Gröbner Deformations Of Hypergeometric Differential Equations
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Gröbner Deformations of Hypergeometric Differential Equations by Mutsumi Saito,Bernd Sturmfels,Nobuki Takayama Pdf
The theory of Gröbner bases is a main tool for dealing with rings of differential operators. This book reexamines the concept of Gröbner bases from the point of view of geometric deformations. The algorithmic methods introduced in this book are particularly useful for studying the systems of multidimensional hypergeometric PDE's introduced by Gelfand, Kapranov, and Zelevinsky. A number of original research results are contained in the book, and many open problems are raised for future research in this rapidly growing area of computational mathematics.
Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers by Kenji Iohara,Philippe Malbos,Masa-Hiko Saito,Nobuki Takayama Pdf
This edited volume presents a fascinating collection of lecture notes focusing on differential equations from two viewpoints: formal calculus (through the theory of Gröbner bases) and geometry (via quiver theory). Gröbner bases serve as effective models for computation in algebras of various types. Although the theory of Gröbner bases was developed in the second half of the 20th century, many works on computational methods in algebra were published well before the introduction of the modern algebraic language. Since then, new algorithms have been developed and the theory itself has greatly expanded. In comparison, diagrammatic methods in representation theory are relatively new, with the quiver varieties only being introduced – with big impact – in the 1990s. Divided into two parts, the book first discusses the theory of Gröbner bases in their commutative and noncommutative contexts, with a focus on algorithmic aspects and applications of Gröbner bases to analysis on systems of partial differential equations, effective analysis on rings of differential operators, and homological algebra. It then introduces representations of quivers, quiver varieties and their applications to the moduli spaces of meromorphic connections on the complex projective line. While no particular reader background is assumed, the book is intended for graduate students in mathematics, engineering and related fields, as well as researchers and scholars.
Harmony of Gröbner Bases and the Modern Industrial Society by Takayuki Hibi Pdf
This volume consists of research papers and expository survey articles presented by the invited speakers of the conference on “Harmony of Gröbner Bases and the Modern Industrial Society”. Topics include computational commutative algebra, algebraic statistics, algorithms of D-modules and combinatorics. This volume also provides current trends on Gröbner bases and will stimulate further development of many research areas surrounding Gröbner bases. Contents:Multidegree for Bifiltered D-modules and Hypergeometric Systems (R Arcadias)Desingularization Algorithms: A Comparison from the Practical Point of View (R Blanco and A Frühbis-Krüger)Computing Localizations Iteratively (F J Castro-Jiménez and A Leykin)KNOPPIX/Math: A Live System for Mathematics (T Hamada and KNOPPIX/Math Committers)Running Markov Chain without Markov Basis (H Hara, S Aoki and A Takemura)Degree Bounds for a Minimal Markov Basis for the Three-state Toric Homogeneous Markov Chain Model (D Haws, A Martín del Campo and R Yoshida)First Steps toward the Geometry of Cophylogeny (P Huggins, M Owen and R Yoshida)Cones of Elementary Imsets and Supermodular Functions: A Review and Some New Results (T Kashimura, T Sei, A Takemura and K Tanaka)Non-vanishingness of Betti Numbers of Edge Ideals (K Kimura)Abstract Tubes Associated with Perturbed Polyhedra with Applications to Multidimensional Normal Probability Computations (S Kuriki, T Miwa and A J Hayter)An Algorithm of Computing Inhomogeneous Difference Equations for a Definite Sum (H Nakayama)Incomplete A-Hypergeometric Systems (K Nishiyama and N Takayama)Implementation of a Primary Decomposition Package (M Noro)On Computation of the Characteristic Polynomials of the Discriminantal Arrangements and the Arrangements Generated by Generic Points (Y Numata and A Takemura)A Dictionary of Gröbner Bases of Toric Ideals (H Ohsugi)Log-linear Model Estimation for Stratified Educational Data (T Otsu)Toric Statistical Models: Ising and Markov (G Pistone and M P Rogantin)Algebraic Reliability Based on Monomial Ideals: A Review (E Sáenz-de-Cabezón and H P Wynn)On Irreducibility of Algebroid Curves over the Complex Number Field (T Shibuta)Polyhedral Approach to Statistical Learning Graphical Models (M Studený, D Haws, R Hemmecke and S Lindner) Readership: Graduates and researchers in the field of Gröbner bases. Keywords:Gröbner Basis;Algebraic Statistics;D-Module;Computational Algebra;AlgorithmKey Features:Comprehensive treatment of Gröbner basesArticles by leading figures in the mathematics worldA guidebook for graduate studentsThe reader can see a panoramic view of Gröbner bases
Partitions, Hypergeometric Systems, and Dirichlet Processes in Statistics by Shuhei Mano Pdf
This book focuses on statistical inferences related to various combinatorial stochastic processes. Specifically, it discusses the intersection of three subjects that are generally studied independently of each other: partitions, hypergeometric systems, and Dirichlet processes. The Gibbs partition is a family of measures on integer partition, and several prior processes, such as the Dirichlet process, naturally appear in connection with infinite exchangeable Gibbs partitions. Examples include the distribution on a contingency table with fixed marginal sums and the conditional distribution of Gibbs partition given the length. The A-hypergeometric distribution is a class of discrete exponential families and appears as the conditional distribution of a multinomial sample from log-affine models. The normalizing constant is the A-hypergeometric polynomial, which is a solution of a system of linear differential equations of multiple variables determined by a matrix A, called A-hypergeometric system. The book presents inference methods based on the algebraic nature of the A-hypergeometric system, and introduces the holonomic gradient methods, which numerically solve holonomic systems without combinatorial enumeration, to compute the normalizing constant. Furher, it discusses Markov chain Monte Carlo and direct samplers from A-hypergeometric distribution, as well as the maximum likelihood estimation of the A-hypergeometric distribution of two-row matrix using properties of polytopes and information geometry. The topics discussed are simple problems, but the interdisciplinary approach of this book appeals to a wide audience with an interest in statistical inference on combinatorial stochastic processes, including statisticians who are developing statistical theories and methodologies, mathematicians wanting to discover applications of their theoretical results, and researchers working in various fields of data sciences.
Ams-IMS-Siam Joint Summer Research Conference on Symbolic Computation Solving Equations I,Ams-Ims-Siam Joint Summer Research Conference on Symbolic Computation
Author : Ams-IMS-Siam Joint Summer Research Conference on Symbolic Computation Solving Equations I,Ams-Ims-Siam Joint Summer Research Conference on Symbolic Computation Publisher : American Mathematical Soc. Page : 240 pages File Size : 44,9 Mb Release : 2001 Category : Mathematics ISBN : 9780821826799
Symbolic Computation by Ams-IMS-Siam Joint Summer Research Conference on Symbolic Computation Solving Equations I,Ams-Ims-Siam Joint Summer Research Conference on Symbolic Computation Pdf
This volume contains papers related to the research conference, 'Symbolic Computation: Solving Equations in Algebra, Analysis, and Engineering', held at Mount Holyoke College (MA). It provides a broad range of active research areas in symbolic computation as it applies to the solution of polynomial systems. The conference brought together pure and applied mathematicians, computer scientists, and engineers, who use symbolic computation to solve systems of equations or who develop the theoretical background and tools needed for this purpose. Within this general framework, the conference focused on several themes: systems of polynomials, systems of differential equations, non commutative systems, and applications.
Noncommutative Gröbner Bases and Filtered-Graded Transfer by Huishi Li Pdf
This self-contained monograph is the first to feature the intersection of the structure theory of noncommutative associative algebras and the algorithmic aspect of Groebner basis theory. A double filtered-graded transfer of data in using noncommutative Groebner bases leads to effective exploitation of the solutions to several structural-computational problems, e.g., an algorithmic recognition of quadric solvable polynomial algebras, computation of GK-dimension and multiplicity for modules, and elimination of variables in noncommutative setting. All topics included deal with algebras of (q-)differential operators as well as some other operator algebras, enveloping algebras of Lie algebras, typical quantum algebras, and many of their deformations.
Algebraic and Algorithmic Aspects of Differential and Integral Operators by Moulay Barkatou,Thomas Cluzeau,Georg Regensburger,Markus Rosenkranz Pdf
This book constitutes the proceedings of the 5th International Meeting on Algebraic and Algorithmic Aspects of Differential and Integral Operators, AADIOS 2012, held at the Applications of Computer Algebra Conference in Sofia, Bulgaria, on June 25-28, 2012. The total of 9 papers presented in this volume consists of 2 invited papers and 7 regular papers which were carefully reviewed and selected from 13 submissions. The topics of interest are: symbolic computation for operator algebras, factorization of differential/integral operators, linear boundary problems and green's operators, initial value problems for differential equations, symbolic integration and differential galois theory, symbolic operator calculi, algorithmic D-module theory, rota-baxter algebra, differential algebra, as well as discrete analogs and software aspects of the above.
The idea of the Gröbner basis first appeared in a 1927 paper by F. S. Macaulay, who succeeded in creating a combinatorial characterization of the Hilbert functions of homogeneous ideals of the polynomial ring. Later, the modern definition of the Gröbner basis was independently introduced by Heisuke Hironaka in 1964 and Bruno Buchberger in 1965. However, after the discovery of the notion of the Gröbner basis by Hironaka and Buchberger, it was not actively pursued for 20 years. A breakthrough was made in the mid-1980s by David Bayer and Michael Stillman, who created the Macaulay computer algebra system with the help of the Gröbner basis. Since then, rapid development on the Gröbner basis has been achieved by many researchers, including Bernd Sturmfels. This book serves as a standard bible of the Gröbner basis, for which the harmony of theory, application, and computation are indispensable. It provides all the fundamentals for graduate students to learn the ABC’s of the Gröbner basis, requiring no special knowledge to understand those basic points. Starting from the introductory performance of the Gröbner basis (Chapter 1), a trip around mathematical software follows (Chapter 2). Then comes a deep discussion of how to compute the Gröbner basis (Chapter 3). These three chapters may be regarded as the first act of a mathematical play. The second act opens with topics on algebraic statistics (Chapter 4), a fascinating research area where the Gröbner basis of a toric ideal is a fundamental tool of the Markov chain Monte Carlo method. Moreover, the Gröbner basis of a toric ideal has had a great influence on the study of convex polytopes (Chapter 5). In addition, the Gröbner basis of the ring of differential operators gives effective algorithms on holonomic functions (Chapter 6). The third act (Chapter 7) is a collection of concrete examples and problems for Chapters 4, 5 and 6 emphasizing computation by using various software systems.
Algebraic Approach to Differential Equations by D?ng Tr ng L Pdf
Mixing elementary results and advanced methods, Algebraic Approach to Differential Equations aims to accustom differential equation specialists to algebraic methods in this area of interest. It presents material from a school organized by The Abdus Salam International Centre for Theoretical Physics (ICTP), the Bibliotheca Alexandrina, and the International Centre for Pure and Applied Mathematics (CIMPA).
The book provides a self-contained account of the formal theory of general, i.e. also under- and overdetermined, systems of differential equations which in its central notion of involution combines geometric, algebraic, homological and combinatorial ideas.
Numerical and Symbolic Scientific Computing by Ulrich Langer,Peter Paule Pdf
The book presents the state of the art and results and also includes articles pointing to future developments. Most of the articles center around the theme of linear partial differential equations. Major aspects are fast solvers in elastoplasticity, symbolic analysis for boundary problems, symbolic treatment of operators, computer algebra, and finite element methods, a symbolic approach to finite difference schemes, cylindrical algebraic decomposition and local Fourier analysis, and white noise analysis for stochastic partial differential equations. Further numerical-symbolic topics range from applied and computational geometry to computer algebra methods used for total variation energy minimization.
Ring Theory And Algebraic Geometry by A. Granja,J.A. Hermida Alonso,A Verschoren Pdf
Focuses on the interaction between algebra and algebraic geometry, including high-level research papers and surveys contributed by over 40 top specialists representing more than 15 countries worldwide. Describes abelian groups and lattices, algebras and binomial ideals, cones and fans, affine and projective algebraic varieties, simplicial and cellular complexes, polytopes, and arithmetics.
This contributed volume is a follow-up to the 2013 volume of the same title, published in honor of noted Algebraist David Eisenbud's 65th birthday. It brings together the highest quality expository papers written by leaders and talented junior mathematicians in the field of Commutative Algebra. Contributions cover a very wide range of topics, including core areas in Commutative Algebra and also relations to Algebraic Geometry, Category Theory, Combinatorics, Computational Algebra, Homological Algebra, Hyperplane Arrangements, and Non-commutative Algebra. The book aims to showcase the area and aid junior mathematicians and researchers who are new to the field in broadening their background and gaining a deeper understanding of the current research in this area. Exciting developments are surveyed and many open problems are discussed with the aspiration to inspire the readers and foster further research.
Defined as solutions of linear differential or difference equations with polynomial coefficients, D-finite functions play an important role in various areas of mathematics. This book is a comprehensive introduction to the theory of these functions with a special emphasis on computer algebra algorithms for computing with them: algorithms for detecting relations from given data, for evaluating D-finite functions, for executing closure properties, for obtaining various kinds of “explicit” expressions, for factoring operators, and for definite and indefinite symbolic summation and integration are explained in detail. The book comes “with batteries included” in the sense that it requires no background in computer algebra as the relevant facts from this area are summarized in the beginning. This makes the book accessible to a wide range of readers, from mathematics students who plan to work themselves on D-finite functions to researchers who want to apply the theory to their own work. Hundreds of exercises invite the reader to apply the techniques in the book and explore further aspects of the theory on their own. Solutions to all exercises are given in the appendix. When algorithms for D-finite functions came up in the early 1990s, computer proofs were met with a certain skepticism. Fortunately, these times are over and computer algebra has become a standard tool for many mathematicians. Yet, this powerful machinery is still not as widely known as it deserves. This book helps to spread the word that certain tasks can be safely delegated to a computer algebra system, and also what the limitations of these techniques are.
