Author : Peter J. Olver
Publisher : Cambridge University Press
Page : 308 pages
File Size : 15,25 MB
Release : 1999-01-13
Category : Mathematics
ISBN : 9780521558211
The book is a self-contained introduction to the results and methods in classical invariant theory.
Author : Shigeru Mukai
Publisher : Cambridge University Press
Page : 528 pages
File Size : 31,43 MB
Release : 2003-09-08
Category : Mathematics
ISBN : 9780521809061
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Author : David M. Jackson
Publisher : Springer
Page : 0 pages
File Size : 17,93 MB
Release : 2019-05-16
Category : Mathematics
ISBN : 9783030052126
This book provides an accessible introduction to knot theory, focussing on Vassiliev invariants, quantum knot invariants constructed via representations of quantum groups, and how these two apparently distinct theories come together through the Kontsevich invariant. Consisting of four parts, the book opens with an introduction to the fundamentals of knot theory, and to knot invariants such as the Jones polynomial. The second part introduces quantum invariants of knots, working constructively from first principles towards the construction of Reshetikhin-Turaev invariants and a description of how these arise through Drinfeld and Jimbo's quantum groups. Its third part offers an introduction to Vassiliev invariants, providing a careful account of how chord diagrams and Jacobi diagrams arise in the theory, and the role that Lie algebras play. The final part of the book introduces the Konstevich invariant. This is a universal quantum invariant and a universal Vassiliev invariant, and brings together these two seemingly different families of knot invariants. The book provides a detailed account of the construction of the Jones polynomial via the quantum groups attached to sl(2), the Vassiliev weight system arising from sl(2), and how these invariants come together through the Kontsevich invariant.
Author : Tomotada Ohtsuki
Publisher : World Scientific
Page : 516 pages
File Size : 37,98 MB
Release : 2002
Category : Invariants
ISBN : 9789812811172
This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The ChernOCoSimons field theory and the WessOCoZuminoOCoWitten model are described as the physical background of the invariants. Contents: Knots and Polynomial Invariants; Braids and Representations of the Braid Groups; Operator Invariants of Tangles via Sliced Diagrams; Ribbon Hopf Algebras and Invariants of Links; Monodromy Representations of the Braid Groups Derived from the KnizhnikOCoZamolodchikov Equation; The Kontsevich Invariant; Vassiliev Invariants; Quantum Invariants of 3-Manifolds; Perturbative Invariants of Knots and 3-Manifolds; The LMO Invariant; Finite Type Invariants of Integral Homology 3-Spheres. Readership: Researchers, lecturers and graduate students in geometry, topology and mathematical physics."
Author : Daniel Zingaro
Publisher :
Page : 188 pages
File Size : 19,11 MB
Release : 2008
Category : Computers
ISBN : 9781904987833
Algorithms are central to all areas of computer science, from compiler construction to numerical analysis to artificial intelligence. Throughout your academic and professional careers, you may be required to construct new algorithms, analyze existing algorithms, or modify algorithms to suit new purposes. How do we know that such algorithms are correct? One method involves making claims about how we expect our programs to operate, and then constructing code that carries out these tasks. The key component of such reasoning is the invariant, and is the topic of this book. In these pages, you will study how invariants are developed, how they are used to construct correct algorithms, and how they are helpful in analyzing existing programs. Along the way, you'll be introduced to some classic sorting, searching and mathematical algorithms, and even some solutions to games and logic puzzles. These examples, though, are only conduits for the loftier goal: understanding why algorithms work.
Author : Roe Goodman
Publisher : Springer Science & Business Media
Page : 731 pages
File Size : 47,86 MB
Release : 2009-07-30
Category : Mathematics
ISBN : 0387798528
Symmetry is a key ingredient in many mathematical, physical, and biological theories. Using representation theory and invariant theory to analyze the symmetries that arise from group actions, and with strong emphasis on the geometry and basic theory of Lie groups and Lie algebras, Symmetry, Representations, and Invariants is a significant reworking of an earlier highly-acclaimed work by the authors. The result is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups, in a new presentation that is more accessible to students and includes a broader range of applications. The philosophy of the earlier book is retained, i.e., presenting the principal theorems of representation theory for the classical matrix groups as motivation for the general theory of reductive groups. The wealth of examples and discussion prepares the reader for the complete arguments now given in the general case. Key Features of Symmetry, Representations, and Invariants: (1) Early chapters suitable for honors undergraduate or beginning graduate courses, requiring only linear algebra, basic abstract algebra, and advanced calculus; (2) Applications to geometry (curvature tensors), topology (Jones polynomial via symmetry), and combinatorics (symmetric group and Young tableaux); (3) Self-contained chapters, appendices, comprehensive bibliography; (4) More than 350 exercises (most with detailed hints for solutions) further explore main concepts; (5) Serves as an excellent main text for a one-year course in Lie group theory; (6) Benefits physicists as well as mathematicians as a reference work.
Author : Leonard Eugene Dickson
Publisher :
Page : 122 pages
File Size : 37,12 MB
Release : 1914
Category : Forms (Mathematics)
ISBN :
Author : Peter J. Olver
Publisher : Cambridge University Press
Page : 546 pages
File Size : 49,67 MB
Release : 1995-06-30
Category : Mathematics
ISBN : 9780521478113
Drawing on a wide range of mathematical disciplines, including geometry, analysis, applied mathematics and algebra, this book presents an innovative synthesis of methods used to study problems of equivalence and symmetry which arise in a variety of mathematical fields and physical applications. Systematic and constructive methods for solving equivalence problems and calculating symmetries are developed and applied to a wide variety of mathematical systems, including differential equations, variational problems, manifolds, Riemannian metrics, polynomials and differential operators. Particular emphasis is given to the construction and classification of invariants, and to the reductions of complicated objects to simple canonical forms. This book will be a valuable resource for students and researchers in geometry, analysis, algebra, mathematical physics and other related fields.
Author : Patrick Viafore
Publisher : "O'Reilly Media, Inc."
Page : 365 pages
File Size : 39,21 MB
Release : 2021-07-12
Category : Computers
ISBN : 1098100611
Does it seem like your Python projects are getting bigger and bigger? Are you feeling the pain as your codebase expands and gets tougher to debug and maintain? Python is an easy language to learn and use, but that also means systems can quickly grow beyond comprehension. Thankfully, Python has features to help developers overcome maintainability woes. In this practical book, author Patrick Viafore shows you how to use Python's type system to the max. You'll look at user-defined types, such as classes and enums, and Python's type hinting system. You'll also learn how to make Python extensible and how to use a comprehensive testing strategy as a safety net. With these tips and techniques, you'll write clearer and more maintainable code. Learn why types are essential in modern development ecosystems Understand how type choices such as classes, dictionaries, and enums reflect specific intents Make Python extensible for the future without adding bloat Use popular Python tools to increase the safety and robustness of your codebase Evaluate current code to detect common maintainability gotchas Build a safety net around your codebase with linters and tests
Author : Mara D. Neusel
Publisher : American Mathematical Soc.
Page : 326 pages
File Size : 40,99 MB
Release : 2007
Category : Mathematics
ISBN : 0821841327
This book presents the characteristic zero invariant theory of finite groups acting linearly on polynomial algebras. The author assumes basic knowledge of groups and rings, and introduces more advanced methods from commutative algebra along the way. The theory is illustrated by numerous examples and applications to physics, engineering, numerical analysis, combinatorics, coding theory, and graph theory. A wide selection of exercises and suggestions for further reading makes the book appropriate for an advanced undergraduate or first-year graduate level course.
