Related papers: Effective difference elimination and Nullstellensa…
The need to smoothly cover a computational domain of interest generically requires the adoption of several grids. To solve the problem of interest under this grid-structure one must ensure the suitable transfer of information among the…
Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new…
We study the interplay between geometry and partial differential equations. We show how the fundamental ideas we use require the ability to correctly calculate the dimensions of spaces associated to the varieties of zeros of the symbols of…
The authors show that the round-off error can break the consistency which is the premise of using the difference equation to replace the original differential equations. We therefore proposed a theoretical approach to investigate this…
A Nullstellensatz is a theorem providing information on polynomials that vanish on a certain set: David Hilbert's Nullstellensatz (1893) is a cornerstone of algebraic geometry, and Noga Alon's Combinatorial Nullstellensatz (1999) is a…
The construction of the general solution sequence of row-finite linear systems is accomplished by implementing -ad infinitum- the Gauss-Jordan algorithm under a rightmost pivot elimination strategy. The algorithm generates a basis (finite…
We study approximation of non-autonomous linear differential equations with variable delay over infinite intervals. We use piecewise constant argument to obtain a corresponding discrete difference equation. The study of numerical…
We show that constant-depth Frege systems with counting axioms modulo $m$ polynomially simulate Nullstellensatz refutations modulo $m$. Central to this is a new definition of reducibility from formulas to systems of polynomials with the…
We prove a Nullstellensatz for the ring of polynomial functions in n non-commuting variables over Hamilton's ring of real quaternions. We also characterize the generalized polynomial identities in n variables which hold over the…
In this paper, we revisit the problem of classifying real algebraic and semialgebraic sets by their topological types, focusing on establishing the effectiveness of bounds rather than deriving new quantitative estimates. Building on Hardt's…
In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretisation method. Gradient schemes are defined for the optimality…
We consider a class of systems of difference equations defined on an elementary quadrilateral of the ${\mathbb{Z}}^2$ lattice, define their eliminable and dynamical variables, and demonstrate their use. Using the existence of infinite…
The notions of equivalence and strict equivalence for order one differential equations are introduced. The more explicit notion of strict equivalence is applied to examples and questions concerning autonomous equations and equations having…
We prove an effective integrability criterion for differential-algebraic Pfaffian systems leading to a decision method of consistency with a triple exponential complexity bound. As a byproduct, we obtain an upper bound for the order of…
We develop numerical algorithms to approximate positive solutions of elliptic boundary value problems with superlinear subcritical nonlinearity on the boundary of the form $-\Delta u + u = 0$ in $\Omega$ with $\frac{\partial u}{\partial…
We consider a system of integer polynomials of the same degree with non-singular local zeros and in many variables. Generalising the work of Birch (1962) we find quantitative asymptotics (in terms of the maximum of the absolute value of the…
We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose…
In this work we derive equivalence relations between mimetic finite difference schemes on simplicial grids and modified N\'ed\'elec-Raviart-Thomas finite element methods for model problems in $\mathbf{H}(\operatorname{\mathbf{curl}})$ and…
We obtain unconditional, effective number-field analogues of the three Mertens' theorems, all with explicit constants and valid for $x\geq 2$. Our error terms are explicitly bounded in terms of the degree and discriminant of the number…
In analogy with the Riemann zeta function at positive integers, for each finite field F_p^r with fixed characteristic p we consider Carlitz zeta values zeta_r(n) at positive integers n. Our theorem asserts that among the zeta values in…