Related papers: On wild frieze patterns

Pilz's conjecture states that for any finite set $A=\{a_1,a_2,\dots,a_k\}$ of positive integers and positive integer $n$ in the union of the sets $\{a_1,2a_1,\dots,na_1\},\dots, \{a_k,2a_k,\dots,na_k\}$ (considered as a multiset) at least…

It is known that any infinite frieze comes from a triangulation of an annulus by Baur, Parsons and Tschabold. In this paper we show that each periodic infinite frieze determines a triangulation of an annulus in essentially a unique way.…

We show the first known example for a pattern $q$ for which $\lim_{n\to \infty} \sqrt[n]{S_n(q)}$ is not an integer. We find the exact value of the limit and show that it is irrational. Then we generalize our results to an infinite sequence…

In this paper, we study strongly coupled elliptic systems in non-variational form with negative exponents involving fractional Laplace operators. We investigate the existence, nonexistence, and uniqueness of the positive classical solution.…

For a fixed positive integer $m$ and any partition $m = m_1 + m_2 + \cdots + m_e$ , there exists a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that $$m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots+\frac{1}{n_{k}},$$ with the property…

In this paper, for a square-free integer l>1, a even positive integer k and a positive integer N, we give a trace formula of the Hecke operator T(l) on the space S_k^0(N) of all newforms of weight k and level \Gamma_0(N). Moreover, we give…

This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers $n$ such that $n, n+h$ and $n+k$ are all sums of two squares where $h$ and $k$…

We study modular forms for $\textrm{SL}_2(\mathbb{Z})$ with no negative Fourier coefficients. Let $A(k)$ be the positive integer where if the first $A(k)$ Fourier coefficients of a modular form of weight $k$ for $\textrm{SL}_2(\mathbb{Z})$…

Let A={a_s+n_sZ}_{s=1}^k be a finite system of arithmetic sequences which forms an m-cover of Z (i.e., every integer belongs at least to m members of A). In this paper we show the following sharp result: For any positive integers…

We consider the class of integer rectifiable currents without boundary satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.

Tropical friezes are the tropical analogues of Coxeter-Conway frieze patterns. In this note, we study them using triangulated categories. A tropical frieze on a 2-Calabi-Yau triangulated category $\mathcal{C}$ is a function satisfying a…

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.

We consider symbolic flows over finite alphabets and study certain kinds of repetitions in these sequences. Positive and negative results for the existence of such repetitions are given for codings of interval exchange transformations and…

We study the number $\nu(n)$ of representations of a positive integer $n$ by the form $x^3+y^3+z^3-3xyz$ in the conditions $0\leq x\leq y\leq z; z\geq x+1.$ We proved the following results: (i) for every positive $n,$ except for…

Given a positive integer $n$, we let ${\rm sfp}(n)$ denote the squarefree part of $n$. We determine all positive integers $n$ for which $\max \{ {\rm sfp}(n), {\rm sfp}(n+1), {\rm sfp}(n+2) \} \leq 150$ by relating the problem to finding…

On the space of rhythms of arbitrary length with a fixed number of onsets, a self map $F$ is constructed. It is shown that for any rhythm $\mathbf{r}$ of the space there exists a nonnegative integer $k$ such that $F^k(\mathbf{r})$ falls…

An s-tuple of positive integers are k-wise relatively prime if any k of them are relatively prime. Exact formula is obtained for the probability that s positive integers are k-wise relatively prime.

Let $S= \{ p_1, \ldots, p_s\}$ be a finite, non-empty set of distinct prime numbers and $(U_{n})_{n \geq 0}$ be a linear recurrence sequence of integers of order $r$. For any positive integer $k,$ we define $(U_j^{(k)})_{j\geq 1}$ an…

A (positive definite and non-classic integral) quadratic form is called strongly $s$-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this article, we prove that for any…

Linek's 1989 problem asks whether the numbers of independent sets of trees avoid infinitely many positive integers. We show that the set of natural numbers realized as the number of independent sets of a tree has a lower growth exponent of…