Dr Gregory Chaitin, one of the world’s leading mathematicians, is best known for his discovery of the remarkable Ω number, a concrete example of irreducible complexity in pure mathematics which shows that mathematics is infinitely complex. In this volume, Chaitin discusses the evolution of these ideas, tracing them back to Leibniz and Borel as well as Godel and Turing.
This book contains 23 non-technical papers by Chaitin, his favorite tutorial and survey papers, including Chaitin's three Scientific American articles. These essays summarize a lifetime effort to use the notion of program-size complexity or algorithmic information content in order to shed further light on the fundamental work of Godel and Turing on the limits of mathematical methods, both in logic and in computation. Chaitin argues here that his information-theoretic approach to metamathematics suggests a quasi-empirical view of mathematics that emphasizes the similarities rather than the differences between mathematics and physics. He also develops his own brand of digital philosophy, which views the entire universe as a giant computation, and speculates that perhaps everything is discrete software, everything is 0's and 1's.
Chaitin's fundamental mathematical work will be of interest to philosophers concerned with the limits of knowledge and to physicists interested in the nature of complexity.
Contents:
On the Difficulty of Computations
Information-Theoretic Computational Complexity
Randomness and Mathematical Proof
Godel's Theorem and Information
Randomness in Arithmetic
Paradoxes of Randomness
Complexity and Leibniz
The Limits of Reason
How Real Are Real Numbers?
Is Incompleteness a Serious Problem?
How Much Information Can There Be in a Real Number?
and other papers
Readership: Students and professors of mathematics, computer science, philosophy and physics.
368pp Pub. date: Aug 2007
ISBN 978-981-270-895-3
981-270-895-2
ISBN 978-981-270-896-0(pbk)
981-270-896-0(pbk)
Phase-Locked Loops (PLLs) are electronic systems that can be used as a synchronized oscillator, a driver or multiplier of frequency, a modulator or demodulator and as an amplifier of phase modulated signals. This book updates the methods used in the analysis of PLLs by drawing on the results obtained in the last 40 years. Many are published for the first time in book form. Nonlinear and deterministic mathematical models of continuous-time and discrete-time PLLs are considered and their basic properties are given in the form of theorems with rigorous proofs. The book exhibits very beautiful dynamics, and shows various physical phenomena observed in synchronized oscillators described by complete (not averaged) equations of PLLs. Specially selected mathematical tools are used ? the theory of differential equations on a torus, the phase-plane portraits on a cyclinder, a perturbation theory (Melnikov’s theorem on heteroclinic trajectories), integral manifolds, iterations of one-dimensional maps of a circle and two-dimensional maps of a cylinder. Using these tools, the properties of PLLs, in particular the regions of synchronization are described. Emphasis is on bifurcations of various types of periodic and chaotic oscillations. Strange attractors in the dynamics of PLLs are considered, such as those discovered by Rossler, Henon, Lorenz, May, Chua and others.
Contents:
Introduction:
What Is Phase-Locked Loop?
PLL and Differential or Recurrence Equations
Averaging Method
Organization of the Book
The First Order Continuous-Time Phase-Locked Loops:
Equations of the System
The Averaged Equation
Solutions of the Basic Frequency
Differential Equation on the Torus
Fractional Synchronization
The System with Rectangular Waveform Signals
The Mapping f(p)=p+2pm+a sin(p)
The Second Order Continuous-Time Phase-Locked Loops:
The System with a Low-Pass Filter
Phase-Plane Portrait of the Averaged System
Perturbation of the Phase Difference j(wt)
Stable Integral Manifold
The PLL System Reducible to the First Order One
Homoclinic Structures
Boundaries of Attractive Domains
The Smale Horseshoe, Transient Chaos
Higher Order Systems Reducible to the Second Order Ones
One-Dimensional Discrete-Time Phase-Locked Loop:
Recurrence Equations of the System
Periodic Output Signals
Rotation Interval and Frequency Locking Regions
Stable Orbits, Hold-In Regions
The Number of Stable Orbits
Bifurcations of Periodic Orbits
Bifurcation of the Rotation Interval
Two-Dimensional Discrete-Time Phase-Locked Loop:
Description of the DPLL System by a Two-Dimensional Map
Stable Periodic Orbits
Reduction to a One-Dimensional System: Strange Attractors and Chaotic Steady-States
Readership: Graduate students and researchers in nonlinear science and applied physics; mathematically inclined engineers and mathematicians.
236pp Pub. date: Aug 2007
ISBN 978-981-277-090-5
981-277-090-9
The quantum Hall effect (QHE) is one of the most fascinating and beautiful phenomena in all branches of physics. Tremendous theoretical and experimental developments are still being made in this sphere. In the original edition of this book, composite bosons, composite fermions and fractional charged excitations (anyons) were among the distinguished ideas presented. This new edition includes many novel ideas according to recent progress. Fantastic phenomena associated with the interlayer phase coherence and SU(4) quantum Hall ferromagnets in the bilayer system are extensively reviewed. The microscopic theory of the QHE is formulated based on noncommutative geometry, the underlying mathematical structure. Quasiparticles are described as noncommutative solitons. The coverage also includes the recent development of the unconventional QHE in graphene (a single atomic layer graphite), where the electron dynamics can be treated as relativistic Dirac fermions and even the supersymmetric quantum mechanics plays a key role. An instructive and comprehensive overview of the QHE, this book is also suitable as an introduction to quantum field theory with vivid applications. Only a knowledge of quantum mechanics is assumed.
Contents:
Quantum Field Theory:
Canonical Quantization, Spontaneous Symmetry Breaking
Electromagnetic Field
Dirac Equations
Topological Solitons
Anyons
Monolayer QH Effects:
Landau Quantization
Quasiparticles and Activation Energy
Composite Bosons and Composite Fermions
Semiclassical Analysis
Quantum Hall Ferromagnets
Spin Textures
Hierarchy of Fractional QH States
Edge Effects
QH Effects in Graphene
Bilayer QH Effects:
SU(2) Pseudospin Structure
Interlayer Phase Coherence
SU(4) QH Ferromagnets
Bilayer QH Junction
Microscopic Theory of QH Effects:
Noncommutative Geometry
Lowest Landau Level Projection
Noncommutative Solitons
Effective Theory on Noncummutative Plane
Readership: Advanced undergraduates, graduates and researchers in condensed matter physics and particle physics.
700pp (approx.) Pub. date: Scheduled Winter 2007
ISBN 978-981-270-032-2
981-270-032-3
This textbook explains the fundamental concepts and techniques of group theory by making use of language familiar to physicists. Application methods to physics are emphasized. New materials drawn from the teaching and research experience of the author are included. This book can be used by graduate students and young researchers in physics, especially theoretical physics. It is also suitable for some graduate students in theoretical chemistry.
Contents:
A Review of Linear Algebras
Group and Its Subsets
Theory of Linear Representations of Groups
Three-Dimensional Rotation Groups
Symmetry of Crystals
Permutation Groups
Lie Groups and Lie Algebras
Unitary Groups
Real Orthogonal Groups
The Symplectic Groups
Readership: Graduate students and young researchers in physics, especially theoretical physics.
500pp (approx.) Pub. date: Scheduled Winter 2007
ISBN 978-981-277-141-4
981-277-141-7
ISBN 978-981-277-142-1(pbk)
981-277-142-5(pbk)