Related papers: Day's Theorem is sharp for $n$ even
It is well known that there may be many causal explanations that are consistent with a given set of data. Recent work has been done to represent the common aspects of these explanations into one representation. In this paper, we address…
If V is a finitely generated variety such that the first-order theory of the finite members of V is decidable, we show that V is residually finite, and in fact has a finite bound on the sizes of subdirectly irreducible algebras. This result…
We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao's solution to Varchenko's conjecture on complements of hyperplane…
Let $\alpha$, $\beta$, $\gamma, \dots$ $\Theta$, $\Psi, \dots$ $R$, $S$, $T, \dots$ be variables for, respectively, congruences, tolerances and reflexive admissible relations. Let juxtaposition denote intersection. We show that if the…
Strong bounds - going beyond Sarnak's density hypothesis - are obtained for the number of automorphic forms for the congruence subgroup Gamma_0(q) of SL_n(Z) violating the Ramanujan conjecture at any given unramified place. The proof is…
In the study of infinite words, various notions of balancedness provide quantitative measures for how regularly letters or factors occur, and they find applications in several areas of mathematics and theoretical computer science. In this…
In 2003, H\'{e}thelyi and K\"{u}lshammer proposed that if $G$ is a finite group and $p$ is a prime dividing the group order, then $k(G)\geq 2\sqrt{p-1}$, and they proved this conjecture for solvable $G$ and showed that it is sharp for those…
Dilworth's theorem. Every finite distributive lattice $D$ can be represented as the congruence lattice of a finite lattice $L$. We want: Every finite distributive lattice $D$ can be represented as the congruence lattice of a nice finite…
Different directed acyclic graphs (DAGs) may be Markov equivalent in the sense that they entail the same conditional independence relations among the observed variables. Meek (1995) characterizes Markov equivalence classes for DAGs (with no…
Given a rank 3 real arrangement $\mathcal A$ of $n$ lines in the projective plane, the Dirac-Motzkin conjecture (proved by Green and Tao in 2013) states that for $n$ sufficiently large, the number of simple intersection points of $\mathcal…
We prove a version of the Khinchine--Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This…
Let $\mathbf{A}$ be a finite algebra generating a finitely decidable variety and having nontrivial strongly solvable radical $\tau$. We provide an improved bound on the number of variables in which a term can be sensitive to changes within…
We investigate unital, subtractive and strongly unital regular categories with enough projectives and give characterizations of their projective covers. The categorical equation "strongly unital = unital + subtractive" is explored: this…
The study of extensions realizing affine datum is specialized to central extensions in varieties with a difference term which leads to generalizations of several classical theorems on central extensions from group theory. We establish a…
Strong compactness results for families of functions in seminormed nonnegative cones in the spirit of the Aubin-Lions-Dubinski\u{\i} lemma are proven, refining some recent results in the literature. The first theorem sharpens slightly a…
We employ combinatorial techniques to present an explicit formula for the coefficients in front of Chern classes involving in the Hattori-Stong integrability conditions. We also give an evenness condition for the signature of stably…
The "Modularity Conjecture" is the assertion that the join of two nonmodular varieties is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning…
We extend Campion's pasting theorem for $(\infty, n)$-categories to a larger class of polygraphs, called the directed complexes with frame-acyclic molecules. It follows, for instance, that this pasting theorem applies to any polygraph…
Solutions to conservation laws satisfy the monotonicity property: the number of local extrema is a non-increasing function of time, and local maximum/minimum values decrease/increase monotonically in time. This paper investigates this…
The partition function is known to exhibit beautiful congruences that are often proved using the theory of modular forms. In this paper, we study the extent to which these congruence results apply to the generalized Frobenius partitions…