相关论文: On the differential geometry of numerical schemes …

We prove an implicit function theorem for Keller C^k_c-maps from arbitrary real or complex topological vector spaces to Frechet spaces, imposing only a certain metric estimate on the partial differentials. As a tool, we show the…

This paper introduces a notion of differential forms on closed, potentially fractal, subsets of Euclidean space by defining pointwise cotangent spaces using the restriction of $C^1$ functions to this set. Aspects of cohomology are…

We introduce a neural implicit framework that exploits the differentiable properties of neural networks and the discrete geometry of point-sampled surfaces to approximate them as the level sets of neural implicit functions. To train a…

Based upon elements of the modern Pseudoanalytic Function Theory, we analyse a new method for numerically approaching the solution of the Dirichlet boundary value problem, corresponding to the two-dimensional Electrical Impedance Equation.…

This paper is concerned with the Cauchy-Dirichlet problem for a doubly nonlinear parabolic equation involving variable exponents and provides some theorems on existence and regularity of strong solutions. In the proof of these results, we…

Sawin recently gave an axiomatic characterization of multiple Dirichlet series over the function field $\mathbb{F}_{q}(T)$ and proved their existence by exhibiting the coefficients as trace functions of specific perverse sheaves. However,…

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…

We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices,…

An equation containing a fractional power of an elliptic operator of second order is studied for Dirichlet boundary conditions. Finite difference approximations in space are employed. The proposed numerical algorithm is based on solving an…

Towards identifying the number of minimal surfaces sharing the same boundary from the geometry of the boundary, we propose a numerical scheme with high speed and high accuracy. Our numerical scheme is based on the method of fundamental…

This article proposes a new discrete framework for approximating solutions to shape optimization problems under convexity constraints. The numerical method, based on the support function or the gauge function, is guaranteed to generate…

We develop a framework for studying variational problems in Banach spaces with respect to gradient relations, which encompasses many of the notions of generalized gradients that appear in the literature. We stress the fact that our approach…

We prove an implicit function theorem for functions on infinite-dimensional Banach manifolds, invariant under the (local) action of a finite dimensional Lie group. Motivated by some geometric variational problems, we consider group actions…

We study an infinite system of ordinary differential equations that models the evolution of coagulating and fragmenting clusters, which we assume to be composed of identical units. Under very mild assumptions on the coefficients we prove…

In this paper, we develop a universal, conceptually simple and systematic method to prove well-posedness to Cauchy problems for weak solutions of parabolic equations with non-smooth, time-dependent, elliptic part having a variational…

The existence of solutions to Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients is considered in a space of integrable functions. First, we consider the existence and uniqueness of…

We construct a sequence that converges to a solution of the Cauchy problem for a singularly perturbed linear inhomogeneous differential equation of an arbitrary order. This sequence is also an asymptotic sequence in the following sense: the…

The present paper is a continuation of our work on curved finitary spacetime sheaves of incidence algebras and treats the latter along Cech cohomological lines. In particular, we entertain the possibility of constructing a non-trivial de…

We introduce a differential structure for the space of weakly geometric p rough paths over a Banach space V for 2<p<3. We begin by considering a certain natural family of smooth rough paths and differentiating in the truncated tensor…