Related papers: Period-Different $m$-Sequences With At Most A Four…

For a degree sequence $d:d_1\geq \cdots \geq d_n$, we consider the smallest chromatic number $\chi_{\min}(d)$ and the largest chromatic number $\chi_{\max}(d)$ among all graphs with degree sequence $d$. We show that if $d_n\geq 1$, then…

Podobnik and Stanley recently proposed a novel framework, Detrended Cross-Correlation Analysis, for the analysis of power-law cross-correlation between two time-series, a phenomenon which occurs widely in physical, geophysical, financial…

The unitary correlation sets defined by the first author in conjunction with tensor products of $\mathcal{U}_{nc}(n)$ are further studied. We show that Connes' embedding problem is equivalent to deciding whether or not two smaller versions…

Let $n \geq 4$. The helm graph $H_n$ on $2n-1$ vertices is obtained from the wheel graph $W_n$ by adjoining a pendant edge to each vertex of the outer cycle of $W_n$. Suppose $n$ is even. Let $D := [d_{ij}]$ be the distance matrix of $H_n$.…

For fixed $\alpha \in [0,1]$, consider the set $S_{\alpha,N}$ of dilated squares $\alpha, 4\alpha, 9\alpha, \dots, N^2\alpha \, $ modulo $1$. Rudnick and Sarnak conjectured that for Lebesgue almost all such $\alpha$ the gap-distribution of…

The problem of efficiently characterizing degree sequences of simple hypergraphs is a fundamental long-standing open problem in Graph Theory. Several results are known for restricted versions of this problem. This paper adds to the list of…

In this paper, we analyze the cross-correlation properties for Chu sequences, which provide information on the distribution of the maximum magnitudes of the cross-correlation function. Furthermore, we can obtain the number of available…

A recently posed question of Haggkvist and Scott's asked whether or not there exists a constant c such that if G is a graph of minimum degree ck then G contains cycles of k consecutive even lengths. In this paper we answer the question by…

In 1977, Borodin and Kostochka conjectured that every graph with maximum degree $\Delta \geq 9$ is $(\Delta-1)$-colourable, unless it contains a clique of size $\Delta$. In 1999, Reed confirmed the conjecture when $\Delta\geq 10^{14}$. We…

The Hamming graph $H(d,n)$ is the Cartesian product of $d$ complete graphs on $n$ vertices. Let $m=d(n-1)$ be the degree and $V = n^d$ be the number of vertices of $H(d,n)$. Let $p_c^{(d)}$ be the critical point for bond percolation on…

Mubayi and Verstraete conjectured that if $T$ is a tree on $t + 1$ vertices, then any $n$-vertex graph $G$ with average degree $d$ contains at least \[ n d(d - 1) \cdots (d - t + 1) \] labeled copies of $T$ as long as $d$ is sufficiently…

We obtain explicit factorizations of reduced period polynomials of degree $2^m$, $m\ge 4$, for finite fields of characteristic $p\equiv 3$ or $5\pmod{8}$. This extends the results of G. Myerson, who considered the cases $m=1$ and $m=2$, and…

Multivariate spatial phenomena are ubiquitous, spanning domains such as climate, pandemics, air quality, and social economy. Cross-correlation between different quantities of interest at different locations is asymmetric in general. This…

Let q>1 and m>0 be relatively prime integers. We find an explicit period $\nu_m(q)$ such that for any integers n>0 and r we have $[n+\nu_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q),…

Let $m\geq 3$, we prove that $(\alpha n^\theta \mod 1)_{n>0}$ has Poissonian $m$-point correlation for all $\alpha>0$, provided $\theta<\theta_m$, where $\theta_m$ is an explicit bound which goes to $0$ as $m$ increases. This work builds on…

The decimal expansion of 1/7 is 0.142857142857..., the block 142857 repeating forever. We call 142857 the period and its length is 6 = 2x3. If the period is broken into 2 pieces each of length 3 which are then added, the result is 142 + 857…

For positive integers $d<k$ and $n$ divisible by $k$, let $m_{d}(k,n)$ be the minimum $d$-degree ensuring the existence of a perfect matching in a $k$-uniform hypergraph. In the graph case (where $k=2$), a classical theorem of Dirac says…

We study the effect of quenched random field disorder on a driven elastic interface close to the depinning transition at the upper critical dimension d_{c}=4 using the functional renormalization group. We have found that the displacement…

We call $(a_1, \dots, a_n)$ an \emph{$r$-partial sequence} if exactly $r$ of its entries are positive integers and the rest are all zero. For ${\bf c} = (c_1, \dots, c_n)$ with $1 \leq c_1 \leq \dots \leq c_n$, let $S_{\bf c}^{(r)}$ be the…

The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting,…