In this paper we study the concept of ph-biatness ofa Banach algebra A, where ph is a continuous homomorphism on A.We prove that if ph is a continuous epimorphism on A and A hasa bounded approximate identity and A is ph- biat, then A is ph-amenable. In the case where ph is an isomorphism on A we showthat the ph- amenability of A implies its ph-biatness.

The literature on Dedekind sums is vast. In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational poly-topes. In particular, there are some natural finite Fourier series which we call Fourier-Dedekind sums, and which form the building blocks of the number of partitions of an integer...

A Dedekind algebra is an ordered pair (B, h) where B is a nonempty set and h is a "similarity transformation" on B. Among the Dedekind algebras is the sequence of positive integers. Each Dedekind algebra can be decomposed into a family of disjointed, countable subalgebras which are called the configurations of the algebra. There are many isomorphic types of configurations. Each Dedekind algebra...

The main aim of this paper is to analyse minimally-coupled scalar-fields -- quintessence and phantom as the candidates explain accelerated expansion universe compare its observables current cosmological observations; a byproduct we present python module. This work includes parameter $\epsilon$ which allows incorporate both fields within same analysis. Examples potentials, so far included, are $...

Vaught’s conjecture says that for any countable (complete) first-order theory T, the number of non-isomorphic countable models of T is either countable or 2, where ω is the first infinite cardinal. Vaught’s conjecture for ω-stable theories of modules was proved by Garavaglia [6, Theorem 6]. Buechler proved that Vaught’s conjecture is correct for modules of U-rank 1 [2]. It has been shown that V...

Abstract Dedekind type DC sums and their generalizations are defined in terms of Euler functions generalization. Recently, Ma et al. (Adv. Differ. Equ. 2021:30 2021) introduced the poly-Dedekind by replacing function appearing sums, they were shown to satisfy a reciprocity relation. In this paper, we consider two kinds new sums. One is unipoly-Dedekind sum associated with 2 unipoly-Euler expres...

Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism between the space of Dedekind symbols with polynomial (Laurent polynomial) reciprocity laws and the space of cusp (modular) forms. In this article we introduce Hecke operators on the space of weighted Dedekind...

Let $mathcal {A} $ and $mathcal {B} $ be C$^*$-algebras. Assume that $mathcal {A}$ is of real rank zero and unital with unit $I$ and $k>0$ is a real number. It is shown that if $Phi:mathcal{A} tomathcal{B}$ is an additive map preserving $|cdot|^k$ for all normal elements; that is, $Phi(|A|^k)=|Phi(A)|^k $ for all normal elements $Ainmathcal A$, $Phi(I)$ is a projection, and there exists a posit...

A Dedekind cut of an ordered abelian group G is a pair (X, Y) of nonempty subsets of G where Y=G−X and every member of X precedes every member of Y. A Dedekind cut (X, Y) is said to be continuous if X has a greatest member or Y has a least member, but not both; if every Dedekind cut of G is a continuous cut, G is said to be (Dedekind) continuous. The ordered abelian group R of real numbers is, ...

The subject of this thesis is the theory of nonholomorphic modular forms of non-integral weight, and its applications to arithmetical functions involving Dedekind sums and Kloosterman sums. As was discovered by Andre Weil, automorphic forms of non-integral weight correspond to invariant funtions on Metaplectic groups. We thus give an explicit description of Meptaplectic groups corresponding to ...