In this paperwe present a fixed point property for amenable hypergroups that is analogous to Rickert’s fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, certain hypergroups are shown to have a left Haar measure.

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

We give an explicit formula for the resultant of Chebyshev polynomials of the first, second, third, and fourth kinds. We also compute the resultant of modified cyclotomic polynomials.

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

Let $A\,:=\,-\,\left( \nabla \,-\,i\overrightarrow{a} \right)\,.\,\left( \nabla \,-\,i\overrightarrow{a} \right)\,+\,V$ be a magnetic Schrödinger operator on ${{\mathbb{R}}^{n}}$, where

$$\vec{a}:=\left( {{a}_{1}},...,{{a}_{n}} \right)\in L_{\text{loc}}^{2}\left( {{\mathbb{R}}^{n}},{{\mathbb{R}}^{n}} \right)\operatorname{and}0\le V\in L_{\text{loc}}^{1}\left( {{\mathbb{R}}^{n}} \right)$$

satisfy some reverse Hölder conditions. Let $\phi :\,{{\mathbb{R}}^{n}}\,\times \,[0,\,\infty )\,\to \,[0,\,\infty )$ be such that $\phi \left( x,\,. \right)$ for any given $x\,\in \,{{\mathbb{R}}^{n}}$ is an Orlicz function, $\phi \left( ^{.}\,,\,t \right)\,\in \,{{\mathbb{A}}_{\infty }}\left( {{\mathbb{R}}^{n}} \right)$ for all $t\,\in \,\left( 0,\,\infty \right)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I\left( \phi \right)\,\in \,(0,\,1]$. In this article, the authors prove that second-order Riesz transforms $V{{A}^{-1}}$ and ${{\left( \nabla \,-\,i\overrightarrow{a} \right)}^{2}}{{A}^{-1}}$ are bounded from the Musielak–Orlicz–Hardy space ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$, associated with $A$, to theMusielak–Orlicz space ${{L}^{\phi }}\left( {{\mathbb{R}}^{n}} \right)$. Moreover, we establish the boundedness of $V{{A}^{-1}}$ on ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$. As applications, some maximal inequalities associated with $A$ in the scale of ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$ are obtained.

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.