Related papers: On the Uniqueness Problem for Quadrature Domains
We study the influence of geometry on semilinear elliptic equations of bistable or nonlinear-field type in unbounded domains. We discover a surprising dichotomy between epigraphs that are bounded from below and those that contain a cone of…
We study uniqueness of solutions to degenerate parabolic problems, posed in bounded domains, where no boundary conditions are imposed. Under suitable assumptions on the operator, uniqueness is obtained for solutions that satisfy an…
The paper deals with a construction of a separating system of rational invariants for finite dimensional generic algebras. In the process of dealing an approach to a rough classification of finite dimensional algebras is offered by…
We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes,…
We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial…
We first show that the tilings of a general domain form a lattice which we then undertake to decompose and generate without any redundance. To this end, we study extensively the relatively simple case of hexagons and their deformations. We…
A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly…
In this work, for an equation of high even order with a fractional derivative in the sense of Caputo, a problem is studied in a rectangular domain with conjugation conditions. A criterion for the uniqueness of a solution is given. The…
This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…
We develop a systematic procedure for constructing quantum many-body problems whose spectrum can be partially or totally computed by purely algebraic means. The exactly-solvable models include rational and hyperbolic potentials related to…
In this work, we present a systematic approach to investigate the existence, multiplicity, and local gradient regularity of solutions for nonlocal quasilinear equations with local gradient degeneracy. Our method involves an interactive…
The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…
Using exhaustion method and finite differences a new method to solve system of partial differential equations and is presented. This method allows design algorithm to solve linear and nonlinear systems in irregular domains. Applying this…
Quantum computing has demonstrated potential for solving complex optimization problems; however, its application to spatial regionalization remains underexplored. Spatial contiguity, a fundamental constraint requiring spatial entities to…
Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of…
We prove a linear upper bound for the number of singular points on the boundary of a quadrature domain, improving a previously known quadratic bound due to Gustafsson \cite{Gus88}. This linear upper bound on the number of boundary double…
In this article, we discuss the numerical solution of diffusion equations on random surfaces within the isogeometric framework. We describe in detail, how diffusion problems on random surfaces can be modelled and how quantities of interest…
This paper explores and proves the one-seventh area triangle using a purely algebraic approach as opposed to a geometric one. A triangle set purely in the complex plane is used so that we can utilise features of the complex number system to…
In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which…
A multi-cube method is developed for solving systems of elliptic and hyperbolic partial differential equations numerically on manifolds with arbitrary spatial topologies. It is shown that any three-dimensional manifold can be represented as…