Related papers: Steiner systems and configurations of points

In recent work of T. Cassidy and the author, a notion of complete intersection was defined for (non-commutative) regular skew polynomial rings, defining it using both algebraic and geometric tools, where the commutative definition is a…

In this paper we elucidate the advantage of examining the connections between Hilbert-Kamke equations and geometric designs, or Chebyshev-type quadrature, for classical orthogonal polynomials. We first establish that if a $5$-design with…

We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines. We pay special attention to a family of power ideals…

Coherent configurations are a generalization of association schemes. In this paper, we introduce the concept of $Q$-polynomial coherent configurations and study the relationship among intersection numbers, Krein numbers, and eigenmatrices.…

Spherical Designs are finite sets of points on the sphere $\mathbb{S}^{d}$ with the property that the average of certain (low-degree) polynomials in these points coincides with the global average of the polynomial on $\mathbb{S}^{d}$. They…

We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a…

We define and study a bigraded knot invariant whose Euler characteristic is the Alexander polynomial, closely connected to knot Floer homology. The invariant is the homology of a chain complex whose generators correspond to Kauffman states…

Let I and J be homogeneous ideals in a standard graded polynomial ring. We study upper bounds of the Hilbert function of the intersection of I and g(J), where g is a general change of coordinates. Our main result gives a generalization of…

This paper answers a fundamental question in the theory of Steiner operations (StOps) as defined and studied in the monograph, "Global Methods for Combinatorial Isoperimetric Problems" (GMCIP). StOps are morphisms for combinatorial…

Superintegrable systems are classical and quantum Hamiltonian systems which enjoy much symmetry and structure that permit their solubility via analytic and even, algebraic means. They include such well-known and important models as the…

We prove that the set of $n$-point configurations for which the solution of the planar Steiner problem is not unique has the Hausdorff dimension at most $2n-1$ (as a subset of $\mathbb{R}^{2n}$). Moreover, we show that the Hausdorff…

Let $K\ < x_i\ >$ be the free associative algebra generated by a finite or countable number of variables $x_i$. The notion of "letterplace correspondence" introduced in [1,2] for the graded (two-sided) ideals of $K\ < x_i\ >$ is extended in…

Let $\F_q^n$ be a vector space of dimension $n$ over the finite field $\F_q$. A $q$-analog of a Steiner system (briefly, a $q$-Steiner system), denoted $S_q[t,k,n]$, is a set $S$ of $k$-dimensional subspaces of $\F_q^n$ such that each…

Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade 2 as those with a Hilbert--Burch matrix that has a maximal symmetric subblock. We also prove that every…

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical…

The concept of Schreier extensions of loops was introduced in the general case in [11] and, more recently, it has been explored in the context of Steiner loops in [6]. In the latter case, it gives a powerful method for constructing Steiner…

In the sub-Riemannian Heisenberg group equipped with its Carnot-Caratheodory metric and with a Haar measure, we consider isodiametric sets, i.e. sets maximizing the measure among all sets with a given diameter. In particular, given an…

In this article we study the interplay between algebro-geometric notions related to $\pi$-points and structural features of the stable Auslander-Reiten quiver of a finite group scheme. We show that $\pi$-points give rise to a number of new…

In this paper, we introduce relative Rota-Baxter systems on Leibniz algebras and give some characterizations and new constructions. Then we construct a graded Lie algebra whose Maurer-Cartan elements are relative Rota-Baxter systems. This…

Aim of this paper is to count $0$-dimensional stable and strongly stable ideals in $2$ and $3$ variables, given their (constant) affine Hilbert polynomial. To do so, we define the Bar Code, a bidimensional structure representing any finite…