
We consider the variance of sums of arithmetic functions over random short intervals in the function field setting. Based on the analogy between factorizations of random elements of $\mathbb{F}_q[T]$ into primes and the factorizations of…
We call an objective function or algorithm symmetric with respect to an input if after swapping two parts of the input in any algorithm, the solution of the algorithm and the output remain the same. More formally, for a permutation $\pi$ of…
Our paper introduces a novel method for calculating the inverse $\mathcal{Z}$-transform of rational functions. Unlike some existing approaches that rely on partial fraction expansion and involve dividing by $z$, our method allows for the…
We define $\overline{\psi}$ to be the multiplicative arithemtic function that satisfies \[\overline{\psi}(p^{\alpha})=\begin{cases} p^{\alpha-1}(p+1), & \mbox{if } p\neq 2; \\ p^{\alpha-1}, & \mbox{if } p=2 \end{cases}\] for all primes $p$…
Linear algebra computations are foundational for neural networks and machine learning, often handled through arrays. While many functional programming languages feature lists and recursion, arrays in linear algebra demand constant-time…