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"Number Systems: An Introduction to Algebra and Analysis",
by Sergei Ovchinnikov, AMS Publications, 2015.
This book offers a rigorous and coherent introduction to the five basic number systems of mathematics, namely natural numbers,
integers, rational numbers, real numbers, and complex numbers. It is a subject that many mathematicians believe should be
learned by any student of mathematics including future teachers. The book starts with the development of Peano arithmetic
in the first chapter which includes mathematical induction and elements of recursion theory. It proceeds to an examination of
integers that also covers rings and ordered integral domains. The presentation of rational numbers includes material on ordered
fields and convergence of sequences in these fields. Cauchy and Dedekind completeness properties of the field of real numbers
are established, together with some properties of real continuous functions. An elementary proof of the Fundamental
Theorem of Algebra is the highest point of the chapter on complex numbers.
Book's web page
"Graphs and Cubes",
by Sergei Ovchinnikov,
Springer, 2011.
Currently, Graphs and Cubes is the only book available on the market that presents a comprehensive coverage of cubical graph
and partial cube theories. Many exercises, along with historical notes, are included at the end of every chapter, and
readers are encouraged to explore the exercises fully, and use them as a basis for research projects.
"Media Theory",
by D.Eppstein, J.Cl.Falmagne and S.Ovchinnikov,
Springer, 2008. Sample chapter.
The focus of this book is a mathematical structure modeling a physical or biological system that can be in any
of a number of 'states'. Each state is characterized by a set of binary features, and differs from some other
neighbor state or states by just one of those feature. The evolution of such a system over time is considered. Such a structure is
analyzed from algebraic and probabilistic (stochastic) standpoints.
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