2011, XV, 324 p. 3 illus., Hardcover
ISBN: 978-1-4419-0494-2
Presents the historical background of various topics in number theory;
Provides a self-contained introduction to classical number theory;
Includes step-by-step proofs of theorems and solutions to exercises;
Designed for undergraduate students, particularly those who would like to prepare for mathematical competitions.
This book is designed to introduce some of the most important theorems
and results from number theory while testing the readerfs understanding
through carefully selected Olympiad-caliber problems. These problems and
their solutions provide the reader with an opportunity to sharpen their
skills and to apply the theory. This framework guides the reader to an
easy comprehension of some of the jewels of number theory The book is self-contained
and rigorously presented. Various aspects will be of interest to graduate
and undergraduate students in number theory, advanced high school students
and the teachers who train them for mathematics competitions, as well as
to scholars who will enjoy learning more about number theory. Michael Th.
Rassias has received several awards in mathematical problem solving competitions
including two gold medals at the Pan-Hellenic Mathematical Competitions
of 2002 and 2003 held in Athens, a silver medal at the Balkan Mathematical
Olympiad of 2002 held in Targu Mures, Romania and a silver medal at the
44th International Mathematical Olympiad of 2003 held in Tokyo, Japan.
Content Level ā Lower undergraduate
Keywords ā Mathematics Competition - Mathematics Olympiad - Number Theory - Problem-Solving
Related subjects ā Number Theory and Discrete Mathematics
- Introduction.- The Fundamental Theorem of Arithmetic.- Arithmetic functions.- Perfect numbers, Fermat numbers.- Basic theory of
congruences.- Quadratic residues and the Law of Quadratic Reciprocity.-
The functions p(x) and li(x).- The Riemann zeta function.- Dirichlet series.-
Partitions of integers.- Generating functions.- Solved exercises and problems.-
The harmonic series of prime numbers.- Lagrange four-square theorem.- Bertrand
postulate.- An inequality for the function p(n).- An elementary proof of
the Prime Number Theorem.- Historical remarks on Fermatfs Last Theorem.-
Bibliography and Cited References.- Author index.- Subject index.
Series: Statistics for Social and Behavioral Sciences,
1st Edition., 2011, XII, 234 p. 13 illus., Hardcover
ISBN: 978-1-4419-9886-6
Due: May 29, 2011
Aside from distribution theory, projections and the singular value decomposition
(SVD) are the two most important concepts for understanding the basic mechanism
of multivariate analysis. The former underlies the least squares estimation
in regression analysis, which is essentially a projection of one subspace
onto another, and the latter underlies principal component analysis, which
seeks to find a subspace that captures the largest variability in the original
space.This book is about projections and SVD. A thorough discussion of
generalized inverse (g-inverse) matrices is also given because it is closely
related to the former. The book provides systematic and in-depth accounts
of these concepts from a unified viewpoint of linear transformations finite
dimensional vector spaces. More specially, it shows that projection matrices
(projectors) and g-inverse matrices can be defined in various ways so that
a vector space is decomposed into a direct-sum of (disjoint) subspaces.
Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition
will be useful for researchers, practitioners, and students in applied
mathematics, statistics, engineering, behaviormetrics, and other fields.
Content Level ā Research
Keywords ā Multivariate analysis
Related subjects ā Life Sciences, Medicine & Health - Statistics
Fundamentals of Linear Algebra.- Projection Matrices.- Generalized Inverse
Matrices.- Explicit Representations.- Singular Value Decomposition (SVD).-
Various Applications.
Series: Progress in Mathematics, Vol. 290
2011, X, 124 p., Hardcover
ISBN: 978-3-0348-0144-7
Due: June 30, 2011
This book is an introduction to the subject of mean curvature flow of hypersurfaces
with special emphasis on the analysis of singularities. This flow occurs
in the description of the evolution of numerous physical models where the
energy is given by the area of the interfaces. These notes provide a detailed
discussion of the classical parametric approach (mainly developed by R.
Hamilton and G. Huisken). They are well suited for a course at PhD/PostDoc
level and can be useful for any researcher interested in a solid introduction
to the technical issues of the field. All the proofs are carefully written,
often simplified, and contain several comments. Moreover, the author revisited
and organized a large amount of material scattered
around in literature in the last 25 years.
Content Level ā Research
Keywords ā geometric analysis - mean curvature flow - parametric approach
Related subjects ā Analysis
Foreword.- Chapter 1. Definition and Short Time Existence.- Chapter 2.
Evolution of Geometric Quantities.- Chapter 3. Monotonicity Formula and
Type I Singularities.- Chapter 4. Type II Singularities.- Chapter 5. Conclusions
and Research Directions.- Appendix A. Quasilinear Parabolic Equations on
Manifolds.- Appendix B. Interior Estimates of Ecker and Huisken.- Appendix
C. Hamiltonfs Maximum Principle for Tensors.- Appendix D. Hamiltonfs
Matrix Li?Yau?Harnack Inequality in Rn.- Appendix E. Abresch and Langer
Classification of Homothetically Shrinking Closed Curves.- Appendix F.
Important Results without Proof in the Book.- Bibliography.- Index.
Series: Lecture Notes in Mathematics, Vol. 2021
2011, X, 160 p., Softcover
ISBN: 978-3-642-20437-1
Due: June 2011
The aim of this research is to develop a systematic scheme that makes it possible to transfer important parts of the by now classical theory of summation of general orthonormal series into a similar theory for series in noncommutative Lp-spaces built over a noncommutative measure space (a von Neumann algebra of operators acting on a Hilbert space together with a faithful normal state on this algebra).
Content Level ā Research
Keywords ā 46-XX; 47-XX - Menchoff-Rademacher type theorems - noncommutative
L_p-spaces - pointwise convergence of orthonormal series - summation methods
of orthonormal series - symmetric spaces of operators
Related subjects ā Analysis - Probability Theory and Stochastic Processes
1 Introduction.- 2 Commutative Theory.- 3 Noncommutative Theory.