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The theory of linear topological dynamics is a rapidly growing field of research over the last three decades or so. This book presents a survey of recent results of the author obtained in this field during the period 2016-2019. Without any doubt, this is the first research monograph concerning the topological dynamics of multivalued operators and binary relations, especially, multivalued linear woperators, simple graphs, digraphs and tournaments (we feel duty bound to say that multivalued topological dynamics is still a very undeveloped field of investigation, full of open problems and possible for further expansion). Asiede from that, the main purpose of this monograph is to consider topologically dynamical properties of linear single-valued operators in Frechet spaces and abstract fractional differential equations in Frechet spaces, which could be degenerate or non-degenerate in time variable. In this monograph, we use only two types of fractional derivatives, namely the Caputo time-fractional derivatives and Weyl time-fractional derivatives. However, most results on dynamics of differential equations are given to the abstract differential equations with integer order derivatives, especially those of first and second order in time. The monograph is consistsed of two chapters; the first chapter is further broken down into nine sections, while the second chapter is broken down into seven sections. It is not of introductory character to linear topological dynamics and it is not written in a traditional manner. As in all my previously published monographs, the numbering of definitions, theorems, propositions, remarks, lemmas, corollaries, definitions, etc., are by chapter and section; the bibliography is by author in alphabetic order. Concerning target audience, wWe deeply believe that the book could be of invaluable help to experts in linear topological dynamics, researchers in abstract partial differential equations but and also to PhD students and advanced graduate students in mathematics as well. A potential reader should be familiar with backgrounds including elementary functional analysis, measure and integration theory as well as the basic theory of abstract (degenerate) Volterra integro-differential equations. At some places, the knowledge of graph theory is preferable but not demandedable. This monograph is not intended to be a comprehensive review of current trends; albeit includes several recent results from the field of linear topological dynamics and has more than 450 titles, our reference list is far from being exhaustively complete.

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

A new class of (not necessarily bounded) operators related to (mainly infinite) directed trees is introduced and investigated. Operators in question are to be considered as a generalization of classical weighted shifts, on the one hand, and of weighted adjacency operators, on the other; they are called weighted shifts on directed trees. The basic properties of such operators, including closedness, adjoints, polar decomposition and moduli are studied. Circularity and the Fredholmness of weighted shifts on directed trees are discussed. The relationships between domains of a weighted shift on a directed tree and its adjoint are described. Hyponormality, cohyponormality, subnormality and complete hyperexpansivity of such operators are entirely characterized in terms of their weights. Related questions that arose during the study of the topic are solved as well. Particular trees with one branching vertex are intensively studied mostly in the context of subnormality and complete hyperexpansivity of weighted shifts on them. A strict connection of the latter with

In this volume we study noncommutative domains Each such a domain has a universal model Free holomorphic functions, Cauchy transforms, and Poisson transforms on noncommutative domains We associate with each We introduce two numerical invariants, the curvature and We present a commutant lifting theorem for pure In the particular case when

The use of mathematical concepts of mapping in cryptography provides advantages in securing text or image data. The qualitative properties of mapping can preserve data that is kept confidential. Two of the essential properties in the mapping are the reversible and the preserving area. In this article, besides constructing a linear mapping derived from a second-order QRT difference equation and examining its qualitative properties, the coding procedure is used to encrypt text and fractal images based on the two-dimensional linear maps. For the digital text and image security algorithms, we developed the pseudo code algorithm implemented in Mathematica®. The proposed encoding technique will be compared with a 2D mKdV linear map to demonstrate its efficacy.

The first two chapters of this book offer a modern, self-contained exposition of the elementary theory of triangulated categories and their quotients. The simple, elegant presentation of these known results makes these chapters eminently suitable as a text for graduate students. The remainder of the book is devoted to new research, providing, among other material, some remarkable improvements on Brown's classical representability theorem. In addition, the author introduces a class of triangulated categories\"--the \"well generated triangulated categories\"--and studies their properties. This exercise is particularly worthwhile in that many examples of triangulated categories are well generated, and the book proves several powerful theorems for this broad class. These chapters will interest researchers in the fields of algebra, algebraic geometry, homotopy theory, and mathematical physics.

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

This article introduces an innovative method for constructing substitution and permutation boxes and their application in image encryption. In encryption, the essential component that creates confusion is the substitution box(S-box). S-boxes can be constructed using different mathematical structures. An S-box with high nonlinearity enhances confusion in encryption. The nonlinearity of the S-box depends on its mathematical structure; the more nonlinear the structure, the higher the quality of the S-box. Quality can be measured using differential, linear, and statistical attacks. Another main component in image encryption is the permutation box(P-box), which creates confusion by scrambling the image pixels. The proposed scheme is based on the algebraic structure of the Galois field. In this scheme, we utilize the group action over an 8-bit finite field and then apply a bi-linear transformation to construct S-boxes and P-boxes. The proposed encryption scheme satisfies standard cryptographic requirements. Experimental results show that, in comparison with other well-known existing schemes, the method successfully generates many unique, uncorrelated, and secure S-boxes. A security study of the proposed scheme indicates that multiple color-encrypted images can swiftly and effectively mitigate numerous risks, rendering them suitable for real-time applications with stringent security demands.

The Drazin inverse is the most important pseudo-inverse of linear operators of finite-dimensional vector spaces. One of its applications is in finite Markov chains theory, where they are used to calculate certain properties such as the average number of times the chain passes through a particular state or the expected time it takes for the chain to enter it. Since it has been possible to generalize Drazin's inverse to linear operators in arbitrary dimension (Core-Nilpotent Operators), the aim of this paper is to use it to obtain the same results for infinite Markov chains. Furthermore, the results will be applied to a concrete example.

Let A and B be two unital prime complex *-algebras such that A has a nontrivial projection. In this paper, we study the structure of the bijective mappings ? ? A ? B preserving sum of products ?1ab* + ?2b*a + ?3ba* (resp., ?1ab* + ?2b*a + ?3a*b), where the scalars ?k3k =1 are rational numbers satisfying some conditions.