Related papers: Universal inequalities in Ehrhart Theory
We show that there exist absolute constants $\Delta > \delta > 0$ such that, for all $n \geqslant 2$, there exists a polynomial $P$ of degree $n$, with $\pm 1$ coefficients, such that $$\delta\sqrt{n} \leqslant |P(z)| \leqslant…
Higson proved that every homotopy invariant, stable and split exact functor from the category of $C^*$-algebras to an additive category factors through Kasparov's $KK$-theory. By adapting a group equivariant generalization of this result by…
The study of the $h$-vectors of graded Gorenstein algebras is an important topic in combinatorial commutative algebra, which despite the large amount of literature produced during the last several years, still presents many interesting open…
By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a…
A universal category-theoretical characterization of groupoid equivariant $KK^G$-theory for ${\mathbb{Z}}_2$-graded $C^*$-algebras is established, by observing the ``$KK$-axiom'' that for each $[s,{\cal E} \oplus B, \mathbb{F}] \in…
We compare ambient and outer Lipschitz geometry of Lipschitz normally embedded H\"older triangles in $\mathbb{R}^4$. In contrast to the case of $\mathbb{R}^3$ there are infinitely many equivalence classes. The equivalence classes are…
Ehrhart theory is the study of the enumeration of lattice points in lattice polytopes. Equivariant Ehrhart theory is a generalization of Ehrhart theory that takes into account the action of a finite group acting via affine transformations…
In this paper we investigate the Ehrhart Theory of the independence matroid polytope of uniform matroids. It is proved that these polytopes have an Ehrhart polynomial with positive coefficients. To do that, we prove that indeed all…
Let $d$ be a nonnegative integer, and let $P \subset \mathbb R^d$ be a $d$-dimensional convex lattice polytope. In this article, we prove that the ratio of the volume of a normal-sized miniature of $P$ to that of $P$ is $1:\binom{2d+1}{d},$…
We prove sharp $L^p-L^q$ estimates for averaging operators along general polynomial curves in two and three dimensions. These operators are translation-invariant, given by convolution with the so-called affine arclength measure of the curve…
A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope with the integer decomposition property (IDP) has a unimodal (Ehrhart) $h^\ast$-polynomial. This conjecture can be viewed as a strengthening of a…
We classify all the possible $delta$-vectors of d-dimensional integral convex polytopes whose volumes are less than or equal to 3/(d!).
Let $\mathcal{P} \subseteq \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a…
In the paper, we provide an effective method for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent…
We prove a Hardy-type inequality for the gradient of the Heisenberg Laplacian on open bounded convex polytopes on the first Heisenberg Group. The integral weight of the Hardy inequality is given by the distance function to the boundary…
In this article Ehrhart quasi-polynomials of simplices are employed to determine isospectral lens spaces in terms of a finite set of numbers. Using the natural lattice associated with a lens space the associated toric variety of a lens…
This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree…
We determine the class of finite T_0-spaces allowing for a universal coefficient theorem computing equivariant KK-theory by filtrated K-theory.
This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that…
A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…