In matrix analysis Stahl's theorem is a theorem proved in 2011 by Herbert Stahl concerning Laplace transforms for special matrix functions. [1] It originated in 1975 as the Bessis-Moussa-Villani (BMV) conjecture by Daniel Bessis, Pierre Moussa, and Marcel Villani. [2] In 2004 Elliott H. Lieb and Robert Seiringer gave two important reformulations of the BMV conjecture. [3] In 2015, Alexandre Eremenko gave a simplified proof of Stahl's theorem. [4]
In 2023, Otte Heinävaara proved a structure theorem for Hermitian matrices introducing tracial joint spectral measures that implies Stahl's theorem as a corollary. [5]
Let denote the trace of a matrix. If and are Hermitian matrices and is positive semidefinite, define , for all real . Then can be represented as the Laplace transform of a non-negative Borel measure on . In other words, for all real ,
for some non-negative measure depending upon and . [6]
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices that are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.
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In mathematics, and especially differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stability was first introduced by Gang Tian and reformulated more algebraically later by Simon Donaldson. The definition was inspired by a comparison to geometric invariant theory (GIT) stability. In the special case of Fano varieties, K-stability precisely characterises the existence of Kähler–Einstein metrics. More generally, on any compact complex manifold, K-stability is conjectured to be equivalent to the existence of constant scalar curvature Kähler metrics.
