Related papers: On local symbolic approximation and resolution of …
We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold $M$. We give several applications of this theory, concerning: 1) differentiability and geometrical properties of the…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
Imposing some conditions on derivatives of the known functions, using the Fiber Contraction Theorem we prove the existence of $C^1$ solutions of a class of iterative functional equations which involves iterates of the unknown functions and…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
We propose a physical analogy between finding the solution of an ordinary differential equation (ODE) and a $N$ particle problem in statistical mechanics. It uses the fact that the solution of an ODE is equivalent to obtain the minimum of a…
In this paper, we study a generalized finite element method for solving second-order elliptic partial differential equations with rough coefficients. The method uses local approximation spaces computed by solving eigenvalue problems on…
We investigate the first order implicit linear difference equation over residue class rings modulo m. We prove an existence criterion and establish the amount of solutions for this equation. We obtain analogous results for the initial…
We prove a duality theorem the computation of certain Bellman functions is usually based on. As a byproduct, we obtain sharp results about the norms of monotonic rearrangements. The main novelty of our approach is a special class of…
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…
We adapt the method of solution regions to prove new existence and localization results for systems of discontinuous differential equations. Some assumptions concerning the definition of a solution region are relaxed and thus our results…
This paper deals with fractional differential equations, with dependence on a Caputo fractional derivative of real order. The goal is to show, based on concrete examples and experimental data from several experiments, that fractional…
This article establishes a complete approximate axiomatization for the real-closed field $\mathbb{R}$ expanded with all differentially-defined functions, including special functions such as $\sin(x), \cos(x), e^x, \dots$. Every true…
We develop a globalized Proximal Newton method for composite and possibly non-convex minimization problems in Hilbert spaces. Additionally, we impose less restrictive assumptions on the composite objective functional considering…
The purpose of this paper is to give an easy to understand with step-by-step explanation to allow interested people to fully appreciate the power of natural deduction for first-order logic. Natural deduction as a proof system can be used to…
In this Letter, we introduce a notion of local fraction for experiments taking place against arbitrary static causal backgrounds -- greatly generalising previous results on no-signalling scenarios -- and we explicitly formulate a linear…
We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: first, we prove that…
Recent research has shown that piecewise smooth (PS) functions can be approximated by piecewise linear functions with second order error in the distance to a given reference point. A semismooth Newton type algorithm based on successive…
This work presents a brief discussion and a plan towards the analytical solving of Partial Differential Equations (PDEs) using symbolic computing, as well as an implementation of part of this plan as the PDEtools software-package of…
We consider the problem of explaining the predictions of an arbitrary blackbox model $f$: given query access to $f$ and an instance $x$, output a small set of $x$'s features that in conjunction essentially determines $f(x)$. We design an…
We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations…