In this paper, we define the first topological (σ, τ)-cohomology group and examine vanishing of the first (σ, τ)-cohomology groups of certain triangular Banach algebras. We apply our results to study the (σ, τ)-weak amenability and (σ, τ)-amenability of triangular Banach algebras.

We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability just of a certain explicit family of groups (“Mother groups”) which is done by analyzing the asymptotic properties of random walks on these groups.

We introduce the notion of amenability for affine algebras. We characterize amenability by Følner-sequences, paradoxicality and the existence of finitely invariant dimension-measures. Then we extend the results of Rowen on ranks, from affine algebras of subexponential growth to amenable affine algebras. AMS Subject Classifications: 43A07, 16P90

We study the concept of α-amenability of commutative hypergroups K. We establish several characterizations of α-amenability by combining results of [1] and [2] and adding the Glicksberg-Reiter property. In addition, as examples compact K and discrete polynomial hypergroups on N0 are discussed. 2000 Mathematics Subject Classification: Primary 43A62, Secondary 46H25

Abstract. For any finite unital commutative idempotent semigroup S, a unital semilattice, we show how to compute the amenability constant of its semigroup algebra l(S), which is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. Our theory applies to certain natural subalgebras of Fourier-Stieltjes alg...

In 2014, Zead Mustafa introduced $b_2$-metric spaces, as a generalization of both $2$-metric and $b$-metric spaces. Then new fixed point results for the classes of  rational Geraghty contractive mappings of type I,II and III in the setup of $b_2$-metric spaces are investigated. Then, we prove some fixed point theorems under various contractive conditions in partially ordered $b_2$-metric spaces...

BACKGROUND Although advances in early detection and treatment of cancer improve overall population survival, these advances may not benefit all population groups equally and may heighten racial/ethnic differences in survival. METHODS We identified cancer cases in the Surveillance, Epidemiology and End Results program, who were ages > or = 20 years and diagnosed with one invasive cancer in 199...

Let the function $$\varphi $$ be holomorphic in unit disk $${\mathbb {D}}$$ of complex plane {C}}$$ and let ({\mathbb {D}})\subset {\mathbb . We study level sets critical points hyperbolic derivative , $$\begin{aligned} |D_{\varphi }(z)|:=\frac{(1-|z|^2)|\varphi '(z)|}{1-|\varphi (z)|^2}. \end{aligned}$$ In particular, we show how Schwarzian reveals nature points.