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¿This book discusses the process by which Ulam's conjecture is proved, aptly detailing how mathematical problems may be solved by systematically combining interdisciplinary theories. It presents the state-of-the-art of various research topics and methodologies in mathematics, and mathematical analysis by presenting the latest research in emerging research areas, providing motivation for further studies. The book also explores the theory of extending the domain of local isometries by introducing a generalized span.For the reader, working knowledge of topology, linear algebra, and Hilbert space theory, is essential. The basic theories of these fields are gently and logically introduced. The content of each chapter provides the necessary building blocks to understanding the proof of Ulam¿s conjecture and are summarized as follows: Chapter 1 presents the basic concepts and theorems of general topology. In Chapter 2, essential concepts and theorems in vector space, normed space, Banach space, inner product space, and Hilbert space, are introduced. Chapter 3 gives a presentation on the basics of measure theory. In Chapter 4, the properties of first- and second-order generalized spans are defined, examined, and applied to the study of the extension of isometries. Chapter 5 includes a summary of published literature on Ulam¿s conjecture; the conjecture is fully proved in Chapter 6.

This book expounds on the recent developments in applications of holomorphic functions in the theory of hypercomplex and anti-Hermitian manifolds as well as in the geometry of bundles. It provides detailed information about holomorphic functions in algebras and discusses some of the areas in geometry with applications. The book proves the existence of a one-to-one correspondence between hyper-complex anti-Kähler manifolds and anti-Hermitian manifolds with holomorphic metrics, and also a deformed lifting to bundles. Researchers and students of geometry, algebra, topology and physics may find the book useful as a self-study guide.

This monograph studies the heat kernel for the spin-tensor Laplacians on Lie groups and maximally symmetric spaces. It introduces many original ideas, methods, and tools developed by the author and provides a list of all known exact results in explicit form ¿ and derives them ¿ for the heat kernel on spheres and hyperbolic spaces. Part I considers the geometry of simple Lie groups and maximally symmetric spaces in detail, and Part II discusses the calculation of the heat kernel for scalar, spinor, and generic Laplacians on spheres and hyperbolic spaces in various dimensions. This text will be a valuable resource for researchers and graduate students working in various areas of mathematics ¿ such as global analysis, spectral geometry, stochastic processes, and financial mathematics ¿ as well in areas of mathematical and theoretical physics ¿ including quantum field theory, quantum gravity, string theory, and statistical physics.

This monograph thoroughly explores the development of the theory of varieties of semigroups and of two related algebras: involution semigroups and monoids. Through this in-depth analysis, readers will attain a deeper understanding of the differences between these three types of varieties, which may otherwise seem counterintuitive. New results with detailed proofs are also presented that answer previously unsolved fundamental problems. Featuring both a comprehensive overview as well as highlighting the author¿s own significant contributions to the area, this book will help establish this subfield as a matter of timely interest. Advances in the Theory of Varieties of Semigroups will appeal to researchers in universal algebra and will be particularly valuable for specialists in semigroups.

This monograph presents new insights into the perturbation theory of dynamical systems based on the Gromov-Hausdorff distance.  In the first part, the authors introduce the notion of Gromov-Hausdorff distance between compact metric spaces, along with the corresponding distance for continuous maps, flows, and group actions on these spaces. They also focus on the stability of certain dynamical objects like shifts, global attractors, and inertial manifolds.  Applications to dissipative PDEs, such as the reaction-diffusion and Chafee-Infante equations, are explored in the second part.  This text will be of interest to graduates students and researchers working in the areas of topological dynamics and PDEs.  

This research monograph develops the theory of relative nonhomogeneous Koszul duality. The thematic example, meaning the classical duality between the ring of differential operators and the de Rham DG algebra of differential forms, involves some of the most important objects of study in the contemporary algebraic and differential geometry.

The first part presents the purely algebraic branch of the theory, the second part presents the functional analytic branch, and the third part discusses various applications.The book is intended for researchers and graduate students in ring theory, Banach algebra theory, and nonassociative algebra.

More precisely, it investigates the existence, multiplicity and qualitative properties of solutions for fractional Schroedinger equations by applying suitable variational and topological methods.The book is mainly intended for researchers in pure and applied mathematics, physics, mechanics, and engineering.

Introduction.- The Solution Theory for a Basic Class of Evolutionary Equations.- Some Applications to Models from Physics and Engineering.- But what about the Main Stream?.- Two Supplements for the Toolbox.- Requisites from Functional Analysis.

In this book the authors use a technique based on recurrence relations to study the convergence of the Newton method under mild differentiability conditions on the first derivative of the operator involved.

Approximation of continuous functions on compact spaces.- Approximation of continuous functions on locally compact spaces.- Approximation of continuous differentiable functions.- Approximations theorems in locally convex lattices.

This book presents recent results on nonlinear evolutionary fluid equations such as the compressible (radiative) magnetohydrodynamics (MHD) equations, compressible viscous micropolar fluid equations, the full non-Newtonian fluid equations and non-autonomous compressible Navier-Stokes equations. These types of partial differential equations arise in many fields of mathematics, but also in other branches of science such as physics and fluid dynamics.This book will be a valuable resource for graduate students and researchers interested in partial differential equations, and will also benefit practitioners in physics and engineering.

Differential equations with delay naturally arise in various applications, such as control systems, viscoelasticity, mechanics, nuclear reactors, distributed networks, heat flows, neural networks, combustion, interaction of species, microbiology, learning models, epidemiology, physiology, and many others. This book systematically investigates the stability of linear as well as nonlinear vector differential equations with delay and equations with causal mappings. It presents explicit conditions for exponential, absolute and input-to-state stabilities. These stability conditions are mainly formulated in terms of the determinants and eigenvalues of auxiliary matrices dependent on a parameter; the suggested approach allows us to apply the well-known results of the theory of matrices. In addition, solution estimates for the considered equations are established which provide the bounds for regions of attraction of steady states. The main methodology presented in the book is based on a combined usage of the recent norm estimates for matrix-valued functions and the following methods and results: the generalized Bohl-Perron principle and the integral version of the generalized Bohl-Perron principle; the freezing method; the positivity of fundamental solutions. A significant part of the book is devoted to the Aizerman-Myshkis problem and generalized Hill theory of periodic systems. The book is intended not only for specialists in the theory of functional differential equations and control theory, but also for anyone with a sound mathematical background interested in their various applications.

This book presents new accurate and efficient exponentially convergent methods for abstract differential equations with unbounded operator coefficients in Banach space. These methods are highly relevant for practical scientific computing since the equations under consideration can be seen as the meta-models of systems of ordinary differential equations (ODE) as well as of partial differential equations (PDEs) describing various applied problems. The framework of functional analysis allows one to obtain very general but at the same time transparent algorithms and mathematical results which then can be applied to mathematical models of the real world. The problem class includes initial value problems (IVP) for first order differential equations with constant and variable unbounded operator coefficients in a Banach space (the heat equation is a simple example), boundary value problems for the second order elliptic differential equation with an operator coefficient (e.g. the Laplace equation), IVPs for the second order strongly damped differential equation as well as exponentially convergent methods to IVPs for the first order nonlinear differential equation with unbounded operator coefficients. For researchers and students of numerical functional analysis, engineering and other sciences this book provides highly efficient algorithms for the numerical solution of differential equations and applied problems.

This book presents the extensions to the quaternionic setting of some of the main approximation results in complex analysis. The book is addressed to researchers in various areas of mathematical analysis, in particular hypercomplex analysis, and approximation theory.

This book is devoted to the study of positive solutions to indefinite problems. The new technique developed in the book gives, as a byproduct, infinitely many subharmonic solutions and globally defined positive solutions with chaotic behaviour.

This book shows the importance of studying semilocal convergence in iterative methods through Newton's method and addresses the most important aspects of the Kantorovich's theory including implicated studies.