A subgroup $H$ of a group $G$ is said to be weakly $s$-supplementedly embedded in $G$ if there exists a subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq H_{se}\leq H$, where $H_{se}$ is an S-permutably embedded subgroup of $G$. In this paper, we investigate the structure of $G$ under the assumption that some subgroups of prime-power order are weakly S-supplementedly embedded in $G$, and some new criteria for $p$-nilpotency are obtained.

The purpose of this paper is to classify all pairs $(\mathcal{D}, G)$, where $\mathcal{D}$ is a non-trivial $2$-$(v, k, \lambda)$ design with $\lambda\leq10$, and $G\leq \mathrm{Aut}(\mathcal{D})$ acts transitively on the set of blocks of $\mathcal{D}$ and primitively on the set of points of $\mathcal{D}$ with sporadic socle. We prove that there are exactly 15 such pairs $(\mathcal{D}, G)$.

Using explicit methods, we provide an upper bound to the norm of the Busemann cocycle of a locally finite regular tree $X$, emphasizing the symmetries of the cocycle. The latter takes value into a submodule of square summable functions on the edges of $X$, 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 $\mathtt{f}_{a, \nu}$, $\mathtt{g}_{a, \nu}$ and $\mathtt{h}_{a, \nu}$ in the class $\mathcal{A}$ consisting of analytic functions $f$ in the unit disk satisfying $f(0)=f'(0)-1=0$. Here $a \in \{1, 2, 3, \ldots\},$ and $\nu$ 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 $\nu$ is determined for a fixed $a$ to ensure the functions $\mathtt{f}_{a, \nu}$ and $\mathtt{g}_{a, \nu}$ are starlike of positive order in the entire unit disk. When $a=1,$ the results obtained reduced to earlier known results.

Let $X$ be a real Banach space, $T$ be a compact metrizable space, $C(T)$ be the real-valued continuous functions space, and $f:X\rightarrow C(T)$ be a standard $\varepsilon$-isometric embedding. Then for any $\lambda>6$ there is an isometric embedding $h: X\rightarrow C(T)$ such that $\|f(u)-h(u)\|\leq\lambda\varepsilon$ for all $u\in X$.

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 $\sigma$-finite) vector measure on a $\delta$-ring. Extending some nowadays well-known results for the Koml\'os property involving Cesaro sums, we prove that the weak $\sigma$-Fatou property for a Banach lattice of integrable functions $E$ is equivalent to the existence for each norm bounded sequence $(f_n)$ in $E$ of a regular method of summability $D$ such that the sequence $(f_n^D)$ 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 $\mathcal{T}$ be the well-known topological $N=2$ superconformal algebra. In this paper, we prove that every super-skewsymmetric super-biderivation of $\mathcal{T}$ is inner. Based on the result of super-biderivations, we show that all the linear super-commuting maps on $\mathcal{T}$ which have the form $\psi(x)=\lambda x+f(x)c$ 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 $H=\dfrac12 (p_x^2+p_y^2+p_z^2) + \dfrac{ v_0^2}{2} \ln(x^2 - \lambda x^3 + \alpha y^2+b z^2 + c_b^2)$, that is relevant in the study of elliptic galactic dynamics. We first introduce a scale transformation in the coordinates and momenta with a parameter $\varepsilon$ and we find, using averaging theory of first order in $\varepsilon$, the existence up to three periodic orbits if $\alpha,\beta$ are irrational, and one periodic orbit if either $\alpha$ is irrational and $\beta$ is rational, or $\beta$ is irrational and $\alpha$ is rational, for $\varepsilon$ 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.