Related papers: Basis-free Solution to General Linear Quaternionic…
We introduce B-splines on the line of quaternionic order $B_q$ ($q$ in the algebra of quaternions) for the purposes of multi-channel signal and image analysis. The functions $B_q$ are defined first by their Fourier transforms, then as the…
Some idea, which leads to a non-trivial solution of the quantum four-simplex equation, is exposed in this paper. We call this idea "pentagonal algebra". Few examples of the realisation of this idea are given here, and thus few examples of…
Within the framework of the theory of quaternion column-row determinants and using determinantal representations of the Moore-Penrose inverse previously obtained by the author, we get explicit determinantal representation formulas of…
The functional equation f(p(z))=g(q(z)) is studied, where p,q are polynomials and f,g are trancendental meromorphic functions in C. We find all the pairs p,q for which there exist nonconstant f,g satisfying our equation and there exist no…
It is a classical problem in algebraic number theory to decide if a number field admits power integral bases and further to calculate all generators of power integral bases. This problem is especially delicate to consider in an infinite…
The problem of finding a nonzero solution of a linear recurrence $Ly = 0$ with polynomial coefficients where $y$ has the form of a definite hypergeometric sum, related to the Inverse Creative Telescoping Problem of [14][Sec. 8], has now…
Baxter's TQ-equation is solved for the six-vertex model using the representation theory of quantum groups at roots of unity. A novel simplified construction of the Q-operator is given depending on a new free parameter. Specializing this…
The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let $n\geq 2$ be an arbitrarily fixed integer, let further $X$ and $Y$ be linear spaces over the field $\mathbb{K}$ and let…
A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. It is described by a system of four quadratic equations with respect to six…
We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we…
The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.
Let $A,B,C,D$ be rational numbers such that $ABC \neq 0$, and let $n_1>n_2>n_3>0$ be positive integers. We solve the equation $$ Ax^{n_1}+Bx^{n_2}+Cx^{n_3}+D = f(g(x)),$$ in $f,g \in \mathbb{Q}[x]$. In sequel we use Bilu-Tichy method to…
A proof of Lagrange's and Jacobi's four-square theorem due to Hurwitz utilizes orders in a quaternion algebra over the rationals. Seeking a generalization of this technique to orders over number fields, we identify two key components: an…
We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n-dimensional quaternionic vector spaces. For quaternionic linear operators the eigenvalue spectrum consists of n complex values. For…
The inversion problem for rational B\'ezier curves is addressed by using resultant matrices for polynomials expressed in the Bernstein basis. The aim of the work is not to construct an inversion formula but finding the corresponding value…
In recent years, expansion-based techniques have been shown to be very powerful in theory and practice for solving quantified Boolean formulas (QBF), the extension of propositional formulas with existential and universal quantifiers over…
Starting from the solution to Bring's equation the root ambiguity is removed from the solution to the quintic equation. This gives the five complex roots of the quintic equation as indicated by Gauss's Fundamental Theorem of Algebra.r
In this paper, we propose a new method to obtain a solution to a single-parameter Bring quintic equation of the form, $x^{5}+x=a$, where $a$ is real. The method transforms the given quintic equation to an infinite but convergent series…
A quaternionic partial differential equation is shown to be a generalisation of the Riccati ordinary differential equation and its relationship with the Schrodinger equation is established. Various approaches to the problem of finding…
Let f be a square-free polynomial in Fq[t][x] where Fq is a field of q elements. We view f as a polynomial in the variable x with coefficients in the ring Fq[t]. We study squarefree values of f in sparse subsets of Fq[t] which are given by…