In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class 1 functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory to the case of Baire class ξ functions, and generalize most of the results from the Baire class 1 case. We also show that their assumption of the compactness ...

Recently, a number of reversible functional programming languages have been proposed. Common to several of these is the assumption of totality, a property that is not necessarily desirable, and certainly not required in order to guarantee reversibility. In a categorical setting, however, faithfully capturing partiality requires handling it as additional structure. Recently, Giles studied invers...

From now on we will work with sets A ⊂ X within a fixed rectangle X ⊂ R. Elementary sets in X make an algebra. Definition 1.8. The outer measure of a setA ⊂ X is μ∗(A) = inf ∑∞ i=1Area(Ri) where the infimum is taken over all countable covers of A by rectangles, A ⊂ ∪i=1Ri. Definition 1.9. The inner measure of a set A ⊂ X is μ∗(A) = Area(X)− μ∗(Ac) where A = X \ A is the complement of A. Lemma. ...

in this paper the concepts of fuzzifying β − irresolute functions and fuzzifying β − compactspaces are characterized in terms of fuzzifying β − open sets and some of their properties are discussed.

In this paper, we obtain the Rényi entropy rate for irreducible-aperiodic Markov chains with countable state space, using the theory of countable nonnegative matrices. We also obtain the bound for the rate of Rényi entropy of an irreducible Markov chain. Finally, we show that the bound for the Rényi entropy rate is the Shannon entropy rate.

Abstract This paper provides four theorems on the existence of a free-disposal equilibrium in Walrasian economy: first with an arbitrary set agents compact consumption sets, next highlighting trade-offs involved relaxation compactness assumption, and last two countable endowed weighting structure. The results generalize antecedent literature pioneered by Shafer–Sonnenschein 1975, currently form...

A Tychonoff space X is called RG if the embedding of C(X) → C(Xδ) is an epimorphism of rings. Compact RG spaces are known and easily described. We study the pseudocompact RG spaces. These must be scattered of finite Cantor Bendixon degree but need not be locally compact. However, under strong hypotheses, (countable compactness, or small cardinality) these spaces must, indeed, be compact. The ma...

where (f + g)(u) = V_,,+,,,f(v) A f(w). Since E is countable, ZcB) can be defined for any countably complete Boolean algebra (ccBa) B where Z is the group of the integers. This kind of group was first (1962) studied by Balcerzyk [l]. However, it seems that not much attention was paid to such groups for a rather long period. Under the point of view in [l, Theorem 51 and [13, Proposition 11, it c...

A separated cell A of a topological space X is a family of pairwise disjoint open sets with the property that, for any partition A= A1 ∪ A2 of A, ⋃ A1 and ⋃ A2 are completely separated. The separated cellularity sc(X) of X is the supremum of all cardinals of separated cells. A space X has finite separation if it has no infinite separated cells. With X compact, every separated cell of X has size...

I use generic embeddings induced by generic normal measures on Pκ(λ) that can be forced to exist if κ is an indestructibly weakly compact cardinal. These embeddings can be used in order to obtain the forcing axioms MA(<μ-closed) in forcing extensions. This has consequences in V: The singular cardinal hypothesis holds above κ, and κ has a useful Jónsson-like property. This, in turn, implies that...