Related papers: Numerical Calculation of a Minimal Surface Using B…

We present sparse interpolation algorithms for recovering a polynomial with $\le B$ terms from $N$ evaluations at distinct values for the variable when $\le E$ of the evaluations can be erroneous. Our algorithms perform exact arithmetic in…

By $B=B(x^{(0)};R)$ we denote the Euclidean ball in ${\mathbb R}^n$ given by the inequality $\|x-x^{(0)}\|\leq R$. Here $x^{(0)}\in{\mathbb R}^n, R>0$, $\|x\|:=\left(\sum_{i=1}^n x_i^2\right)^{1/2}$. We mean by $C(B)$ the space of…

The Bj\"orling problem amounts to the construction of a minimal surface from a real-analytic curve with a given real-analytic normal vector field. We approximate that solution locally by discrete minimal surfaces as special discrete…

We find a system of two polynomial equations in two unknowns, whose solution allows to give an explicit expression of the conformal representation of a simply connected three sheeted compact Riemann surface onto the extended complex plane.…

We consider fractional Sobolev spaces $H^\theta$, $\theta\in (0,1)$, on 2D domains and $H^1$-conforming discretizations by globally continuous piecewise polynomials on a mesh consisting of shape-regular triangles and quadrilaterals. We…

Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation…

The least square solution of minimum norm of a rectangular linear system of equations can be found out iteratively by using matrix splittings. However, the convergence of such an iteration scheme arising out of a matrix splitting is…

Let $\Sigma^2 \subset M^3$ be a minimal surface of index 0 or 1. Assume that a neighborhood of $\Sigma$ can be foliated by constant mean curvature (cmc) hypersurfaces. We use min-max theory and the catenoid estimate to construct…

The challenge to measure exposures regularly forces financial institutions into a choice between an overwhelming computational burden or oversimplification of risk. To resolve this unsettling dilemma, we systematically investigate replacing…

Let $P_N(R)$ be the space of all real polynomials in $N$ variables with the usual inner product $<, >$ on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form…

We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation…

For three dimensional complete Riemannian manifolds with scalar curvature no less than one, we obtain the sharp upper bound of complete stable minimal surfaces' diameter.

We study of the shape of a compact singular minimal surface in terms of the geometry of its boundary, asking what type of {\it a priori} information can be obtained on the surface from the knowledge of its boundary. We derive estimates of…

In this paper, we study the approximation of an unknown quasiconcave function based on limited partial information. Available information includes lower bounds on the values of the target function at a specified set of points, as well as…

The zero-field partition function of two-dimensional nearest neighbor Ising models of square lattices is derived in terms of the generalized hypergeometric series by evaluating the integral in the exact solution of Onsager. An approximate…

Let $f_1,\dots,f_m$ be polynomials in $n$ variables with coefficients in a finite field $\mathbb{F}_q$. We estimate the number of points $\underline{x}$ in $\mathbb{F}_q^n$ such that each value $f_i(\underline{x})$ is a nonzero square in…

We develop a constructive piecewise polynomial approximation theory in weighted Sobolev spaces with Muckenhoupt weights for any polynomial degree. The main ingredients to derive optimal error estimates for an averaged Taylor polynomial are…

We consider a rational elliptic surface with a relatively minimal fibration. We compute the number of integral sections in the above rational elliptic surface. As an application, we obtain an estimate of polynomial solutions of some…

In the numerical treatment of large-scale Sylvester and Lyapunov equations, projection methods require solving a reduced problem to check convergence. As the approximation space expands, this solution takes an increasing portion of the…

An exact conservative remapping scheme requires overlaps between two meshes and a reconstruction scheme on the old cells (Lagrangian mesh). While the are intensive discussion on reconstruction schemes, there are relative sparse discussion…