Description
This contemporary first course focuses on concepts and ideas of
Measure Theory, highlighting the theoretical side of the subject.
Its primary intention is to introduce Measure Theory to a new
generation of students, whether in mathematics or in one of the
sciences, by offering them on the one hand a text with complete,
rigorous and detailed proofs--sketchy proofs have been a
perpetual complaint, as demonstrated in the many Amazon reader
reviews critical of authors who "omit 'trivial' steps"
and "make not-so-obvious 'it is obvious' remarks." On
the other hand, Kubrusly offers a unique collection of fully
hinted problems. On the other hand, Kubrusly offers a unique
collection of fully hinted problems. The author invites the
readers to take an active part in the theory construction,
thereby offering them a real chance to acquire a firmer grasp on
the theory they helped to build. These problems, at the end of
each chapter, comprise complements and extensions of the theory,
further examples and counterexamples, or auxiliary results. They
are an integral part of the main text, which sets them apart from
the traditional classroom or homework exercises. JARGON BUSTER:
measure theory Measure theory investigates the conditions under
which integration can take place. It considers various ways in
which the "size" of a set can be estimated. This topic
is studied in pure mathematics programs but the theory is also
foundational for students of statistics and probability,
engineering, and financial engineering.
Audience
Beginning graduate students and advanced undergraduates in Math,
Statistics, Economics, Engineering, and Physics
Contents
Measurability; Measure; Integral; Integrability; Spaces L^p;
Convergence; Decomposition; Extension; Product
Hardbound, 176 pages, publication date: NOV-2006
ISBN-13: 978-0-12-370899-1
ISBN-10: 0-12-370899-0
Included in series
Mathematics in Science and Engineering, 208
Description
The book is of interest to graduate students in functional
analysis, numerical analysis, and ill-posed and inverse problems
especially. The book presents a general method for solving
operator equations, especially nonlinear and ill-posed. It
requires a fairly modest background and is essentially self-contained.
All the results are proved in the book, and some of the
background material is also included. The results presented are
mostly obtained by the author.
Audience
Mathematicians, numerical analysists, specialists in scientific
computing, engineers and others interested in solving operator
equations, ill-posed and inverse problems.
Contents
Preface Contents 1. Introduction 2. Ill-posed problems 3. DSM for
well-posed problems 4. DSM and linear ill-posed problems 5. Some
inequalities 6. DSM for monotone operators 7. DSM for general
nonlinear operator equations 8 DSM for operators satisfying a
spectral assumption 9. DSM in Banach spaces 10. DSM and Newton-type
methods without inversion of the derivative 11. DSM and unbounded
operators 12. DSM and nonsmooth operators 13. DSM as a
theoretical tool 14. DSM and iterative methods 15. Numerical
problems arising in applications 16. Auxiliary results from
analysis Bibliographical notes Bibliography Index
Hardbound, 304 pages, publication date: SEP-2006
ISBN-13: 978-0-444-52795-0
ISBN-10: 0-444-52795-8
Description
The area of Psychometrics, a field encompassing the statistical
methods used in Psychological and educational testing, has become
a very important and active area of research, evident from the
large body of literature that has been developed in the form of
books, volumes and research papers. Mainstream statisticians also
have found profound interest in the field because of its unique
nature. This book presents a state of the art exposition of
theoretical, methodological and applied issues in Psychometrics.
This book represents a thorough cross section of internationally
renowned thinkers who are inventing methods for dealing with
recent challenging psychometric problems. Key Features/ -
Emphasis on the most recent developments in the field - Plenty of
real, often complicated, data examples to demonstrate the
applications of the statistical techniques - Information on
available software
Contents
Introduction
1. History and overview of psychometrics (L.V. Jones and D.
Thissen)
Some basic ideas of test theory
2. Classical test theory (C. Lewis)
3. Validity: foundational issues and statistical methodology (B.D.
Zumbo)
4. Reliability and generalizability theory (N.M. Webb, R.J.
Shavelson and E.H. Haertel)
5. Differential item functioning and item bias (R.D. Penfield and
G. Camilli)
6. Equating test scores (P.W. Holland, N.J. Dorans and N.S.
Petersen)
7. Electronic essay grading (S.J. Haberman)
A variety of approaches/models to handle psychometric data
8. Some matrix results useful in psychometric research (C.R. Rao)
9. Factor Analysis (H. Yanai and M. Ichikawa)
10. Structural equation modeling (K.-H. Yuan and P.M. Bentler)
11. Applications of multidimensional scaling in psychometrics (Y.
Takane)
12. Multilevel models in psychometrics (F. Steele and H.
Goldstein)
13. Latent class analysis in psychometrics (C.M. Dayton and G.B.
Macready)
14. Random-effects models for preference data (U. Bockenholt and
R.-C. Tsai)
IRT models
15. Item response theory in a general framework (R.D. Bock and I.
Moustaki)
16. Rasch models (G.H. Fischer)
17. Hierarchicam item response theory models (M.S. Johnson, S.
Sinharay and E.T. Bradlow)
18. Multidimensional item response theory (M.D. Reckase)
19. Mixture distribution item response models (M. von Davier and
J. Rost)
20. Scoring open ended questions (G. Maris and T. Bechger)
21. Assessing the fit of item response theory models (H.
Swaminathan, R.K. Hambleton and H.J. Rogers)
22. Nonparametric item response theory and special topics (K.
Sijtsma and R.R. Meijer)
Topics of special interst
23. Automatic item generation and cognitive psychology (S.
Embretson and X. Yang)
24. Statistical inference for causal effects, with emphasis on
applications in psychometrics and education (D.B. Rubin)
25. Statistical aspects of adaptive testing (W.J. van der Linden
and C.A.W. Glas)
26. Bayesian Psychometric modeling from an evidence-centered
design perspective (R.J. Mislevy and R. Levy)
27. Value-added modeling (H. Braun and H. Wainer)
28. Three statistical paradoxes in the interpretation of group
differences: illustrated with medical school admission and
licencing data (H. Wainer and L.M. Brown)
29. Meta-analysis (L.W. Hedges)
30. Vertical scaling: statistical models for measuring growth and
achievement (R.J. Patz and L. Yao)
31. Cognitive Diagnosis - part I: (L.V. Dibello, L.A. Roussos and
W. Stout) - part II: (S.J. Haberman and M. von Davier)
Operational
32. Marginal Estimation of Populalation characteristics: recent
developments and future directions (M. von Davier, S. Sinharay, A.
Oranje and A. Beaton)
33. Statistical procedures used in college admissions testing (J.
Liu, D.J. Harris and A. Schmidt)
34. Integration of models (R.L. Brennan, D.R. Eignor, M.J. Gierl,
J.P. Leighton, I. Lawrence, N. Kingston, P. Sanders and C.
Schmeiser)
Subject Index
Hardbound, 1190 pages, publication date: NOV-2006
ISBN-13: 978-0-444-52103-3
ISBN-10: 0-444-52103-8
"Upon completion of this book a student or researcher would be well prepared
to employ finite elements for an application problem or proceed to the
cutting edge of research in finite element methods. The accuracy and the
thoroughness of the book are excellent." Anthony Kearsley, research
mathematician, National Institute of Standards and Technology
The infinite element method is the most powerful general-purpose
technique for computing accurate solutions to partial
differential equations. Understanding and Implementing the Finite
Element Method is essential reading for those interested in
understanding both the theory and the implementation of the
finite element method for equilibrium problems. This book
contains a thorough derivation of the finite element equations as
well as sections on programming the necessary calculations,
solving the finite element equations, and using a posteriori
error estimates to produce validated solutions. Accessible
introductions to advanced topics, such as multigrid solvers, the
hierarchical basis conjugate gradient method, and adaptive mesh
generation, are provided. Each chapter ends with exercises to
help readers master these topics.
Understanding and Implementing the Finite Element Method includes
a carefully documented collection of MATLAB programs implementing
the ideas presented in the book. Readers will benefit from a
careful explanation of data structures and specific coding
strategies and will learn how to write a fi?nite element code
from scratch. Students can use the MATLAB codes to experiment
with the method and extend them in various ways to learn more
about programming fi?nite elements.
Audience: This practical book should provide an excellent
foundation for those who wish to delve into advanced texts on the
subject, including advanced undergraduates and beginning graduate
students in mathematics, engineering, and the physical sciences.
About the Author: Mark S. Gockenbach is a Professor of
Mathematical Sciences at Michigan Technological University. His
research interests include inverse problems, computational
optimization, and mathematical software. His first book, Partial
Differential Equations: Analytical and Numerical Methods, was
published by SIAM in 2002.
Contents:
Preface
Part I The Basic Framework for Stationary Problems
1 Some Model PDEs
2 The weak form of a BVP
3 The Galerkin method
4 Piecewise polynomials and the finite element method
5 Convergence of the finite element method
Part II Data Structures and Implementation
6 The mesh data structure
7 Programming the finite element method: Linear Lagrange
triangles
8 Lagrange triangles of arbitrary degree
9 The finite element method for general BVPs
Part III Solving the Finite Element Equations
10 Direct solution of sparse linear systems
11 Iterative methods: Conjugate gradients
12 The classical stationary iterations
13 The multigrid method
Part IV Adaptive Methods
14 Adaptive mesh generation
15 Error estimators and indicators
Bibliography
Index
2006 / xvi + 363 pages / Softcover
ISBN-10: 0-89871-614-4 / ISBN-13:978-0-898716-14-6
"No present book comes near this one in the range and
depth of treatment of these two extremely important methods?the
Lanczos algorithm and the method of conjugate gradients."
?Chris Paige, School of Computer Science, McGill University
The Lanczos and conjugate gradient (CG) algorithms are
fascinating numerical algorithms. This book presents the most
comprehensive discussion to date of the use of these methods for
computing eigenvalues and solving linear systems in both exact
and floating point arithmetic. The author synthesizes the
research done over the past 30 years, describing and explaining
the "average" behavior of these methods and providing
new insight into their properties in finite precision. Many
examples are given that show significant results obtained by
researchers in the field.
The author emphasizes how both algorithms can be used efficiently
in finite precision arithmetic, regardless of the growth of
rounding errors that occurs. He details the mathematical
properties of both algorithms and demonstrates how the CG
algorithm is derived from the Lanczos algorithm. Loss of
orthogonality involved with using the Lanczos algorithm, ways to
improve the maximum attainable accuracy of CG computations, and
what modifications need to be made when the CG method is used
with a preconditioner are addressed.
Audience This book is intended for applied mathematicians,
computational scientists, engineers, and physicists who have an
interest in linear algebra, numerical analysis, and partial
differential equations. It will be of interest to engineers and
scientists using the Lanczos algorithm to compute eigenvalues and
the CG algorithm to solve linear systems, and to researchers in
Krylov subspace methods for symmetric matrices, especially those
concerned with floating point error analysis. Moreover, it can be
used in advanced courses on iterative methods or as a
comprehensive presentation of a well-known numerical method in i￢?nite
precision arithmetic.
About the Author Gerard Meurant is Director of Research in the
military applications division at Commissariat a l'Energie
Atomique (CEA) in Bruyeres le Chatel, France. He is the author of
Computer Solution of Large Linear Systems (North?Holland, 1999)
and serves on the editorial boards of the International Journal
of High Speed Computing and Numerical Algorithms. In 1988 Meurant
was awarded the Prix CEA and in 1995 the Palmes Academiques, an
honor presented each year by the French Ministry of Education.
Contents
Preface
Chapter 1: The Lanczos algorithm in exact arithmetic
Chapter 2: The CG algorithm in exact arithmetic
Chapter 3: A historical perspective on the Lanczos algorithm in
finite precision
Chapter 4: The Lanczos algorithm in finite precision
Chapter 5: The CG algorithm in finite precision
Chapter 6: The maximum attainable accuracy
Chapter 7: Estimates of norms of the error in finite precision
Chapter 8: The preconditioned CG algorithm
Chapter 9: Miscellaneous
Appendix
Bibliography
Index
2006 / xvi + 365 pages / Softcover
ISBN-13: 978-0-898716-16-0 / ISBN-10: 0-89871-616-0