Related papers: Herbert Stahl's proof of the BMV conjecture

By finding orthogonal representation for a family of simple connected called $\delta$-graphs it is possible to show that $\delta$-graphs satisfy delta conjecture. An extension of the argument to graphs of the form…

We give a criterion for modular extension of rank-4 hypermodular matroids, and prove a weakening of Kantor's conjecture for rank-4 realizable matroids. This proves the sticky matroid conjecture and Kantor's conjecture for realizable…

In this short note, we state a stable and a $\tau$-reduced version of the second Brauer-Thrall Conjecture. The former is a slight strengthening of a brick version of the second Brauer-Thrall Conjecture raised by Mousavand and…

We give a short proof based on Lusztig's generalized Springer correspondence of some results of [BrCh,BaCr,P].

For two hermitian matrices $A$ and $B$, Golden-Thompson inequality \cite{golden1965lower, thompson1965inequality} states that $$ \mathrm{tr}\left[ \exp{(A+B)} \right] \leq \mathrm{tr}\left[ \exp{(A)}\exp{(B)} \right]. $$ We elaborate here…

When specialized to the context of permutations, Schmerl and Trotter's Theorem states that every simple permutation which is not a parallel alternation contains a simple permutation with one fewer entry. We give an elementary proof of this…

We improve and expand in two directions the theory of norms on complex matrices induced by random vectors. We first provide a simple proof of the classification of weakly unitarily invariant norms on the Hermitian matrices. We use this to…

In (Haglund, Remmel, Wilson 2018) Haglund, Remmel and Wilson introduced their Delta conjectures, which give two different combinatorial interpretations of the symmetric function $\Delta'_{e_{n-k-1}} e_n$ in terms of rise-decorated or…

We prove the following conjecture of Leighton and Moitra. Let $T$ be a tournament on $[n]$ and $S_n$ the set of permutations of $[n]$. For an arc $uv$ of $T$, let $A_{uv}=\{\sigma \in S_n \, : \, \sigma(u)<\sigma(v) \}$. $\textbf{Theorem.}$…

We report the results of our empirical investigations on the Bateman-Horn conjecture. This conjecture, in its commonly known form, produces rather large deviations when the polynomials involved are not monic. We propose a modified version…

In this paper, a conjecture of Mazur, Rubin and Stein concerning certain averages of modular symbols is proved.

This note gives an informal overview of the proof in our paper "Borel Conjecture and Dual Borel Conjecture", see arXiv:1105.0823.

We give again (see also arXiv:1112.0676) a proof of weighted estimate of any Calder\'on-Zygmund operator. This is under a universal sharp sufficient condition that is weaker than the so-called bump condition. Bump conjecture was recently…

In this note we give generalizations and prove 'minimalistic' refinements of the t-birational Section Conjecture (t-BSC), cf. [Be], by doing both: First, by extending the class of base fields over which the t-BSC holds, and second, by…

Let X and Y be two nxn Hermitian matrices. In the article "Proof of a conjectured exponential formula" (Linear and Multilinear Algebra (19) 1986, 187-197) R. C. Thompson proved that there exist two nxn unitary matrices U and V such that $$…

Elementary proofs of Sylvester's, Wolstenholme's, Morley's and Lehmer's congruence theorems

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation $w\in \Sn$ is at most the number of elements below $w$ in the Bruhat order, and (B) that equality…

Let $A=(a_{ij})$ be an $n\times n$ random matrix with i.i.d. entries such that $\mathbb{E} a_{11} = 0$ and $\mathbb{E} {a_{11}}^2 = 1$. We prove that for any $\delta>0$ there is $L>0$ depending only on $\delta$, and a subset $\mathcal{N}$…

The purpose of this note is to explain that the combinatorial local log-concavity conjecture introduced by Gross, Mansour, Tucker and Wang (Eur. J. Comb. 52, 207-222, 2016) in fact follows from a result of Stanley (Eur. J. Comb. 32 (6),…

We revisit Haiman's conjecture on the relations between characters of Kazdhan-Lusztig basis elements of the Hecke algebra over the symmetric group. The conjecture asserts that, for purposes of character evaluation, any Kazhdan-Lusztig basis…