It is well known that rational multiplication preserves normality in base $$. We study related normality preserving operations for the $$-Cantor series expansions. In particular, we show that while integer multiplication preserves $$-distribution normality, it fails to preserve $$-normality in a particularly strong manner. We also show that $$-distribution normality is not preserved by non-integer rational multiplication on a set of zero measure and full Hausdorff dimension.

The main purpose of this paper is to establish the existence of positive solutions to a class of quasilinear elliptic equations involving the ($$-$$)-Laplacian operator. We consider a nonlinearity that can be subcritical at infinity and supercritical at the origin; we also consider potential functions that can vanish at infinity. The approach is based on variational arguments dealing with the mountain-pass lemma and an adaptation of the penalization method. In order to overcome the lack of compactness, we modify the original problem and the associated energy functional. Finally, to show that the solution of the modified problem is also a solution of the original problem we use an estimate obtained by the Moser iteration scheme.

We introduce and study a stability property for submodules of measurable operators and Calkin spaces and characterize the tensor stable singly generated Calkin spaces. Given semifinite von Neumann algebras $$, $$ and corresponding measurable operators $$, $$, we provide a necessary and sufficient condition for the operator $$ to be measurable with respect to $$.

Let $$ be a compact metrizable space equipped with a continuous action of a countable amenable group $$. Suppose that the dynamical system $$ is expansive and is the quotient by a uniformly bounded-to-one factor map of a strongly irreducible subshift. Let $$ be a continuous map commuting with the action of $$. We prove that if there is no pair of distinct $$-homoclinic points in $$ having the same image under $$ then $$ is surjective.

This paper introduces two analogues of the Lusternik–Schnirelmann category from a combinatorial viewpoint. One analogue is defined for finite cell complexes using their subcomplexes and simple homotopy theory; the other is an invariant for finite posets with respect to simple equivalence based on the notion of weak beat points. We examine the relation between these two invariants by taking the face posets of complexes or order complexes of posets.

The linearity defect, introduced by Herzog and Iyengar, is a numerical measure for the complexity of minimal free resolutions. Employing a characterization of the linearity defect due to Şega, we study the behavior of linearity defect along short exact sequences. We point out two classes of short exact sequences involving Koszul modules, along which linearity defect behaves nicely. We also generalize the notion of Koszul filtrations from the graded case to the local setting. Among the applications, we prove that if $$ is a surjection of noetherian local rings such that $$ is a Koszul $$-module, and $$ is a finitely generated $$-module, then the linearity defect of $$ as an $$-module is the same as its linearity defect as an $$-module. In particular, we confirm that specializations of absolutely Koszul algebras are again absolutely Koszul, answering positively a question due to Conca, Iyengar, Nguyen and Römer.

Let $$ be the random increasing sequence of natural numbers which takes each value independently with probability $$, $$, and let $$, $$. We prove that, almost surely, for every measure-preserving system $$ and every $$ the modulated, random averages

$$ converge to 0 pointwise almost everywhere.

A minor is principal means it is defined by the same row and column indices. We study ideals generated by principal minors of size $$ of a generic $$ matrix $$, in the polynomial ring generated over an algebraically closed field by the entries of $$. When $$ the resulting quotient ring is a normal complete intersection domain. We show for any $$, upon inverting $$ the ideals given respectively by the size $$ and the size $$ principal minors become isomorphic. From that we show the algebraic set given by the size $$ principal minors has a codimension $$ component defined by the determinantal ideal, plus a codimension $$ component. When $$ the two components are linked, and in fact, geometrically linked.

We show that the Thurston norm of any irreducible 3-manifold can be detected using twisted Reidemeister torsions corresponding to integral representations and also corresponding to representations over finite fields. In particular, our result holds for all graph manifolds, these are not covered by the earlier work of the first author and Vidussi.

We generalise previous results of the author concerning the compactness of composition operators on the Hardy spaces $$, $$, whose symbol is a universal covering map from the unit disk in the complex plane to general finitely connected domains. We demonstrate that the angular derivative criterion for univalent symbols extends to this more general case. We further show that compactness in this setting is equivalent to compactness of the composition operator induced by a univalent mapping onto the interior of the outer boundary component of the multiply connected domain.

We relax the conditions for measures in our previous paper [Analytic characterizations of gaugeability for generalized Feynman–Kac functionals (2016) Preprint] on analytic characterizations of (conditional) gaugeability for generalized Feynman–Kac functionals in the framework of symmetric Markov processes. The analytic characterization is also equivalent to the maximum principle for generalized Feynman–Kac semigroups, extending the result by Takeda [The bottom of the spectrum of time-changed processes and the maximum principle of Schrödinger operators (2015) Preprint].

We consider a class of Hilbert–Schmidt integral operators with an isotropic, stationary kernel acting on square integrable functions defined on flat tori. For any fixed kernel which is positive and decreasing, we show that among all unit-volume flat tori, the equilateral torus maximizes the operator norm and the Hilbert–Schmidt norm.

We observe that Kirchberg’s QWEP conjecture is equivalent to the statement that $$ is elementarily equivalent to a QWEP C$$ algebra. We also make a few other model-theoretic remarks about WEP and LLP C$$ algebras.

We study operator algebraic properties of the reduced and von Neumann algebraic versions of the free wreath products $$, where $$ is a compact matrix quantum group. Based on recent results on their corepresentation theory by Lemeux and Tarrago in [Lemeux and Tarrago (2014)], we prove that $$ is of Kac type whenever $$ is, and that the reduced version of $$ is simple with unique trace state whenever $$. Moreover, we prove that the reduced von Neumann algebra of $$ does not have property $$.

The decompositions of an element of a finite von Neumann algebra into the sum of a normal operator plus an s.o.t.-quasinilpotent operator, obtained using the Haagerup–Schultz hyperinvariant projections, behave well with respect to holomorphic functional calculus.