Here, η(b, f) = 2αb = 2βf is the total probability of cross-space transition, φ = min{Φ∗(QBF ),Φ∗(QFB)}. Corollary 1. The joint chain P+ is ergodic when the collapsed chains (QBF and QFB) are both ergodic. Lemma 3. Let P be a reversible transition matrix over X, such that P(x, x) ≥ ξ > 0 for each x ∈ X then its smallest eigenvalue λn has λn ≥ 2ξ − 1. Theorem 4. The hierarchically bridging Marko...

Convex programming duality is usually stated in its most general form, with convex objective functions and convex constraints. (The book by Boyd and Vandenberghe is an excellent reference [2].) At this level of generality the process of constructing a dual of a convex program is not so simple, in contrast to LP duality where there is a simple set of rules one can use for the same purpose. In th...

Abstract We evaluate the classic sum $$\sum _{n\in {\mathbb {Z}}} e^{-\pi n^2}$$ ∑n∈Ze-πn2</mm...

in this paper, we show that the conditional transform with respect to the gaussian process involving the first variation can be expressed in terms of the conditional transform without the first variation. we then use this result to obtain various integration formulas involving the conditional $diamond$-product and the first variation.