Quasi Controlled K-Metric Spaces over C^(*)-Algebras with an Application to Stochastic Integral Equations

Generally,the field of fixed point theory has attracted the attention of researchers in different fields of science and engineering due to its use in proving the existence and uniqueness of solutions of real-world dynamic models.C^(∗)-algebra is being continually used to explain a physical system in quantum field theory and statistical mechanics and has subsequently become an important area of research.The concept of a C^(∗)-algebra-valued metric space was introduced in 2014 to generalize the concept of metric space.In fact,It is a generalization by replacing the set of real numbers with a C^(∗)-algebra.After that,this line of research continued,where several fixed point results have been obtained in the framework of C^(∗)-algebra valued metric,aswell as(more general)C^(∗)-algebra-valued b-metric spaces andC^(∗)-algebra-valued extended b-metric spaces.Very recently,based on the concept and properties of C^(∗)-algebras,we have studied the quasi-case of such spaces to give a more general notion of relaxing the triangular inequality in the asymmetric case.In this paper,we first introduce the concept of C^(∗)-algebra-valued quasi-controlledK-metric spaces and prove some fixed point theorems that remain valid in this setting.To support our main results,we also furnish some exampleswhichdemonstrate theutility of ourmainresult.Finally,as an application,we useour results to prove the existence and uniqueness of the solution to a nonlinear stochastic integral equation.

A Determinant Criterion of Invertible Elements in Group Algebras

An embedding from a group algebra to a matrix algebra is given in this paper. By using it, a criterion for an invertible element in a group algebra is proven.

T-local Derivations of von Neumann Algebras

In this paper, we introduce the concept of T-local derivations and obtain the main result: each T-local derivation of a von Neumann algebra A into a dual A-bimodule M is a T-derivation, where T is an endomorphism of A to A.

Generalized Szasz Problems in Normal Classes of Algebras

Let A be a normal class of algebras. In the present paper, we characterize the following four problems for A: for which radical class R, there holds that(1) R(i1∧i2) = R(i1)∧R(i2);(2) R(i1∨i2) = R(i1)∨R(i2);(3) (i1∧i2) =i1∧i2;(4) (i1∨i2) = i1∨i2, for arbitrary algebra a∈A and any i1,i2∈La,where j denotes the ideal of a uniquely determined by R(a/j) = j/j?

THE BANACH-LIE GROUP OF LIE TRIPLE AUTOMORPHISMS OF AN H^＊-ALGEBRA^＊

We study the Banach-Lie group Ltaut（A） of Lie triple automorphisms of a complex associative H^＊-algebra A. Some consequences about its Lie algebra, the algebra of Lie triple derivations of A, Ltder（A）, are obtained. For a topologically simple A, in the infinite-dimensional case we have Ltaut（A）0 = Aut（A） implying Ltder（A） = Der（A）. In the finite-dimensional case Ltaut（A）0 is a direct product of Aut（A） and a certain subgroup of Lie derivations δ from A to its center, annihilating commutators.

Algebraic algorithm for reliability index weighted capacity of communication network

Communication network has communication capacity and connection reliability of the links. They canbe independently defined and can be used separately, and when the reliability of a communication network isanalyzed from a macroscopical angle of view, it is more objective to express the performance index of a commu-nication network as a whole. The reliability index weighted capacity is just obtained by integrating these two pa-rameters. It is necessary to further study the algorithm to calculate the reliability index of the communicationnetwork with a complicated topologic structure and a whole algebraic algorithm is therefore proposed for calcula-tion of the reliability index weighted capacity of a communication network with a topologic structure. The wholecomputational procedure of the algorithm is illustrated with a typical example.

The Maximal Graded Left Quotient Algebra of a Graded Algebra

We construct the maximal graded left quotient algebra of every graded algebra A without homogeneous total right zero divisors as the direct limit of graded homomorphisms （of left A-modules） from graded dense left ideals of A into a graded left quotient algebra of A. In the case of a superalgebra, and with some extra hypothesis, we prove that the component in the neutral element of the group of the maximal graded left quotient algebra coincides with the maximal left quotient algebra of the component in the neutral element of the group of the superalgebra.

Distance Between Unitary Orbits of Self-Adjoint Elements in C^(*)-Algebras of Tracial Rank One

The note studies certain distance between unitary orbits.A result about Riesz interpolation property is proved in the first place.Weyl(1912) shows that dist(U(x),U(y))= δ(x,y) for self-adjoint elements in matrixes.The author generalizes the result to C*-algebras of tracial rank one.It is proved that dist(U(x),U(y)) = D_(c)(x,y) in unital AT-algebras and in unital simple C*-algebras of tracial rank one,where x,y are self-adjoint elements and D_(C)(x,y) is a notion generalized from δ(x,y).

On Evolution Algebras

The structural constants of an evolution algebra are given by a quadratic matrix. In this work we establish an equivalence between nil, right nilpotent evolution algebras and evolution algebras defined by upper triangular matrices. The classification of 2-dimensional complex evolution algebras is obtained. For an evolution algebra with a special form of the matrix, we describe all its isomorphisms and their compositions. We construct an algorithm running under Mathematica which decides if two finite dimensional evolution algebras are isomorphic.

Twisted bounded-dilation group C*-algebras as C*-metric algebras

We construct a class of C*-metric algebras. We prove that for a discrete group Γ with a 2-cocycle σ,the closure of the seminorm ||[Ml1,·]|| on Cc(Γ, σ) is a Leibniz Lip-norm on the twisted reduced group C*-algebra C*r(Γ, σ) for the pointwise multiplication operator Mlon l2(Γ), induced by a proper length function l on Γ with the property of bounded θ-dilation. Moreover, the compact quantum metric space structures depend only on the cohomology class of 2-cocycles in the Lipschitz isometric sense.

On Some Properties of Graph of Prefix Code

We investigate decomposition of codes and finite languages. A prime decomposition is a decomposition of a code or languages into a concatenation of nontrivial prime codes or languages. A code is prime if it cannot be decomposed into at least two nontrivial codes as the same for the languages. In the paper, a linear time algorithm is designed, which finds the prime decomposition. If codes or finite languages are presented as given by its minimal deterministic automaton, then from the point of view of abstract algebra and graph theory, this automaton has special properties. The study was conducted using system for computational Discrete Algebra GAP. .

The double Ringel-Hall algebra on a hereditary abelian finitary length category

In this paper, we study the category H (ρ) of semi-stable coherent sheaves of a fixed slope ρ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of H (ρ) and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.

Gorenstein Projective Dimensions of Modules over Minimal Auslander-Gorenstein Algebras

In this article we investigate the relations between the Gorenstein projective dimensions of Λ-modules and their socles for re-minimal Auslander-Gorenstein algebras Λ.First we give a description of projective-injective Λ-modules in terms of their socles.Then we prove that a Λ-module N has Gorenstein projective dimension at most n if and only if its socle has Gorenstein projective dimension at most n if and only if N is cogenerated by a projective Λ-module.Furthermore,we show that n-minimal Auslander-Gorenstein algebras can be characterised by the relations between the Gorenstein projective dimensions of modules and their socles.

Finite Generation of Lie Derived Powers of Skew Lie Algebras

Let A be a finitely generated associative algebra over a field of characteristic different from 2.Herstein asked when the Lie algebra[A,A]is finitely generated.Recently,it was shown that for a finitely generated nil algebra A all derived powers of A are finitely generated Lie algebras.Let K be the Lie algebra of skew-symmetric elements of an associative algebra with involution.We consider all derived powers of the Lie algebra K and prove that for any finitely generated associative nil algebra with an involution,all derived powers of K are finitely generated Lie algebras.

ON ZEROS OF ALGEBRAIC EQUATIONS——AN APPLICATION OF RITT PRINCIPLE

Let K be a basic field of characteristic 0, and fi,i=1,…,r, be polynomials in K(x1,…Xn)Consider the system of algebraic equations which defines an algebraic variety V consisting of zeros of the system in an arbitrary extension field of K. The study of the structure of V or that of the set

Lychrel Numbers in Base 10: A Probabilistic Approach

For decades, Lychrel numbers have been studied on many bases. Their existence has been proven in base 2, 11 or 17. This paper presents a probabilistic proof of the existence of Lychrel number in base 10 and provides some properties which enable a mathematical extraction of new Lychrel numbers from existing ones. This probabilistic approach has the advantage of being extendable to other bases. The results show that palindromes can also be Lychrel numbers.

可信联邦学习进化优化算法综述

With the development of edge devices and cloud computing,the question of how to accomplish machine learning and optimization tasks in a privacy-preserving and secure way has attracted increased attention over the past decade.As a privacy-preserving distributed machine learning method,federated learning(FL)has become popular in the last few years.However,the data privacy issue also occurs when solving optimization problems,which has received little attention so far.This survey paper is concerned with privacy-preserving optimization,with a focus on privacy-preserving data-driven evolutionary optimization.It aims to provide a roadmap from secure privacy-preserving learning to secure privacy-preserving optimization by summarizing security mechanisms and privacy-preserving approaches that can be employed in machine learning and optimization.We provide a formal definition of security and privacy in learning,followed by a comprehensive review of FL schemes and cryptographic privacy-preserving techniques.Then,we present ideas on the emerging area of privacy-preserving optimization,ranging from privacy-preserving distributed optimization to privacy-preserving evolutionary optimization and privacy-preserving Bayesian optimization(BO).We further provide a thorough security analysis of BO and evolutionary optimization methods from the perspective of inferring attacks and active attacks.On the basis of the above,an in-depth discussion is given to analyze what FL and distributed optimization strategies can be used for the design of federated optimization and what additional requirements are needed for achieving these strategies.Finally,we conclude the survey by outlining open questions and remaining challenges in federated data-driven optimization.We hope this survey can provide insights into the relationship between FL and federated optimization and will promote research interest in secure federated optimization.

ON SUFFICIENT ALGEBRAIC CRITERIA FOR TRANSCENDENTAL FUNCTION det(a_(ij)+b_(ij)e-λτ—δ_(ij)λ)_(n×n) ONLY WITH NULL POINTS HAVING NEGATIVE REAL PART

Ⅰ. INTRODUCTION In the theory of stability for difference differential equation and functional differential equation, the null points distribution of the transcendental function det (aij+bije-λτ—δijλ)n×n is a basic problem to be studied so far as we know the transcendental criteria for null points distribution of the polynomial H(λ, e-λ) were given by L. S. Pontriagin. Qin Yuanxun et al. offered the equivalent algebraic criteria. All these are sufficient and necessary conditions. But when n≥2, it is rather difficult to check the conditions of these criteria.

THE NECESSARY AND SUFFICIENT CONDITION OF A COMPLETELY REDUCIBLE MODULE OF THE LIE ALGEBRAS G(A)

Suppose that A is an n×n complex matrix. Denote by G(A)the contragradient Lie algebra corresponding to the matrix A. If A is a generalized Cartan matrix, we call G(A) a Kac-Moody Lie algebra.

On the “Onion Husk” Algorithm for Approximate Solution of the Traveling Salesman Problem

The paper describes some implementation aspects of an algorithm for approximate solution of the traveling salesman problem based on the construction of convex closed contours on the initial set of points (“cities”) and their subsequent combination into a closed path (the so-called contour algorithm or “onion husk” algorithm). A number of heuristics related to the different stages of the algorithm are considered, and various variants of the algorithm based on these heuristics are analyzed. Sets of randomly generated points of different sizes (from 4 to 90 and from 500 to 10,000) were used to test the algorithms. The numerical results obtained are compared with the results of two well-known combinatorial optimization algorithms, namely the algorithm based on the branch and bound method and the simulated annealing algorithm. .