Related papers: Tight universal triangular forms

We present general formulas for transverse and transverse-traceless (TT) symmetric tensors in flat spaces. TT tensors in conformally flat spaces can be obtained by means of a conformal transformation.

For any integer $m\ge0$, we recall that triangular numbers are those $\mathbf{T}(m)=\frac{m(m+1)}{2}$. A conjecture of Sun Zhi-Wei states that an integer $2^n\pm n$ with any $n>2$ can not be a triangular number. The motivation of this work…

Let $S=\{s_{1},s_{2},\ldots,s_{n}\}$ be an ordered set of $n$ distinct positive integers. The $m$th-order $n$-dimensional tensor $T_{[S]}=(t_{i_{1}i_{2}\ldots i_{m}}),$ where $t_{i_{1}i_{2}\ldots…

We compute the algebraic K-theory of the non-commutative ring k<x_1,...,x_n>/(m^a) when k is a perfect field of positive characteristic and m=(x_1,...,x_n). We express the answer in terms of the truncation poset Witt vectors developed in…

In this paper, we give a necessary and sufficient condition for an even order three dimensional strongly symmetric circulant tensor to be positive semi-definite. In some cases, we show that this condition is also sufficient for this tensor…

A positive quadratic form is $(k,\ell)$-universal if it represents all the numbers $kx+\ell$ where $x$ is a non-negative integer, and almost $(k,\ell)$-universal if it represents all but finitely many of them. We prove that for any $k,\ell$…

Let $S$ be a Scott set, or even an $\omega$-model of $\mathsf{WWKL}$. Then for each $A\in S$, either there is $X \in S$ that is weakly 2-random relative to $A$, or there is $X\in S$ that is 1-generic relative to $A$. It follows that if…

A general construction is found for `topological' singular vectors of the twisted N=2 superconformal algebra. It demonstrates many parallels with the known construction for sl(2) singular vectors due to Malikov--Feigin--Fuchs, but is…

A sequence of integers $ \{ s_n \}_{n \in \mathbb{N}} $ is called a T-sequence if there exists a Hausdorff group topology on $ \mathbb{Z} $ such that $ \{ s_n \}_{n \in \mathbb{N}} $ converges to zero. For every finite set of primes $ S $…

An S-type eigenvalue localization set for a tensor is given by breaking N={1,2,...,n} into disjoint subsets S and its complement. It is shown that the new set is tighter than those provided by L. Qi (Journal of Symbolic Computation 40…

Let k be an algebraically closed field and A be a finite-dimensional associative basic k-algebra of the form A=kQ/I where Q is a quiver without oriented cycles or double arrows and I is an admissible ideal of kQ. We consider roots of the…

A triangular form is defined to be an integer-valued quadratic polynomial of the form $a_1P_3(x_1)+a_2P_3(x_2)+\cdots+a_kP_3(x_k)$ where $a_i's$ are positive integers and $P_3(x)=x(x+1)/2$. A triangular form is called regular if it…

For a set $A$ of nonnegative integers, let $R_2(A,n)$ and $R_3(A,n)$ denote the number of solutions to $n=a+a'$ with $a,a'\in A$, $a<a'$ and $a\leq a'$, respectively. In this paper, we prove that, if $A\subseteq \mathbb{N}$ and $N$ is a…

Let A={a_s(mod n_s)}_{s=0}^k be a system of residue classes. With the help of cyclotomic fields we obtain a theorem which unifies several previously known results concerning system A. In particular, we show that if every integer lies in…

We study determinant functors which are defined on a triangulated category and take values in a Picard category. The two main results are the existence of a universal determinant functor for every small triangulated category, and a…

For a set $A$ of positive integers with $\gcd(A)=1$, let $\langle A \rangle$ denote the set of all finite linear combinations of elements of $A$ over the non-negative integers. The it is well known that only finitely many positive integers…

The sets of strongly supercyclic, weakly l-sequentially supercyclic, weakly sequentially supercyclic, and weakly supercyclic vectors for an arbitrary normed-space operator are all dense in the normed space, regardless the notion of…

We study the problem of finding a triangulation T of a planar point set S such as to minimize the expected distance between two points x and y chosen uniformly at random from S. By distance we mean the length of the shortest path between x…

In analogy with the 290-Theorem of Bhargava-Hanke, a criterion set is a finite subset $C$ of the totally positive integers in a given totally real number field such that if a quadratic form represents all elements of $C$, then it…

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$…