Related papers: Amending the Lonely Runner Spectrum Conjecture

Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o(1)) \frac{3n^2}{2\pi^2}$. This conjecture can be rephrased in terms of the spectral gap as follows: the…

A centipede made of $N$ quantum walkers on a one-dimensional lattice is considered. The distance between two consecutive legs is either one or two lattice spacings, and a global constraint is imposed: the maximal distance between the first…

We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect…

Let $n$ be a large integer and $M_n$ be a random $n$ by $n$ matrix whose entries are i.i.d. Bernoulli random variables (each entry is $\pm 1$ with probability 1/2). We show that the probability that $M_n$ is singular is at most $(3/4…

We show that the Subgroup distance problem regarding the Hamming distance, the Cayley distance and the $l_\infty$ distance is NP-complete when the input group is cyclic. When we restrict the $l_\infty$ distance to fixed values we show that…

For a non-square positive integer x, let k_x denote the distance between x^3 and the perfect square closest to x^3. A conjecture of Marshall Hall states that the ratios r_x = (x^(1/2))/k_x, are bounded above. (Elkies has shown that any such…

In theories, whose Lorentz invariance is violated by involvement of an external any-rank tensor, we show that the standard relativistic rule still holds true for summing the signal speed, understood as the group velocity of a wave, with the…

Let $A_n$ be a random symmetric matrix with Bernoulli $\{\pm 1\}$ entries. For any $\kappa>0$ and two real numbers $\lambda_1,\lambda_2$ with a separation $|\lambda_1-\lambda_2|\geq \kappa n^{1/2}$ and both lying in the bulk…

Given a compact $E \subset \mathbb{R}^n$ and $s > 0$, the maximum distance problem seeks a compact and connected subset of $\mathbb{R}^n$ of smallest one dimensional Hausdorff measure whose $s$-neighborhood covers $E$. For $E\subset…

This work presents the solution of the Signal Sleuths team for the 2024 SHL recognition challenge. The challenge involves detecting transportation modes using shuffled, non-overlapping 5-second windows of phone movement data, with exactly…

We revise and extend the algorithm provided in [1] to compute the finite Connes' distance between normal states. The original formula in [1] contains an error and actually only provides a lower bound. The correct expression, which we…

The speed of light is a complicated synthesizer quantity with distinctive origins which lead to coincident values in the standard theory. Due to the fact that different aspects of speed of light do not coincide in the local inertial frame…

We give an optimal upper bound for the maximum-norm distance from a vertex of a knapsack polyhedron to its nearest feasible lattice point. In a randomised setting, we show that the upper bound can be significantly improved on average. As a…

It is known that the Scholz conjecture on addition chains is true for all integers $n$ with $\ell(2n) = \ell(n)+1$. There exists infinitely many integers with $\ell(2n) \leq \ell(n)$ and we don't know if the conjecture still holds for them.…

We consider the convergence of the empirical spectral measures of random $N \times N$ unitary matrices. We give upper and lower bounds showing that the Kolmogorov distance between the spectral measure and the uniform measure on the unit…

Given an integer $k$, deciding whether a graph has a clique of size $k$ is an NP-complete problem. Wilf's inequality provides a spectral bound for the clique number of simple graphs. Wilf's inequality is stated as follows: $\frac{n}{n -…

We study the spectral gap of subgraphs of the hypercube induced by monotone subsets of vertices. For a monotone subset $A\subseteq\{0,1\}^{n}$ of density $\mu(A)$, the previous best lower bound on the spectral gap, due to Cohen, was…

In any spatially discrete model of pedestrian motion which uses a regular lattice as basis, there is the question of how the symmetry between the different directions of motion can be restored as far as possible but with limited…

The Wiener index of a connected graph is defined as the sum of distances between all its unordered pairs of vertices. Characterising graphs on $n$ vertices with a fixed diameter that maximise the Wiener index is a long-standing open…

Let $\a$ be a complex random variable with mean zero and bounded variance $\sigma^{2}$. Let $N_{n}$ be a random matrix of order $n$ with entries being i.i.d. copies of $\a$. Let $\lambda_{1}, ..., \lambda_{n}$ be the eigenvalues of…