Multipliers on some weighted Lp‐spaces

Weighted First-Order Model Counting (WFOMC) computes the weighted sum of the models of a first-order logic theory on a given finite domain. First-Order Logic theories that admit polynomial-time WFOMC w.r.t domain cardinality are called domain liftable. We introduce the concept of lifted interpretations as a tool for formulating closed forms for WFOMC. Using lifted interpretations, we reconstruct the closed-form formula for polynomial-time FOMC in the universally quantified fragment of FO2, earlier proposed by Beame et al. We then expand this closed-form to incorporate cardinality constraints, existential quantifiers, and counting quantifiers (a.k.a C2) without losing domain-liftability. Finally, we show that the obtained closed-form motivates a natural definition of a family of weight functions strictly larger than symmetric weight functions.

An original boundary integral formulation is proposed for the problem of a semi-infinite crack at the interface between two dissimilar elastic materials in the presence of heat flows and mass diffusion. Symmetric and skew-symmetric weight function matrices are used together with a generalized Betti's reciprocity theorem in order to derive a system of integral equations that relate the applied loading, the temperature and mass concentration fields, the heat and mass fluxes on the fracture surfaces and the resulting crack opening. The obtained integral identities can have many relevant applications, such as for the modelling of crack and damage processes at the interface between different components in electrochemical energy devices characterized by multi-layered structures (solid oxide fuel cells and lithium ions batteries).

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

In a recent article [K. H. Hofmann and F. G. Russo, ‘The probability that$x$and$y$commute in a compact group’,Math. Proc. Cambridge Phil Soc., to appear] we calculated for a compact group$G$the probability$d(G)$that two randomly selected elements$x, y\in G$satisfy$xy=yx$, and we discussed the remarkable consequences on the structure of$G$which follow from the assumption that$d(G)$is positive. In this note we consider two natural numbers$m$and$n$and the probability$d_{m,n}(G)$that for two randomly selected elements$x, y\in G$the relation$x^my^n=y^nx^m$holds. The situation is more complicated whenever$n,m\gt 1$. If$G$is a compact Lie group and if its identity component$G_0$is abelian, then it follows readily that$d_{m,n}(G)$is positive. We show here that the following condition suffices for the converse to hold in an arbitrary compact group$G$: for any nonopen closed subgroup$H$of$G$, the sets$\{g\in G: g^k\in H\}$for both$k=m$and$k=n$have Haar measure$0$. Indeed, we show that if a compact group$G$satisfies this condition and if$d_{m,n}(G)\gt 0$, then the identity component of$G$is abelian.

We study Pontryagin reflexivity in the class of precompact topological Abelian groups. We find reflexive groups among precompact not pseudocompact and among pseudocompact not compact groups. Making use of Martin's axiom we give an example of a reflexive countably compact not compact Abelian group. We also prove that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.

Let $G$ be a locally compact group, let $(\varphi , \psi )$ be a complementary pair of Young functions, and let $L^\varphi (G)$ and $L^\psi (G)$ be the corresponding Orlicz spaces. Under some conditions on $\varphi $, we will show that for a Banach $L^\va

Given a compact group G and r ∈ N, let sr(G) denote the least upper bound of real ϵ > 0 such that for every measurable r-coloring of G, there exists a monochrome symmetric subset of measure ≥ ϵ. A subset A ⊆ G is symmetric if there exists g ∈ G such that gA −1 g = A. We give a general picture of asymptotic behaviour of the function sr(G) for compact Abelian groups.

In Pontrjagin's theory of duality for compact abelian groups, the following theorem is known : Let G be a compact abelian group, G* the dual group. Then the topological dimension of G, in the sense of Lebesgue, is equal to the rank of discrete abelian group G*. It was Prof. T. Tannaka who has called my attention to the lack of corresponding theorem in non-commutative case. I intend to give, in this note, a theorem of this kind in the following form: THEOREM A. Let G be an arbitrary compact group, G the aggregate of continuous finite dimensional representstions of G, C[ G ] the algebra over the complex numbers C generated by the coefficients of representations in G, i.e., the representative ring of G in the sense of C. Chevalley^. Then the topological dimension of G, in the sense of Lebesgue, is equal to the transcendental degree of C[G] over C.

On square roots of the Haar state on compact quantum groups

We study weighted (PLB)-spaces of ultradifferentiable functions defined via a weight function (in the sense of Braun, Meise and Taylor) and a weight system. We characterize when such spaces are ultrabornological in terms of the defining weight system. This generalizes Grothendieck’s classical result that the space $${\mathcal {O}}_M$$ of slowly increasing smooth functions is ultrabornological to the context of ultradifferentiable functions. Furthermore, we determine the multiplier spaces of Gelfand-Shilov spaces and, by using the above result, characterize when such spaces are ultrabornological. In particular, we show that the multiplier space of the space of Fourier ultrahyperfunctions is ultrabornological, whereas the one of the space of Fourier hyperfunctions is not.

Given a locally compact group G, G L U C is the largest semigroup compactification of G and G * = G L U C ∖ G . We show that (i) for every locally compact compactly generated Abelian group G and for every p , q ∈ G * , the multiplication in G L U C is discontinuous at ( p , q ) , (ii) there is a locally compact σ-compact torsion-free Abelian group G for which, assuming Martin's Axiom, there are p , q ∈ G * such that the multiplication in G L U C is continuous at ( p , q ) , and (iii) it is consistent with ZFC that for every locally compact Abelian group G and for every p , q ∈ G * , the left translation by p in G * is discontinuous at q.

In this paper the structure of all open semigroups in any compact topological group is determined. We can then extend this result to a classification of the angular semigroups of any abelian topological group H which contains a compact open subgroup K. Coupling this with the results for Euclidean n-space, we obtain an essentially complete description of the angular semigroups of an arbitrary locally compact group. For compact groups, the result is remarkably simple.

We study the existence of many nonclosed pure subgroups of nondiscrete locally compact abelian groups. It is shown that every nondiscrete locally compact abelian group has uncountably many nonclosed pure subgroups. This in particular solves a question of Armacost. It is also shown that, if G G is a nondiscrete locally compact abelian group and if either G G is a compact group or the torsion part t ( G ) t\left ( G \right ) of G G is nonopen, then G G has 2 c {2^c} proper dense pure subgroups, where c c denotes the power of the continuum. This in particular gives a partial answer to another question of Armacost.

In this note we point out two properties (Theorems 1 and 2) of the group algebra of the direct product of a compact group and a locally compact abelian group and indicate briefly how these properties enable one to extend to these group algebras theorems which are known in the compact and abelian cases. Our interest in these properties is two-fold; they establish the applicability of known results in [4]1 to the group algebras in question, and they provide an interesting example of a type of B-algebra, a sort of Kroliecker product algebra, which may very well deserve independenlt systematic study. The reader is referred to results of Kaplansky [1 ] and Segal [2 ] of which this note is essentially a continuation. Let C be a compact group, A be a locally compact abelian group, and G = CXA be their direct product group. We are interested in the group algebras L1(C), L1(A), and L1(G). In particular, we ask how the latter is related to the first two. We will show (1) how the B-algebra L1(G) is formed out of the B-algebras L1(C) and L1(A) and (2) how the structure space of L1(G) is formed from the structure spaces of L1(C) and L1(A). DEFINITION 1. The complex algebra R is the Kronecker product of the complex algebras R1 and R2 (written R = R1 XR2) if there exists a map (xl, X2)>X1 X X2 of (R1, R2) into R such that

Notes on compact Discrete Analysis 2017:16, 57 pp. This is the second paper in a two-part series. The first paper, [also published in this journal](http://discreteanalysisjournal.com/article/2105-notes-on-nilspaces-algebraic-aspects), developed the algebraic theory of nilspaces, which are an abstraction both of the notion of a nilmanifold, and of that of an abelian group (both with their attendant parallelogram structures). In this paper, the author applies this algebraic theory to study compact which can be viewed as a generalization of the notion of a compact abelian group. The main motivation for this is that such compact nilspaces arise naturally when trying to attack additive combinatorial problems such as the inverse conjecture for the Gowers norms via the methods of nonstandard analysis (or ultraproducts), as was demonstrated in work of Szegedy ([On higher order Fourier analysis](https://arxiv.org/abs/1203.2260)). The algebraic theory in the first part lets one describe $k$-step compact nilspaces as abelian extensions of $k-1$-step with the extension given by a cocycle. Among other things, this can be used to assign to each nilspace a probability measure that is analogous to the Haar probability measure one can assign to a compact group. However, it is not immediately obvious that this cocycle has any good measurability or continuity properties. Much of the delicate analysis in this paper (particularly those concerning the properties of continuous systems of measures) is devoted to ensuring that such good properties can be imposed (locally at least) on the cocycle, perhaps after modifying the cocycle by a suitable coboundary that does not affect the isomorphism class of the extension. Indeed, one can view the paper as developing some basic for although the formidable machinery of modern cohomology is not really deployed here. As with the first paper in this series, the paper closely follows the previous work of Camarena and Szegedy, but with many more details provided in the arguments. (An alternative approach to this theory has also recently been worked out in a series of papers by Gutman, Manners, and Varju.) A full structural description of compact nilspaces seems to be quite difficult, but in this paper two important partial descriptions are obtained. The first is to describe every compact nilspace as an inverse limit of compact nilspaces of rank. This is somewhat analogous to how the Peter-Weyl theorem can be used to describe a compact group as the inverse limit of finite-dimensional compact Lie groups. The second is to show that every connected compact nilspace of finite rank is isomorphic to a nilmanifold. These results were previously obtained by Camarena and Szegedy (and later used by Szegedy to provide a new proof of the inverse conjecture for the Gowers uniformity norms), but the author here reproves these results with much more detailed proofs, in particular treating with some care certain steps that were only briefly discussed in the prior paper of Camarena and Szegedy.