Related papers: The algebra of the box spline
In this paper we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, "mixed smoothness" refers to the choice of different orders of smoothness across different edges of the mesh. To study the…
This is basically the text of a survey talk (entitled 'Painleve, Klein and the icosahedron') given at Hitchin's 60th birthday conference. It discusses the search for and construction of algebraic solutions of the sixth Painleve differential…
As continuation of the study of polynomial approximation and composition operators on Dirichlet spaces of unit disk, which has settled a problem posed by Cima in 1976, the present paper aims to consider the case of the unbounded domains,…
It is well-known that the convex and concave envelope of a multilinear polynomial over a box are polyhedral functions. Exponential-sized extended and projected formulations for these envelopes are also known. We consider the convexification…
We generalize a result of M. Kapranov, O. Schiffmann, and E. Vasserot by showing that, for a number field $K$ with class number one, the spherical Hall algebra of $\overline{\operatorname{Spec}(\mathcal{O}_K)}$, where $\mathcal{O}_K$ is the…
We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact "multiline queue" formula for Macdonald polynomials due to Corteel, Mandelshtam and Williams. We also…
In this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on Bergman-type spaces. In particular, the results in this paper apply to the weighted Bergman space on the unit ball, the unit…
In this article, we start to recall the inversion formula for the convolution with the Box spline. The equivariant cohomology and the equivariant K-theory with respect to a compact torus G of various spaces associated to a linear action of…
We formulate a type B extended nilHecke algebra, following the type A construction of Naisse and Vaz. We describe an action of this algebra on extended polynomials and describe some results on the structure on the extended symmetric…
We consider spaces of multivariate splines defined on a particular type of simplicial partitions that we call (generalized) oranges. Such partitions are composed of a finite number of maximal faces with exactly one shared medial face. We…
We employ appropriate realizations of the affine Hecke algebra and we recover previously known non-diagonal solutions of the reflection equation for the $U_{q}(\hat{gl_n})$ case. With the help of linear intertwining relations involving the…
This paper makes precise the close connection between the affine Hecke algebra, the path model, and the theory of crystals. Section 2 is a basic pictorial exposition of Weyl groups and affine Weyl groups and Section 5 is an exposition of…
In this paper, we propose a new construction of vertex algebras using the Deligne category. This approach provides a rigorous framework for defining the so-called large $N$ vertex algebra, which has appeared in recent physics literatures.…
Spline functions have long been used in numerically solving differential equations. Recently it revives as isogeometric analysis, which uses NURBS for both parametrization and element functions. In this paper, we introduce some multivariate…
General results on multiplicative lattices found recently by Facchini, Finocchiaro and Janelidze have been studied in the particular case of groups by Facchini, de Giovanni and Trombetti. In this paper we prove that these results hold not…
This article explains basic constructions and results on group algebras and their cohomology, starting from the point of view of commutative algebra. It provides the background necessary for a novice in this subject to begin reading Dave…
We try to understand which morphisms of complex analytic spaces come from algebraic geometry. We start with a series of conjectures, and then give some partial solutions.
We consider solutions of the 2x2 matrix Hamiltonians of the physical systems within the context of the su(2) and su(1,1) Lie algebra. Our technique is relatively simple when compared with the others and treats those Hamiltonians which can…
Brackets by another name, Whitehead or Samelson products, have a history parallel to that in Kosmann-Schwarzbach's From Schouten to Mackenzie: notes on brackets. Here I sketch the development of these and some of the other brackets and…
In a recent paper with Sahi and Stokman, we introduced quasi-polynomial generalizations of Macdonald polynomials for arbitrary root systems via a new class of representations of the double affine Hecke algebra. These objects depend on a…