Synthesis Lectures on Mathematics and Statistics
2008, 194 pages, (doi:10.2200/S00175ED1V01Y200901MAS004)
The mere mention of hyperbolic geometry is enough to strike fear in the heart of the undergraduate mathematics and physics student. Some regard themselves as excluded from the profound insights of hyperbolic geometry so that this enormous portion of human achievement is a closed door to them. The mission of this book is to open that door by making the hyperbolic geometry of Bolyai and Lobachevsky, as well as the special relativity theory of Einstein that it regulates, accessible to a wider audience in terms of novel analogies that the modern and unknown share with the classical and familiar. These novel analogies that this book captures stem from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Remarkably, the mere introduction of Thomas gyration turns Euclidean geometry into hyperbolic geometry, and reveals mystique analogies that the two geometries share. Accordingly, Thomas gyration gives rise to the prefix "gyro" that is extensively used in the gyrolanguage of this book, giving rise to terms like gyrocommutative and gyroassociative binary operations in gyrogroups, and gyrovectors in gyrovector spaces. Of particular importance is the introduction of gyrovectors into hyperbolic geometry, where they are equivalence classes that add according to the gyroparallelogram law in full analogy with vectors, which are equivalence classes that add according to the parallelogram law. A gyroparallelogram, in turn, is a gyroquadrilateral the two gyrodiagonals of which intersect at their gyromidpoints in full analogy with a parallelogram, which is a quadrilateral the two diagonals of which intersect at their midpoints.
Gyrogroups / Gyrocommutative Gyrogroups / Gyrovector Spaces / Gyrotrigonometry
Synthesis Lectures on Mathematics and Statistics
2008, 85 pages, (doi:10.2200/S00146ED1V01Y200808MAS002)
Jordan Canonical Form (JCF) is one of the most important, and useful, concepts in linear algebra. In this book we develop JCF and show how to apply it to solving systems of differential equations. We first develop JCF, including the concepts involved in it*eigenvalues, eigenvectors, and chains of generalized eigenvectors. We begin with the diagonalizable case and then proceed to the general case, but we do not present a complete proof. Indeed, our interest here is not in JCF per se, but in one of its important applications. We devote the bulk of our attention in this book to showing how to apply JCF to solve systems of constant-coefficient first order differential equations, where it is a very effective tool. We cover all situations*homogeneous and inhomogeneous systems; real and complex eigenvalues. We also treat the closely related topic of the matrix exponential. Our discussion is mostly confined to the 2-by-2 and 3-by-3 cases, and we present a wealth of examples that illustrate all the possibilities in these cases (and of course, exercises for the reader).
Jordan Canonical Form / Solving Systems of Linear Differential Equations / Background Results: Bases, Coordinates, and Matrices / Properties of the Complex Exponential
Synthesis Lectures on Mathematics and Statistics
2008, 82 pages, (doi:10.2200/S00142ED1V01Y200807MAS001)
Statistics is the activity of inferring results about a population given a sample. Historically, statistics books assume an underlying distribution to the data (typically, the normal distribution) and derive results under that assumption. Unfortunately, in real life, one cannot normally be sure of the underlying distribution. For that reason, this book presents a distribution-independent approach to statistics based on a simple computational counting idea called resampling.
This book explains the basic concepts of resampling, then systematically presents the standard statistical measures along with programs (in the language Python) to calculate them using resampling, and finally illustrates the use of the measures and programs in a case study. The text uses junior high school algebra and many examples to explain the concepts. The ideal reader has mastered at least elementary mathematics, likes to think procedurally, and is comfortable with computers.
The Basic Idea / Bias Corrected Confidence Intervals / Pragmatic Considerations When Using Resampling / Terminology / The Essential Stats / Case Study: New Mexico's 2004 Presidential Ballots / References
Synthesis Lectures on Mathematics and Statistics
2009, 108 pages, (doi:10.2200/S00218ED1V01Y200908MAS006)
Jordan Canonical Form (JCF) is one of the most important, and useful, concepts in linear algebra. The JCF of a linear transformation, or of a matrix, encodes all of the structural information about that linear transformation, or matrix. This book is a careful development of JCF. After beginning with background material, we introduce Jordan Canonical Form and related notions: eigenvalues, (generalized) eigenvectors, and the characteristic and minimum polynomials. We decide the question of diagonalizability, and prove the Cayley-Hamilton theorem. Then we present a careful and complete proof of the fundamental theorem: Let V be a finite-dimensional vector space over the field of complex numbers C, and let T : V → V be a linear transformation. Then T has a Jordan Canonical Form. This theorem has an equivalent statement in terms of matrices: Let A be a square matrix with complex entries. Then A is similar to a matrix J in Jordan Canonical Form, i.e., there is an invertible matrix P and a matrix J in Jordan Canonical Form with A = PJP-1. We further present an algorithm to find P and J, assuming that one can factor the characteristic polynomial of A. In developing this algorithm we introduce the eigenstructure picture (ESP) of a matrix, a pictorial representation that makes JCF clear. The ESP of A determines J, and a refinement, the labeled eigenstructure picture (*ESP) of A, determines P as well. We illustrate this algorithm with copious examples, and provide numerous exercises for the reader.
Fundamentals on Vector Spaces and Linear Transformations / The Structure of a Linear Transformation / An Algorithm for Jordan Canonical Form and Jordan Basis
Series: Contemporary Mathematicians
2010, X, 760 p. 12 illus., 6 in color., Hardcover
ISBN: 978-0-8176-4967-8
Due: July 29, 2010
Published in honor of his 70th birthday, this volume explores and celebrates the work of G.W. (Pete) Stewart, a world-renowned expert in computational linear algebra. It is widely accepted that Stewart is the successor to James Wilkinson, the first giant in the field, taking up the perturbation theory research that Wilkinson so ably began and using it as a foundation for algorithmic insights. Stewart’s results on rounding error in numerical computations provided basic understanding of floating- point computation. His results on perturbation of eigensystems, pseudo-inverses, least-squares problems, and matrix factorizations are fundamental to numerical practice today. His algorithms for the singular value decomposition, updating and downdating matrix factorizations, and the eigenproblem broke new ground and are still widely used in an increasing number of applications. Stewart’s papers, widely cited, are characterized by elegance in theorems and algorithms and clear, concise, and beautiful exposition. His six popular textbooks are excellent sources of knowledge and history. Stewart is a member of the National Academy of Engineering and has received numerous additional honors, including the Bauer Prize. Key features of this volume include: * Forty-four of Stewart’s most influential research papers in two subject areas: matrix algorithms and rounding and perturbation theory * A biography of Stewart * A complete list of Stewart’s publications, students, and honors * Selected photographs * Commentaries on Stewart’s works in collaboration with leading experts in the field G.W. Stewart: Selected Works with Commentaries will appeal to graduate students, practitioners, and researchers in computational linear algebra and the history of mathematics.
Content Level ≫ Research
Keywords ≫ Krylov methods - LINPACK - computational linear algebra - eigenproblem - floating point computation - invariant subspaces - least squares - matrix algorithms - matrix decompositions - matrix factorizations - perturbation theory - rounding theory - singular value decomposition - updating and downdating
Related subjects ≫ Engineering - HistoryofScience - Mathematics
Foreword.- Part I. G.W. Stewart.- Biography of G.W. Stewart.- Publications, Honors, and Students.- Part II. Commentaries.- Introduction to the Commentaries.- Matrix Decompositions: LINPACK and Beyond.- Updating and Downdating Matrix Decompositions.- Least Squares, Projections, and Psuedo-Inverses.- The Eigenproblem and Invariant Subspaces: Perturbation Theory.- The SVD, Eigenproblem, and Invariant Subspaces: Algorithms.- The Generalized Eigenproblem.- Krylov Subspace Methods for the Eigenproblem.- Other Contributions.- References.- Index.- Part III. Reprints.- Papers on Matrix Decompositions.- Papers on Updating and Downdating Matrix Decompositions.- Papers on Least Squares, Projections, and Generalized Inverses.- Papers on the Eigenproblem and Invariant Subspaces: Perturbation Theory.- Papers on the SVD, Eigenproblem and Invariant Subspaces: Algorithms.- Papers on the Generalized Eigenproblem.- Papers on Krylov Subspace Methods for the Eigenproblem.
2010, VIII, 160 p., Softcover
ISBN: 978-3-0346-0564-9
Due: August 2, 2010
Homotopy theory and C* algebras are central topics in contemporary mathematics. This book introduces a modern homotopy theory for C*-algebras. One basic idea of the setup is to merge C*-algebras and spaces studied in algebraic topology into one category comprising C*-spaces. These objects are suitable fodder for standard homotopy theoretic moves, leading to unstable and stable model structures. With the foundations in place one is led to natural definitions of invariants for C*-spaces such as homology and cohomology theories, K-theory and zeta-functions. The text is largely self-contained. It serves a wide audience of graduate students and researchers interested in C*-algebras, homotopy theory and applications.
Progress in Mathematical Physics, Vol. 59
VIII, 176 p., Hardcover
ISBN: 978-0-8176-4983-8
Due: August 29, 2010
Without using the customary Clifford algebras frequently studied in connection
with the representations of orthogonal groups, this book gives an elementary
introduction to the two component spinor formalism for four-dimensional
spaces with any signature. Some of the useful applications of four-dimensional
spinors, such as Yang*Mills theory, are derived in detail using illustrative
examples. Key topics and features: * Uniform treatment of the spinor formalism
for four-dimensional spaces of any signature, not only the usual signature
(+ + + *) employed in relativity * Examples taken from Riemannian geometry
and special or general relativity are discussed in detail, emphasizing
the usefulness of the two-component spinor formalism * Exercises in each
chapter * The relationship of Clifford algebras and Dirac four-component
spinors is established * Applications of the two-component formalism, focusing
mainly on general relativity, are presented in the context of actual computations
Spinors in Four-Dimensional Spaces is aimed at graduate students and researchers
in mathematical and theoretical physics interested in the applications
of the two-component spinor formalism in any four-dimensional vector space
or Riemannian manifold with a definite or indefinite metric tensor. This
systematic and self-contained book is suitable as a seminar text, a reference
book, and a self-study guide. Reviews from the author's previous book,
3-D Spinors, Spin-Weighted Functions and their Applications: In summary…the
book gathers much of what can be done with 3-D spinors in an easy-to-read,
self-contained form designed for applications that will supplement many
available spinor treatments. The book…should be appealing to graduate
students and researchers in relativity and mathematical physics. *Mathematical
Reviews The present book provides an easy-to-read and unconventional presentation
of the spinor formalism for three-dimensional spaces with a definite or
indefinite metric...Following a nice and descriptive introduction…the
final chapter contains some applications of the formalism to general relativity.
*Monatshefte fur Mathematik
Content Level ≫ Research
Keywords ≫ Conformal Curvature - Curvature Spinors - Dirac Spinors - Einstein’s
Equations - Killing Bispinors - Self-Dual Yang-Mills Fields - Spinor Algebra
Related subjects ≫ Mathematics - Physics
1 Spinor Algebra.-1.1 Orthogonal Groups.-1.2 Null Tetrads and the Spinor Equivalent of a Tensor.-1.3 Spinorial Representation of the Orthogonal Transformations.-1.3.1 Euclidean Signature.-1.3.2 Lorentzian Signature.-1.3.3 Ultrahyperbolic Signature.-1.4 Reflections.-1.5 Clifford Algebra. Dirac Spinors.-1.6 Inner Products. Mate of a Spinor.-1.7 Principal Spinors. Algebraic Classification.-Exercises.-2 Connection and Curvature.-2.1 Covariant Differentiation .- 2.2 Curvature.-2.2.1 Curvature Spinors.-2.2.2 Algebraic Classification of the Conformal Curvature.-2.3 Conformal Rescalings.-2.4 Killing Vectors. Lie Derivative of Spinors.-Exercises.- 3 Applications to General Relativity.-3.1 Maxwell’s Equations.-3.2 Dirac’s Equation .-3.3 Einstein’s Equations.-3.3.1 The Goldberg*Sachs Theorem.-3.3.2 Space-Times with Symmetries. Ernst Potentials.-3.4 Killing Spinors.-Exercises.-4 Further Applications.-4.1 Self-Dual Yang*Mills Fields.-4.2 H and H H Spaces.-4.3 Killing Bispinors. The Dirac Operator.-Exercises.-A Bases Induced by Coordinate Systems.-References.
Hardback (ISBN-13: 9780521761598)
69 b/w illus. 104 tables 118 exercises
Page extent: 300 pages
Size: 253 x 215 mm
This textbook for a second course in basic statistics for undergraduates or first-year graduate students introduces linear regression models and describes other linear models including Poisson regression, logistic regression, proportional hazards regression, and nonparametric regression. Numerous examples drawn from the news and current events with an emphasis on health issues illustrate these concepts. Assuming only a pre-calculus background, the author keeps equations to a minimum and demonstrates all computations using SAS. Most of the programs and output are displayed in a self-contained way, with an emphasis on the interpretation of the output in terms of how it relates to the motivating example. Plenty of exercises conclude every chapter. All of the datasets and SAS programs are available from the book’s website, along with other ancillary material.
* Minimum of mathematics and equation knowledge required * Emphasis on
interpretation and use of statistical methods, with many examples from
current events * Use of the computer language SAS with a minimal knowledge
of SAS needed * All of the datasets and SAS programs are available from
the book’s website along with other ancillary material
1. Introduction; 2. Principles of statistics; 3. Introduction to linear regression; 4. Assessing the regression; 5. Multiple linear regression; 6. Indicators, interactions, and transformations; 7. Nonparametric statistics; 8. Logistic regression; 9. Diagnostics for logistic regression; 10. Poisson regression; 11. Survival analysis; 12. Proportional hazards regression; 13. Review of methods; Appendix: statistical tables.