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On continuity of measurable group representations and homomorphisms

Yulia Kuznetsova (2012)

Studia Mathematica

Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space ℒ(H) of bounded linear operators on H with the weak operator topology. We prove that if U is a measurable map from G to ℒ(H) then it is continuous. This result was known before for separable H. We also prove that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous.

On continuity of the entropy-based differently implicational algorithm

Yiming Tang, Witold Pedrycz (2019)

Kybernetika

Aiming at the previously-proposed entropy-based differently implicational algorithm of fuzzy inference, this study analyzes its continuity. To begin with, for the FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens) problems, the continuous as well as uniformly continuous properties of the entropy-based differently implicational algorithm are demonstrated for the Tchebyshev and Hamming metrics, in which the R-implications derived from left-continuous t-norms are employed. Furthermore, four numerical...

On convergence of integrals in o-minimal structures on archimedean real closed fields

Tobias Kaiser (2005)

Annales Polonici Mathematici

We define a notion of volume for sets definable in an o-minimal structure on an archimedean real closed field. We show that given a parametric family of continuous functions on the positive cone of an archimedean real closed field definable in an o-minimal structure, the set of parameters where the integral of the function converges is definable in the same structure.

On Core XPath with Inflationary Fixed Points

Loredana Afanasiev, Balder Ten Cate (2013)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We prove the undecidability of Core XPath 1.0 (CXP) [G. Gottlob and C. Koch, in Proc. of 17th Ann. IEEE Symp. on Logic in Computer Science, LICS ’02 (Copenhagen, July 2002). IEEE CS Press (2002) 189–202.] extended with an Inflationary Fixed Point (IFP) operator. More specifically, we prove that the satisfiability problem of this language is undecidable. In fact, the fragment of CXP+IFP containing only the self and descendant axes is already undecidable.

On countable cofinality and decomposition of definable thin orderings

Vladimir Kanovei, Vassily Lyubetsky (2016)

Fundamenta Mathematicae

We prove that in some cases definable thin sets (including chains) of Borel partial orderings are necessarily countably cofinal. This includes the following cases: analytic thin sets, ROD thin sets in the Solovay model, and Σ¹₂ thin sets under the assumption that ω ₁ L [ x ] < ω ₁ for all reals x. We also prove that definable thin wellorderings admit partitions into definable chains in the Solovay model.

On definably proper maps

Mário J. Edmundo, Marcello Mamino, Luca Prelli (2016)

Fundamenta Mathematicae

In this paper we work in o-minimal structures with definable Skolem functions, and show that: (i) a Hausdorff definably compact definable space is definably normal; (ii) a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if and only if it is a proper morphism in the category of definable spaces. We give several other characterizations of definably proper, including one involving the existence of limits of definable types. We also prove the...

On d-finite tuples in random variable structures

Shichang Song (2013)

Fundamenta Mathematicae

We prove that the d-finite tuples in models of ARV are precisely the discrete random variables. Then, we apply d-finite tuples to the work by Keisler, Hoover, Fajardo, and Sun concerning saturated probability spaces. In particular, we strengthen a result in Keisler and Sun's recent paper.

On d-finiteness in continuous structures

Itaï Ben Yaacov, Alexander Usvyatsov (2007)

Fundamenta Mathematicae

We observe that certain classical results of first order model theory fail in the context of continuous first order logic. We argue that this happens since finite tuples in a continuous structure may behave as infinite tuples in classical model theory. The notion of a d-finite tuple attempts to capture some aspects of the classical finite tuple behaviour. We show that many classical results involving finite tuples are valid in continuous logic upon replacing "finite" with "d-finite". Other results,...

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