Related papers: Farey determinants matrix
We consider Hadamard fractional derivatives and integrals of variable fractional order. A new type of fractional operator, which we call the Hadamard-Marchaud fractional derivative, is also considered. The objective is to represent these…
A procedure to obtain differentiation matrices is extended straightforwardly to yield new differentiation matrices useful to obtain derivatives of complex rational functions. Such matrices can be used to obtain numerical solutions of some…
Fracterms are introduced as a proxy for fractions. A precise definition of fracterms is formulated and on that basis reasonably precise definitions of various classes of fracterms are given. In the context of the meadow of rational numbers…
Higher order derivatives of functions are structured high dimensional objects which lend themselves to many alternative representations, with the most popular being multi-index, matrix and tensor representations. The choice between them…
We define quantum determinants in Quantum Matrix Algebras, related to couples of compatible braidings following the scheme from [G]. We establish relations between these determinants and the so-called column-(row-)determinants, often used…
Matrix determinants play an important role in data analysis, in particular when Gaussian processes are involved. Due to currently exploding data volumes, linear operations - matrices - acting on the data are often not accessible directly…
Due to their rich algebraic structures and various applications, circulant matrices have been of interest and continuously studied. In this paper, the notions of Binomial-related matrices have been introduced. Such matrices are circulant…
Rationals are known to form interesting and computationally rich structures, such as Farey sequences and infinite trees. Little attention is being paid to more general, systematic exposition of the basic properties of fractions as a set.…
In the present article, a new method for the evaluation of fractional derivatives of arbitrary real order is proposed. Numerous but inequivalent formulations have been given in the past. Some of them exhibit unsatisfactory properties such…
Recently it has been found that some special subsequences within a Farey sequence play a crucial role in determining the ranges of coupling constant for which quantum soliton states can exist for an integrable derivative nonlinear…
In this article, we study the pair correlation of Farey fractions by proving that the limiting pair correlation function of the sequence of Farey fractions with square-free denominators exists and provide an explicit formula for the…
An identity is proven that evaluates the determinant of a block tridiagonal matrix with (or without) corners as the determinant of the associated transfer matrix (or a submatrix of it).
Let $\cP$ be a system of $n$ linear nonhomogeneous ordinary differential polynomials in a set $U$ of $n-1$ differential indeterminates. Differential resultant formulas are presented to eliminate the differential indeterminates in $U$ from…
How could the Fourier and other transforms be naturally discovered if one didn't know how to postulate them? In the case of the Discrete Fourier Transform (DFT), we show how it arises naturally out of analysis of circulant matrices. In…
Just as knowing some roots of a polynomial allows one to factor it, a well-known result provides a factorization of any scalar differential operator given a set of linearly independent functions in its kernel. This note provides a…
We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors and in term of them we construct…
We associate with a matrix over an arbitrary field an infinite family of matrices whose sizes vary from one to infinity; their entries are traces of powers of the original matrix. We explicitly evaluate the determinants of matrices in our…
Circulant matrices play a central role in a recently proposed formulation of three-way data computations. In this setting, a three-way table corresponds to a matrix where each "scalar" is a vector of parameters defining a circulant. This…
Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new…
After analyzing the 4x4 determinant of a matrix, a shortcut was obtained to find such a determinant. Similarly to the Sarrus method for 2x2 or 3x3 determinants, the method consists of laying 19 columns of size 4 each and adding and…