Kernel theorems and nuclearity in idempotent mathematics. An algebraic approach

幂等/热带分析，Hamilton-Jacobi和Bellman方程

Idempotent/tropical analysis, the Hamilton-Jacobi and Bellman equations

本文讨论幂等/热带分析及其与Hamilton-Jacobi方程和矩阵Bellman方程的关系，通过Maslov去量子化将Schrödinger方程转化为Hamilton-Jacobi-Bellman方程，揭示其在热带代数上的线性性质，并研究幂等数学中的对应原理、叠加原理、矩阵Bellman方程在图优化中的应用以及通用数值算法。

The idempotent Radon--Nikodym theorem has a converse statement

Idempotent integration is an analogue of the Lebesgue integration where $σ$-additive measures are replaced by $σ$-maxitive measures. It has proved useful in many areas of mathematics such as fuzzy set theory, optimization, idempotent analysis, large deviation theory, or extreme value theory. Existence of Radon--Nikodym derivatives, which turns out to be crucial in all of these applications, was proved by Sugeno and Murofushi. Here we show a converse statement to this idempotent version of the Radon--Nikodym theorem, i.e. we characterize the $σ$-maxitive measures that have the Radon--Nikodym pr…

What is the role of continuity in continuous linear forms representation?

The recent extensions of domain theory have proved particularly efficient to study lattice-valued maxitive measures, when the target lattice is continuous. Maxitive measures are defined analogously to classical measures with the supremum operation in place of the addition. Building further on the links between domain theory and idempotent analysis highlighted by Lawson (2004), we investigate the concept of domain-valued linear forms on an idempotent (semi)module. In addition to proving representation theorems for continuous linear forms, we address two applications: the idempotent Radon--Nikod…