Related papers: Explicit Rational Solution of the KZ Equation (exa…
An irreducible quintic equation is solvable by radicals if and only if its Galois group is solvable. In this work, we provide necessary and sufficient conditions for solvability, expressed in terms of invariants of the quintic.
In 1997 we proved that if $n$ is of the form $$ 4k, \quad 8k-1\quad {\rm or} \quad 2^{2m+1}(2k-1)+3, $$ where $k,m\in \mathbb N,$ then there are no positive rational numbers $x,y,z$ satisfying $$ xyz = 1, \quad x+y+z = n. $$ Recently, N. X.…
Motivated by the introduction of the Zakharov-Kuznetsov equation as a higher dimensional generalization of the Korteweg-de Vries equation, in this paper we introduce the modified Zakharov-Kuznetsov (mZK) equation as a 2-dimensional…
In this article, we derive multiple polylogarithms from multiple zeta values by using a recursive Riemann-Hilbert problem of additive type. Furthermore we show that this Riemann-Hilbert problem is regarded as an inverse problem for the…
An explicit analytic solution to the nonlinear differential equation d^k y (--) ^n = y^l dx^kk is obtained for arbitrary integer values of k, l and n.
A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical…
We give a method to determine an explicit solution to a system of two inhomogeneous linear recursive sequences of higher order. Our approach can be used efficiently in solving certain combinatorial problems. We finish the paper by…
We introduce the notion of a regular quadratic equation and a regular NTQ system over a free group. We prove the results that can be described as Implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ…
We give description of rational solutions of polynomial-equations.
We find all polynomials f,g,h over a field K such that g and h are linear and f(g(x))=h(f(x)). We also solve the same problem for rational functions f,g,h, in case the field K is algebraically closed.
We study systems of $n \geq 1$ discrete differential equations of order $k\geq1$ in one catalytic variable and provide a constructive and elementary proof of algebraicity of their solutions. This yields effective bounds and a systematic…
In this paper, a linear system of equations with crisp coefficients and fuzzy right-hand sides is investigated. All possible cases pertaining to the number of variables, n, and the number of equations, m, are dealt with. A solution is…
In this paper we strengthen the results of [SV] by presenting their derived version. Namely, we define a "derived Knizhnik - Zamolodchikov connection"\ and identify it with a "derived Gauss - Manin connection".
We consider a fuzzy linear system with crisp coefficient matrix and with an arbitrary fuzzy number in parametric form on the right-hand side. It is known that the well-known existence and uniqueness theorem of a strong fuzzy solution is…
In this note we discuss the approach which was given by Wazwaz for the proof of the complete integrability to the system of nonlinear differential equations. We show that his method presented in [Wazwaz A.M. Completely integrable coupled…
We prove the local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^s(\mathbb{R}^2)$, for $s\in [1,2]$, on the background of an $L^\infty(\mathbb{R}^3)$-function $\Psi(t,x,y)$, with $\Psi(t,x,y)$ satisfying some…
We derive the generalization of the Knizhnik-Zamolodchikov equation (KZE) associated with the Ding-Iohara-Miki (DIM) algebra U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1). We demonstrate that certain refined topological string amplitudes…
We provide an alternative proof that the finite rational linear combination of radicals, under certain constraint, are linearly independent over $\mathbb{Q}$.
In this paper, we study the long time behavior of solutions of Klein-Gordon-Zakharov system. We show that there exists a solution with special characteristics, which we shall refer to as a dipole solution, that is, there exists a solution…
Let $A$ be a real $n\times n$ matrix and $z,b\in \mathbb R^n$. The piecewise linear equation system $z-A\vert z\vert = b$ is called an \textit{absolute value equation}. We consider two solvers for this problem, one direct, one…