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Author: Jorge L. Ramírez Alfonsín
Publisher: Oxford University Press, USA
ISBN: 0198568207
Category : Mathematics
Languages : en
Pages : 260
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Book Description
During the early part of the last century, Ferdinand Georg Frobenius (1849-1917) raised he following problem, known as the Frobenius Problem (FP): given relatively prime positive integers a1,...,an, find the largest natural number (called the Frobenius number and denoted by g(a1,...,an) that is not representable as a nonnegative integer combination of a1,...,an, . At first glance FP may look deceptively specialized. Nevertheless it crops up again and again in the most unexpected places and has been extremely useful in investigating many different problems. A number of methods, from several areas of mathematics, have been used in the hope of finding a formula giving the Frobenius number and algorithms to calculate it. The main intention of this book is to highlight such methods, ideas, viewpoints and applications to a broader audience.
Author: Jorge L. Ramírez Alfonsín
Publisher: Oxford University Press, USA
ISBN: 0198568207
Category : Mathematics
Languages : en
Pages : 260
Get Book Here
Book Description
During the early part of the last century, Ferdinand Georg Frobenius (1849-1917) raised he following problem, known as the Frobenius Problem (FP): given relatively prime positive integers a1,...,an, find the largest natural number (called the Frobenius number and denoted by g(a1,...,an) that is not representable as a nonnegative integer combination of a1,...,an, . At first glance FP may look deceptively specialized. Nevertheless it crops up again and again in the most unexpected places and has been extremely useful in investigating many different problems. A number of methods, from several areas of mathematics, have been used in the hope of finding a formula giving the Frobenius number and algorithms to calculate it. The main intention of this book is to highlight such methods, ideas, viewpoints and applications to a broader audience.
Author: Jorge L. Ramírez Alfonsin
Publisher:
ISBN: 9780191718229
Category : Diophantine analysis
Languages : en
Pages : 243
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Book Description
A number of methods, from several areas of mathematics have been used in the hope of finding a formula giving the Frobenius number and algorithms to calculate it. The main intention of this book is to highlight these viewpoints, ideas and applications to a broader audience.
Author: Matthias Beck
Publisher: Springer
ISBN: 1493929690
Category : Mathematics
Languages : en
Pages : 295
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Book Description
This richly illustrated textbook explores the amazing interaction between combinatorics, geometry, number theory, and analysis which arises in the interplay between polyhedra and lattices. Highly accessible to advanced undergraduates, as well as beginning graduate students, this second edition is perfect for a capstone course, and adds two new chapters, many new exercises, and updated open problems. For scientists, this text can be utilized as a self-contained tooling device. The topics include a friendly invitation to Ehrhart’s theory of counting lattice points in polytopes, finite Fourier analysis, the Frobenius coin-exchange problem, Dedekind sums, solid angles, Euler–Maclaurin summation for polytopes, computational geometry, magic squares, zonotopes, and more. With more than 300 exercises and open research problems, the reader is an active participant, carried through diverse but tightly woven mathematical fields that are inspired by an innocently elementary question: What are the relationships between the continuous volume of a polytope and its discrete volume? Reviews of the first edition: “You owe it to yourself to pick up a copy of Computing the Continuous Discretely to read about a number of interesting problems in geometry, number theory, and combinatorics.” — MAA Reviews “The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the mate rial, exercises, open problems and an extensive bibliography.” — Zentralblatt MATH “This beautiful book presents, at a level suitable for advanced undergraduates, a fairly complete introduction to the problem of counting lattice points inside a convex polyhedron.” — Mathematical Reviews “Many departments recognize the need for capstone courses in which graduating students can see the tools they have acquired come together in some satisfying way. Beck and Robins have written the perfect text for such a course.” — CHOICE
Author: Stefan Matthias Ritter
Publisher:
ISBN:
Category :
Languages : en
Pages : 11
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Book Description
Author: H. Krawczyk
Publisher:
ISBN:
Category :
Languages : en
Pages : 4
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Book Description
Author: Curtis Kifer
Publisher: LAP Lambert Academic Publishing
ISBN: 9783845405131
Category :
Languages : en
Pages : 64
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Book Description
Given integer-valued relatively prime coins' a1; a2; : ak, the Frobenius number is the largest integer n such that the linear diophantine equation a1m1 + a2m2 +::: + akmk = n has no solution in non-negative integers m1;m2; : mk. We denote by g(a1; : ak) the largest integer value not attainable by this coin system. That is to say that any integer x greater than the Frobenius number g(a1; : ak) has a representation x = a1x1 + a2x2 +::: + akxk by a1; a2; : ak for some non-negative integers x1; x2; : xk. We say x is representable by a1; a2; : ak. While it is obvious that there are representable positive integers and non-representable positive integers, must there be a largest non-representable integer? Maybe there are indefinitely large non-representable integers for a1; a2; : ak with gcd (a1; a2; : ak) = 1. This notion of whether or not the Frobenius number is well-defined will be the first bit of mathematics we look at in this paper. Proposition 1.1. The Frobenius number g(a1; : ak) is well-defined. Proof. Given a1; a2; : ak with gcd (a1; a2; : ak) = 1, the extended Euclidean algorithm gives that there exist m1;m2; : mk 2 Z such that...
Author: Bharti Temkin
Publisher:
ISBN:
Category : Diophantine equations
Languages : en
Pages : 238
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Book Description
Author: Titu Andreescu
Publisher: Springer Science & Business Media
ISBN: 0817645497
Category : Mathematics
Languages : en
Pages : 350
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Book Description
This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The presentation features some classical Diophantine equations, including linear, Pythagorean, and some higher degree equations, as well as exponential Diophantine equations. Many of the selected exercises and problems are original or are presented with original solutions. An Introduction to Diophantine Equations: A Problem-Based Approach is intended for undergraduates, advanced high school students and teachers, mathematical contest participants — including Olympiad and Putnam competitors — as well as readers interested in essential mathematics. The work uniquely presents unconventional and non-routine examples, ideas, and techniques.
Author: Curtis Kifer
Publisher:
ISBN:
Category :
Languages : en
Pages : 100
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Book Description
Author: Masami Ito
Publisher: Springer
ISBN: 354085780X
Category : Mathematics
Languages : en
Pages : 555
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Book Description
This book constitutes the refereed proceedings of the 12th International Conference on Developments in Language Theory, DLT 2008, held in Kyoto, Japan, September 2008. The 36 revised full papers presented together with 6 invited papers were carefully reviewed and selected from 102 submissions. All important issues in language theory are addressed including grammars, acceptors and transducers for words, trees and graphs; algebraic theories of automata; algorithmic, combinatorial and algebraic properties of words and languages; variable length codes; symbolic dynamics; cellular automata; polyominoes and multidimensional patterns; decidability questions; image manipulation and compression; efficient text algorithms; relationships to cryptography, concurrency, complexity theory and logic; bio-inspired computing; quantum computing.
