Related papers: Stahl's Theorem (aka BMV Conjecture): Insights and…
We prove the BMV (Bessis, Moussa, Villani, 1975) conjecture, which states that the function t -> Tr exp(A-tB), t \geq 0, is the Laplace transform of a positive measure on [0,\infty) if A and B are n x n Hermitian matrices and B is positive…
We find further implications of the BMV conjecture, which states that for hermitian matrices A and B, the function Tr exp(A - t B) is the Laplace transform of a positive measure.
The BMV conjecture for traces, which states that $Tr exp(A -\lambda B)$ is the Laplace transform of a positive measure, is shown to be equivalent to two other statements: (i) The polynomial $\lambda\mapsto Tr(A+\lambda B)^p$ has only…
The paper contains a simplified version of Stahl's proof of a conjecture of Bessis, Moussa and Villani on the trace of matrices A+tB with Hermitean A and B.
We study trace functions on the form $ t\to\tr f(A+tB) $ where $ f $ is a real function defined on the positive half-line, and $ A $ and $ B $ are matrices such that $ A $ is positive definite and $ B $ is positive semi-definite. If $ f $…
A long-standing conjecture asserts that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has nonnegative coefficients whenever $m$ is a positive integer and $A$ and $B$ are any two $n \times n$ positive semidefinite Hermitian matrices. The…
The BMV conjecture states that for $n\times n$ Hermitian matrices $A$ and $B$ the function $f_{A,B}(t)=trace{\, } e^{tA+B}$ is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is…
The Bessis-Moussa-Villani (BMV) conjecture, originating in quantum statistical mechanics, was proved by Stahl after an influential reformulation by Lieb and Seiringer. A later refinement asks whether the normalized average over all words…
Consider the polynomial $tr (A + tB)^m$ in $t$ for positive hermitian matrices $A$ and $B$ with $m \in \N$. The Bessis-Moussa-Villani conjecture (in the equivalent form of Lieb and Seiringer) states that this polynomial has nonnegative…
The reformulation of the Bessis-Moussa-Villani conjecture given by Lieb and Seiringer asserts that the coefficient of t^r in the polynomial Trace[(A+tB)^p], with A and B positive semidefinite matrices, is nonnegative for all p and r. We…
The BMV conjecture states that for \(n\times n\) Hermitian matrices \(A\) and \(B\) the function \(f_{A,B}(t)=\tr e^{tA+B}\) is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is…
It is shown that the polynomial $\lambda(t) = {\rm Tr}[(A + tB)^p]$ has nonnegative coefficients when $p \leq 7$ and A and B are any two complex positive semidefinite $n \times n$ matrices with arbitrary $n$. This proofs a general…
Consider the polynomial $tr (A + tB)^m$ in $t$ for positive hermitian matrices $A$ and $B$ with $m \in \N$. The Bessis-Moussa-Villani conjecture (in the equivalent form of Lieb and Seiringer) states that this polynomial has nonnegative…
We obtain a solution to the Bessis-Moussa-Villani conjecture for a trace-class perturbation of a semi-bounded operator and answer affirmatively the question on positivity of higher order spectral shift functions in the setting of…
The Shimura-Taniyama conjecture states that the Mellin transform of the Hasse-Weil L-function of any elliptic curve defined over the rational numbers is a modular form. Recent work of Wiles, Taylor-Wiles and Breuil-Conrad-Diamond-Taylor has…
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 analyze why the refined Bessis--Moussa--Villani conjecture fails. The refined conjecture proposed that the normalized trace average over all words with prescribed numbers of letters \(A\) and \(B\) should be bounded above by the…
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 $A_{uv}=\{\sigma \in S_n \, : \, \sigma(u)<\sigma(v) \}$. $\textbf{Theorem.}$…
In the study of Zeilberger's conjecture on an integer sequence related to the Catalan numbers, Lassalle proposed the following conjecture. Let $(t)_n$ denote the rising factorial, and let $\Lambda_{\mathbb{R}}$ denote the algebra of…
In this paper we prove a conjecture regarding the form of the Born-Infeld Lagrangian with a U(1)^2n gauge group after the elimination of the auxiliary fields. We show that the Lagrangian can be written as a symmetrized trace of Lorentz…