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In this paper, the result of applying iterative univariate resultant constructions to multivariate polynomials is analyzed. We consider the input polynomials as generic polynomials of a given degree and exhibit explicit decompositions into…
We construct stable periodic solutions for a simple form nonlinear delay differential equation (DDE) with a periodic coefficient. The equation involves one underlying nonlinearity with the multiplicative periodic coefficient. The well-known…
In this paper, we study non-linear differential equations associated with Legendre polynomials and their applications. From our study of non- linear differential equations, we derive some new and explicit identities for Legendre…
A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
Motion polynomials are a specific type of polynomial over a Clifford algebra that can conveniently describe rational motions. There exists an algorithm for the factorization of motion polynomials that works in generic cases. It hinges on…
This article is concerned with the existence and uniqueness of solutions to some fractional order boundary value problems. Our results are based on some fixed point theorems. For the applicability of our results, we provide an example.
We describe a way of solving a partial differential equation using the differential invariants of its point symmetries. By first solving its quotient PDE, which is given by the differential syzygies in the algebra of differential…
We represent a version of multidimensional quasilinear partial differential equation (PDE) together with large manifold of particular solutions given in an integral form. The dimensionality of constructed PDE can be arbitrary. We call it…
Certain computable polynomials are described whose leading coefficients are equal to multiplicities in the tensor product decomposition for representations of a Lie algebra of ADE type.
The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering.…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
Solutions of nonlinear multi-component Euler-Monge partial differential equations are constructed in n spatial dimensions by dimension-doubling, a method that completely linearizes the problem. Nonlocal structures are an essential feature…
A polynomial is expansive if all of its roots lie outside the unit circle. We define some special determinants involving the coefficients of a real polynomial and formulate necessary and sufficient conditions for expansivity using these…
In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially…
In this paper, we consider modular local polynomials. These functions satisfy modularity while they are locally defined as polynomials outside of an exceptional set. We prove an inequality for the dimension of the space of such forms when…
We prove an existence and uniqueness result for Neumann boundary problem of a parabolic partial differential equation (PDE for short) with a singular nonlinear divergence term which can only be understood in a weak sense. A probabilistic…
The last decades saw growing interest across multiple disciplines in nonlinear phenomena described by partial differential equations (PDE). Integrability of such equations is tightly related with the Painleve property - solutions being free…
For a prime $p$ and an integer $x$, the $p$-adic valuation of $x$ is denoted by $\nu_{p}(x)$. For a polynomial $Q$ with integer coefficients, the sequence of valuations $\nu_{p}(Q(n))$ is shown to be either periodic or unbounded. The first…