De Finetti would claim that we can make sense of a draw in which each positive integer has equal probability of winning. This requires a uniform probability distribution over the natural numbers, violating countable additivity. Countable additivity thus appears not to be a fundamental constraint on subjective probability. It does, however, seem mandated by Dutch Book arguments similar to those ...

We prove that a Borel equivalence relation is classifiable by countable structures if and only if it is Borel reducible to a countable level of the hereditarily countable sets. We also prove the following result which was originally claimed in [FS89]: the zero density ideal of sets of natural numbers is not classifiable by countable structures. Introduction. The Borel reducibility theory of Pol...

We prove that a space is a sequentially-quotient (pseudo-sequencecovering), π-image of a separable metric space if and only if it has a point-star network consisting of countable-cs∗-covers. We also investigate spaces with count-

In this article, we present a method for constructing wavelet frames of L2(R) of the type {| detAj |ψ(Ajx − xj,k) : j ∈ J, k ∈ K} on irregular lattices of the form X = {xj,k ∈ R : j ∈ J, k ∈ K}, and with an arbitrary countable family of invertible d × d matrices {Aj ∈ GLd(R) : j ∈ J}. Possible applications include image and video compression, speech coding, image and digital data transmission, ...

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.

This work explores the linguistic distinction between count and mass nouns in the visual modality. Since the former class typically refers to well-defined, countable objects, with the latter prototypically including less countable substances, we explore to which extent the linguistic distinction is grounded in the visual representations of the entities denoted by count/mass nouns. Using visual ...

We show that even for subsetsX of the real line which do not contain perfect sets, the Hurewicz property does not imply the property S1(Γ,Γ), asserting that for each countable family of open γ-covers of X , there is a choice function whose image is a γcover of X . This settles a problem of Just, Miller, Scheepers, and Szeptycki. Our main result also answers a question of Bartoszyński and the se...

We prove that the relation of bisimilarity between countable labelled transition systems is Σ1-complete (hence not Borel), by reducing the set of non-wellorders over the natural numbers continuously to it. This has an impact on the theory of probabilistic and nondeterministic processes over uncountable spaces, since logical characterizations of bisimilarity (as, for instance, those based on the...