Related papers: Solution Theory for Systems of Bilinear Equations
Reparable systems are systems that are characterized by their ability to undergo maintenance actions when failures occur. These systems are often described by transport equations, all coupled through an integro-differential equation. In…
We develop a systematic way to solve linear equations involving tensors of arbitrary rank. We start off with the case of a rank $3$ tensor, which appears in many applications, and after finding the condition for a unique solution we derive…
In this article we determine several theorems and methods for solving linear congruences and systems of linear congruences, and we find the number of distinct solutions. Many examples of solving congruences are given.
We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…
A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…
Linear systems are the bedrock of virtually all numerical computation. Machine learning poses specific challenges for the solution of such systems due to their scale, characteristic structure, stochasticity and the central role of…
A quadratic dynamical system with practical applications is taken into considered. This system is transformed into a new bilinear system with Hadamard products by means of the implicit matrix structure. The corresponding quadratic bilinear…
Some superlinear fourth order elliptic equations are considered. Ground states are proved to exist and to concentrate at a point in the limit. The proof relies on variational methods, where the existence and concentration of nontrivial…
We give an explicit description of the basic solutions of max-linear systems with two inequalities.
Bilinear dynamical systems are ubiquitous in many different domains and they can also be used to approximate more general control-affine systems. This motivates the problem of learning bilinear systems from a single trajectory of the…
The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.
Lie's linearizability criteria for scalar second-order ordinary differential equations had been extended to systems of second-order ordinary differential equations by using geometric methods. These methods not only yield the linearizing…
In this paper we will give a unified proof of several results on the sovability of systems of certain equations over finite fields, which were recently obtained by Fourier analytic methods. Roughly speaking, we show that almost all systems…
Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of…
It is shown that every scalar linear quadrilateral lattice equation lies within a family of similar equations, members of which are compatible between one another on a higher dimensional lattice. There turn out to be two such families, a…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.
It is known a method for converting a system of Boolean polynomial equations to a single Boolean polynomial equation with less variables. In this paper, we show a formula for systems of Boolean polynomial equations which is based on the…
The goal of this paper is to provide computational tools able to find a solution of a system of polynomial inequalities. The set of inequalities is reformulated as a system of polynomial equations. Three different methods, two of which…
We tersely review a recently introduced technique to identify systems of two nonlinearly-coupled Ordinary Di{\S}erential Equations (ODEs) solvable by algebraic operations; and we report some specifc examples of this kind, namely systems of…