Bisymmetric functionals revisited or a converse of the Fubini theorem

We observe that bisymmetry is in fact the assertion of the Fubini theorem and we describe the form of general bisymmetric operations on some function spaces.

Stability of reaction–diffusion systems with stochastic switching

In this paper, we investigate the stability for reaction systems with stochastic switching. Two types of switched models are considered: (i) Markov switching and (ii) independent and identically distributed switching. By means of the ergodic property of Markov chain, Dynkin formula and Fubini theorem, together with the Lyapunov direct method, some sufficient conditions are obtained to ensure that the zero solution of reaction–diffusion systems with Markov switching is almost surely exponential stable or exponentially stable in the mean square. By using Theorem 7.3 in [R. Durrett, Probability: Theory and Examples, Duxbury Press, Belmont, CA, 2005], we also investigate the stability of reaction–diffusion systems with independent and identically distributed switching. Meanwhile, an example with simulations is provided to certify that the stochastic switching plays an essential role in the stability of systems.

Some Applications Idempotent Probability Measures Space

For idempotent probability measures, max–plus variant of Fubini theorem is established. Further, it is proven metrizability of a given compact if the space of idempotent probability measures defined on the compact, is hereditary normal.

Fubini’s Theorem

Summary Fubini theorem is an essential tool for the analysis of high-dimensional space [8], [2], [3], a theorem about the multiple integral and iterated integral. The author has been working on formalizing Fubini’s theorem over the past few years [4], [6] in the Mizar system [7], [1]. As a result, Fubini’s theorem (30) was proved in complete form by this article.