Gradient Inequalities
Gradient Inequalities

Author: Sen-Zhong Huang

Publisher: American Mathematical Soc.

Published: 2006

Total Pages: 194

ISBN-13: 0821840703

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This book presents a survey of the relatively new research field of gradient inequalities and their applications. The exposition emphasizes the powerful applications of gradient inequalities in studying asymptotic behavior and stability of gradient-like dynamical systems. It explains in-depth how gradient inequalities are established and how they can be used to prove convergence and stability of solutions to gradient-like systems. This book will serve as an introduction for furtherstudies of gradient inequalities and their applications in other fields, such as geometry and computer sciences. This book is written for advanced graduate students, researchers and applied mathematicians interested in dynamical systems and mathematical modeling.

Sobolev Gradients and Differential Equations
Sobolev Gradients and Differential Equations

Author: John Neuberger

Publisher: Springer Science & Business Media

Published: 2009-12-01

Total Pages: 287

ISBN-13: 3642040403

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A Sobolev gradient of a real-valued functional on a Hilbert space is a gradient of that functional taken relative to an underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. For discrete versions of partial differential equations, corresponding Sobolev gradients are seen to be vastly more efficient than ordinary gradients. In fact, descent methods with these gradients generally scale linearly with the number of grid points, in sharp contrast with the use of ordinary gradients. Aside from the first edition of this work, this is the only known account of Sobolev gradients in book form. Most of the applications in this book have emerged since the first edition was published some twelve years ago. What remains of the first edition has been extensively revised. There are a number of plots of results from calculations and a sample MatLab code is included for a simple problem. Those working through a fair portion of the material have in the past been able to use the theory on their own applications and also gain an appreciation of the possibility of a rather comprehensive point of view on the subject of partial differential equations.

Finite-Dimensional Variational Inequalities and Complementarity Problems
Finite-Dimensional Variational Inequalities and Complementarity Problems

Author: Francisco Facchinei

Publisher: Springer Science & Business Media

Published: 2007-06-04

Total Pages: 698

ISBN-13: 0387218157

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This is part two of a two-volume work presenting a comprehensive treatment of the finite-dimensional variational inequality and complementarity problem. It details algorithms for solving finite dimensional variational inequalities and complementarity problems. Coverage includes abundant exercises as well as an extensive bibliography. The book will be an enduring reference on the subject and provide the foundation for its sustained growth.

Geometric Analysis of Quasilinear Inequalities on Complete Manifolds
Geometric Analysis of Quasilinear Inequalities on Complete Manifolds

Author: Bruno Bianchini

Publisher: Springer Nature

Published: 2021-01-18

Total Pages: 291

ISBN-13: 3030627047

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This book demonstrates the influence of geometry on the qualitative behaviour of solutions of quasilinear PDEs on Riemannian manifolds. Motivated by examples arising, among others, from the theory of submanifolds, the authors study classes of coercive elliptic differential inequalities on domains of a manifold M with very general nonlinearities depending on the variable x, on the solution u and on its gradient. The book highlights the mean curvature operator and its variants, and investigates the validity of strong maximum principles, compact support principles and Liouville type theorems. In particular, it identifies sharp thresholds involving curvatures or volume growth of geodesic balls in M to guarantee the above properties under appropriate Keller-Osserman type conditions, which are investigated in detail throughout the book, and discusses the geometric reasons behind the existence of such thresholds. Further, the book also provides a unified review of recent results in the literature, and creates a bridge with geometry by studying the validity of weak and strong maximum principles at infinity, in the spirit of Omori-Yau’s Hessian and Laplacian principles and subsequent improvements.

Zeroing Dynamics, Gradient Dynamics, and Newton Iterations
Zeroing Dynamics, Gradient Dynamics, and Newton Iterations

Author: Yunong Zhang

Publisher: CRC Press

Published: 2018-10-09

Total Pages: 310

ISBN-13: 1498753787

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Neural networks and neural dynamics are powerful approaches for the online solution of mathematical problems arising in many areas of science, engineering, and business. Compared with conventional gradient neural networks that only deal with static problems of constant coefficient matrices and vectors, the authors’ new method called zeroing dynamics solves time-varying problems. Zeroing Dynamics, Gradient Dynamics, and Newton Iterations is the first book that shows how to accurately and efficiently solve time-varying problems in real-time or online using continuous- or discrete-time zeroing dynamics. The book brings together research in the developing fields of neural networks, neural dynamics, computer mathematics, numerical algorithms, time-varying computation and optimization, simulation and modeling, analog and digital hardware, and fractals. The authors provide a comprehensive treatment of the theory of both static and dynamic neural networks. Readers will discover how novel theoretical results have been successfully applied to many practical problems. The authors develop, analyze, model, simulate, and compare zeroing dynamics models for the online solution of numerous time-varying problems, such as root finding, nonlinear equation solving, matrix inversion, matrix square root finding, quadratic optimization, and inequality solving.

Inequalities in Respiratory Health
Inequalities in Respiratory Health

Author: Ian P. Sinha

Publisher: European Respiratory Society

Published: 2023-03-01

Total Pages: 264

ISBN-13: 1849841586

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Health inequalities have long been deeply engrained in society. If we are to address these inequalities, we need to reflect on what has driven them, and critically review the approaches that do and do not work. This Monograph brings together leading experts and up-and-coming researchers, in a collection of state-of-the-art articles, discussing the drivers and consequences of respiratory inequality.

Inequalities in Health
Inequalities in Health

Author: Nir Eyal

Publisher: Oxford University Press, USA

Published: 2013-10

Total Pages: 348

ISBN-13: 0199931399

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Which inequalities in longevity and health among individuals, groups, and nations are unfair? And what priority should health policy attach to narrowing them? These essays by philosophers, economists, epidemiologists, and physicians attempt to determine how health inequalities should be conceptualized, measured, ranked, and evaluated.

Functional Inequalities Markov Semigroups and Spectral Theory
Functional Inequalities Markov Semigroups and Spectral Theory

Author: Fengyu Wang

Publisher: Elsevier

Published: 2006-04-06

Total Pages: 391

ISBN-13: 0080532071

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In this book, the functional inequalities are introduced to describe:(i) the spectrum of the generator: the essential and discrete spectrums, high order eigenvalues, the principle eigenvalue, and the spectral gap;(ii) the semigroup properties: the uniform intergrability, the compactness, the convergence rate, and the existence of density;(iii) the reference measure and the intrinsic metric: the concentration, the isoperimetic inequality, and the transportation cost inequality.

Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare Conjecture
Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare Conjecture

Author: Qi S. Zhang

Publisher: CRC Press

Published: 2010-07-02

Total Pages: 434

ISBN-13: 1439834601

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Focusing on Sobolev inequalities and their applications to analysis on manifolds and Ricci flow, Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare Conjecture introduces the field of analysis on Riemann manifolds and uses the tools of Sobolev imbedding and heat kernel estimates to study Ricci flows, especially with surgeries. The