Let $\Gamma$ be a $d$-bounded distance-regular graph with $d\geq 3$. Suppose that $P(x)$ is a set of all strongly closed subgraphs containing $x$ and that $P(x,i)$ is a subset of $P(x)$ consisting of the elements of $P(x)$ with diameter $i$. Let ${\mathcal{L}}'(x,i)$ be the set generated by the join of the elements in $P(x,i)$. By ordering ${\mathcal{L}}'(x,i)$ by inclusion or reverse inclusion...

The problem of minimax detection of Gaussian random signal vector in White Gaussian additive noise is considered. It is supposed that an unknown vector $\boldsymbol{\sigma}$ of the signal vector intensities belong to the given set ${\mathcal E}$. It is investigated when it is possible to replace the set ${\mathcal E}$ by a smaller set ${\mathcal E}_{0}$ without loss of quality (and, in particul...

In his technical report~\cite[sec. 6]{barrontech}, Barron states that the de Bruijn's identity for Gaussian perturbations holds for any RV having a finite variance. In this report, we follow Barron's steps as we prove the existence of $J_{\alpha}\left(X + \sqrt[\alpha]{\eta}N\right)$, $\eta>0$ for any Radom Variable (RV) $X \in \mathcal{L}$ where \begin{equation*} \mathcal{L} = \left\{ \text{RV...

We propose the notion of {\it resistance of a graph} as an accompanying notion of the structure entropy to measure the force of the graph to resist cascading failure of strategic virus attacks. We show that for any connected network $G$, the resistance of $G$ is $\mathcal{R}(G)=\mathcal{H}^1(G)-\mathcal{H}^2(G)$, where $\mathcal{H}^1(G)$ and $\mathcal{H}^2(G)$ are the one- and two-dimensional s...

A compact metric space $(X, \rho)$ is given. Let $\mu$ be a Borel measure on $X$. By $r$-cluster we mean a measurable subset of $X$ with diameter at most $r$. A family of $k$ $2r$-clusters is called a $r$-cluster structure of order $k$ if any two clusters from the family are separated by a distance at least $r$. By measure of a cluster structure we mean a sum of clusters measures from the clust...

We show that call-by-need is observationally equivalent to weak-head needed reduction. The proof of this result uses a semantical argument based on a (non-idempotent) intersection type system called $\mathcal{V}$. Interestingly, system $\mathcal{V}$ also allows to syntactically identify all the weak-head needed redexes of a term.

We extend the notion of $P_4$-sparse graphs previously introduced by {\scshape Ho\`ang} by considering $\mathcal{F}$-sparse graphs were $\mathcal{F}$ denotes a finite set of graphs on $p$ vertices. Thus we obtain some results on $(P_5,\bar{P_5})$-sparse graphs already known on $(P_5,\bar{P_5})$-free graphs. Finally we completely describe the structure of $(P_5,\bar{P_5}, bull$)-sparse graphs, i...

We revisit the topic of polynomial kernels for Vertex Cover relative to structural parameters. Our starting point is a recent paper due to Fomin and Str{\o}mme [WG 2016] who gave a kernel with $\mathcal{O}(|X|^{12})$ vertices when $X$ is a vertex set such that each connected component of $G-X$ contains at most one cycle, i.e., $X$ is a modulator to a pseudoforest. We strongly generalize this re...

Given a finite family $\mathcal U$ of subsets $\mathbb Z^d\setminus \{0\}$, the U$-$voter\ dynamics$ in space configurations $\{+,-\}^{\mathbb Z^d}$ is defined as follows: every $v\in\mathbb Z^d$ has an independent exponential random clock, and when clock at $v$ rings, vertex chooses $X\in\mathcal uniformly random. If set $v+X$ entirely state $+$ (resp. $-$), then updates to otherwise nothing h...