We prove a finite model theorem and infinitary completeness result for the propositional μ-calculus. The construction establishes a link between finite model theorems for propositional program logics and the theory of wellquasi-orders.

The double-controlled metric-type space (X,D) is a metric in which the triangle inequality has form D(η,μ)≤ζ1(η,θ)D(η,θ)+ζ2(θ,μ)D(θ,μ) for all η,θ,μ∈X. maps ζ1,ζ2:X×X→[1,∞) are called control functions. In this paper, we introduce novel generalization of double-composed space, where D(η,μ)≤αD(η,θ)+βD(θ,μ) our new functions α,β:[0,∞)→[0,∞) composed D inequality, multiplied with D. We establish s...

The aim of these notes is to illustrate a proof of the following remarkable Theorem of Alberti (first proved in [1]). Here, when μ is a Radon measure on Ω ⊂ R, we denote by μ its absolutely continuous part (with respect to the Lebesgue measure L ), by μ := μ− μ its singular part, and by |μ| its total variation measure. Clearly, |μ|a = |μa| and |μ|s = μ. When μ = Du for some u ∈ BV (Ω,R), we wil...

Reactions of [MoReCp(μ-PMes*)(CO)6] with internal alkynes RC≡CR yielded the phosphapropenylidene-bridged complexes [MoReCp(μ-κ2P,C:η3-PMes*CRCR)(CO)5] (Mes* = 2,4,6-C6H2tBu3; R CO2Me, Ph). Terminal HC≡CR1 gave mixtures isomers [MoReCp(μ-κ2P,C:η3-PMes*CHCR1)(CO)5] and [MoReCp(μ-κ2P,C:η3-PMes*CR1CH)(CO)5], first isomer being major (R1 CO2Me) or unique tBu), indicating relevance steric repulsions ...

(a) Write down the probability distribution of x. Solution: p(x = 0|μ) = 1− μ, so p(x|μ) = μx(1− μ)1−x, this is known as the Bernoulli distribution. (b) Show that this is a proper probability distribution, i.e. the probability sum up to 1. What is the expectation and variance of this distribution? Solution: ∑ x∈{0,1} p(x|μ) = p(x = 0|μ) + p(x = 1|μ) = 1− μ+ μ = 1 E(x) = ∑ x∈{0,1} xp(x|μ) = 0× p...