Introduces $p$-adic numbers from the point of view of number theory, topology, and analysis. Covering several topics from real analysis and elementary topology, this book includes totally disconnected spaces and Cantor sets, points of discontinuity of maps

The common thread throughout this work is aperiodic tilings; the best-known example is the kite and dart tiling. This tiling has been widely discussed, particularly since 1984 when it was adopted to model quasicrystals.

This English translation of a Russian book presents the basic notions of differential and algebraic topology, which are indispensable for specialists and useful for research mathematicians and theoretical physicists. In particular, ideas and results are introduced related to such areas as manifolds, cell spaces, coverings and fibrations, homotopy groups, homology and cohomology, and intersection indexes.

Presents a discussion of algebraic operations on the points in the plane and rigid motions in the Euclidean plane. This work introduces the notions of a transformation group and of an abstract group. It gives an elementary exposition of the basic ideas of Sophus Lie about symmetries of differential equations.

Offers an introduction to the theory of frames in a Hilbert space. This book teaches linear algebra, using frames as an application of the theoretical concepts. It provides a resource for students doing undergraduate research projects using frames. It pres

Elementary geometry provides the foundation of modern geometry. For the most part, the standard introductions end at the formal Euclidean geometry of high school. This book revisits geometry, but from the higher viewpoint of university mathematics. It is s

Nature tries to minimize the surface area of a soap film through the action of surface tension. The process can be understood mathematically by using differential geometry, complex analysis, and the calculus of variations. This book employs ingredients from each of these subjects to tell the mathematical story of soap films. The text is fully self-contained, bringing together a mixture of types of mathematics along with a bit of the physics that underlies the subject. The development is primarily from first principles, requiring no advanced background material from either mathematics or...

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. Written for students with some mathematical maturity and an interest in abstraction and formal reason

Diophantine analysis, an area of number theory that helps to discover hidden treasures and truths within the numbers by exploring rational numbers, comprises two different but interconnected domains - diophantine approximation and diophantine equations. This book presents the fundamental ideas and theorems from diophantine approximation.

It is remarkable that so much about Lie groups could be packed into this small book. But after reading it, students will be well-prepared to continue with more advanced, graduate-level topics in differential geometry or the theory of Lie groups. The theory of Lie groups involves many areas of mathematics: algebra, differential geometry, algebraic geometry, analysis, and differential equations. In this book, Arvanitoyeorgos outlines enough of the prerequisites to get the reader started. He then chooses a path through this rich and diverse theory that aims for an understanding of the geometry...