Related papers: Odd Vector Cycles in $\mathbb{Z}^m$

We give a polynomial improvement to the cycle-complete Ramsey numbers \[ r(C_{\ell},K_k) \geq k^{1+1/(\ell- 2) + \varepsilon_{\ell} + o(1)}, \] for all fixed odd $\ell > 7$ with $k \rightarrow \infty$, for some $\varepsilon_{\ell} > 0$.

We define integral geometric analogues of the Chern classes for real vector bundle on a smooth real variety. More precisely, we define the Chern densities of a real bundle. These densities are analogues of the Chern forms of a complex…

Variants of the coordinate descent approach for minimizing a nonlinear function are distinguished in part by the order in which coordinates are considered for relaxation. Three common orderings are cyclic (CCD), in which we cycle through…

J.P. Serre showed that for any integer $m,~a(n)\equiv 0 \pmod m$ for almost all $n,$ where $a(n)$ is the $n^{\text{th}}$ Fourier coefficient of any modular form with rational coefficients. In this article, we consider a certain class of…

The $k$-mismatch problem consists in computing the Hamming distance between a pattern $P$ of length $m$ and every length-$m$ substring of a text $T$ of length $n$, if this distance is no more than $k$. In many real-world applications, any…

We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$…

Let $n,m$ be positive integers and $c \in \mathbb{Z}_n$, where $\mathbb{Z}_n$ is the ring of integers modulo $n$. We almost complete providing the answer to the following problem, partially solved by N. Alon. Does any infinite sequence over…

In this paper, we mainly establish a congruence for a sum involving Ap\'{e}ry numbers, which was conjectured by Z.-W. Sun. Namely, for any prime $p>3$ and positive odd integer $m$, we prove that there is a $p$-adic integer $c_m$ only…

The main object of this paper is to determine the maximum number of $\{0,\pm 1\}$-vectors subject to the following condition. All vectors have length $n$, exactly $k$ of the coordinates are $+1$ and one is $-1$, $n \geq 2k$. Moreover, there…

In a recent paper (2024) M. Buratti and M.E:Muzychuck have established some lower bounds on the number of non isomorphic cyclic Steiner Triple Systems of order $v\equiv 1$ (mod $6$). We complete their result to the case $v\equiv 3$ (mod…

If V is the vertex sequence of a symmetric 2t-cycle in the hypercube graph with the vertices {1,-1}^t, then for any vertex T of the graph there exists a unique inclusion-minimal subset of V such that T is the sum of its elements. We present…

Let $\log^{2+\varepsilon} n \le d \le n/2$ for some fixed $\varepsilon \in (0,1)$, and let $M_n$ be an $n\times n$ random matrix with entries in ${0,1}$, where each row is independently and uniformly sampled from the set of all vectors in…

The Ramsey number $r(C_{\ell},K_n)$ is the smallest natural number $N$ such that every red/blue edge-colouring of a clique of order $N$ contains a red cycle of length $\ell$ or a blue clique of order $n$. In 1978, Erd\H{o}s, Faudree,…

For the minimal O(N) sigma model, which is defined to be generated by the O(N) scalar auxiliary field alone, all n-point functions, till order 1/N included, can be expressed by elementary functions without logarithms. Consequently, the…

CVT and XOR are two binary operations together used to calculate the sum of two non-negative integers on using a recursive mechanism. In this present study the convergence behaviors of this recursive mechanism has been captured through a…

This paper addresses the problem of finding $Q_{m,t}\left(n\right)$, the number of possible ways to partition any member $n$ of the cyclic group $\mathbb{Z}/m\mathbb{Z}$ into $t$ distinct parts. When $m$ is odd, it was previously known that…

In this paper we contribute to the construction of families of arithmetically Cohen-Macaulay (aCM) indecomposable vector bundles on a wide range of polarized surfaces $(X,\Oo_X(1))$ for $\Oo_X(1)$ an ample line bundle. In many cases, we…

The central charge $C_T$ is computed for scalar and Dirac fields propagating according to GJMS-type kinetic operators acting on odd $d$-dimensional spheres in the presence of a spherical monodromy. The relation of $C_T$ to the derivatives…

One can recover vectors from $\mathbb{R}^m$ with arbitrary precision, using only $\lceil \log_2(m+1)\rceil +1$ continuous measurements that are chosen adaptively. This surprising result is explained and discussed, and we present…

Given a closed Riemannian manifold $(M^m,g)$ and a vector field $v$ on $M$, we form the Sasaki metric $g_S$ on $TM$, and restrict it to the image of the cross section map of $M$ into $TM$ defined by $v$, whose pull back to $M$ defines a new…