Related papers: Quadrature rules and distribution of points on man…

Stolarsky's invariance principle quantifies the deviation of a subset of a metric space from the uniform distribution. Classically derived for spherical sets, it has been recently studied in a number of other situations, revealing a general…

This paper is devoted to the study of a newly introduced tool, projectional coderivatives and the corresponding calculus rules in finite dimensions. We show that when the restricted set has some nice properties, more specifically, is a…

This article presents a novel approach to enhance the accuracy of classical quadrature rules by incorporating correction terms. The proposed method is particularly effective when the position of an isolated discontinuity in the function and…

We study manifolds $M$ equipped with a quadrature rule $$ \int_{M}{\phi(x) dx} \simeq \sum_{i=1}^{n}{a_i \phi(x_i)}.$$ We show that $n-$point quadrature rules with nonnegative weights on a compact $d-$dimensional manifold cannot integrate…

We examine the total mixed scalar curvature of a fixed distribution as a functional of a pseudo-Riemannian metric. We develop variational formulas for quantities of extrinsic geometry of the distribution to find the critical points of this…

We prove various inequalities measuring how far from an isometry a local map from a manifold of high curvature to a manifold of low curvature must be. We consider the cases of volume-preserving, conformal and quasi-conformal maps. The…

A method of deriving quadrature rules has been developed which gives nodes and weights for a Gaussian-type rule which integrates functions of the form: f(x,y,t) = a(x,y,t)/((x-t)^2+y^2) + b(x,y,t)/([(x-t)^2+y^2]^{1/2}) +…

We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules…

The standard design principle for quadrature formulas is that they should be exact for integrands of a given class, such as polynomials of a fixed degree. We show how this principle fails to predict the actual behavior in four cases:…

This paper is a self-contained presentation of certain aspects of the theory of weighted Sobolev spaces and elliptic operators on non-compact Riemannian manifolds. Specifically, we discuss (i) the standard and weighted Sobolev Embedding…

We obtain sharp Hardy inequalities on antisymmetric functions where antisymmetry is understood for multi-dimensional particles. Partially it is an extension of the previously published paper \cite{HL}, where Hardy's inequalities were…

This is basically a review of the field of Quasi-Monte Carlo intended for computational physicists and other potential users of quasi-random numbers. As such, much of the material is not new, but is presented here in a style hopefully more…

The sectional curvature of a compact Riemannian manifold M can be seen as a random variable on the Grassmann bundle of 2-planes in TM endowed with the Fubini-Study volume density. In this article we calculate the moments of this random…

We study a new simple quadrature rule based on integrating a $C^1$ quadratic spline quasi-interpolant on a bounded interval. We give nodes and weights for uniform and non-uniform partitions. We also give error estimates for smooth functions…

Some results on the approximation of functions from the Sobolev spaces on metric graphs by step functions are obtained. The estimates are uniform with respect to all graphs of a given finite length, and the constant factors in the…

In the present paper, we define Morse-Bott functions on manifolds with boundary which are generalizations of Morse functions and show Morse-Bott inequalities for these manifolds.

In this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms, pyramids, and hexahedra. The methodology is free from manual intervention and is capable…

For convex univalent functions we give instances where the sharp bound for various coefficient functionals are identical to those for the corresponding bound for the inverse function. We give instances where the sharp bounds differ and also…

We follow the method of ABP estimate in \cite{brendle2021} and apply it to spacelike submanifolds in $\mathbb R^{n,1}$. We then obtain Michael-Simon type inequalities. Surprisingly, our investigation leads to a Sobolev inequality without a…

In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown…