Arithmetic properties of ℓ-regular partitions

For a given prime p, by studying p  -dissection identities for Ramanujan?s theta functions ψ(q)$\psi (q)$ and f(−q)$f(-q)$, we derive infinite families of congruences modulo 2 for some ?  -regular partition functions, where ?=2,4,5,8,13,16$?=2,4,5,8,13,16$.     

γ-Bounded representations of amenable groups

Let G be an amenable group, let X be a Banach space and let π:G→B(X) be a bounded representation. We show that if the set is γ-bounded then π extends to a bounded homomorphism w:C^∗(G)→B(X) on the group C^∗-algebra of G. Moreover w is necessarily γ-bounded. This extends to the Banach space setting a theorem of Day and Dixmier saying that any bounded representation of an amenable group on Hilbert space is unitarizable. We obtain additional results and complements when G=Z, R or T, and/or when X has property (α).     

Maximal ℓ-Frattini quotients of ℓ-Poincaré duality groups of dimension 2

Any morphism of profinite groups has maximal ℓ-Frattini quotients

On thick subgroups of uncountable σ-compact locally compact commutative groups

We show that, for any uncountable commutative group (G,+), there exists a countable covering where each Gj is a subgroup of G satisfying the equality card(G/Gj)=card(G). This purely algebraic fact is used in certain constructions of thick and nonmeasurable subgroups of an uncountable σ-compact locally compact commutative group equipped with the completion of its Haar measure.     

Ordinal remainders of ψ-spaces

We consider which ordinals, with the order topology, can be Stone-?ech remainders of which spaces of the form ψ(κ,M), where ω?κ is a cardinal number and M⊂ω[κ] is a maximal almost disjoint family of countable subsets of κ (MADF). The cardinality of the continuum, denoted c, and its successor cardinal, c⁺, play important roles. We show that if κ>c⁺, then no ψ(κ,M) has any ordinal as a Stone-?ech remainder. If κ?c then for every ordinal δ<κ⁺ there exists Mδ⊂ω[κ], a MADF, such that βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1. For κ=c⁺, βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1 if and only if c⁺?δ<c⁺⋅ω.     

The σ-isotypic decomposition and the σ-index of reversible-equivariant systems

This work is concerned with dynamical systems in presence of symmetries and reversing symmetries. We describe a construction process of subspaces that are invariant by linear Γ-reversible-equivariant mappings, where Γ is the compact Lie group of all the symmetries and reversing symmetries of such systems. These subspaces are the σ-isotypic components, first introduced by Lamb and Roberts in (1999) [10] and that correspond to the isotypic components for purely equivariant systems. In addition, by representation theory methods derived from the topological structure of the group Γ, two algebraic formulae are established for the computation of the σ-index of a closed subgroup of Γ. The results obtained here are to be applied to general reversible-equivariant systems, but are of particular interest for the more subtle of the two possible cases, namely the non-self-dual case. Some examples are presented.     

Factorizing τ-spectra of mappings

In this paper by a spectrum of mappings we mean a morphism of spectra of spaces. However, using the notion of a mapping of mappings, we give the definition of a spectrum of mappings similar to that of a spectrum of spaces. In this case, the formulations of the given results are also similar to the formulations of the corresponding results concerning the spectra of spaces.For the spectra of mappings we define the notion of a τ-spectrum of mappings factorizing in a special sense and prove a version of the Spectral Theorem for such spectra. Furthermore, to a given indexed collection F of mapping we associate a τ-spectrum factorizing in the above special sense whose mappings are Containing Mappings for F constructed in Iliadis (2005) [4]. These associated τ-spectra and the corresponding version of the Spectral Theorem imply that for a given indexed collection F of mappings any so-called “natural” τ-spectrum for F factorizing in the special sense contains a cofinal and τ-closed subspectrum whose mappings are Containing Mapping for F. Thus, Containing Mappigs for F appear here without any concrete construction. The associated τ-spectra are used also in order to define and characterize the so-called second-type saturated classes of mappings (which are “saturated” by universal elements).     

Monotone versions of δ-normality

According to Mack a space is countably paracompact if and only if its product with [0,1] is δ-normal, i.e. any two disjoint closed sets, one of which is a regular Gδ-set, can be separated. In studying monotone versions of countable paracompactness, one is naturally led to consider various monotone versions of δ-normality. Such properties are the subject of this paper. We look at how these properties relate to each other and prove a number of results about them, in particular, we provide a factorization of monotone normality in terms of monotone δ-normality and a weak property that holds in monotonically normal spaces and in first countable Tychonoff spaces. We also discuss the productivity of these properties with a compact metrizable space.     

On the nonexistence of a relation between σ-left-porosity and σ-right-porosity

Given an arbitrarily weak notion of left-〈f〉$〈f〉$-porosity and an arbitrarily strong notion of right-〈g〉$〈g〉$-porosity, we construct an example of closed subset of R$\mathbb{R}$ which is not σ  -left-〈f〉$〈f〉$-porous and is right-〈g〉$〈g〉$-porous. We also briefly summarize the relations between three different definitions of porosity controlled by a function; we then observe that our construction gives the example for any combination of these definitions of left-porosity and right-porosity.     

The structure of analytic τ-sheaves

Let R be a complete discrete valuation -algebra whose residue field is algebraic over , and let K denote its fraction field. In this paper, we study the structure of τ-sheaves M without good reduction on the curve , seen as a rigid analytic space. One motivation is the Tate uniformization theorem for t-motives of Drinfeld modules, which we want to extend to general τ-sheaves. On the other hand, we are interested in the action of inertia on a generic Tate module T_?(M) of M.For a given τ-sheaf M on , we prove the existence of a maximal model for M on , an R-model of , and, over a finite separable extension R′ of R, of nondegenerate models for M.We prove the following ‘semistability’ theorem: there exists a finite extension K′ of K, a nonempty open subscheme C′⊂C, and a filtration

     

Composition-closed ?-groups of almost-piecewise-linear functions

In the present work, we shall construct some non-essential H-closed epireflections of W that are not comparable with any other known H-closed epireflections of W other than the divisible hull and the epicompletion. We show first that the free objects in any H-closed epireflective subcategory must be closed under composition (see Section 2 for a precise definition), and that any epic extension of a free W-object on n generators that is closed under composition is actually the free object on n generators in some H-closed epireflective subcategory of W. We then apply these results to certain ?-groups of almost-piecewise-linear Baire functions on R. By definition, a function f:R→R is almost-piecewise-linear if there is a finite point set S⊂R such that f is piecewise-linear on the complement of any neighborhood of S.     

Quotients and colimits of κ-quantales

Let κQnt be the category of κ-quantales, quantales closed under κ-joins in which the monoid identity is the largest element. (κ is an infinite regular cardinal.) Although the lack of lattice completeness in this setting would seem to mitigate against the techniques which lend themselves so readily to the calculation of frame quotients, we show how to easily compute κQnt quotients by applying generalizations of the frame techniques to suitable extensions of this category.The second major tool in the analysis is the free κ-quantale over a λ-quantale, κ?λ. Surprisingly, these can be characterized intrinsically, and the generating sub-κ-quantale can even be identified. The result that the λ-free κ-quantales coincide with the λ-coherent κ-quantales directly generalizes Madden?s corresponding result for κ-frames.These tools permit a direct and intuitive construction of κQnt colimits. We provide two applications: an intrinsic characterization of κQnt colimits, and of free (over sets) κ-quantales. The latter is a direct generalization of Whitman?s condition for distributive lattices.     

On Γ-convergence of pairs of dual functionals

The paper considers a slightly modified notion of the Γ-convergence of convex functionals in uniformly convex Banach spaces and establishes that under standard coercitivity and growth conditions the Γ-convergence of a sequence of functionals {Fj} to implies that the corresponding sequence of dual functionals converges in an analogous sense to the dual to functional .     

ω-hypoelliptic differential operators of constant strength

We study ω-hypoelliptic differential operators of constant strength. We show that any operator with constant strength and coefficients in which is homogeneous ω-hypoelliptic is also σ-hypoelliptic for any weight function σ=O(ω). We also present a sufficient condition in order to ensure that a differential operator admits a parametrix and, as a consequence, we obtain some conditions on the weights (ω,σ) to conclude that, for any operator P(x,D) with constant strength, the σ-hypoellipticity of the frozen operator P(x₀,D) implies the ω-hypoellipticity of P(x,D). This requires the use of pseudodifferential operators.     

Rings of continuous functions on σ-frames

This paper deals with the ?-rings RS of all real-valued continuous functions on a completely regular σ-frame. It shows that, in marked contrast with the situation for frames, any ?-ring homomorphism RS→RT results from a σ-frame homomorphism S→T. Further, it proves the analogue of this for integer-valued continuous functions and 0-dimensional σ-frames. In all, this demonstrates that the important classical difference between Alexandroff spaces and Tychonoff spaces with respect to the real-valued continuous functions carries over fully to the pointfree setting - indeed, it adds the integer-valued case which seems to be new in this context.     

Some integrals of the Dedekind η-function

This note presents selected values of the class of integrals

     

On a σ-ideal of compact sets

We recall from [T. Mátrai, Kenilworth, Proc. Amer. Math. Soc. 137 (3) (2009) 1115-1125] a Gδσ-ideal of compact subsets of ω² and prove that it is not Tukey reducible to the ideal . This result answers a question of S. Solecki and S. Todor?evi? in the negative.