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In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
This paper aims at reviewing and analysing the method of reflections. The latter is an iterative procedure designed to linear boundary value problems set in multiply connected domains. Being based on a decomposition of the domain boundary,…
The aim of this paper is twofold. First, we introduce a new class of linearizations, based on the generalization of a construction used in polynomial algebra to find the zeros of a system of (scalar) polynomial equations. We show that one…
Using the character expansion method, we generalize several well-known integrals over the unitary group to the case where general complex matrices appear in the integrand. These integrals are of interest in the theory of random matrices and…
We introduce a method for finding general solutions of third-order nonlinear differential equations by extending the modified Prelle-Singer method. We describe a procedure to deduce all the integrals of motion associated with the given…
We propose a new method to generate a sequence of random numbers with long-range power-law correlations that overcomes known difficulties associated with large systems. The new method presents an improvement on the commonly-used methods. We…
The systems of nonlinear Volterra integral equations of the first kind with jump discontinuous kernels are studied. The iterative numerical method for such nonlinear systems is proposed. Proposed method employs the modified…
Given a set of integers with no three in arithmetic progression, we construct a Stanley sequence by adding integers greedily so that no arithmetic progression is formed. This paper offers two main contributions to the theory of Stanley…
We present a framework for the construction of linearizations for scalar and matrix polynomials based on dual bases which, in the case of orthogonal polynomials, can be described by the associated recurrence relations. The framework…
Let $\{x_1, x_2, ..., x_n\}$ be a vector of real numbers. An integer relation algorithm is a computational scheme to find the $n$ integers $a_k$, if they exist, such that $a_1 x_1 + a_2 x_2 + ... + a_n x_n= 0$. In the past few years,…
We will derive a function that eliminates any sequence of equidistant numbers from the integer numbers, then we will derive its inverse. Then we will use the Sequence elimination function to eliminate the multiples of the prime numbers from…
A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers.
A simple iteration methodology for the solution of a set of a linear algebraic equations is presented. The explanation of this method is based on a pure geometrical interpretation and pictorial representation. Convergence using this method…
In this paper, we provide properties and applications of some special integer sequences. We generalize and give some properties of Pisano period. Moreover, we provide a new application in Cryptography and applications of some quaternion…
We present here some results of applying the Cayley-Dickson process to certain alternative algebras (notably built upon Galois fields and congruence rings), in a manner which might yield new building blocks for cryptographic systems. We…
In this work, we present an algorithm for the diagonalization of the Integration-by-Parts (IBP) equations. Diagonalized IBP equations are indispensable for reducing loop integrals with high numerator powers to master integrals and for…
A new method for solving systems of linear algebraic equations of a special type arising in solving problems of image reconstruction has been proposed. This method, due to a certain symmetry of the matrix and the choice of the voxel…
We generalize an operation described by Sloane on the binary representation of an integer to other bases, thus finding several new sequences.
We provide an algorithm for computing semi-Fourier sequences for expressions constructed from arithmetic operations, exponentiations and integrations. The semi-Fourier sequence is a relaxed version of Fourier sequence for polynomials…
This paper discusses new analytic algorithms and software for the enumeration of all integer flows inside a network. Concrete applications abound in graph theory \cite{Jaeger}, representation theory \cite{kirillov}, and statistics…