Series: Undergraduate Texts in Mathematics
2005, XIV, 394 p. 50 illus., Hardcover
ISBN: 0-387-24637-1
About this textbook
The predictive power of mathematics in quantum phenomena is one
of the great intellectual successes of the 20th century. This
textbook, aimed at undergraduate or graduate level students (depending
on the college or university), concentrates on how to make
predictions about the numbers of each kind of basic state of a
quantum system from only two ingredients: the symmetry and the
linear model of quantum mechanics. This method, involving the
mathematical area of representation theory or group theory,
combines three core mathematical subjects, namely, linear
algebra, analysis and abstract algebra. Wide applications of this
method occur in crystallography, atomic structure, classification
of manifolds with symmetry, and other areas.
The topics unfold systematically, introducing the reader first to
an important example of a quantum system with symmetry, the
single electron in a hydrogen atom. Then the reader is given just
enough mathematical tools to make predictions about the numbers
of each kind of electronic orbital based solely on the physical
spherical symmetry of the hydrogen atom. The final chapters
address the related ideas of quantum spin, measurement and
entanglement.
This user-friendly exposition, driven by numerous examples and
exercises, requires a solid background in calculus and
familiarity with either linear algebra or advanced quantum
mechanics. Linearity, Symmetry, and Prediction in the Hydrogen
Atom will benefit students in mathematics, physics and chemistry,
as well as a literate general readership.
Table of contents
Setting the Stage.- Linear Algebra over the Complex Numbers.-
Complex Scalar Product Spaces (a.k.a. Hilbert Spaces).- Lie
Groups and Lie Group Representations.- New Representations from
Old.- Irreducible Representations and Invariant Integration.-
Represenatations and the Hydrogen Atom.- The Algebra so(4)
Symmetry of the Hydrogen Atom.- The Group SO(4) Symmetry of the
Hydrogen Atom.- Projective Representations and Spin.- Independent
Events and Tensor Products.- A. Spherical Harmonics.- B. Proof of
the Correspondence.- C. Suggested Paper Topics.- Bibliography.-
Index.- References.
Series: Progress in Mathematical Physics, Vol. 44
2005, X, 219 p., Hardcover
ISBN: 0-8176-4366-4
About this book
The spectral theory of Schrodinger operators, in particular those
with random potentials, continues to be a very active field of
research. This work focuses on various known criteria in the
spectral theory of selfadjoint operators in order to identify the
spectrum and its components a la Lebesgue decomposition.
Key features and topics:
* Well-developed exposition of criteria that are especially
useful in determining the spectra of deterministic and random
Schrodinger operators occurring in quantum theory
* Systematically uses measures and their transforms (Fourier,
Borel, wavelet) to present a unifying theme
* Establishes criteria for identifying the spectrum
* Examines a series of applications to show point spectrum and
continuous spectrum in some models of random operators
* Presents a series of spectral-theoretic results for the
perturbed operators introduced in the earlier chapters with
examples of localization and delocalization in the theory of
disordered systems
* Presents modern criteria (using wavelet transform,
eigenfunction decay) that could be used to do spectral theory
* Unique work in book form combining the presentation of the
deterministic and random cases, which will serve as a platform
for further research activities
This concise unified presentation is aimed at graduate students
and researchers working in the spectral theory of Schrodinger
operators with either fixed or random potentials in particular.
However, given the large gap that this book fills in the
literature, it will serve a wider audience of mathematical
physicists in its contribution to works in spectral theory.
Table of contents
Series: Springer Series in Statistics
1st ed. 2003. 2nd printing 2005., 2005, Approx. 550 p., Softcover
ISBN: 0-387-26142-7
About this book
This book presents the contemporary statistical methods and
theory of nonlinear time series analysis. The principal focus is
on nonparametric and semiparametric techniques developed in the
last decade. It covers the techniques for modelling in state-space,
in frequency-domain as well as in time-domain. To reflect the
integration of parametric and nonparametric methods in analyzing
time series data, the book also presents an up-to-date exposure
of some parametric nonlinear models, including ARCH/GARCH models
and threshold models. A compact view on linear ARMA models is
also provided. Data arising in real applications are used
throughout to show how nonparametric approaches may help to
reveal local structure in high-dimensional data. Important
technical tools are also introduced. The book will be useful for
graduate students, application-oriented time series analysts, and
new and experienced researchers. It will have the value both
within the statistical community and across a broad spectrum of
other fields such as econometrics, empirical finance, population
biology and ecology. The prerequisites are basic courses in
probability and statistics. Jianqing Fan, coauthor of the highly
regarded book Local Polynomial Modeling, is Professor of
Statistics at the University of North Carolina at Chapel Hill and
the Chinese University of Hong Kong. His published work on
nonparametric modeling, nonlinear time series, financial
econometrics, analysis of longitudinal data, model selection,
wavelets and other aspects of methodological and theoretical
statistics has been recognized with the Presidents' Award from
the Committee of Presidents of Statistical Societies, the
Hettleman Prize for Artistic and Scholarly Achievement from the
University of North Carolina, and by his election as a fellow of
the American Statistical Association and the Institute of
Mathematical Statistics. Qiwei Yao is Professor of Statistics at
the London School of Economics and Political Science. He is an
elected member of the International Statistical Institute, and
has served on the editorial boards for the Journal of the Royal
Statistical Society (Series B) and the Australian and New Zealand
Journal of Statistics.
Table of contents
Introduction.- Characteristics of Time Series.- ARMA Modeling and
Forecasting.- Parametric Nonlinear Time Series Models.-
Nonparametric Density Estimation.- Smoothing in Time Series.-
Spectral Density Estimation and Its Applications.- Nonparametric
Models.- Model Validation.- Nonlinear Prediction
Series: Springer Series in Statistics
2005, XXII, 690 p. 61 illus., Hardcover
ISBN: 0-387-25144-8
About this book
This book provides a comprehensive treatment on modeling
approaches for non-Gaussian repeated measures, possibly subject
to incompleteness. The authors begin with models for the full
marginal distribution of the outcome vector. This allows model
fitting to be based on maximum likelihood principles, immediately
implying inferential tools for all parameters in the models. At
the same time, they formulate computationally less complex
alternatives, including generalized estimating equations and
pseudo-likelihood methods. They then briefly introduce
conditional models and move on to the random-effects family,
encompassing the beta-binomial model, the probit model and, in
particular the generalized linear mixed model. Several frequently
used procedures for model fitting are discussed and differences
between marginal models and random-effects models are given
attention.
The authors consider a variety of extensions, such as models for
multivariate longitudinal measurements, random-effects models
with serial correlation, and mixed models with non-Gaussian
random effects. They sketch the general principles for how to
deal with the commonly encountered issue of incomplete
longitudinal data. The authors critique frequently used methods
and propose flexible and broadly valid methods instead, and
conclude with key concepts of sensitivity analysis.
Without putting too much emphasis on software, the book shows how
the different approaches can be implemented within the SAS
software package. The text is organized so the reader can skip
the software-oriented chapters and sections without breaking the
logical flow.
Table of contents
2nd ed., 2005, XVI, 600 p. 4 illus., Hardcover
ISBN: 0-387-25282-7
About this book
Prime numbers beckon to the beginner, the basic notion of
primality being accessible to a child. Yet, some of the simplest
questions about primes have stumped humankind for millennia. In
this book, the authors concentrate on the computational aspects
of prime numbers, such as recognizing primes and discovering the
fundamental prime factors of a given number. Over 100 explicit
algorithms cast in detailed pseudocode are included in the book.
Applications and theoretical digressions serve to illuminate,
justify, and underscore the practical power of these algorithms.
The 2nd edition adds new material on primality and algorithms and
updates all the numerical records, such as the largest prime, etc.
It has been revised throughout.
From the reviews of the first edition:
"cThe exercises are a gold mine of interesting examples,
pointers to the literature and potential research projects. c
Prime Numbers is a welcome addition to the literature of number
theory?comprehensive, up-to-date and written with style. It will
be useful to anyone interested in algorithms dealing with the
arithmetic of the integers and related computational issues."
American Scientist
"Destined to become a definitive textbook conveying the most
modern computational ideas about prime numbers and factoring,
this book will stand as an excellent reference for this kind of
computation, and thus be of interest to both educators and
researchers. It is also a timely book, since primes and factoring
have reached a certain vogue, partly because of cryptography. c"
LfEnseignement Mathematique
"The book is an excellent resource for anyone who wants to
understand these algorithms, learn how to implement them, and
make them go fast. It's also a lot of fun to read! It's rare to
say this of a math book, but open Prime Numbers to a random page
and it's hard to put down. Crandall and Pomerance have written a
terrific book." Bulletin of the AMS
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