Related papers: Tight universal triangular forms

We study solutions of a quadratic matrix equation arising in Riemannian geometry. Let $S$ be a real symmetric $n\times n$-matrix with zeros on the diagonal and let $\theta$ be a real number. We construct nonzero solutions $(S,\theta)$ of…

We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 5/8, then all sufficiently large odd positive integers can be written as the sum of three primes in A. The constant 5/8 in this statement…

Let ${\mathbb K}={\mathbb Q}(\sqrt{m})$ be a real quadratic number field, where $m>1$ is a squarefree integer. Suppose that $0 < \theta< \pi $ has rational cosine, say $\cos (\theta)=s/r$ with $0< |s|<r$ and $\gcd(r,s)=1$. A positive…

In this article, we develop an algorithm to calculate the set of all integers $m$ for which there exists a linear operator $T$ on ${\mathbb R}^n$ such that ${\mathbb R}^n$ has exactly $m$ $T$-invariant subspaces. A brief discussion is…

In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and…

Let (k(n)) n=1,2,... be a strictly increasing sequence of positive integers . We consider a specific sequence of differential operators Tk(n),{\lambda} , n=1,2,... on the space of entire functions , that depend on the sequence (k(n))…

We give a geometric interpretation of all the $m$-th elliptic integrable systems associated to a $k'$-symmetric space $N=G/G_0$ (in the sense of C.L. Terng). It turns out that we have to introduce the integer $m_{k'}$ defined by m_{1}=0 and…

We study the following generalization of Roth's theorem for 3-term arithmetic progressions. For s>1, define a nontrivial s-configuration to be a set of s(s+1)/2 integers consisting of s distinct integers x_1,...,x_s as well as all the…

A symmetric tensor, which has a symmetric nonnegative decomposition, is called a completely positive tensor. We consider the completely positive tensor decomposition problem. A semidefinite algorithm is presented for checking whether a…

Let F be a finite extension of Qp and G be GL(2,F). When V is the tensor product of three infinite dimensional, irreducible, admissible representations of G, the space of G-invariant linear forms has dimension 0 or 1. When a non-zero linear…

Let Gamma < SL(2,Z) be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive non-zero vectors v0, w0 in Z^2. We consider the set S of all integers occurring in <v0 gamma,w0>, for gamma in…

For integers $n,k,s$, we give a formula for the number $T(n,k,s)$ of order $k$ subsets of the ring $\mathbb{Z}/n\mathbb{Z}$ whose sum of elements is $s$ modulo $n$. To do so, we describe explicitly a sequence of matrices $M(k)$, for…

Generalized $m$-gonal numbers are those $p_m(x)= [ (m - 2)x^2 - (m - 4)x ]/2 $ where $x$ and $m$ are integers with $m \geq 3$. If any nonnegative integer can be written in the form $ap_r(h)+bp_s(l)+cp_t(m)+dp_u(n)$, where $a,b,c,d$ are…

A set $S$ of vertices in a hypergraph is \textit{strongly independent} if every hyperedge shares at most one vertex with $S$. We prove a sharp result for the number of maximal strongly independent sets in a $3$-uniform hypergraph analogous…

A triangulation of a simplicial complex $\Delta$ is called uniform if the $f$-vector of its restriction to a face of $\Delta$ depends only on the dimension of that face. This paper proves that the entries of the $h$-vector of a uniform…

We generalize the Stirling numbers of the first kind $s(a,k)$ to the case where $a$ may be an arbitrary real number. In particular, we study the case in which $a$ is an integer. There, we discover new combinatorial properties held by the…

A $(t, s, v)$-all-or-nothing transform is a bijective mapping defined on $s$-tuples over an alphabet of size $v$, which satisfies the condition that the values of any $t$ input co-ordinates are completely undetermined, given only the values…

A subset of positive integers $F$ is a Schreier set if it is non-empty and $|F|\leqslant \min F$ (here $|F|$ is the cardinality of $F$). For each positive integer $k$, we define $k\mathcal{S}$ as the collection of all the unions of at most…

Let $m\leq n$ be positive integers and $\mathfrak X$ a class of groups which is closed for subgroups, quotient groups and extensions. Suppose that a finite group $G$ satisfies the condition that for every two subsets $M$ and $N$ of…

We search for triangular numbers that are multiples of other triangular numbers. It is found that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers that are triangular numbers and…