Related papers: On positive cubature rules on the simplex and isom…

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 construct isospectral pairs of Riemannian metrics on S^5 and on B^6, thus lowering by three the dimension of spheres and balls on which such metrics have been constructed previously (S^{n\ge 8} and B^{n\ge 9}). We also construct…

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 give several descriptions of positive quadrature formulas which are exact for trigonometric -, respectively, Laurent polynomials of degree less or equal $n-1-m$, $0\leq m\leq n-1$. A complete and simple description is obtained with the…

Let $X$ be a real Banach lattice with a unit, let $Y \subseteq X$ be a closed subspace containing the unit. In this paper we study the order theoretic (also isometric) structure of $Y$ that it may inherit from $X$ under some additional…

We introduce a new isomorphic quantity for Banach spaces, the index $\Theta_X$, based on finite convex coverings of the unit ball. This index is closely related to the asymptotic moduli of uniform convexity and uniform smoothness, so that…

We introduce an asymptotic notion of positivity in algebraic geometry that turns out to be related to some high-dimensional convex sets. The dimension of the convex sets grows with the number of birational operations. In the case of complex…

In this paper we provide a framework for the study of isoperimetric problems in finitely generated group, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions,…

Motivated by recent studies of superconformal mechanics extended by spin degrees of freedom, we construct minimally superintegrable models of spinning particles on 2-sphere, the spin degrees of freedom of which are represented by a 3-vector…

Composed of square particles, the tetratic phase is characterised by a four-fold symmetry with quasi-long-range orientational order but no translational order. We construct the elastic free energy for tetratics and find a closed form…

We continue the study of computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a…

This paper introduces the scaled boundary cubature (SBC) scheme for accurate and efficient integration of functions over polygons and two-dimensional regions bounded by parametric curves. Over two-dimensional domains, the SBC method reduces…

Node elimination is a numerical approach to obtain cubature rules for the approximation of multivariate integrals. Beginning with a known cubature rule, nodes are selected for elimination, and a new, more efficient rule is constructed by…

The main purpose of the paper is to prove the following results: Let $A$ be a locally finite metric space whose finite subsets admit uniformly bilipschitz embeddings into a Banach space $X$. Then $A$ admits a bilipschitz embedding into $X$.…

Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with…

Numerical integration formulas in $n$-dimensional Euclidean space of degree three are discussed. For the integrals with permutation symmetry we present a method to construct its third-degree integration formulas with $2n$ real points. We…

We study two modifications of the trapezoidal product cubature formulae, approximating double integrals over the square domain $[a,b]^2=[a,b]\times [a,b]$. Our modified cubature formulae use mixed type data: except evaluations of the…

We consider a general notion of snowflake of a metric space by composing the distance by a nontrivial concave function. We prove that a snowflake of a metric space $X$ isometrically embeds into some finite-dimensional normed space if and…

We classify, up to homeomorphisms, the closed simply-connected 4-manifolds that admit a Riemannian metric for which averages of pairs of sectional curvatures of orthogonal planes are positive.

In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also…