Related papers: Tight universal triangular forms

In the conventional formulation of N=1 supersymmetry, a vector multiplet is supposed to be in the adjoint representation of a given gauge group. We present a new formulation with a vector multiplet in the non-adjoint representation of SO(N)…

This paper has been withdrawn and replaced by arXiv:1309.5035. In this paper we describe some examples of so called spherical functors between triangulated categories, which generalize the notion of a spherical object. We also give…

We give an efficient algorithm that, given a graph $G$ and a partition $V_1,\ldots,V_m$ of its vertex set, finds either an independent transversal (an independent set $\{v_1,\ldots,v_m\}$ in $G$ such that $v_i\in V_i$ for each $i$), or a…

Let $n,k$ be fixed natural numbers with $1\le k\le n$ and let $A_{n+1,k,2k,\dots,sk}$ denote an $(n+1)\times (n+1)$ complex multidiagonal matrix having $s=[n/k]$ sub- and superdiagonals at distances $k,2k,\dots,sk$ from the main diagonal.…

Let $A$ be a set of positive integers. We define a positive integer $n$ as an $A$-practical number if every positive integer from the set $\left\{1,\ldots ,\sum_{d\in A, d\mid n}d\right\}$ can be written as a sum of distinct divisors of $n$…

In this paper we provide characterizing properties of TDI systems, among others the following: a system of linear inequalities is TDI if and only if its coefficient vectors form a Hilbert basis, and there exists a test-set for the system's…

Let us fix a prime $p$ and a homogeneous system of $m$ linear equations $a_{j,1}x_1+\dots+a_{j,k}x_k=0$ for $j=1,\dots,m$ with coefficients $a_{j,i}\in\mathbb{F}_p$. Suppose that $k\geq 3m$, that $a_{j,1}+\dots+a_{j,k}=0$ for $j=1,\dots,m$…

Let $K = \mathbb{R}$ or $\mathbb{C}$. An $n$-element subset $A$ of $K$ is a $B_h$-set if every element of $K$ has at most one representation as the sum of $h$ not necessarily distinct elements of $A$. Associated to the $B_h$ set $A =…

Given an inhomogeneous quadratic form $Q_\xi(v)=Q(v+\xi)$ with $Q$ an indefinite $\mathbb{Q}$-isotropic rational ternary form and $\xi\in \mathbb{R}^3$ irrational, we prove an effective lower bound for the number of integer vectors $v\in…

In this paper we discuss face numbers of generalised triangulations of manifolds in arbitrary dimensions. This is motivated by the study of triangulations of simply connected $4$-manifolds: We observe that, for a triangulation $\mathcal{T}$…

Let A= (a_{ij}) be a non-negative integer k x k matrix. A is a homogeneous matrix if a_{ij} + a_{kl}=a_{il} + a_{kj} for any choice of the four indexes. We ask: If A is a homogeneous matrix and if F is a form in C[x_1, \dots x_n] with…

Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $\alpha$ ($\leq k$), let \[\Sigma_{\alpha}…

An integer of the form $p_m(x)= \frac{(m-2)x^2-(m-4)x}{2} \ (m\ge 3)$, for some integer $x$ is called a generalized polygonal number of order $m$. A ternary sum $\Phi_{i,j,k}^{a,b,c}(x,y,z)=ap_{i+2}(x)+bp_{j+2}(y)+cp_{k+2}(z)$ of…

Let $\Gamma$ be the unit circle, $A(\Gamma)$ the Wiener algebra of continuous functions whose series of Fourier coefficients are absolutely convergent, and $A^+$ the subalgebra of $A(\Gamma)$ of functions whose negative coefficients are…

We define lower triangular tensors, and show that all diagonal entries of such a tensor are eigenvalues of that tensor. We then define lower triangular sub-symmetric tensors, and show that the number of independent entries of a lower…

We say a positive integer is a sum of three nonunit squares if it is a sum of three squares of integers other than one. In this article, we find all integers which are sums of three nonunit squares assuming that the Generalized Riemann…

We define a category $v\mathcal{T}$ of tangles diagrams drawn on surfaces with boundaries. On the one hand we show that there is a natural functor from the category of virtual tangles to $v\mathcal{T}$ which induces an equivalence of…

A perfect number is a positive integer $N$ such that the sum of all the positive divisors of $N$ equals $2N$, denoted by $\sigma(N) = 2N$. The question of the existence of odd perfect numbers (OPNs) is one of the longest unsolved problems…

The support of a vector is the number of nonzero-components. We show that given an integral $m\times n$ matrix $A$, the integer linear optimization problem $\max\left\{\boldsymbol{c}^T\boldsymbol{x} : A\boldsymbol{x} = \boldsymbol{b}, \,…

We study the general rational trigonometry of a tetrahedron, based on quadrances, spreads and solid spreads, using vector products associated to an arbitrary symmetric bilinear form over a general field, not of characteristic two. This…