scholarly journals Characterization of ideals of rhotrices over a ring and its applications

2021 ◽  
Vol 27 (1) ◽  
pp. 138-147
Author(s):  
Kailash M. Patil ◽  
Keyword(s):  
Banach Algebra ◽  
Complex Plane ◽  
Higher Order ◽  
Unital Ring ◽  
Maximal Ideals ◽  
Unital Banach Algebra

We define higher order rhotrices over a commutative unital ring S and obtain a ring \mathcal{R}_n(S) of rhotrices of the order n \in \mathbb{N}. We characterize the ideals and maximal ideals of \mathcal{R}_n(S). As a particular case, we record ideals of rhotrix rings over integers and rhotrix algebras over complex plane \mathbb{C}. As an application, we characterize the maximal ideals of the commutative unital Banach algebra \mathcal{R}_n(\mathbb{C}).

A characterization of Jordan and 5-Jordan homomorphisms between Banach algebras

2018 ◽  
Vol 11 (02) ◽  
pp. 1850021 ◽  
Author(s):  
A. Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms ◽  
Unital Banach Algebra

We prove that each surjective Jordan homomorphism from a Banach algebra [Formula: see text] onto a semiprime commutative Banach algebra [Formula: see text] is a homomorphism, and each 5-Jordan homomorphism from a unital Banach algebra [Formula: see text] into a semisimple commutative Banach algebra [Formula: see text] is a 5-homomorphism.

Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜

2018 ◽  
Vol 17 (09) ◽  
pp. 1850169 ◽  
Author(s):  
Hossein Javanshiri ◽  
Mehdi Nemati
Keyword(s):  
Banach Algebra ◽  
Cohomology Group ◽  
Banach Algebras ◽  
Multiplier Algebra ◽  
Topological Center ◽  
Maximal Ideals ◽  
Open Questions ◽  
Arens Regularity ◽  
Amalgamated Duplication

Let [Formula: see text] and [Formula: see text] be Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with compatible actions. We define the product [Formula: see text], which is a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of [Formula: see text], we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of [Formula: see text] in relation to that of the algebras [Formula: see text], [Formula: see text] and the action of [Formula: see text] on [Formula: see text]. We also compute the first cohomology group [Formula: see text] for all [Formula: see text] as well as the first-order cyclic cohomology group [Formula: see text], where [Formula: see text] is the [Formula: see text]th dual space of [Formula: see text] when [Formula: see text] and [Formula: see text] itself when [Formula: see text]. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and [Formula: see text]-weak amenability of [Formula: see text]. In this context, several open questions arise.

On a Characterization of Maximal Ideals

1970 ◽  
Vol 13 (2) ◽  
pp. 219-220
Author(s):  
Jamil A. Siddiqi
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Analytic Functions ◽  
Entire Functions ◽  
Factorization Theorem ◽  
Codimension One ◽  
Complex Banach Algebra ◽  
Maximal Ideals ◽  
Invertible Elements

Let A be a commutative complex Banach algebra with identity e. Gleason [1] (cf. also Kahane and Żelazko [2]) has given the following characterization of maximal ideals in A.Theorem. A subspace X ⊂ A of codimension one is a maximal ideal in A if and only if it consists of non-invertible elements.The proofs given by Gleason and by Kahane and Żelazko are both based on the use of Hadamard's factorization theorem for entire functions. In this note we show that this can be avoided by using elementary properties of analytic functions.

scholarly journals Algebras and differential equations

1977 ◽  
Vol 68 ◽  
pp. 59-122 ◽  
Author(s):  
Helmut Röhrl
Keyword(s):  
Differential Equations ◽  
Banach Algebra ◽  
Ordinary Differential Equations ◽  
Polynomial Ring ◽  
Matrix Multiplication ◽  
Complex Numbers ◽  
Unital Ring ◽  
Right Hand ◽  
The Right ◽  
Unital Banach Algebra

One purpose of this paper is a purely algebraic study of (systems of) ordinary differential equations of the typewhere the coefficients are taken from a fixed associative, commutative, unital ring R, such as the field R of real or C of complex numbers or a commutative, unital Banach algebra. The right hand sides of D are considered to be elements in the polynomial ring R[X1, …, Xn] of associating but non-commuting variables X1, …, Xn. An algebraic study calls for maps between such differential equations and, in fact, morphisms are defined between differential equations having the same arity m but not necessarily the same dimension n. These morphisms are rectangular matrices with entries in R which satisfy certain relations. This leads to a category RDiffm whose objects are precisely the differential equations of arity m and in which the composition of the morphisms is the usual matrix multiplication.

scholarly journals A closed-loop optogenetic screen for neurons controlling feeding in Drosophila

2021 ◽  
Author(s):  
Celia K S Lau ◽  
Meghan Jelen ◽  
Michael D Gordon
Keyword(s):  
Drosophila Melanogaster ◽  
Expression Pattern ◽  
Sensory Neurons ◽  
Closed Loop ◽  
Higher Order ◽  
Taste Detection ◽  
Proof Of Principle ◽  
Feeding Circuits

Abstract Feeding is an essential part of animal life that is greatly impacted by the sense of taste. Although the characterization of taste-detection at the periphery has been extensive, higher order taste and feeding circuits are still being elucidated. Here, we use an automated closed-loop optogenetic activation screen to detect novel taste and feeding neurons in Drosophila melanogaster. Out of 122 Janelia FlyLight Project GAL4 lines preselected based on expression pattern, we identify six lines that acutely promote feeding and 35 lines that inhibit it. As proof of principle, we follow up on R70C07-GAL4, which labels neurons that strongly inhibit feeding. Using split-GAL4 lines to isolate subsets of the R70C07-GAL4 population, we find both appetitive and aversive neurons. Furthermore, we show that R70C07-GAL4 labels putative second-order taste interneurons that contact both sweet and bitter sensory neurons. These results serve as a resource for further functional dissection of fly feeding circuits.

scholarly journals A Characterization of the Dynamics of Schröder’s Method for Polynomials with Two Roots

2021 ◽  
Vol 5 (1) ◽  
pp. 25
Author(s):  
Víctor Galilea ◽  
José M. Gutiérrez
Keyword(s):  
Nonlinear Equations ◽  
Complex Plane ◽  
Iterative Process ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Schröder’S Method

The purpose of this work is to give a first approach to the dynamical behavior of Schröder’s method, a well-known iterative process for solving nonlinear equations. In this context, we consider equations defined in the complex plane. By using topological conjugations, we characterize the basins of attraction of Schröder’s method applied to polynomials with two roots and different multiplicities. Actually, we show that these basins are half-planes or circles, depending on the multiplicities of the roots. We conclude our study with a graphical gallery that allow us to compare the basins of attraction of Newton’s and Schröder’s method applied to some given polynomials.

Near‐Field Characterization of Higher‐Order Topological Photonic States at Optical Frequencies

2021 ◽  
pp. 2004376
Author(s):  
Anton Vakulenko ◽  
Svetlana Kiriushechkina ◽  
Mingsong Wang ◽  
Mengyao Li ◽  
Dmitry Zhirihin ◽  
...  
Keyword(s):  
Near Field ◽  
Higher Order ◽  
Photonic States

scholarly journals Extended lubrication theory: improved estimates of flow in channels with variable geometry

2017 ◽  
Vol 473 (2206) ◽  
pp. 20170234 ◽  
Author(s):  
Behrouz Tavakol ◽  
Guillaume Froehlicher ◽  
Douglas P. Holmes ◽  
Howard A. Stone
Keyword(s):  
Stokes Equations ◽  
Perturbation Expansion ◽  
Accurate Estimate ◽  
Higher Order ◽  
Lubrication Theory ◽  
Channel Height ◽  
Fluid Films ◽  
Flow Characterization ◽  
Systematic Perturbation

Lubrication theory is broadly applicable to the flow characterization of thin fluid films and the motion of particles near surfaces. We offer an extension to lubrication theory by starting with Stokes equations and considering higher-order terms in a systematic perturbation expansion to describe the fluid flow in a channel with features of a modest aspect ratio. Experimental results qualitatively confirm the higher-order analytical solutions, while numerical results are in very good agreement with the higher-order analytical results. We show that the extended lubrication theory is a robust tool for an accurate estimate of pressure drop in channels with shape changes on the order of the channel height, accounting for both smooth and sharp changes in geometry.

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