MAA Press: An Imprint of the American Mathematical Society
Dolciani Mathematical Expositions, Volume: 52
2019; 394 pp; Hardcover
MSC: Primary 11;
Print ISBN: 978-1-4704-4737-3
Quadratic Number Theory is an introduction to algebraic number theory for readers with a moderate knowledge of elementary number theory and some familiarity with the terminology of abstract algebra. By restricting attention to questions about squares the author achieves the dual goals of making the presentation accessible to undergraduates and reflecting the historical roots of the subject. The representation of integers by quadratic forms is emphasized throughout the text.
Lehman introduces an innovative notation for ideals of a quadratic domain that greatly facilitates computation and he uses this to particular effect. The text has an unusual focus on actual computation. This focus, and this notation, serve the author's historical purpose as well; ideals can be seen as number-like objects, as Kummer and Dedekind conceived of them. The notation can be adapted to quadratic forms and provides insight into the connection between quadratic forms and ideals. The computation of class groups and continued fraction representations are featured?the author's notation makes these computations particularly illuminating.
Quadratic Number Theory, with its exceptionally clear prose, hundreds of exercises, and historical motivation, would make an excellent textbook for a second undergraduate course in number theory. The clarity of the exposition would also make it a terrific choice for independent reading. It will be exceptionally useful as a fruitful launching pad for undergraduate research projects in algebraic number theory.
Undergraduate and graduate students interested in number theory and algebraic number theory.
MAA Press: An Imprint of the American Mathematical Society
AMS/MAA Textbooks, Volume: 49
2019; 232 pp; Hardcover
MSC: Primary 91; 97; Secondary 60; 65; 35
Print ISBN: 978-1-4704-4839-4
Mathematical Modeling in Economics and Finance is designed as a textbook for an upper-division course on modeling in the economic sciences. The emphasis throughout is on the modeling process including post-modeling analysis and criticism. It is a textbook on modeling that happens to focus on financial instruments for the management of economic risk. The book combines a study of mathematical modeling with exposure to the tools of probability theory, difference and differential equations, numerical simulation, data analysis, and mathematical analysis.
Students taking a course from Mathematical Modeling in Economics and Finance will come to understand some basic stochastic processes and the solutions to stochastic differential equations. They will understand how to use those tools to model the management of financial risk. They will gain a deep appreciation for the modeling process and learn methods of testing and evaluation driven by data. The reader of this book will be successfully positioned for an entry-level position in the financial services industry or for beginning graduate study in finance, economics, or actuarial science.
The exposition in Mathematical Modeling in Economics and Finance is crystal clear and very student-friendly. The many exercises are extremely well designed. Steven Dunbar is Professor Emeritus of Mathematics at the University of Nebraska and he has won both university-wide and MAA prizes for extraordinary teaching. Dunbar served as Director of the MAA's American Mathematics Competitions from 2004 until 2015. His ability to communicate mathematics is on full display in this approachable, innovative text.
Undergraduate and graduate students interested in mathematical finance.
MAA Press: An Imprint of the American Mathematical Society
Dolciani Mathematical Expositions, Volume: 53
2019; Hardcover
MSC: Primary 11; 00; 70;
Print ISBN: 978-1-4704-4795-3
There is a nineteen-year recurrence in the apparent position of the sun and moon against the background of the stars, a pattern observed long ago by the Babylonians. In the course of those nineteen years the Earth experiences 235 lunar cycles. Suppose we calculate the ratio of Earth's period about the sun to the moon's period about Earth. That ratio has 235/19 as one of its early continued fraction convergents, which explains the apparent periodicity.
Exploring Continued Fractions explains this and other recurrent phenomena?astronomical transits and conjunctions, lifecycles of cicadas, eclipses?by way of continued fraction expansions. The deeper purpose is to find patterns, solve puzzles, and discover some appealing number theory. The reader will explore several algorithms for computing continued fractions, including some new to the literature. He or she will also explore the surprisingly large portion of number theory connected to continued fractions: Pythagorean triples, Diophantine equations, the Stern-Brocot tree, and a number of combinatorial sequences.
The book features a pleasantly discursive style with excursions into music (The Well-Tempered Clavier), history (the Ishango bone and Plimpton 322), classics (the shape of More's Utopia) and whimsy (dropping a black hole on Earth's surface). Andy Simoson has won both the Chauvenet Prize and Polya Award for expository writing from the MAA and his Voltaire's Riddle was a Choice magazine Outstanding Academic Title. This book is an enjoyable ramble through some beautiful mathematics. For most of the journey the only necessary prerequisites are a minimal familiarity with mathematical reasoning and a sense of fun.
Undergraduate students interested in number theory.
Throughout the book, results are carefully proved using the inference rules introduced at the beginning
A particular focus is on the means-end approach to the justification of inductive inference rules
The instructor's manual contains the solutions to the 50 exercises as well as suggested exam questions
Description
A Logical Introduction to Probability and Induction is a textbook on the mathematics of the probability calculus and its applications in philosophy.
On the mathematical side, the textbook introduces these parts of logic and set theory that are needed for a precise formulation of the probability calculus. On the philosophical side, the main focus is on the problem of induction and its reception in epistemology and the philosophy of science. Particular emphasis is placed on the means-end approach to the justification of inductive inference rules.
In addition, the book discusses the major interpretations of probability. These are philosophical accounts of the nature of probability that interpret the mathematical structure of the probability calculus. Besides the classical and logical interpretation, they include the interpretation of probability as chance, degree of belief, and relative frequency. The Bayesian interpretation of probability as degree of belief locates probability in a subject's mind. It raises the question why her degrees of belief ought to obey the probability calculus. In contrast to this, chance and relative frequency belong to the external world. While chance is postulated by theory, relative frequencies can be observed empirically.
A Logical Introduction to Probability and Induction aims to equip students with the ability to successfully carry out arguments. It begins with elementary deductive logic and uses it as basis for the material on probability and induction. Throughout the textbook results are carefully proved using the inference rules introduced at the beginning, and students are asked to solve problems in the form of 50 exercises. An instructor's manual contains the solutions to these exercises as well as suggested exam questions.
The book does not presuppose any background in mathematics, although sections 10.3-10.9 on statistics are technically sophisticated and optional. The textbook is suitable for lower level undergraduate courses in philosophy and logic.
Hardcover ISBN: 9781785482502
Imprint: ISTE Press - Elsevier
Published Date: 1st December 2019
Page Count: 230
Fundamentals of Advanced Mathematics, Volume Three begins with the study of differential and analytic infinite-dimensional manifolds, then progresses into fibered bundles, in particular, tangent and cotangent bundles. In addition, subjects covered include the tensor calculus on manifolds, differential and integral calculus on manifolds (general Stokes formula, integral curves and manifolds), an analysis on Lie groups, the Haar measure, the convolution of functions and distributions, and the harmonic analysis over a Lie group. Finally, the theory of connections is (linear connections, principal connections, and Cartan connections) covered, as is the calculus of variations in Lagrangian and Hamiltonian formulations.
This volume is the prerequisite to the analytic and geometric study of nonlinear systems.
Includes sections on differential and analytic manifolds, vector bundles, tensors, Lie derivatives, applications to algebraic topology, and more
Presents an ideal prerequisite resource on the analytic and geometric study of nonlinear systems
Provides theory as well as practical information
Graduate students in Systems Theory, Robotics, Physics or Mathematics, research engineers in Automatic control and/or robotics, assistant professors and professors in Automatic control and/or robotics