Bounded sets structure of CpX and quasi‐(DF)‐spaces

For wide classes of locally convex spaces, in particular, for the space $C_p(X)$ of continuous real‐valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of $(DF)$‐spaces have led us to introduce quasi‐ $(DF)$‐spaces, a class of locally convex spaces containing $(DF)$‐spaces that preserves subspaces, countable direct sums and countable products. Regular $(LM)$‐spaces as well as their strong duals are quasi‐ $(DF)$‐spaces. Hence the space of distributions $D^{\prime }(\mathrm{\Omega })$ provides a concrete example of a quasi‐ $(DF)$‐space not being a $(DF)$‐space. We show that $C_p(X)$ has a fundamental bounded resolution if and only if $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if the strong dual of $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if X is countable. If X is metrizable, then $C_k(X)$ is a quasi‐ $(DF)$‐space if and only if X is a σ‐compact Polish space.     

Isomorphisms of spaces with large distortion

Let K and S be locally compact Hausdorff spaces and let X be a strictly convex Banach space of finite dimension at least 2. In this paper, we prove that if there exists an isomorphism T from 038:mana201800038-math-0005" class="section_image" src="/cms/asset/dad00cb2-a4b2-4997-8d42-bcfa70975e9c/mana201800038-math-0005.png"> onto 038:mana201800038-math-0006" class="section_image" src="/cms/asset/03d8f47f-b5ab-4c0a-be8e-5e77495bae65/mana201800038-math-0006.png"> satisfying 038-math-0007.png" alt="urn:x-wiley:0025584X:media:mana201800038:mana201800038-math-0007" title="urn:x-wiley:0025584X:media:mana201800038:mana201800038-math-0007"> then K and S are homeomorphic. Here 038:mana201800038-math-0008" class="section_image" src="/cms/asset/43e0b239-c1b0-4950-9bcb-4eacff4d8d65/mana201800038-math-0008.png"> denotes the Schäffer constant of X. Even for the classical cases 038:mana201800038-math-0009" class="section_image" src="/cms/asset/89b31add-9f01-4a3b-9486-5ae2bf767bc9/mana201800038-math-0009.png">, 038:mana201800038-math-0010" class="section_image" src="/cms/asset/2d502066-0d3d-4288-abc0-73e0cc0621be/mana201800038-math-0010.png"> and 038:mana201800038-math-0011" class="section_image" src="/cms/asset/20787805-80b3-4cb2-b217-07eb08ff1c1a/mana201800038-math-0011.png">, this result is the X‐valued Banach–Stone theorem via isomorphism with the largest distortion that is known so far, namely 038:mana201800038-math-0012" class="section_image" src="/cms/asset/d7a605c2-d207-42b7-b935-ce052fa624cb/mana201800038-math-0012.png">. On the other hand, it is well known that this result is not true for 038:mana201800038-math-0013" class="section_image" src="/cms/asset/ad6cae7f-ac71-46ac-943c-a3547960f774/mana201800038-math-0013.png">, even though K and S are compact Hausdorff spaces.     

Existential monadic second order logic of undirected graphs: The Le Bars conjecture is false

In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) on undirected graphs. He proved that there exists an EMSO sentence ? such that $\mathsf{P}(G_n??)$ does not converge as $n\to \infty$ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices $\{1,\dots ,n\}$). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law holds. In this paper, we disprove this conjecture.     

Reasoning about proof and knowledge

In previous work [15], we presented a hierarchy of classical modal systems, along with algebraic semantics, for the reasoning about intuitionistic truth, belief and knowledge. Deviating from Gödel's interpretation of IPC in S4, our modal systems contain IPC in the way established in [13]. The modal operator can be viewed as a predicate for intuitionistic truth, i.e. proof. Epistemic principles are partially adopted from Intuitionistic Epistemic Logic IEL [4]. In the present paper, we show that the S5-style systems of our hierarchy correspond to an extended Brouwer–Heyting–Kolmogorov interpretation and are complete w.r.t. a relational semantics based on intuitionistic general frames. In this sense, our S5-style logics are adequate and complete systems for the reasoning about proof combined with belief or knowledge. The proposed relational semantics is a uniform framework in which also IEL can be modeled. Verification-based intuitionistic knowledge formalized in IEL turns out to be a special case of the kind of knowledge described by our S5-style systems.     

Cofinal elementary extensions

We investigate some properties of ordered structures that are related to their having cofinal elementary extensions. Special attention is paid to models of some very weak fragments of Peano Arithmetic.     

A certain critical density property for invariant Harnack inequalities in H-type groups

We consider second order linear degenerate elliptic operators which are elliptic with respect to horizontal directions generating a stratified algebra of H-type. Extending a result by Gutiérrez and Tournier (2011) for the Heisenberg group, we prove a critical density estimate by assuming a condition of Cordes–Landis type. We then deduce an invariant Harnack inequality for the non-negative solutions from a result by Di Fazio, Gutiérrez, and Lanconelli (2008).     

Modules over quantaloids: Applications to the isomorphism problem in algebraic logic and π-institutions

We solve the isomorphism problem in the context of abstract algebraic logic and of π-institutions, namely the problem of when the notions of syntactic and semantic equivalence among logics coincide. The problem is solved in the general setting of categories of modules over quantaloids. We introduce closure operators on modules over quantaloids and their associated morphisms. We show that, up to isomorphism, epis are morphisms associated with closure operators. The notions of (semi-)interpretability and (semi-)representability are introduced and studied. We introduce cyclic modules, and provide a characterization for cyclic projective modules as those having a g-variable. Finally, we explain how every π-institution induces a module over a quantaloid, and thus the theory of modules over quantaloids can be considered as an abstraction of the theory of π-institutions.