Related papers: Quaternion quadratic equations in characteristic 2
Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\dots+x_n^{2^m}=0$ over a finite field of characteristic $p\equiv\pm 3\pmod{8}$. All of the…
In this paper, we prove a conjecture which was presented in a recent paper [Linear Algebra Appl. 2016; 496: 549--593]. We derive some practical necessary and sufficient conditions for the existence of a solution to a system of coupled…
The main purpose of this paper is solve polynomial equations that are satisfied by (generalized) polynomials. More exactly, we deal with the following problem: let $\mathbb{F}$ be a field with $\mathrm{char}(\mathbb{F})=0$ and $P\in…
The aim of this study is to investigate the precise form of finite-order entire solutions to the following system of Fermat-type partial differential-difference equations: \beas \begin{cases} \left(\frac{\partial f_1\left(z_1, z_2, \ldots,…
{\small In this paper, we find a class of division quaternion algebras over the field }$\mathbb{Q}\left( i\right) ${\small \ and a class of division symbol algebras over a cyclotomic field.}
In this paper we give a novel solution to a classical completion problem for square matrices. This problem was studied by many authors through time, and it is completely solved in [2, 3]. In this paper we relate this classical problem to a…
We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can…
With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[(-\Delta+m^2)^\sigma u+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $0<\sigma<1$, $V$ is a…
We present some simple relations between the absolute minimizers of the functional $||Du+Xu||$, where $X$ is a vector field on $R^n$, and ground state solutions to the (non-relativistic) Schroedinger equation. This article is a byproduct of…
The anomaly cancellation equations for the $U(1)$ gauge group can be written as a cubic equation in $n-1$ integer variables, where $n$ is the number of Weyl fermions carrying the $U(1)$ charge. We solve this Diophantine cubic equation by…
We investigate an economical explanation for the $(g-2)_\mu$ anomaly with a neutral vector boson from a spontaneously broken $\mathrm{U}(1)_X$ gauge symmetry. The Standard Model fermion content is minimally extended by 3 right-handed…
In the present paper we give the general solution of the functional equation $$ (f(x)+g(y)+h(z))^2 = F(x)+G(y)+H(z), x+y+z=0 $$ which is related to the exact factorized ground-state wave function for the quantum one-dimensional problem of…
In this paper, we derive some necessary and sufficient solvability conditions for some systems of one sided coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. We also give the…
We present a renormalizable fermion mass model based on the symmetry $Q_4$ that accommodates all fermion masses and mixing angles in both the quark and lepton sectors. It requires the presence of only four SU(2) doublet scalar fields…
We consider a new class of quaternionic mappings, associated with the spatial partial differential equations. We describe all mappings from this class using four analytic functions of the complex variable.
We present new techniques for finding anomaly-free sets of fermions. Although the anomaly cancellation conditions typically include cubic equations with integer variables that cannot be solved in general, we prove by construction that any…
In our work we study the equations of the form $aX^4+bX^2 Y^2+cY^4=dZ^2$ over Gaussian integers by a method of the resolvents. We study as a new equations $X^4+6X^2 Y^2+Y^4=Z^2$ (Mordell's equation over $\mathbb{Z}[i]$),…
The well-known Heun equation has the form: Q(z)S''(z)+P(z)S'(z)+V(z)S(z)=0 where Q(z) is a cubic complex polynomial, P(z) and V(z) are polynomials of degrees at most 2 and 1 resp. One of the classical problems about the Heun equation is for…
We present a new classification of Clifford algebra elements. Our classification is based on the notion of quaternion type. Using this classification we develop a method for analyzing of commutators and anticommutators of Clifford algebra…
This study establishes consistency conditions and a general solution for a coupled system that consists of five two-sided Sylvester-like tensor equations in ten quaternion variables throughout the Einstein tensor product. Certain specific…