相关论文: Hilbert Bases for Orthogonal Arrays

Orthogonal arrays are a type of combinatorial design that were developed in the 1940s in the design of statistical experiments. In 1947, Rao proved a lower bound on the size of any orthogonal array, and raised the problem of constructing…

We prove a closed formula for the generating function $\mathsf Z_d(t)$ of the motives $[\mathrm{Hilb}^d(\mathbb A^n)_0] \in K_0(\mathrm{Var}_{\mathbb C})$ of punctual Hilbert schemes, summing over $n$, for fixed $d>0$. The result is an…

Given a set of $m$ points and a set of $n$ lines in the plane, we consider the problem of computing the faces of the arrangement of the lines that contain at least one point. In this paper, we present an $O(m^{2/3}n^{2/3}+(n+m)\log n)$ time…

We show that every frame for a Hilbert space H can be written as a (multiple of a) sum of three orthonormal bases for H. A result of N.J. Kalton is included which shows that this is best possible in that: A frame can be represented as a…

Deciding the existence of an $l\times m\times n$ integer threeway table with given line-sums is NP-complete already for fixed $l=3$, but is in P with both $l,m$ fixed. Here we consider {\em huge} tables, where the variable dimension $n$ is…

A frame in an $n$-dimensional Hilbert space $H_n$ is a possibly redundant collection of vectors $\{f_i\}_{i\in I}$ that span the space. A tight frame is a generalization of an orthonormal basis. A frame $\{f_i\}_{i\in I}$ is said to be…

We study the problem of $2$-dimensional orthogonal range counting with additive error. Given a set $P$ of $n$ points drawn from an $n\times n$ grid and an error parameter $\eps$, the goal is to build a data structure, such that for any…

The aim of this paper is to give some properties of Hilbert genus fields and construct the Hilbert genus fields of the fields $L_{m,d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$, where $m\geq 3$ is a positive integer and $d$ is a square-free…

Recently [Karimipour and Memarzadeh, Phys. Rev. A 73, 012329 (2006)] studied the problem of finding a family of orthonormal bases in a bipartite space each of dimension $D$ with the following properties: (i) The family continuously…

Given a complex smooth algebraic variety X, we compute the generating function of the stringy motives of its symmetric powers as a function of motive of X. In dimension two we recover the Goettsche formulas for Hilbert schemes. We use the…

With a computer-aided approach based on the connection with equitable partitions, we establish the uniqueness of the orthogonal array OA$(1536,13,2,7)$, constructed in [D.G.Fon-Der-Flaass. Perfect $2$-Colorings of a Hypercube, Sib. Math. J.…

We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2x2-matrix…

We prove that for any fixed d the generating function of the projection of the set of integer points in a rational d-dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice…

This paper presents a new extension of the classical Heron problem, termed the generalized $(k,m)$-Heron problem, which seeks an optimal configuration among $k$ feasible and $m$ target non-empty closed convex sets in $\mathbb{R}^n$. The…

We solve three enumerative problems concerning families of planar maps. More precisely, we establish algebraic equations for the generating function of non-separable triangulations in which all vertices have degree at least d, for a certain…

In recent developments, a general approach for solving Riemann--Hilbert problems numerically has been developed. We review this numerical framework, and apply it to the calculation of orthogonal polynomials on the real line. Combining this…

In this article, we construct a generating set of rational invariants for the action of the orthogonal group $\text{O}(n)$ on the space $\mathbb{R}[x_1,\dots,x_n]_{2d}$ of real homogeneous polynomials of even degree $2d$. This generalizes a…

A new systematic method for the explicit construction of (basis-)invariants is introduced and employed to construct the full ring of basis invariants of the Two-Higgs-Doublet-Model (2HDM) scalar sector. Co- and invariant quantities are…

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame which encompasses families of arrangements. Surprisingly the proof is based on the study of finite sets of vectors in a…

Let $G$ be a complex classical group, and let $V$ be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of…