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Register a new userUniformly antisymmetric function with bounded range
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The goal of this note is to construct a uniformly antisymmetric function f:R-> R with a bounded countable range. This answers Problem 1(b) of Ciesielski and Larson. (See also list of problems in Thomson and Problem 2(b) from Ciesielskis survey.) A problem of existence of uniformly antisymmetric function f:R-> R with finite range remains open.
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In this paper, we introduce the concept of uniformly bounded fibred coarse embeddability of metric spaces, generalizing the notion of fibred coarse embeddability defined by X. Chen, Q. Wang and G. Yu. Moreover, we show its relationship with uniformly bounded a-T-menability of groups. Finally, we give some examples to illustrate the differences between uniformly bounded fibred coarse embeddability and fibred coarse embeddability.
We show that for acylindrically hyperbolic groups $Gamma$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $rho$ of $Gamma$ in a (nonzero) uniformly convex Banach space the vector space $H^2_b(Gamma;rho)$ is infinite dimensional. The result was known for the regular representations on $ell^p(Gamma)$ with $1<p<infty$ by a different argument. But our result is new even for a non-abelian free group in this great generality for representations, and also the case for acylindrically hyperbolic groups follows as an application.
We define the bounded jump of A by A^b = {x | Exists i <= x [phi_i (x) converges and Phi_x^[A|phi_i(x)](x) converges} and let A^[nb] denote the n-th bounded jump. We demonstrate several properties of the bounded jump, including that it is strictly increasing and order preserving on the bounded Turing (bT) degrees (also known as the weak truth-table degrees). We show that the bounded jump is related to the Ershov hierarchy. Indeed, for n > 1 we have X <=_[bT] 0^[nb] iff X is omega^n-c.e. iff X <=_1 0^[nb], extending the classical result that X <=_[bT] 0 iff X is omega-c.e. Finally, we prove that the analogue of Shoenfield inversion holds for the bounded jump on the bounded Turing degrees. That is, for every X such that 0^b <=_[bT] X <=_[bT] 0^[2b], there is a Y <=_[bT] 0^b such that Y^b =_[bT] X.
The theory of finitely supported algebraic structures represents a reformulation of Zermelo-Fraenkel set theory in which every construction is finitely supported according to the action of a group of permutations of some basic elements named atoms. In this paper we study the properties of finitely supported sets that contain infinite uniformly supported subsets, as well as the properties of finitely supported sets that do not contain infinite uniformly supported subsets. For classical atomic sets, we study whether they contain or not infinite uniformly supported subsets.
We study strong types and Galois groups in model theory from a topological and descriptive-set-theoretical point of view, leaning heavily on topological dynamical tools. More precisely, we give an abstract (not model theoretic) treatment of problems related to cardinality and Borel cardinality of strong types, quotients of definable groups and related objecets, generalising (and often improving) essentially all hitherto known results in this area. In particular, we show that under reasonable assumptions, strong type spaces are locally quotients of compact Polish groups. It follows that they are smooth if and only if they are type-definable, and that a quotient of a type-definable group by an analytic subgroup is either finite or of cardinality at least continuum.
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