Student Mathematical Library, Volume: 40
2007; 295 pp; softcover
ISBN-10: 0-8218-4212-9
ISBN-13: 978-0-8218-4212-6
Expected publication date is December 23, 2007.
Frames for Undergraduates is an undergraduate-level introduction to the theory of frames in a Hilbert space. This book can serve as a text for a special-topics course in frame theory, but it could also be used to teach a second semester of linear algebra, using frames as an application of the theoretical concepts. It can also provide a complete and helpful resource for students doing undergraduate research projects using frames.
The early chapters contain the topics from linear algebra that students need to know in order to read the rest of the book. The later chapters are devoted to advanced topics, which allow students with more experience to study more intricate types of frames. Toward that end, a Student Presentation section gives detailed proofs of fairly technical results with the intention that a student could work out these proofs independently and prepare a presentation to a class or research group. The authors have also presented some stories in the Anecdotes section about how this material has motivated and influenced their students.
Readership
Undergraduate and graduate students interested in linear algebra and applications, and the theory of frames.
Table of Contents
Introduction
Linear algebra review
Finite-dimensional operator theory
Introduction to finite frames
Frames in mathbb{R}^2
The dilation property of frames
Dual and orthogonal frames
Frame operator decompositions
Harmonic and group frames
Sampling theory
Student presentations
Anecdotes on frame theory projects by undergraduates
Bibliography
List of symbols
Index
University Lecture Series, Volume: 41
2008; 167 pp; softcover
ISBN-10: 0-8218-4411-3
ISBN-13: 978-0-8218-4411-3
Expected publication date is January 11, 2008.
This book contains detailed descriptions of the many exciting recent developments in the combinatorics of the space of diagonal harmonics, a topic at the forefront of current research in algebraic combinatorics. These developments led in turn to some surprising discoveries in the combinatorics of Macdonald polynomials, which are described in Appendix A. The book is appropriate as a text for a topics course in algebraic combinatorics, a volume for self-study, or a reference text for researchers in any area which involves symmetric functions or lattice path combinatorics.
The book contains expository discussions of some topics in the theory of symmetric functions, such as the practical uses of plethystic substitutions, which are not treated in depth in other texts. Exercises are interspersed throughout the text in strategic locations, with full solutions given in Appendix C.
Readership
Graduate students and research mathematicians interested in combinatorics.
Table of Contents
Introduction to q-analogues and symmetric functions
Macdonald polynomials and the space of diagonal harmonics
The q, t-Catalan numbers
The q, t-Schroder polynomial
Parking functions and the Hilbert series
The shuffle conjecture
The proof of the q, t-Schroder theorem
The combinatorics of Macdonald polynomials
The Loehr-Warrington conjecture
Solutions to exercises
Bibliography
Student Mathematical Library, Volume: 42
2008; 262 pp; softcover
ISBN-10: 0-8218-4420-2
ISBN-13: 978-0-8218-4420-5
Expected publication date is January 10, 2008.
This book is an introduction to basic concepts in ergodic theory such as
recurrence, ergodicity, the ergodic theorem, mixing, and weak mixing. It
does not assume knowledge of measure theory; all the results needed from
measure theory are presented from scratch. In particular, the book includes
a detailed construction of the Lebesgue measure on the real line and an
introduction to measure spaces up to the Caratheodory extension theorem.
It also develops the Lebesgue theory of integration, including the dominated
convergence theorem and an introduction to the Lebesgue L^pspaces.
Several examples of a dynamical system are developed in detail to illustrate various dynamical concepts. These include in particular the baker's transformation, irrational rotations, the dyadic odometer, the Hajian-Kakutani transformation, the Gauss transformation, and the Chacon transformation. There is a detailed discussion of cutting and stacking transformations in ergodic theory. The book includes several exercises and some open questions to give the flavor of current research. The book also introduces some notions from topological dynamics, such as minimality, transitivity and symbolic spaces; and develops some metric topology, including the Baire category theorem.
Readership
Undergraduate and graduate students interested in ergodic theory and measure theory.
Table of Contents
Introduction
Lebesgue measure
Recurrence and ergodicity
The Lebesgue integral
The ergodic theorem
Mixing notions
Set theory
The completeness property of mathbb{R}
Topology of mathbb{R} and metric spaces
Bibliographic notes
Bibliography
Index
Student Mathematical Library, Volume: 43
2008; approx. 246 pp; softcover
ISBN-10: 0-8218-4347-8
ISBN-13: 978-0-8218-4347-5
Expected publication date is January 18, 2008.
Elementary geometry provides the foundation of modern geometry. For the most part, the standard introductions end at the formal Euclidean geometry of high school. Agricola and Friedrich revisit geometry, but from the higher viewpoint of university mathematics. Plane geometry is developed from its basic objects and their properties and then moves to conics and basic solids, including the Platonic solids and a proof of Euler's polytope formula. Particular care is taken to explain symmetry groups, including the description of ornaments and the classification of isometries by their number of fixed points. Complex numbers are introduced to provide an alternative, very elegant approach to plane geometry. The authors then treat spherical and hyperbolic geometries, with special emphasis on their basic geometric properties.
This largely self-contained book provides a much deeper understanding of familiar topics, as well as an introduction to new topics that complete the picture of two-dimensional geometries. For undergraduate mathematics students the book will be an excellent introduction to an advanced point of view on geometry. For mathematics teachers it will be a valuable reference and a source book for topics for projects.
The book contains over 100 figures and scores of exercises. It is suitable for a one-semester course in geometry for undergraduates, particularly for mathematics majors and future secondary school teachers.
Readership
Undergraduate students interested in plane geometry.
Table of Contents
Introduction: Euclidean space
Elementary geometrical figures and their properties
Symmetries of the plane and of space
Hyperbolic geometry
Spherical geometry
Bibliography
List of symbols
Index
Student Mathematical Library, Volume: 44
2008; 215 pp; softcover
ISBN-10: 0-8218-4419-9
ISBN-13: 978-0-8218-4419-9
Expected publication date is January 13, 2008.
Linear algebra is the study of vector spaces and the linear maps between them. It underlies much of modern mathematics and is widely used in applications.
A (Terse) Introduction to Linear Algebra is a concise presentation of the core material of the subject--those elements of linear algebra that every mathematician, and everyone who uses mathematics, should know. It goes from the notion of a finite-dimensional vector space to the canonical forms of linear operators and their matrices, and covers along the way such key topics as: systems of linear equations, linear operators and matrices, determinants, duality, and the spectral theory of operators on inner-product spaces.
The last chapter offers a selection of additional topics indicating directions in which the core material can be applied.
The Appendix provides all the relevant background material.
Written for students with some mathematical maturity and an interest in abstraction and formal reasoning, the book is self-contained and is appropriate for an advanced undergraduate course in linear algebra.
Readership
Undergraduate and graduate students interested in linear algebra.
Table of Contents
Vector spaces
Linear operators and matrices
Duality of vector spaces
Determinants
Invariant subspaces
Operators on inner-product spaces
Structure theorems
Additional topics
Appendix
Index
Symbols