All double generalized Petersen graphs are Hamiltonian

The double generalized Petersen graph $DP(n,t)$, $n\ge 3$ and $t\in {\mathbb{Z}}_n?\left\{0\right\}$, $2\le 2t<n$, has vertex-set $\{x_i,y_i,u_i,v_i\mid i\in {\mathbb{Z}}_n\}$, edge-set $\{\{x_i,x_{i+1}\},\{y_i,y_{i+1}\},\{u_i,v_{i+t}\},\{v_i,u_{i+t}\},\{x_i,u_i\},\{y_i,v_i\}\mid i\in {\mathbb{Z}}_n\}$. These graphs were first defined by Zhou and Feng as examples of vertex-transitive non-Cayley graphs. Then, Kutnar and Petecki considered the structural properties, Hamiltonicity properties, vertex-coloring and edge-coloring of $DP(n,t)$, and conjectured that all $DP(n,t)$ are Hamiltonian. In this paper, we prove this conjecture.     

On tiling the integers with 4-sets of the same gap sequence

Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set $\{x_1,\dots ,x_n\}$ of integers where $x_1<?<x_n$, let the gap sequence of this set be the unordered multiset $\{d_1,\dots ,d_{n?1}\}=\{x_{i+1}?x_i:i\in \{1,\dots ,n?1\}\}$. This paper addresses the following question, which was explicitly asked by Nakamigawa: can the set of integers be partitioned into sets with the same gap sequence? The question is known to be true for any set where the gap sequence has length at most two. This paper provides evidence that the question is true when the gap sequence has length three. Namely, we prove that given positive integers $p$ and $q$, there is a positive integer $r_0$ such that for all $r\ge r_0$, the set of integers can be partitioned into 4-sets with gap sequence $p,q$, $r$.     

Inférence fondée sur la vraisemblance pour des modèles de records

In a time series $\{X_t,t\ge 1\}$, $X_j$ is said to be an upper record if $X_j>\max ?\{X_1,\dots ,X_{j?1}\}$. Some popular models for records are the Yang–Nevzorov and the Linear Drift models. In this note, we introduce for these models the joint likelihood of the record sequence and the indicators of their occurrence. This likelihood can then be used to obtain estimators of the unknown parameters in the models. It can also be used to derive inferential procedures associated with the selection of a proper model for such data.     

Matrix completion problems over integral domains: The case with a diagonal of prescribed blocks

Let $\mathfrak{R}$ be an arbitrary integral domain, let $\wedge =\{{\lambda }_1\text{,}\dots \text{,}{\lambda }_n\}$ be a multiset of elements of $\mathfrak{R}$, let $\sigma$ be a permutation of $\{1\text{,}\dots \text{,}k\}$ let $n_1\text{,}\dots \text{,}{n}_k$ be positive integers such that $n_1+?+n_k=n$, and for $r=1\text{,}\dots \text{,}k$ let $A_r\in {\mathfrak{R}}^{n_r\times n_{\sigma (r)}}$. We are interested in the problem of finding a block matrix $Q={\left[Q_{\mathit{rs}}\right]}_{r\text{,}s=1}^k\in {\mathfrak{R}}^{n\times n}$ with spectrum $\mathrm{\Lambda }$ and such that $Q_{r\sigma (r)}=A_r$ for $r=1\text{,}\dots \text{,}k$. Cravo and Silva completely characterized the existence of such a matrix when $\mathfrak{R}$ is a field. In this work we construct a solution matrix $Q$ that solves the problem when $\mathfrak{R}$ is an integral domain with two exceptions: (i) $k=2$; (ii) $k\ge 3$, $\sigma (r)=r$ and $n_r>n/2$ for some $r$.What makes this work quite unique in this area is that we consider the problem over the more general algebraic structure of integral domains, which includes the important case of integers. Furthermore, we provide an explicit and easy to implement finite step algorithm that constructs an specific solution matrix (we point out that Cravo and Silva’s proof is not constructive).     

Fredholmness and smooth dependence for linear time-periodic hyperbolic systems

This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type${?}_t{u}_j+a_j(x){?}_x{u}_j+\sum _{k=1}^n{b}_{jk}(x)u_k=f_j(x,t),j=1,\dots ,n,$ with periodicity conditions in time and reflection boundary conditions in space. We state conditions on the data $a_j$ and $b_{jk}$ and the reflection coefficients such that the system is Fredholm solvable. Moreover, we state conditions on the data such that for any right-hand side there exists exactly one solution, that the solution survives under small perturbations of the data, and that the corresponding data-to-solution map is smooth with respect to appropriate function space norms. In particular, those conditions imply that no small denominator effects occur.We show that perturbations of the coefficients $a_j$ lead to essentially different results than perturbations of the coefficients $b_{jk}$, in general. Our results cover cases of non-strictly hyperbolic systems as well as systems with discontinuous coefficients $a_j$ and $b_{jk}$, but they are new even in the case of strict hyperbolicity and of smooth coefficients.