Using Krasnoselskii’s fixed point theorem, we deduce the existence of periodic solutions of nonlinear system of integro-dynamic equations on periodic time scales. These equations are studied under a set of assumptions on the functions involved in the equations. The equations will be called almost linear when these assumptions hold. The results of this paper are new for the continuous and discrete time scales.

In this note we study the finite groups whose subgroup lattices are dismantlable.

We derive telescoping majorants and minorants for some classes of series and give applications of these results.

In this paper we prove that there is only one conjugacy class of dihedral group of order $2p$ in the $2\left( p\,-\,1 \right)\,\times \,2\left( p\,-\,1 \right)$ integral symplectic group that can be realized by an analytic automorphism group of compact connected Riemann surfaces of genus $p\,-\,1$. A pair of representative generators of the realizable class is also given.

Extending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra $\mathbb{B}\left( H \right)$ of bounded linear operators acting on a Hilbert space $H$. Among other things, we investigate the relationship between the approximate parallelism and norm of inner derivations on $\mathbb{B}\left( H \right)$. We also characterize the parallel elements of a ${{C}^{*}}$-algebra by using states. Finally we utilize the linking algebra to give some equivalent assertions regarding parallel elements in a Hilbert ${{C}^{*}}$-module.