Related papers: Discrete Polynomials and Discrete Holomorphic Appr…

We extend recent work of Gurel-Gurevich--Jerison--Nachmias (2020) and Bou-Rabee--Gwynne (2024) by showing that as the mesh of our lattice tends to $0$, we have a polynomial rate of convergence for the Dirichlet problem on orthodiagonal maps…

This paper investigates the universal approximation capabilities of Hamiltonian Deep Neural Networks (HDNNs) that arise from the discretization of Hamiltonian Neural Ordinary Differential Equations. Recently, it has been shown that HDNNs…

Let F be a holomorphic map whose components satisfy some polynomial relations. We present an algorithm for constructing Nash maps locally approximating F, whose components satisfy the same relations.

This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables.…

This paper develops a discrete theory of real Riemann surfaces based on quadrilateral cellular decompositions (quad-graphs) and a linear discretization of the Cauchy-Riemann equations. We construct a discrete analogue of an antiholomorphic…

To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite-dimensional space of trigonometric polynomials and construct a matrix representation of the…

In some fields such as Mathematics Mechanization, automated reasoning and Trustworthy Computing etc., exact results are needed. Symbolic computations are used to obtain the exact results. Symbolic computations are of high complexity. In…

Estimating the coefficient functionals on various classes of holomorphic functions traditionally forms an important field of geometric complex analysis and its mathematical and physical applications. These coefficients reflect fundamental…

We recall the theory of linear discrete Riemann surfaces and show how to use it in order to interpret a surface embedded in R^3 as a discrete Riemann surface and compute its basis of holomorphic forms on it. We present numerical examples,…

Singular and oscillatory functions feature in numerous applications. The high-accuracy approximation of such functions shall greatly help us develop high-order methods for solving applied mathematics problems. This paper demonstrates that…

We introduce polystar bodies: compact starshaped sets whose gauge or radial functions are expressible by polynomials, enabling tractable computations, such as that of intersection bodies. We prove that polystar bodies are uniformly dense in…

We prove that univalent harmonic mappings can be approximated by univalent Fourier series of step functions.

Inspired by the Weierstrass representation of smooth affine minimal surfaces with indefinite metric, we propose a constructive process producing a large class of discrete surfaces that we call discrete affine minimal surfaces. We show that…

We define holonomic measures to be certain analogues of varifolds that keep track of the local parameterization and orientation of the submanifold they represent. They are Borel measures on the direct sum of several copies of the tangent…

Submodular functions are a fundamental object of study in combinatorial optimization, economics, machine learning, etc. and exhibit a rich combinatorial structure. Many subclasses of submodular functions have also been well studied and…

We construct automorphisms of $\C^n$ which map certain discrete sequences one onto another with prescribed finite jet at each point, thus solving a general Mittag-Leffler interpolation problem for automorphisms. Under certain circumstances,…

It was recently conjectured that every component of a discrete-time rational dynamical system is a solution to an algebraic difference equation that is linear in its highest-shift term (a quasi-linear equation). We prove that the conjecture…

We investigate a way to approximate the maximum of a polynomial over a polytopal region by using Handelman's polynomial decomposition and continuous multivariate generating functions. The maximization problem is NP-hard, but our…

Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the…

The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on…