Approximately locating an invisible agent in a graph with relative distance queries

In a pursuit evasion game on a finite, simple, undirected, and connected graph $G$, a first player visits vertices $m_1,m_2,\dots$ of $G$, where $m_{i+1}$ is in the closed neighborhood of $m_i$ for every $i$, and a second player probes arbitrary vertices $c_1,c_2,\dots$ of $G$, and learns whether or not the distance between $c_{i+1}$ and $m_{i+1}$ is at most the distance between $c_i$ and $m_i$. Up to what distance $d$ can the second player determine the position of the first? For trees of bounded maximum degree and grids, we show that $d$ is bounded by a constant. We conjecture that $d=O(logn)$ for every graph $G$ of order $n$, and show that $d=0$ if $m_{i+1}$ may differ from $m_i$ only if $i$ is a multiple of some sufficiently large integer.     

Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $b(G_1,G_2,\dots ,G_k)$ is the least positive integer $b$ so that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a copy of $G_i$ in the $i$th color for some $i$. In this paper, our main focus will be to bound the following numbers: $b(C_{2t_1},C_{2t_2})$ and $b(C_{2t_1},C_{2t_2},C_{2t_3})$ for all $t_i\ge 3,$$b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4})$ for $3\le t_i\le 9,$ and $b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4},C_{2t_5})$ for $3\le t_i\le 5.$ Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.     

Generating regular elements

Let R be a prime right Goldie ring. A useful fact is that, if $a,b\in R$ are such that $aR+bR$ contains a regular element, then there exists $\lambda \in R$ such that $a+b\lambda$ is regular. We show that the analogous result holds for $n?1$ pairs of elements: if R contains a field of cardinality at least $n+1$, and if $a_i,b_i\in R$ are such that $a_{i}R+b_{i}R$ contains a regular element for $1?i?n$, then there exists a single element $\lambda \in R$ such that $a_i+b_i\lambda$ is regular for each i.     

Dyadic representations of graphs

A graph $G$ is dyadic provided it has a representation $v\to S_v$ from vertices $v$ of $G$ to subtrees $S_v$ of a host tree $T$ with maximum degree 3 such that $\left(i\right)$$v$ and $w$ are adjacent in $G$ if and only if $S_v$ and $S_w$ share at least three nodes and $\left(ii\right)$ each edge of $T$ is used by exactly two representing subtrees. We show that a connected graph is dyadic if and only if it can be constructed from edges and cycles by gluing vertices to vertices and edges to edges.     

Disjoint sparsity for signal separation and applications to hybrid inverse problems in medical imaging

The main focus of this work is the reconstruction of the signals f and $g_i$, $i=1,\dots ,N$, from the knowledge of their sums $h_i=f+g_i$, under the assumption that f and the $g_i$'s can be sparsely represented with respect to two different dictionaries $A_f$ and $A_g$. This generalizes the well-known “morphological component analysis” to a multi-measurement setting. The main result of the paper states that f and the $g_i$'s can be uniquely and stably reconstructed by finding sparse representations of $h_i$ for every i with respect to the concatenated dictionary $[A_f,A_g]$, provided that enough incoherent measurements $g_i$ are available. The incoherence is measured in terms of their mutual disjoint sparsity.This method finds applications in the reconstruction procedures of several hybrid imaging inverse problems, where internal data are measured. These measurements usually consist of the main unknown multiplied by other unknown quantities, and so the disjoint sparsity approach can be directly applied. As an example, we show how to apply the method to the reconstruction in quantitative photoacoustic tomography, also in the case when the Grüneisen parameter, the optical absorption and the diffusion coefficient are all unknown.     

On even (unitary) perfect polynomials over F2

We give all even perfect (resp. unitary perfect) polynomials over the prime field ${\mathbb{F}}_2$ of the form $x^a{(x+1)}^b{M}_1^{h_1}?M_r^{h_r}$, where each $M_i$ is a Mersenne irreducible polynomial, $h_i=2^{n_i}?1$ (resp. $h_i=2^{n_i}$) and $a,b,r,n_i\in \mathbb{N}$. In particular, we characterize nine of the eleven known “sporadic” even perfect polynomials over ${\mathbb{F}}_2$ that have the above form.