Pages: 768
ISBN: 978-981-98-1081-9 (hardcover)
This book presents the theory of harmonic analysis for noncommutative compact groups. If G is a commutative locally compact group, there is a well-understood theory of harmonic analysis as discussed in Aspects of Harmonic Analysis on Locally Compact Abelian Groups. If G is not commutative, things are a lot tougher. In the special case of a compact group, there is a deep interplay between analysis and representation theory which was first discovered by Hermann Weyl and refined by Andre Weil. This book presents these seminal results of Weyl and Weil.
Starting with the basics of representations theory, it presents the famous Petereyl theorems and discusses Fourier analysis on compact groups. This book also introduces the reader to induced representations of locally compact groups, induced representations of G-bundles, and the theory of Gelfand pairs. A special feature is the chapter on equivariant convolutional neural networks (CNNs), a chapter which shows how many of the abstract concepts of representations, analysis on compact groups, Peter?Weyl theorems, Fourier transform, induced representations are used to tackle very practical, modern-day problems.
Preface
Acknowledgments
Introduction
Representations of Algebras and Hilbert Algebras
Unitary Representations of Locally Compact Groups
Analysis on Compact Groups and Representations
Matrix Representations of SL(2,?E, SU(2) and SO(3)
Induced Representations
Constructing Induced Representations a la Mackey
Equivariant Convolutional Neural Networks
Harmonic Analysis on Gelfand Pairs
Bibliography
Index
Symbol Index
Second-year graduate masters/PhD students in mathematics, engineering, or computer vision who are interested in learning about harmonic analysis. Appropriate for mathematical courses on classical and functional analysis, medical imaging and measurement, advanced applied mathematics, mathematical analysis, representation theory of continuous groups, and mathematics for engineering. Supplementary reading for courses in signal/image processing, deep learning/equivariant convolutional neural networks, heat conduction, automatic control, acoustics, optics, and structural analysis.
Pages: 344
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ISBN: 978-981-98-0633-1 (hardcover)
The monograph focuses on the analytical challenges of describing incompressible fluid flows using Lagrangian variables. The Lagrangian method is presented as a complementary approach to the Eulerian method for studying fluid motion. The fundamental nature of fluid dynamics is linked to tracing the trajectories of individual fluid particles, which is precisely what the Lagrangian method facilitates.
The purpose of this publication is to reflect as fully as possible the achievements of analytical fluid dynamics in Lagrangian variables. A significant part of it consists of original articles by the author. Considerable attention is paid to the subtleties of the Lagrangian description and its popularization among experts in fluid dynamics, students, postgraduates, and anyone interested in problems of fluid mechanics.
The book can be considered, among other things, as a tutorial on Lagrangian fluid dynamics. The book covers issues of the general description of the movement of incompressible fluid, problems of classical hydrodynamics and geophysical hydrodynamics. Fundamental topics include free-boundary motion, vortices in fluids, and water waves. The book includes a number of problems that take into account the stratification and viscosity of fluids, as well as the rotation of the Earth.
Description of Basics:
Ideal Fluid Equations
Potential Motion
Vortex Motion
Waves on Water:
Potential Approximation
Gerstner Wave
Weakly Vortex Waves
Exact Solutions in Vortex Dynamics:
Two-Dimensional Vortex Flow
Localized Two-Dimensional Vortices in the External Flow
Spatial Vortex Flows
Generalized Gerstner Waves:
Waves on Water at Unstable Pressure
Vortex Model of Rogue Waves
Breaking of the Surface Gravity Wave
Non-Stationary Edge Waves
Viscous Fluid Flows:
Basic Equations and Examples
Gravity Waves on the Surface of a Viscous Fluid
The Earth's Rotation Effect:
Exact Solutions for Waves
Vortices in a Rotating Fluid
This book can be targeted by a various levels of researchers from undergraduate
and beginner to graduate and professional levels for both industry and
academic sectors in green and sustainable energy.
Pages: 536
ISBN: 978-981-12-7651-4 (hardcover)
At some point in their careers, most physicists make an attempt to read and understand Newton's Principia. Unfortunately, it is an extremely difficult book ?Eit quickly becomes clear that one does not simply "read" the Principia. Even for a professional physicist, Newton's prose (written in Latin and translated to English) is difficult to follow. His diagrams and figures are complicated and confusing. To understand fully what Newton had done, the problems he posed would have to be solved by the reader.
Newton's geometric methods and techniques, and the geometry and vocabulary that passed for common knowledge in the late 17th century, are now arcane and all but inaccessible to a modern reader. The contents of the Principia are not. Most physicists and physics students, and many scientists in general, would find the physics in the Principia interesting, illuminating, and useful.
This book presents all the wonderful physics in the Principia in a manner that a modern reader can recognize and understand, using physics and mathematics as we understand them in the 21st century.
Preface
Acknowledgements
Introduction to the Principia
Newton's Principia:
Definitions
The Laws of Motion
The Motion of Bodies:
The Method of Limits and Ratios
Finding Central Forces
Conicection Orbits
Finding Conicection Orbits Given a Focus
Finding Orbits When no Focus is Given
Finding Motion in Given Orbits
Rectilinear Ascent and Descent
Finding Orbits for any Central Force
Precessing Orbits
Constrained Motion
Objects Attracted by Central Forces
Attractive Forces of Spherical Bodies
Attractive Forces of Non-Spherical Bodies
Motion of Minimally Small Bodies
Motion in Resisting Media:
The Linear Resistive Force
The Quadratic Resisting Force
The Resisting Force
Circular Orbits in Resisting Media
The Density and Compression of Fluids
Simple Pendulums with Resisting Forces
The Motion of Fluids
Wave Motion through Fluids
The Circular Motion of Fluids
The System of the World:
Introduction
Rules for the Study of Science
Phenomena
Propositions of Book III
General Scholium
Epilogue:
Newton's Principia for the Modern Student
References
Index
Undergraduate students of physics and mathematics, as well as academics
and engineers.
Pages: 664
ISBN: 978-981-98-0579-2 (hardcover)
If there were no Coulomb interaction among electrons, it would be relatively straightforward to solve the many-electron Schrodinger equation. It is, however, precisely this interaction that is at the heart of numerous fascinating phenomena in condensed matter physics such as superconductivity, Kondo physics, magnetism, etc. Due to the large number of electrons in a material being of the order of Avogadro's number, it is at present ?Eand perhaps in the foreseeable future ?Enot feasible or even desirable to solve the Schrodinger equation to obtain the many-electron wavefunction. Fortunately, a large number of important physical properties can be calculated without explicit knowledge of the wavefunction.
Two of the most important formalisms for dealing with the many-electron problem which avoid a direct use of the many-electron wavefunction are the Green function and the density functional theory. Within the Kohn-Sham scheme the latter is used to calculate ground-state properties whereas the former for excitation spectra. The book presents the fundamentals of these two theories in detail with essential many-body tools, such as the occupation number representation and Grassmann algebra developed from scratch. Prior knowledge of many-body theory is not a prerequisite so that it is readable for final-year undergraduates and graduate students in physics and chemistry as well as researchers in the field of electronic structure and many-body theory. The book includes in the last chapter an exposition of a density-functional path for determining the Green function, a new formalism recently proposed by the author. The book should be a valuable companion for those embarking in the field of many-electron physics.
Introduction
Occupation-Number Representation
Static Mean-Field Methods
Density-Functional Theory
Bandstructure Methods
Green Function and Self-Energy: Beyond Static Mean-Field Theories
Linear-Response Theory
The GW Approximation
Diagrammatic Approaches Beyond the GW Approximation
The Hubbard Model
Coherent States and Grassmann Variables
Dynamical Mean-Field Theory
Magnetism
Dynamical Exchange-Correlation Field in Lieu of Self-Energy
Advanced undergraduates and graduate students as well as researchers in the field of electronic structure and many-body theory. Courses in electronic structure and many-body theory, with emphasis on first-principle approaches, will find this book suitable as well.
Pages: 336
ISBN: 978-981-12-9359-7 (hardcover)
The unique compendium starts with a simplified version of the information theory which allows any beginner in math to embrace the mysterious concept of entropy without hitting hard the wall of probability theory. The volume continues with the full description of the information theory, enlightening source coding in the heart of the theory with data compression and self-prediction to get into the information age. Then it concludes with artificial intelligence and quantum information.
This useful reference text benefits professionals, researchers, academics and graduate students in the fields of information theory, probability theory, electromagnetism, machine learning and quantum theory.
What is Information?
The Basic Mathematics Inside Information Theory
Probabilistic Information Theory, the Paradoxes of Data Compression and Event Prediction
The Challenge of Information Networks, the Triumph of the Algorithms over the Complexity
The Performance Paradoxes of Wireless Networks Caused by Physics
The Limit of Artificial Intelligence Imposed by Information Theory
Quantum Information Theory
Non-unitary Quantum Information
Answer to Exercises
Researchers, professionals, academics and graduate students in information theory, probability theory, electromagnetism, machine learning and quantum theory.
Pages: 536
ISBN: 978-981-12-8575-2 (hardcover)
The theory of relativity was created by Einstein in two stages, extending over a decade from 1905 to 1915. General relativity is said to be the most powerful tool that can be used to explain the behavior of the universe.
In this book, we try to comprehend the universe with a fundamental formula
known as the Pythagorean theorem, used as a vehicle to review the essence
of Euclidean geometry and non-Euclidean geometry, then move on to Newtonian
mechanics, and review the historical development of electromagnetism, setting
the stage for special relativity. Next, we describe Einstein's efforts
to generalize his theory to include gravitation, which led to a geometric
theory of spacetime: the gravitational field equations. The German astronomer
Schwarzschild quickly solved these equations for a special case. Also presented
are the numerical graphical results of the planetary orbits and light trajectories
using the Python code that we created. Then the reader is taken on an excursion
to the physics of the microcosm, describing how special relativity was
instrumental in the development of quantum theory, and how several Japanese
physicists contributed to atomic and particle physics. Finally, we end
the book by introducing the work of Roger Penrose on black holes, which
is closely related to Schwarzschild's solution, and the existence of intrinsic
singularity at the center of black holes. In his intriguing theory of Conformal
Cyclic Cosmology, our universe may be one in a never-ending birth-and-death
cycle of universes.
Geometry on the Grand
Spherical and Hyperbolic Geometries
Newtonian Dynamics
Electromagnetism
Advent of Einstein
Special Theory of Relativity
General Theory of Relativity
Analysis of Schwarzschild Spacetime
From Macrocosm to Microcosm
General Relativistic Geometry to Conformal Cyclic Cosmology
Undergraduate students of science and engineering.
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