Discrete Mathematics

Discrete Mathematics PDF

Author: Oscar Levin

Publisher: Createspace Independent Publishing Platform

Published: 2018-07-30

Total Pages: 238

ISBN-13: 9781724572639

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Note: This is a custom edition of Levin's full Discrete Mathematics text, arranged specifically for use in a discrete math course for future elementary and middle school teachers. (It is NOT a new and updated edition of the main text.)This gentle introduction to discrete mathematics is written for first and second year math majors, especially those who intend to teach. The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the "introduction to proof" course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this.Four main topics are covered: counting, sequences, logic, and graph theory. Along the way proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs.While there are many fine discrete math textbooks available, this text has the following advantages: - It is written to be used in an inquiry rich course.- It is written to be used in a course for future math teachers.- It is open source, with low cost print editions and free electronic editions.

Recurrent Sequences

Recurrent Sequences PDF

Author: Dorin Andrica

Publisher: Springer Nature

Published: 2020-09-23

Total Pages: 410

ISBN-13: 3030515028

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This self-contained text presents state-of-the-art results on recurrent sequences and their applications in algebra, number theory, geometry of the complex plane and discrete mathematics. It is designed to appeal to a wide readership, ranging from scholars and academics, to undergraduate students, or advanced high school and college students training for competitions. The content of the book is very recent, and focuses on areas where significant research is currently taking place. Among the new approaches promoted in this book, the authors highlight the visualization of some recurrences in the complex plane, the concurrent use of algebraic, arithmetic, and trigonometric perspectives on classical number sequences, and links to many applications. It contains techniques which are fundamental in other areas of math and encourages further research on the topic. The introductory chapters only require good understanding of college algebra, complex numbers, analysis and basic combinatorics. For Chapters 3, 4 and 6 the prerequisites include number theory, linear algebra and complex analysis. The first part of the book presents key theoretical elements required for a good understanding of the topic. The exposition moves on to to fundamental results and key examples of recurrences and their properties. The geometry of linear recurrences in the complex plane is presented in detail through numerous diagrams, which lead to often unexpected connections to combinatorics, number theory, integer sequences, and random number generation. The second part of the book presents a collection of 123 problems with full solutions, illustrating the wide range of topics where recurrent sequences can be found. This material is ideal for consolidating the theoretical knowledge and for preparing students for Olympiads.

Recurrence Sequences

Recurrence Sequences PDF

Author: Graham Everest

Publisher: American Mathematical Soc.

Published: 2015-09-03

Total Pages: 338

ISBN-13: 1470423154

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Recurrence sequences are of great intrinsic interest and have been a central part of number theory for many years. Moreover, these sequences appear almost everywhere in mathematics and computer science. This book surveys the modern theory of linear recurrence sequences and their generalizations. Particular emphasis is placed on the dramatic impact that sophisticated methods from Diophantine analysis and transcendence theory have had on the subject. Related work on bilinear recurrences and an emerging connection between recurrences and graph theory are covered. Applications and links to other areas of mathematics are described, including combinatorics, dynamical systems and cryptography, and computer science. The book is suitable for researchers interested in number theory, combinatorics, and graph theory.

Mathematical Recurrence Relations: Visual Mathematics Series

Mathematical Recurrence Relations: Visual Mathematics Series PDF

Author: Kiran R. Desai, Ph.d.

Publisher: CreateSpace

Published: 2013-04-29

Total Pages: 100

ISBN-13: 9781481219273

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This book is about arranging numbers in a two dimensional space. It illustrates that it is possible to create many different regular patterns of numbers on a grid that represent meaningful summations. It uses a color coding scheme to enhance the detection of the underlying pattern for the numbers. Almost all arrangements presented are scalable or extensible, in that the matrix can be extended to larger size without the need to change existing number placements. The emphasis in this book is about the placement and summation of all the numbers for recursive embeddings. In many cases, visual charts are used to provide a higher level view of the topography, and to make the recurrence relations come alive. Number arrangements are represented for many well known multi-dimensional numbers, polygonal numbers, and various polynomials defined by recurrence relations based on equations that are a function of an integer variable n. The solutions for the recurrence relations can also be checked by adding the numbers in the arrangements presented. It is also possible to create a recurrence relation by starting with any polynomial equation using induction principles. Studying the terms in the recurrence relation helps design of the matrix and the number arrangement. This book has shown arrangements for exact powers of two, three, four, and five. Higher powers are indeed conceivable in two or three dimensional space and could be a topic for further study. Number arrangements for equations with different polynomial degree are seen to differ in the rate of change between values at adjacent levels. These have been elaborated at various places in the book. The study of recurrence relations is then steered towards arrangements for multiplication tables and linear equations in two variables. When enumerated on a coordinate graph, linear equations are seen as planar surfaces in space, and also allow solving a system of such equations visually. Although intended for college or advanced high school level students, for the majority audience this book serves as a treatise on the beauty inherent in numbers.

Analytic Combinatorics

Analytic Combinatorics PDF

Author: Philippe Flajolet

Publisher: Cambridge University Press

Published: 2009-01-15

Total Pages: 825

ISBN-13: 1139477161

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Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines, including probability theory, statistical physics, computational biology, and information theory. With a careful combination of symbolic enumeration methods and complex analysis, drawing heavily on generating functions, results of sweeping generality emerge that can be applied in particular to fundamental structures such as permutations, sequences, strings, walks, paths, trees, graphs and maps. This account is the definitive treatment of the topic. The authors give full coverage of the underlying mathematics and a thorough treatment of both classical and modern applications of the theory. The text is complemented with exercises, examples, appendices and notes to aid understanding. The book can be used for an advanced undergraduate or a graduate course, or for self-study.

An Introduction to the Analysis of Algorithms

An Introduction to the Analysis of Algorithms PDF

Author: Robert Sedgewick

Publisher: Addison-Wesley

Published: 2013-01-18

Total Pages: 734

ISBN-13: 0133373487

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Despite growing interest, basic information on methods and models for mathematically analyzing algorithms has rarely been directly accessible to practitioners, researchers, or students. An Introduction to the Analysis of Algorithms, Second Edition, organizes and presents that knowledge, fully introducing primary techniques and results in the field. Robert Sedgewick and the late Philippe Flajolet have drawn from both classical mathematics and computer science, integrating discrete mathematics, elementary real analysis, combinatorics, algorithms, and data structures. They emphasize the mathematics needed to support scientific studies that can serve as the basis for predicting algorithm performance and for comparing different algorithms on the basis of performance. Techniques covered in the first half of the book include recurrences, generating functions, asymptotics, and analytic combinatorics. Structures studied in the second half of the book include permutations, trees, strings, tries, and mappings. Numerous examples are included throughout to illustrate applications to the analysis of algorithms that are playing a critical role in the evolution of our modern computational infrastructure. Improvements and additions in this new edition include Upgraded figures and code An all-new chapter introducing analytic combinatorics Simplified derivations via analytic combinatorics throughout The book’s thorough, self-contained coverage will help readers appreciate the field’s challenges, prepare them for advanced results—covered in their monograph Analytic Combinatorics and in Donald Knuth’s The Art of Computer Programming books—and provide the background they need to keep abreast of new research. "[Sedgewick and Flajolet] are not only worldwide leaders of the field, they also are masters of exposition. I am sure that every serious computer scientist will find this book rewarding in many ways." —From the Foreword by Donald E. Knuth

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