Related papers: A note on the run length function for intermittenc…

We prove under the Bombieri-Lang conjecture for surfaces that there is an absolute bound on the length of sequences of integer squares with constant second differences, for sequences which are not formed by the squares of integers in…

Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We…

In this short note we observe that recent results of Abert and Hubai and of Csikvari and Frenkel about Benjamini--Schramm continuity of the holomorphic moments of the roots of the chromatic polynomial extend to the theory of dense graph…

This paper considers a game in which a single cop and a single robber take turns moving along the edges of a given graph $G$. If there exists a strategy for the cop which enables it to be positioned at the same vertex as the robber…

The periods of polynomials can be used to characterize discrete structures such as algebraic error control codes and feedback shift registers. We study trinomial $x^h+x+1$ over GF(2), which has the maximum number of consecutive zero…

This is essentially a note on Section 7 of Perelman's first paper on Ricci flow. We list some basic properties of the index form for Perelman's $ \mathcal{L} $-length, which are analogous to the ones in Riemannian case (with fixed metric),…

To solve a linear program, the simplex method follows a path in the graph of a polytope, on which a linear function increases. The length of this path is an key measure of the complexity of the simplex method. Numerous previous articles…

Given a finite graph $G$, the maximum length of a sequence $(v_1,\ldots,v_k)$ of vertices in $G$ such that each $v_i$ dominates a vertex that is not dominated by any vertex in $\{v_1,\ldots,v_{i-1}\}$ is called the Grundy domination number,…

For a graph $G$, $n$ denotes the order of $G$, $p$ the order of a longest path in $G$ and $c$ the order of a longest cycle. We show that if $G$ is a 2-connected graph such that $d(x)+d(y)+d(z)\ge p+2$ for all triples $x,y,z$ of independent…

We study the simple random walk on the Uniform Infinite Half-Plane Map, which is the local limit of critical Boltzmann planar maps with a large and simple boundary. We prove that the simple random walk is recurrent, and that the resistance…

It is now well known that the performance of a linear code $C$ under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for $C$. Several recent…

We discuss asymptotics of the zeros of orthogonal polynomials on the real line and on the unit circle when the recursion coefficients are periodic. The zeros on or near the absolutely continuous spectrum have a clock structure with spacings…

We show that if $\varphi \colon \mathbb R\rightarrow\mathbb R$ is a continuous mapping and the set of nonlinearity of $\varphi$ has nonzero Lebesgue measure, then $\varphi$ maps bijectively a certain set that contains arbitrarily long…

Lattice reduction algorithms have numerous applications in number theory, algebra, as well as in cryptanalysis. The most famous algorithm for lattice reduction is the LLL algorithm. In polynomial time it computes a reduced basis with…

In a generic dynamical system chaos and regular motion coexist side by side, in different parts of the phase space. The border between these, where trajectories are neither unstable nor stable but of marginal stability, manifests itself…

We give an explicit formula for the logarithmic potential of the asymptotic zero-counting measure of the sequence $\left\{\frac{\mathrm{d}^n}{\mathrm{d}z^n}\left(R(z)\exp{T(z)}\right)\right\}$. Here, $R(z)$ is a rational function with at…

By using a jump transformation associated to the Romik map, we define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. Among…

We study a variant of the R\'enyi parking problem in which car length is repeatedly halved and determine the rate at which the remaining space decays.

P-time event graphs are discrete event systems able to model cyclic production systems where tasks need to be performed within given time windows. Consistency is the property of admitting an infinite execution of such tasks that does not…

In this article, we study the structure of the graph implied by a given map on the set $S_p=\{1,2,\dots,p-1\}$, where $p$ is an odd prime. The consecutive applications of the map generate an integer sequence, or in graph theoretical context…