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Author: Pablo A. Parrilo
Publisher: American Mathematical Soc.
ISBN: 1470450259
Category : Education
Languages : en
Pages : 142
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Book Description
This volume is based on lectures delivered at the 2019 AMS Short Course “Sum of Squares: Theory and Applications”, held January 14–15, 2019, in Baltimore, Maryland. This book provides a concise state-of-the-art overview of the theory and applications of polynomials that are sums of squares. This is an exciting and timely topic, with rich connections to many areas of mathematics, including polynomial and semidefinite optimization, real and convex algebraic geometry, and theoretical computer science. The six chapters introduce and survey recent developments in this area; specific topics include the algebraic and geometric aspects of sums of squares and spectrahedra, lifted representations of convex sets, and the algorithmic and computational implications of viewing sums of squares as a meta algorithm. The book also showcases practical applications of the techniques across a variety of areas, including control theory, statistics, finance and machine learning.
Author: Pablo A. Parrilo
Publisher: American Mathematical Soc.
ISBN: 1470450259
Category : Education
Languages : en
Pages : 142
Get Book Here
Book Description
This volume is based on lectures delivered at the 2019 AMS Short Course “Sum of Squares: Theory and Applications”, held January 14–15, 2019, in Baltimore, Maryland. This book provides a concise state-of-the-art overview of the theory and applications of polynomials that are sums of squares. This is an exciting and timely topic, with rich connections to many areas of mathematics, including polynomial and semidefinite optimization, real and convex algebraic geometry, and theoretical computer science. The six chapters introduce and survey recent developments in this area; specific topics include the algebraic and geometric aspects of sums of squares and spectrahedra, lifted representations of convex sets, and the algorithmic and computational implications of viewing sums of squares as a meta algorithm. The book also showcases practical applications of the techniques across a variety of areas, including control theory, statistics, finance and machine learning.
Author: Carlos J. Moreno
Publisher: CRC Press
ISBN: 1584884568
Category : Mathematics
Languages : en
Pages : 368
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Book Description
Sums of Squares of Integers covers topics in combinatorial number theory as they relate to counting representations of integers as sums of a certain number of squares. The book introduces a stimulating area of number theory where research continues to proliferate. It is a book of "firsts" - namely it is the first book to combine Liouville's elementary methods with the analytic methods of modular functions to study the representation of integers as sums of squares. It is the first book to tell how to compute the number of representations of an integer n as the sum of s squares of integers for any s and n. It is also the first book to give a proof of Szemeredi's theorem, and is the first number theory book to discuss how the modern theory of modular forms complements and clarifies the classical fundamental results about sums of squares. The book presents several existing, yet still interesting and instructive, examples of modular forms. Two chapters develop useful properties of the Bernoulli numbers and illustrate arithmetic progressions, proving the theorems of van der Waerden, Roth, and Szemeredi. The book also explains applications of the theory to three problems that lie outside of number theory in the areas of cryptanalysis, microwave radiation, and diamond cutting. The text is complemented by the inclusion of over one hundred exercises to test the reader's understanding.
Author: Carlos J. Moreno
Publisher: CRC Press
ISBN: 1420057235
Category : Computers
Languages : en
Pages : 363
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Book Description
Sums of Squares of Integers covers topics in combinatorial number theory as they relate to counting representations of integers as sums of a certain number of squares. The book introduces a stimulating area of number theory where research continues to proliferate. It is a book of "firsts" - namely it is the first book to combine Liouville's element
Author: T.H Jackson
Publisher: CRC Press
ISBN: 1000948781
Category : Mathematics
Languages : en
Pages : 200
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Book Description
From Polynomials to Sums of Squares describes a journey through the foothills of algebra and number theory based around the central theme of factorization. The book begins by providing basic knowledge of rational polynomials, then gradually introduces other integral domains, and eventually arrives at sums of squares of integers. The text is complemented with illustrations that feature specific examples. Other than familiarity with complex numbers and some elementary number theory, very little mathematical prerequisites are needed. The accompanying disk enables readers to explore the subject further by removing the tedium of doing calculations by hand. Throughout the text there are practical activities involving the computer.
Author: Grigoriy Blekherman
Publisher: SIAM
ISBN: 1611972280
Category : Mathematics
Languages : en
Pages : 487
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Book Description
An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.
Author: Michal Křížek
Publisher: Springer Nature
ISBN: 3030838994
Category : Mathematics
Languages : en
Pages : 342
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Book Description
This book provides an overview of many interesting properties of natural numbers, demonstrating their applications in areas such as cryptography, geometry, astronomy, mechanics, computer science, and recreational mathematics. In particular, it presents the main ideas of error-detecting and error-correcting codes, digital signatures, hashing functions, generators of pseudorandom numbers, and the RSA method based on large prime numbers. A diverse array of topics is covered, from the properties and applications of prime numbers, some surprising connections between number theory and graph theory, pseudoprimes, Fibonacci and Lucas numbers, and the construction of Magic and Latin squares, to the mathematics behind Prague’s astronomical clock. Introducing a general mathematical audience to some of the basic ideas and algebraic methods connected with various types of natural numbers, the book will provide invaluable reading for amateurs and professionals alike.
Author: Nathan Jacob Fine
Publisher: American Mathematical Soc.
ISBN: 0821815245
Category : Mathematics
Languages : en
Pages : 142
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Book Description
The theory of partitions, founded by Euler, has led in a natural way to the idea of basic hypergeometric series, also known as Eulerian series. These series were first studied systematically by Heine, but many early results are attributed to Euler, Gauss, and Jacobi. This book provides a simple approach to basic hypergeometric series.
Author: Bogdan Dumitrescu
Publisher: Springer
ISBN: 3319536885
Category : Technology & Engineering
Languages : en
Pages : 282
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Book Description
This book gathers the main recent results on positive trigonometric polynomials within a unitary framework. The book has two parts: theory and applications. The theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The applications part is organized as a collection of related problems that use systematically the theoretical results.
Author: Marco Fontana
Publisher: CRC Press
ISBN: 9780203910627
Category : Mathematics
Languages : en
Pages : 524
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Book Description
Featuring presentations from the Fourth International Conference on Commutative Algebra held in Fez, Morocco, this reference presents trends in the growing area of commutative algebra. With contributions from nearly 50 internationally renowned researchers, the book emphasizes innovative applications and connections to algebraic number theory, geome
Author: Stephen C. Milne
Publisher: Springer Science & Business Media
ISBN: 1475754620
Category : Mathematics
Languages : en
Pages : 150
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Book Description
The problem of representing an integer as a sum of squares of integers is one of the oldest and most significant in mathematics. It goes back at least 2000 years to Diophantus, and continues more recently with the works of Fermat, Euler, Lagrange, Jacobi, Glaisher, Ramanujan, Hardy, Mordell, Andrews, and others. Jacobi's elliptic function approach dates from his epic Fundamenta Nova of 1829. Here, the author employs his combinatorial/elliptic function methods to derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's (1829) 4 and 8 squares identities to 4n2 or 4n(n+1) squares, respectively, without using cusp forms such as those of Glaisher or Ramanujan for 16 and 24 squares. These results depend upon new expansions for powers of various products of classical theta functions. This is the first time that infinite families of non-trivial exact explicit formulas for sums of squares have been found. The author derives his formulas by utilizing combinatorics to combine a variety of methods and observations from the theory of Jacobi elliptic functions, continued fractions, Hankel or Turanian determinants, Lie algebras, Schur functions, and multiple basic hypergeometric series related to the classical groups. His results (in Theorem 5.19) generalize to separate infinite families each of the 21 of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions in sections 40-42 of the Fundamental Nova. The author also uses a special case of his methods to give a derivation proof of the two Kac and Wakimoto (1994) conjectured identities concerning representations of a positive integer by sums of 4n2 or 4n(n+1) triangular numbers, respectively. These conjectures arose in the study of Lie algebras and have also recently been proved by Zagier using modular forms. George Andrews says in a preface of this book, `This impressive work will undoubtedly spur others both in elliptic functions and in modular forms to build on these wonderful discoveries.' Audience: This research monograph on sums of squares is distinguished by its diversity of methods and extensive bibliography. It contains both detailed proofs and numerous explicit examples of the theory. This readable work will appeal to both students and researchers in number theory, combinatorics, special functions, classical analysis, approximation theory, and mathematical physics.
