Related papers: On the J\'onsson distributivity spectrum

Let $F_k$ be the set of graphs on $k$ vertices. For a graph $G$, a $k$-decomposition is a set of induced subgraphs of $G$, each isomorphic to an element of $F_k$, such that each pair of vertices of $G$ is in exactly one element of the set.…

Let $M_n$ be a class of symmetric sparse random matrices, with independent entries $M_{ij} = \delta_{ij} \xi_{ij}$ for $i \leq j$. $\delta_{ij}$ are i.i.d. Bernoulli random variables taking the value $1$ with probability $p \geq…

Consider the Hales-Jewett theorem. The $k$-dimensional version of it tells us that the combinatorial space $\mathcal{U}_{M, \Lambda} = \{ \eta \mid \eta: M \to \Lambda \}$ has, under suitable assumptions, monochromatic $k$-dimensional…

Let $\mathbf X=(X_{jk})$ denote a Hermitian random matrix with entries $X_{jk}$, which are independent for $1\le j\le k$. We consider the rate of convergence of the empirical spectral distribution function of the matrix $\mathbf X$ to the…

Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chv\'{a}tal's theorem says that for any fixed $n\geq 2$, as $m$ ranges over $\{0,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when $m$ is…

An identity s=t is linear if each variable occurs at most once in each of the terms s and t. Let T be a tolerance relation of an algebra A in a variety defined by a set of linear identities. We prove that there exist an algebra B in the…

Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: $$c_{\delta,\beta}^{(n)} \prod_{1\leq…

Let $n>1$ and $k>0$ be fixed integers. A matrix is said to be level if all its column sums are equal. A level matrix with $m$ rows is called reducible if we can delete $j$ rows, $0<j<m$, so that the remaining matrix is level. We ask if…

Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove…

For Bernoulli percolation on a given graph $G = (V,E)$ we consider the cluster of some fixed vertex $o \in V$. We aim at comparing the number of vertices of this cluster in the set $V_+$ and in the set $V_-$, where $V_+,V_- \subset V$ have…

In the local, characteristic 0, non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We prove that such distributions are invariant by transposition. This implies that an…

Motivated by a question of Grinblat, we study the minimal number $\mathfrak{v}(n)$ that satisfies the following. If $A_1,\ldots, A_n$ are equivalence relations on a set $X$ such that for every $i\in[n]$ there are at least $\mathfrak{v}(n)$…

For the complete five-valued cross-correlation distribution between two $m$-sequences ${s_t}$ and ${s_{dt}}$ of period $2^m-1$ that differ by the decimation $d={{2^{2k}+1}\over {2^k+1}}$ where $m$ is odd and $\mbox{gcd}(k,m)=1$, Johansen…

Let $\mathfrak g$ be a reductive Lie algebra, and $m$ a positive integer. There is a natural density of irreducible representations of $\mathfrak g$, whose degrees are not divisible by $m$. For $\mathfrak g=\mathfrak{gl}_n$, this density…

We study rectangular Vandermonde matrices $\mathbf{V}$ with $N+1$ rows and $s$ irregularly spaced nodes on the unit circle, in cases where some of the nodes are "clustered" together -- the elements inside each cluster being separated by at…

For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge n/k$…

In (B-Gran, 2004), was given a categorical formulation of the Shifting Lemma which is a characterization of the Congruence Modular Varieties among all the variety of Universal Algebra, introduced in (Gumm, 1983). Starting from a…

We classify all pairs (G,V) with G a closed subgroup in a classical group with natural module V over the complex numbers such that G has the same composition factors on the kth tensor power of V, for a fixed (small) k. In particular, we…

Let $G=(V,E)$ be a simple connected graph. A set of vertices $S\subseteq V$ is said to be a dominating set if for any vertex in $V\setminus S$ is adjacent to at least one vertex in $S$. The domination number $\gamma(G)$ of $G$ is the…

Meet semidistributive varieties are in a sense the last of the most important classes in universal algebra for which it is unknown whether it can be characterized by a strong Maltsev condition. We present a new, relatively simple Maltsev…