We study the isomorphism relation on Borel classes of locally compact Polish metric structures. We prove that isomorphism on such classes is always classifiable by countable structures (equivalently: Borel reducible to graph isomorphism), which implies, in particular, that isometry of locally compact Polish metric spaces is Borel reducible to graph isomorphism. We show that potentially $\boldsymbol {\Pi }^{0}_{\alpha + 1}$ isomorphism relations are Borel reducible to equality on hereditarily countable sets of rank $\alpha $, (...) $\alpha \geq 2$. We also study approximations of the Hjorth-isomorphism game, and formulate a condition ruling out classifiability by countable structures. (shrink)

For a locally compact group G, we show that it is possible to present the class of continuous unitary representations of G as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how nondegenerate ∗-representations of a general ∗-algebra A (with some mild assumptions) can be viewed as an elementary class, in a many-sorted language, and use the correspondence between continuous unitary representations of G and nondegenerate ∗-representations of L1(G). We (...) relate the notion of ultraproduct of logical structures, under this presentation, with other notions of ultraproduct of representations appearing in the literature, and we characterize property (T) for G in terms of the definability of the sets of fixed points of L1 functions on G. (shrink)

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o‐minimal structure. This fact together with the results in a previous paper implies a tame dimension theory and a decomposition theorem into good‐shaped definable subsets called quasi‐special submanifolds. Using this fact, we investigate definably complete locally o‐minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o‐minimal expansions (...) of ordered fields, Łojasiewicz's inequality, Tietze's extension theorem and affiness of pseudo‐definable spaces hold true for such structures under the extra assumption that the domains of definition and the pseudo‐definable spaces are definably compact. Here, a pseudo‐definable space is a topological space having finite definable atlases. We also demonstrate Michael's selection theorem for definable set‐valued functions with definably compact domains of definition. (shrink)

We prove a structure theorem for compact inverse categories. The Ehresmann-Schein-Nambooripad theorem gives a structure theorem for inverse monoids: they are inductive groupoids. A particularly nice case due to Clifford is that commutative inverse monoids become semilattices of abelian groups. It has also been categorified by Hoehnke and DeWolf-Pronk to a structure theorem for inverse categories as locally complete inductive groupoids. We show that in the case of compact inverse categories, this takes the particularly nice form of (...) a semilattice of compact groupoids. Moreover, one-object compact inverse categories are exactly commutative inverse monoids. Compact groupoids, in turn, are determined in particularly simple terms of 3-cocycles by Baez-Lauda. (shrink)

The research presented in this paper was motivated by our aim to study a problem due to J. Bourgain [3]. The problem in question concerns the uniform boundedness of the classical separation rank of the elements of a separable compact set of the first Baire class. In the sequel we shall refer to these sets (separable or non-separable) as Rosenthal compacta and we shall denote by ∝(f) the separation rank of a real-valued functionfinB1(X), withXa Polish space. Notice that in (...) [3], Bourgain has provided a positive answer to this problem in the case ofKsatisfyingwithXa compact metric space. The key ingredient in Bourgain's approach is that whenever a sequence of continuous functions pointwise converges to a functionf, then the possible discontinuities of the limit function reflect a local ℓ1-structure to the sequence (fn)n. More precisely the complexity of this ℓ1-structure increases as the complexity of the discontinuities offdoes. This fruitful idea was extensively studied by several authors (c.f. [5], [7], [8]) and for an exposition of the related results we refer to [1]. It is worth mentioning that A.S. Kechris and A. Louveau have invented the rank rND(f) which permits the link between thec0-structure of a sequence (fn)nof uniformly bounded continuous functions and the discontinuities of its pointwise limit. Rosenthal'sc0-theorem [11] and thec0-index theorem [2] are consequences of this interaction.Passing to the case where either (fn)nare not continuous orXis a non-compact Polish space, this nice interaction is completely lost. (shrink)

We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbf {\Sigma }^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth–Kechris (...) about discrete structures. As a different application, we also give a new proof of Kechris’s theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable. (shrink)

We present and study the category of formal topologies and some of its variants. Two main results are proven. The first is that, for any inductively generated formal cover, there exists a formal topology whose cover extends in the minimal way the given one. This result is obtained by enhancing the method for the inductive generation of the cover relation by adding a coinductive generation of the positivity predicate. Categorically, this result can be rephrased by saying that inductively generated formal (...) topologies are coreflective into inductively generated formal covers. The second result is that unary formal covers are exponentiable in the category of inductively generated formal covers and hence, thanks to the coreflection, unary formal topologies are exponentiable in the category of inductively generated formal topologies. From a localic point of view the exponentiability of unary formal topologies means that algebraic dcpos are exponentiable in the category of open locales. But, the coreflection theorem states that open locales are coreflective in locales and hence, as a consequence of well-known impredicative results on exponentiable locales, it allows to prove that locally compact open locales are exponentiable in the category of open locales. (shrink)

We present and study the category of formal topologies and some of its variants. Two main results are proven. The first is that, for any inductively generated formal cover, there exists a formal topology whose cover extends in the minimal way the given one. This result is obtained by enhancing the method for the inductive generation of the cover relation by adding a coinductive generation of the positivity predicate. Categorically, this result can be rephrased by saying that inductively generated formal (...) topologies are coreflective into inductively generated formal covers. The second result is that unary formal covers are exponentiable in the category of inductively generated formal covers and hence, thanks to the coreflection, unary formal topologies are exponentiable in the category of inductively generated formal topologies. From a localic point of view the exponentiability of unary formal topologies means that algebraic dcpos are exponentiable in the category of open locales. But, the coreflection theorem states that open locales are coreflective in locales and hence, as a consequence of well-known impredicative results on exponentiable locales, it allows to prove that locally compact open locales are exponentiable in the category of open locales. (shrink)

We show that it is relatively consistent with ZFC to have PCF structures of heightδ, for all ordinalsδ<ω3.

The implementation of corporate social responsibility is crucial for the legitimacy of an organization in today’s globalized economy. This study aims to enrich our knowledge of the implementation of the largest voluntary CSR initiative—the UN Global Compact. Drawing on insights from stakeholder, network, and institutional theory, I derive a positive impact of UNGC participation duration on the implementation level of the UNGC principles, despite potential weaknesses in the initiative’s accountability structure. Moreover, I scrutinize the validity of the newly introduced (...) UNGC “Differentiation Programme” before applying this framework in the empirical analysis. Results from ordinal, linear, and instrumental variable regression models suggest that, contrary to claims made by UNGC critics, the duration of UNGC participation does affect the level of UNGC implementation. However, this effect appears to be much smaller than previous practitioner studies have suggested. Moreover, strong local UNGC networks affect the implementation level of the UNGC positively. Their hypothesized moderating role between UNGC participation duration and UNGC implementation level, however, is only significant in networks with activities of high quality rather than high quantity. (shrink)

Probabilistic Relational Models (PRMs) are a framework for compactly representing uncertainties (actually probabilities). They result from the combination of Bayesian Networks (BNs), Object-Oriented languages, and relational models. They are specifically designed for their efficient construction, maintenance and exploitation for very large scale problems, where BNs are known to perform poorly. Actually, in large-scale problems, it is often the case that BNs result from the combination of patterns (small BN fragments) repeated many times. PRMs exploit this feature by defining these patterns (...) only once (the so-called PRM’s classes) and using them through multiple instances, as prescribed by the Object-Oriented paradigm. This design induces low construction and maintenance costs. In addition, by exploiting the classes’ structures, PRM’s state-of-the-art inference algorithm “Structured Variable Elimination” (SVE) significantly outperforms BN’s classical inference algorithms (e.g., Variable Elimination, VE; Local Conditioning, LC). SVE is actually an extension of VE that simply exploits classes to avoid redundant computations. In this article, we show that SVE can be enhanced using LC. Although LC is often thought as being outperformed by VE-like algorithms in BNs, we do think that it should play an important role for PRMs because its features are very well suited for best exploiting PRM classes. In this article, relying on Faÿ and Jaffray’s works, we show how LC can be used in conjunction with VE and deduce an extension of SVE that outperforms it for large-scale problems. Numerical experiments highlight the practical efficiency of our algorithm. (shrink)

For a signature L with at least one constant symbol, an L‐structure is called minimal if it has no proper substructures. Let be the set of isomorphism types of minimal L‐structures. The elements of can be identified with ultrafilters of the Boolean algebra of quantifier‐free L‐sentences, and therefore one can define a Stone topology on. This topology on generalizes the topology of the space of n‐marked groups. We introduce a natural ultrametric on, and show that the Stone topology on (...) coincides with the topology of the ultrametric space iff the ultrametric space is compact iff L is locally finite (that is, L contains finitely many n‐ary symbols for any ). As one of the applications of compactness of the Stone topology on, we prove compactness of certain classes of metric spaces in the Gromov‐Hausdorff topology. This slightly refines the known result based on Gromov's ideas that any uniformly totally bounded class of compact metric spaces is precompact. (shrink)

The H‐NS protein is a major component of the nucleoid in Gram‐negative bacterial cells. It is a global regulator of transcription that affects the expression of many genes, including virulence genes in pathogenic species. At a local level, it facilitates the formation of nucleoprotein structures that repress transcriptional promoter function. H‐NS can form bridges between different DNA molecules or between different sections of the same molecule, allowing it to compact and impose structure on the nucleoid. A recent paper (...) by Dame et al.1 reports new insights into H‐NS‐mediated DNA bridging that were obtained using an optical tweezers device. BioEssays 29: 212–216, 2007. © 2007 Wiley Periodicals, Inc. (shrink)

This paper uses the locale theory approach to topology. Two descriptions are given of all locale limits, the first description using suplattice constructions and the second preframe constructions. The symmetries between these two approaches to locale theory are explored. Given an informal assumption that open locale maps are parallel to proper maps we argue that various pairs of locale theory results are ‘parallel’, that is, identical in structure but prove facts about proper maps on one side of the pair and (...) about open maps on the other. The pairs of results are: pullback stability of proper/open maps, regularity of the category of compact Hausdorff/discrete locales, and theorems on information systems. Some remarks are included on a possible formalization of this parallel as a duality. (shrink)

The locale corresponding to the real interval [ − 1, 1] is an interval object, in the sense of Escardó and Simpson, in the category of locales. The map, mapping a stream s of signs ±1 to, is a proper localic surjection; it is also expressed as a coequalizer. The proofs are valid in any elementary topos with natural numbers object.

Although classically every open subspace of a locally compact space is also locally compact, constructively this is not generally true. This paper provides a locally compact remetrization for an open set in a compact metric space and constructs a one-point compactification. MSC: 54D45, 03F60, 03F65.

Given a Polish groupG, let$E(G)$be the right coset equivalence relation$G^{\omega }/c(G)$, where$c(G)$is the group of all convergent sequences inG. The connected component of the identity of a Polish groupGis denoted by$G_0$.Let$G,H$be locally compact abelian Polish groups. If$E(G)\leq _B E(H)$, then there is a continuous homomorphism$S:G_0\rightarrow H_0$such that$\ker (S)$is non-archimedean. The converse is also true whenGis connected and compact.For$n\in {\mathbb {N}}^+$, the partially ordered set$P(\omega )/\mbox {Fin}$can be embedded into Borel equivalence relations between$E({\mathbb {R}}^n)$and$E({\mathbb {T}}^n)$.

We provide a complete classification of the possible cofinal structures of the families of precompact (totally bounded) sets in general metric spaces, and compact sets in general complete metric spaces. Using this classification, we classify the cofinal structure of local bases in the groups C(X, R) of continuous real-valued functions on complete metric spaces X, with respect to the compact-open topology.

We prove that the modal logic of a crowded locally compact generalized ordered space is $\textsf {S4}$. This provides a version of the McKinsey–Tarski theorem for generalized ordered spaces. We then utilize this theorem to axiomatize the modal logic of an arbitrary locally compact generalized ordered space.

We show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of (...) Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces. (shrink)

It is argued that the `problem of time' in quantum gravity necessitates a refinement of the local inertial structure of the world, demanding a replacement of the usual Minkowski line element by a 4+2n dimensional pseudo-Euclidean line element, with the extra 2n being the number of internal phase space dimensions of the observed system. In the refined structure, the inverse of the Planck time takes over the role of observer-independent conversion factor usually played by the speed of light, which now (...) emerges as an invariant but derivative quantity. In the relativistic theory based on the refined structure, energies and momenta turn out to be invariantly bounded from above, and lengths and durations similarly bounded from below, by their respective Planck scale values. Along the external timelike world-lines, the theory naturally captures the `flow of time' as a genuinely structural attribute of the world. The theory also predicts expected deviations---suppressed quadratically by the Planck energy---from the dispersion relations for free fields in the vacuum. The deviations from the special relativistic Doppler shifts predicted by the theory are also suppressed quadratically by the Planck energy. Nonetheless, in order to estimate the precision required to distinguish the theory from special relativity, an experiment with a binary pulsar emitting TeV range gamma-rays is considered in the context of the predicted deviations from the second-order shifts. (shrink)

The paper has two aims: (1) it sets out to show that it is well motivated to seek for an account of quantum non-locality in the framework of ontic structural realism (OSR), which integrates the notions of holism and non-separability that have been employed since the 1980s to achieve such an account. However, recent research shows that OSR on its own cannot provide such an account. Against this background, the paper argues that by applying OSR to the primitive ontology theories (...) of quantum physics, one can accomplish that task. In particular, Bohmian mechanics offers the best prospect for doing so. (2) In general, the paper seeks to bring OSR and the primitive ontology theories of quantum physics together: on the one hand, in order to be applicable to quantum mechanics, OSR has to consider what the quantum ontology of matter distributed in space-time is. On the other hand, as regards the primitive ontology theories, OSR provides the conceptual tools for these theories to answer the question of what the ontological status of the wave-function is. (shrink)

It is known that peer group games are a special class of games with a permission structure. However, peer group games are also a special class of digraph games. To be specific, they are digraph games in which the digraph is the transitive closure of a rooted tree. In this paper we first argue that some known results on solutions for peer group games hold more general for digraph games. Second, we generalize both digraph games as well as games with (...) a permission structure into a model called games with a local permission structure, where every player needs permission from its predecessors only to generate worth, but does not need its predecessors to give permission to its own successors. We introduce and axiomatize a Shapley value-type solution for these games, generalizing the conjunctive permission value for games with a permission structure and the β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-measure for weighted digraphs. (shrink)

The existence of either a maximum or a minimum for a uniformly continuous mapping f of a compact interval into ${\mathbb{R}}$ is established constructively under the hypotheses that f′ is sequentially continuous and f has at most one critical point.

In this dissertation we explore projective Fraïssé theory and its applications, as well as limitations, to the study of compact metrizable spaces. The goal of projective Fraïssé theory is to approximate spaces via classes of finite structures and glean topological or dynamical properties of a space by relating them to combinatorial features of the associated class of structures. Using the framework of compact metrixable structures, we establish general results which expand and help contextualize previous works (...) in the field. Many proofs in the domain of projective Fraïssé theory are carried out in a context dependent fashion and have thus far eluded clean generalizations. A reason is to be found in the lack of a clear understanding of which spaces are amenable to be studied via projective Fraïssé limits. We give both positive and negative results in this direction. We isolate a combinatorial condition which entails a correspondence between finite structures and regular quasi-partitions of compact metrizable spaces. This correspondence greatly aids the combinatorial-toplogical translation. We apply this machinery to study a class of one-dimensional compact metrizable spaces, which we call smooth fences. To this end, we isolate a class of finite structures—finite partial orders whose Hasse diagram is a forest—whose projective Fraïssé limit approximates a distinctive smooth fence with remarkable properties. We call it the Fraïssé fence and characterize it topologically by carefully exploiting the bridge between the combinatorial and topological worlds. We explore homogeneity and universality features of the Fraïssé fence and the properties of its spaces and endpoints, and provide some results on the dynamics of its group of homeomorphisms. The dissertation is partially based on [1, 2]. (shrink)

If a locale is presented by a “flat site”, it is shown how its frame can be presented by generators and relations as a dcpo. A necessary and sufficient condition is derived for compactness of the locale . Although its derivation uses impredicative constructions, it is also shown predicatively using the inductive generation of formal topologies. A predicative proof of the binary Tychonoff theorem is given, including a characterization of the finite covers of the product by basic opens. The discussion (...) is then related to the double powerlocale. (shrink)

We say that a Hausdorff locale is compactly generated if it is the colimit of the diagram of its compact sublocales connected by inclusions. We show that this is the case if and only if the natural map of its frame of opens into the second Lawson dual is an isomorphism. More generally, for any Hausdorff locale, the second dual of the frame of opens gives the frame of opens of the colimit. In order to arrive at this conclusion, (...) we generalize the Hofmann–Mislove–Johnstone theorem and some results regarding the patch construction for stably locally compact locales. (shrink)

We prove the Compact Domination Conjecture for groups definable in linear o-minimal structures. Namely, we show that every definably compact group G definable in a saturated linear o-minimal expansion of an ordered group is compactly dominated by (G/G 00, m, π), where m is the Haar measure on G/G 00 and π : G → G/G 00 is the canonical group homomorphism.

DNA with a curved trajectory of its helix axis is called bent DNA, or curved DNA. Interestingly, biologically important DNA regions often contain this structure, irrespective of the origin of DNA. In the last decade, considerable progress has been made in clarifying one role of bent DNA in prokaryotic transcription and its mechanism of action. However, the role of bent DNA in eukaryotic transcription remains unclear. Our recent study raises the possibility that bent DNA is implicated in the “functional packaging” (...) of transcriptional regulatory regions into chromatin. In this article, I review recent progress in bent DNA research in eukaryotic transcription, and summarize the history of bent DNA research and several subjects relevant to this theme. Finally, I propose a hypothesis that bent DNA structures that mimic a negative supercoil, or have a right‐handed superhelical writhe, organize local chromatin infrastructure to help the very first interaction between cis‐acting DNA elements and activators that trigger transcription. BioEssays 23:708–715, 2001. © 2001 John Wiley & Sons, Inc. (shrink)

A metric space is said to be locally non-compact if every neighborhood contains a sequence that is eventually bounded away from every element of the space, hence contains no accumulation point. We show within recursive mathematics that a nonvoid complete metric space is locally non-compact iff it is without isolated points.The result has an interesting consequence in computable analysis: If a complete metric space has a computable witness that it is without isolated points, then every neighborhood (...) contains a computable sequence that is eventually computably bounded away from every computable element of the space. (shrink)

We introduce and study the framework of compact metric structures and their associated notions of isomorphisms such as homeomorphic and bi-Lipschitz isomorphism. This is subsequently applied to model various classification problems in analysis such as isomorphism ofC*-algebras and affine homeomorphism of Choquet simplices, where among other things we provide a simple proof of the completeness of the isomorphism relation of separable, simple, nuclearC*-algebras recently established by M. Sabok.