A subgroup $$ of a group $$ is said to be weakly $$-supplementedly embedded in $$ if there exists a subgroup $$ of $$ such that $$ and $$, where $$ is an S-permutably embedded subgroup of $$. In this paper, we investigate the structure of $$ under the assumption that some subgroups of prime-power order are weakly S-supplementedly embedded in $$, and some new criteria for $$-nilpotency are obtained.

The purpose of this paper is to classify all pairs $$, where $$ is a non-trivial $$-$$ design with $$, and $$ acts transitively on the set of blocks of $$ and primitively on the set of points of $$ with sporadic socle. We prove that there are exactly 15 such pairs $$.

Using explicit methods, we provide an upper bound to the norm of the Busemann cocycle of a locally finite regular tree $$, emphasizing the symmetries of the cocycle. The latter takes value into a submodule of square summable functions on the edges of $$, which corresponds to the Steinberg representation for rank one groups acting on their Bruhat-Tits tree. The norm of the Busemann cocycle is asymptotically linear with respect to square root of the distance between any two vertices. Independently, Gournay and Jolissaint [10] proved an exact formula for harmonic 1-cocycles covering the present case.

This paper investigates three functions $$, $$ and $$ in the class $$ consisting of analytic functions $$ in the unit disk satisfying $$. Here $$ and $$ is real. Each function is related to the generalized Bessel function. The radius of starlikeness of positive order is obtained for each of the three functions. Further, the best range on $$ is determined for a fixed $$ to ensure the functions $$ and $$ are starlike of positive order in the entire unit disk. When $$ the results obtained reduced to earlier known results.

Let $$ be a real Banach space, $$ be a compact metrizable space, $$ be the real-valued continuous functions space, and $$ be a standard $$-isometric embedding. Then for any $$ there is an isometric embedding $$ such that $$ for all $$.

Consider an abstract Banach lattice. Under some mild assumptions, it can be identified with a Banach ideal of integrable functions with respect to a (non necessarily $$-finite) vector measure on a $$-ring. Extending some nowadays well-known results for the Koml\'os property involving Cesaro sums, we prove that the weak $$-Fatou property for a Banach lattice of integrable functions $$ is equivalent to the existence for each norm bounded sequence $$ in $$ of a regular method of summability $$ such that the sequence $$ converges.

A new characterization for the boundedness, compactness and essential norm for the differences of weighted composition operators on weighted type spaces is given in this paper.

The notion of two-numbers of connected Riemannian manifolds was introduced about 35 years ago in [Un invariant geometrique riemannien, C. R. Acad. Sci. Paris Math. 295 (1982), 389--391] by B.-Y. Chen and T. Nagano. Later, two-numbers have been studied by a number of mathematicians and it was then proved that two-numbers related closely with several important areas in mathematics. The main purpose of this article is to survey on two-numbers and their applications. At the end of this survey, we present several open problems and conjectures on two-numbers.

Let $$ be the well-known topological $$ superconformal algebra. In this paper, we prove that every super-skewsymmetric super-biderivation of $$ is inner. Based on the result of super-biderivations, we show that all the linear super-commuting maps on $$ which have the form $$ are not standard.

We apply the averaging theory of first order to study analytically families of periodic orbits for a three dimensional logarithmic galactic potential $$, that is relevant in the study of elliptic galactic dynamics. We first introduce a scale transformation in the coordinates and momenta with a parameter $$ and we find, using averaging theory of first order in $$, the existence up to three periodic orbits if $$ are irrational, and one periodic orbit if either $$ is irrational and $$ is rational, or $$ is irrational and $$ is rational, for $$ sufficiently small.

We prove that the automorphism group of a topological parallelism on real projective 3-space is compact. This settles a conjecture stated in [1], where it was proved that at least the connected component of the identity is compact.