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Symmetry is one of the most important organising principles in the
natural sciences. The mathematical theory of symmetry has long been
associated with group theory, but it is a basic premise of this
book that there are aspects of symmetry which are more faithfully
represented by a generalization of groups called inverse
semigroups. The theory of inverse semigroups is described from its
origins in the foundations of differential geometry through to its
most recent applications in combinatorial group theory, and the
theory tilings.
Algebra & Geometry: An Introduction to University Mathematics,
Second Edition provides a bridge between high school and
undergraduate mathematics courses on algebra and geometry. The
author shows students how mathematics is more than a collection of
methods by presenting important ideas and their historical origins
throughout the text. He incorporates a hands-on approach to proofs
and connects algebra and geometry to various applications. The text
focuses on linear equations, polynomial equations, and quadratic
forms. The first few chapters cover foundational topics, including
the importance of proofs and a discussion of the properties
commonly encountered when studying algebra. The remaining chapters
form the mathematical core of the book. These chapters explain the
solutions of different kinds of algebraic equations, the nature of
the solutions, and the interplay between geometry and algebra. New
to the second edition Several updated chapters, plus an all-new
chapter discussing the construction of the real numbers by means of
approximations by rational numbers Includes fifteen short 'essays'
that are accessible to undergraduate readers, but which direct
interested students to more advanced developments of the material
Expanded references Contains chapter exercises with solutions
provided online at www.routledge.com/9780367563035
Algebra & Geometry: An Introduction to University Mathematics,
Second Edition provides a bridge between high school and
undergraduate mathematics courses on algebra and geometry. The
author shows students how mathematics is more than a collection of
methods by presenting important ideas and their historical origins
throughout the text. He incorporates a hands-on approach to proofs
and connects algebra and geometry to various applications. The text
focuses on linear equations, polynomial equations, and quadratic
forms. The first few chapters cover foundational topics, including
the importance of proofs and a discussion of the properties
commonly encountered when studying algebra. The remaining chapters
form the mathematical core of the book. These chapters explain the
solutions of different kinds of algebraic equations, the nature of
the solutions, and the interplay between geometry and algebra. New
to the second edition Several updated chapters, plus an all-new
chapter discussing the construction of the real numbers by means of
approximations by rational numbers Includes fifteen short 'essays'
that are accessible to undergraduate readers, but which direct
interested students to more advanced developments of the material
Expanded references Contains chapter exercises with solutions
provided online at www.routledge.com/9780367563035
Interest in finite automata theory continues to grow, not only
because of its applications in computer science, but also because
of more recent applications in mathematics, particularly group
theory and symbolic dynamics. The subject itself lies on the
boundaries of mathematics and computer science, and with a balanced
approach that does justice to both aspects, this book provides a
well-motivated introduction to the mathematical theory of finite
automata. The first half of Finite Automata focuses on the computer
science side of the theory and culminates in Kleene's Theorem,
which the author proves in a variety of ways to suit both computer
scientists and mathematicians. In the second half, the focus shifts
to the mathematical side of the theory and constructing an
algebraic approach to languages. Here the author proves two main
results: Schutzenberger's Theorem on star-free languages and the
variety theorem of Eilenberg and Schutzenberger. Accessible even to
students with only a basic knowledge of discrete mathematics, this
treatment develops the underlying algebra gently but rigorously,
and nearly 200 exercises reinforce the concepts. Whether your
students' interests lie in computer science or mathematics, the
well organized and flexible presentation of Finite Automata
provides a route to understanding that you can tailor to their
particular tastes and abilities.
The theories of V. V. Wagner (1908-1981) on abstractions of systems
of binary relations are presented here within their historical and
mathematical contexts. This book contains the first translation
from Russian into English of a selection of Wagner's papers, the
ideas of which are connected to present-day mathematical research.
Along with a translation of Wagner's main work in this area, his
1953 paper 'Theory of generalised heaps and generalised groups,'
the book also includes translations of three short precursor
articles that provide additional context for his major work.
Researchers and students interested in both algebra (in particular,
heaps, semiheaps, generalised heaps, semigroups, and groups) and
differential geometry will benefit from the techniques offered by
these translations, owing to the natural connections between
generalised heaps and generalised groups, and the role played by
these concepts in differential geometry. This book gives examples
from present-day mathematics where ideas related to Wagner's have
found fruitful applications.
Interest in finite automata theory continues to grow, not only because of its applications in computer science, but also because of more recent applications in mathematics, particularly group theory and symbolic dynamics. The subject itself lies on the boundaries of mathematics and computer science, and with a balanced approach that does justice to both aspects, this book provides a well-motivated introduction to the mathematical theory of finite automata.
The first half of Finite Automata focuses on the computer science side of the theory and culminates in Kleene's Theorem, which the author proves in a variety of ways to suit both computer scientists and mathematicians. In the second half, the focus shifts to the mathematical side of the theory and constructing an algebraic approach to languages. Here the author proves two main results: Schützenberger's Theorem on star-free languages and the variety theorem of Eilenberg and Schützenberger.
Accessible even to students with only a basic knowledge of discrete mathematics, this treatment develops the underlying algebra gently but rigorously, and nearly 200 exercises reinforce the concepts. Whether your students' interests lie in computer science or mathematics, the well organized and flexible presentation of Finite Automata provides a route to understanding that you can tailor to their particular tastes and abilities.
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