Related papers: Waring Problem For Triangular Matrix Algebra

In 1987 Harris proved (Proc. Amer. Math. Soc., 101) - among others- that for each $1\le p<2$ there exists a two-dimensional function $f\in L^p$ such that its triangular Walsh-Fourier series diverges almost everywhere. In this paper we…

Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A and B be upper triangular n-by-n matrices that (i) are not similar to direct sums of matrices of smaller…

In this paper, we prove that for $d=3,\dots,8$, every natural number can be written as $t_x+t_y+3t_z+dt_w$, where $x$, $y$, $z$, and $w$ are nonnegative integers and $t_k=k(k+1)/2$ $(k=0,1,2,\ldots)$ is a triangular number. Furthermore, we…

Triangular factorizations are an important tool for solving integral equations and partial differential equations with hierarchical matrices ($\mathcal{H}$-matrices). Experiments show that using an $\mathcal{H}$-matrix LR factorization to…

In 2014 Marques and Lengyel gave all of the solutions to the equation $T_n=m!$, where $T_n$ is the $n$th term of the Tribonacci sequence $0,1,1,2,4,7,13,24,\ldots$. In 2023 Alahmadi and Luca generalized their result to the equation…

In this paper, we obtain a general expression for the entries of the r. (r is integer) power of a certain n-square complex tridiagonal matrix. In addition, we get the complex factorizations of Fibonacci polynomials, Fibonacci and Pell…

The computational function of a matchgate is represented by its character matrix. In this article, we show that all nonsingular character matrices are closed under matrix inverse operation, so that for every $k$, the nonsingular character…

Let $s_d(p,a) = \min \{k | a = \sum_{i=1}^{k}a_i^d, a_i\in \ff_p^*\}$ be the smallest number of d-th powers in the finite field F_p, sufficient to represent the number a in F_p^*. Then $$g_d(p) = max_{a in F_p^*} s_d(p,a)$$ gives an answer…

For any $n\ge 2$ and fixed $k\ge 1$, we give necessary and sufficient conditions for an arbitrary nonzero square matrix in the matrix ring $\mathbb{M}_n(\mathbb{F})$ to be written as a sum of an invertible matrix $U$ and a nilpotent matrix…

We give a characterization of all pairs $(k,n)$ of positive integers for which the ratio $$ \frac{1^k-2^k+3^k-\dots+(-1)^{n+1} n^k}{1^k-2^k+3^k-\dots+(-1)^{n}(n-1)^k} $$ of two consecutive alternating power sums is an integer.

The McCarty Conjecture states that any McCarty Matrix (an $n\times n$ matrix $A$ with positive integer entries and each of the $2n$ row and column sums equal to $n$), can be additively decomposed into two other matrices, $B$ and $C$, such…

We study some sum-product problems over matrix rings. Firstly, for $A, B, C\subseteq M_n(\mathbb{F}_q)$, we have $$ |A+BC|\gtrsim q^{n^2}, $$ whenever $|A||B||C|\gtrsim q^{3n^2-\frac{n+1}{2}}$. Secondly, if a set $A$ in $M_n(\mathbb{F}_q)$…

We reconsider the classical problem of representing a finite number of forms of degree d in n+1 variables as sums of powers of linear forms. We define a geometric construct called a `grove', which, in a number of cases allows us to…

The set of n by n upper-triangular nilpotent matrices with entries in a finite field F_q has Jordan canonical forms indexed by partitions lambda of n. We present a combinatorial formula for computing the number F_\lambda(q) of matrices of…

We will present solutions to the constant Yang-Baxter equation, in any dimension $n$. More precisely, for any $n$, we will create an infinite family of $n^2$ by $n^2$ matrices which are solutions to the constant Yang-Baxter equation. The…

Let $k \geq 2$ and $b \geq 3$ be integers, and suppose that $d_1, d_2 \in \{0,1,\dots, b - 1\}$ are distinct and coprime. Let $\mathcal{S}$ be the set of non-negative integers, all of whose digits in base $b$ are either $d_1$ or $d_2$. Then…

Let $M_{*}({\bf C})$ denote the C$^{*}$-algebra defined as the direct sum of all matrix algebras $\{M_{n}({\bf C}):n\geq 1\}$. It is known that $M_{*}({\bf C})$ has a non-cocommutative comultiplication $\Delta_{\varphi}$. We show that the…

In this paper, we study a density version of the Waring-Goldbach problem. Suppose that A is a subset of the primes, and the lower density of A in the primes is larger than 1-1/2k. We prove that every sufficiently large natural number n…

Fischer provided a new type of binomial determinant for the number of alternating sign matrices involving the third root of unity. In this paper we prove that her formula, when replacing the third root of unity by an indeterminate $q$, is…

Let $\mathbb{F}_q$ be a finite field with $q$ elements, $f \in \mathbb{F}_q[x_1, \dots, x_n]$ a polynomial in $n$ variables and let us denote by $N(f)$ the number of roots of $f$ in $\mathbb{F}_q^n$. %Many authors, such as Wei Cao and Kung…