In earlier papers C. Dartyge and A. Sárközy presented large families of pseudorandom subsets with prime moduli. In this paper we extend two large families of subsets to the case when the moduli is composite.

If $W$ is a finite Coxeter group and $\varphi$ is a weight function, Lusztig has defined {\it $\\varphi$-constructible characters} of $W$, as well as a partition of the set of irreducible characters of $W$ into {\it Lusztig $\varphi$-families}. We prove that every Lusztig $\varphi$-family contains a unique character with minimal $b$-invariant, and that every $\varphi$-constructible character has a unique irreducible constituent with minimal $b$-invariant. This generalizes Lusztig's result about {\it special characters} to the case where $\varphi$ is not constant. This is compatible with some conjectures of Rouquier and the author about {\it Calogero-Moser families} and {\it Calogero-Moser cellular characters}.

New cyclic Brunn-Minkowski inequalities for dual quermassintegrals of star bodies are established. These inequalities are obtained with respect to radial Minkowski addition as well as radial Blaschke and harmonic Blaschke addition.

We give a contractive Schur multiplier characterization of locally compact groups coarsely embeddable into Hilbert spaces. Consequently, all locally compact groups whose weak Haagerup constant is 1 embed coarsely into Hilbert spaces, and hence the Baum-Connes assembly map with coefficients is split-injective for such groups.

In the present paper, considering a recent technique, which is used by Wardowski [24] for fixed points of single-valued mappings, we give a new and general class of multivalued weakly Picard operators on complete metric spaces and show that the class of multivalued almost contractions (both linear and nonlinear cases in the sense of Berinde and Berinde [9]) is a proper subset of this new class. We also give a nontrivial example showing this fact.

We study nuclearity of the $C^*$-algebra $\mathbb B(\mathcal E)$ of adjointable operators on a full Hilbert $C^*$-module $\mathcal E$ over a $C^*$-algebra $\mathcal A$. When $\mathcal A$ is a von Neumann algebra and $\mathcal E$ is full and self dual, we show that $\mathbb B(\mathcal E)$ is nuclear if and only if $\mathcal A$ is nuclear and $\mathcal E$ is finitely generated. In particular, when $\mathcal A$ is a factor, then nuclearity of $\mathbb B(\mathcal E)$ implies that $\mathcal E$, $\mathcal A$ and $\mathbb B(\mathcal E)$ are finite dimensional.

The note establishes a new weak mean ergodic theorem (Theorem A) for $1$-cocycles associated to weakly mixing representations of amenable groups.

We prove the existence of an additive semigroup of cardinality $2^\mathfrak c$ contained in the intersection of the classes of Hamel functions ($\rm HF$) and Sierpiński-Zygmund functions ($\rm SZ$). In addition, we show that under certain set-theoretic assumptions the lineability of the class of Sierpiński-Zygmund functions ($\rm SZ$) is equal to the lineability of the class of almost continuous Sierpiński-Zygmund functions ($\rm AC\cap\rm SZ$).

We give simple upper bounds for rational sectional category and use them to compute invariants of the type of Farber's topological complexity of rational spaces. In particular we show that the sectional category of formal morphisms reaches its cohomological lower bound and give a method to compute higher topological complexity of formal spaces in terms of their cohomology.

We prove in a uniform way that all ultradifferentiable function classes $\mathcal{E}_{\{\mathcal{M}\}}$ of Roumieu-type and $\mathcal{E}_{(\mathcal{M})}$ of Beurling-type defined in terms of a weight matrix $\mathcal{M}$ admit a convenient setting if $\mathcal{M}$ satisfies some mild regularity conditions. For $\mathcal{C}$ denoting either $\mathcal{E}_{\{\mathcal{M}\}}$ or $\mathcal{E}_{(\mathcal{M})}$ the category $\mathcal{C}$ is cartesian closed, i.e. $\mathcal{C}(E\times F,G)\cong\mathcal{C}(E,\mathcal{C}(F,G))$ for $E,F,G$ convenient vector spaces. As special cases one obtains the classes $\mathcal{E}_{\{M\}}$ and $\mathcal{E}_{(M)}$ respectively $\mathcal{E}_{\{\omega\}}$ and $\mathcal{E}_{(\omega)}$ defined by a weight sequence $M$ respectively a weight function $\omega$.

In this paper, we give the value of the approximation exponent of the hyperquadratic power series satisfying the equation $$Cx^{r}-Ax^{r-1}-1=0$$ where $r> 2$ is a power of a prime number $p$, $A$ and $C$ are nonzero polynomials over a finite field $\mathbf{K}$ of characteristic $p$ and $\deg A> \deg C$. Further, we exhibit explicitly its continued fraction expansion when $C$ divides $A$.