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Properties of solutions of generic hyperbolic systems with multiple characteristics with diagonalizable principal part are investigated. Solutions are represented as a Picard series with terms in the form of iterated Fourier integral…
We present a study of complex singularities of a two-parameter family of solutions for the two-dimensional Euler equation with periodic boundary conditions and initial conditions F(p) cos p z + F(q) cos q z in the short-time asymptotic…
It is shown that if the system of the Euler equations has a special global in time smooth solution with the linear profile of velocity, then another solutions with Cauchy data, close in the Sobolev norm to the initial data of the given…
The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable…
Given an integer linear recurrence sequence $\langle X_n \rangle_n$, the Skolem Problem asks to determine whether there is a natural number $n$ such that $X_n = 0$. Recent work by Lipton, Luca, Nieuwveld, Ouaknine, Purser, and Worrell…
We study the local behavior of weak solutions, with possible singularities, of nonlocal nonlinear equations. We first prove that sets of capacity zero are removable for weak solutions under certain integrability conditions. We then…
This paper mainly discuss the regularity behavior of the hyperbolic magnetic Schroedinger equation with singular coefficients near the origin. We apply the techniques from the microlocal analysis to explore the upper bound of loss of…
This paper first proves the existence, uniqueness and regularity of the solution to a class of linear backward parabolic equations on Carnot tori, namely the periodic linear parabolic equation on Carnot groups. Such groups are…
In this paper, we develop a new deflation technique for refining or verifying the isolated singular zeros of polynomial systems. Starting from a polynomial system with an isolated singular zero, by computing the derivatives of the input…
This article proves the existence and regularity of weak solutions for a class of mixed local-nonlocal problems with singular nonlinearities. We examine both the purely singular problem and perturbed singular problems. A central…
Differential equations with the Painlev\'e property have been studied extensively due to their appearance in many branches of mathematics and their applicability in physics. Although a modern, differential algebraic treatment of the order…
We discuss the effective computation of geometric singularities of implicit ordinary differential equations over the real numbers using methods from logic. Via the Vessiot theory of differential equations, geometric singularities can be…
The regularized Maxwell theory is a recently discovered theory of non-linear electrodynamics that admits many important gravitating solutions within the Einstein theory. Namely, it was originally derived as the unique non-linear…
This paper establishes a regularity theory for the magnetohydrodynamics (MHD) equations with external forces through scaling analysis. Inspired by the existing methodology, we utilize linearized approximations and the monotonicity property…
Regular resolution is a refinement of the resolution proof system requiring that no variable be resolved on more than once along any path in the proof. It is known that there exist sequences of formulas that require exponential-size proofs…
A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre…
Metric regularity is among the central concepts of nonlinear and variational analysis, constrained optimization, and their numerous applications. However, metric regularity can be elusive for some important ill-posed classes of problems…
Efficient characteristic set methods for computing solutions of polynomial equation systems in a finite field are proposed. The concept of proper triangular sets is introduced and an explicit formula for the number of solutions of a proper…
We study a class of elliptic problems, involving a $k$-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the…
The aim of this article is to introduce standard bases of ideals in polynomial rings with respect to a class of orderings which are not necessarily semigroup orderings. Our approach generalises the concept of standard bases with respect to…