Harmony Of Grobner Bases And The Modern Industrial Society

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Harmony of Gröbner Bases and the Modern Industrial Society

Author : Takayuki Hibi
Publisher : World Scientific
ISBN 13 : 9814452947
Total Pages : 388 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Harmony of Gröbner Bases and the Modern Industrial Society by : Takayuki Hibi

Download or read book Harmony of Gröbner Bases and the Modern Industrial Society written by Takayuki Hibi and published by World Scientific. This book was released on 2012-03-21 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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:GrÃ¶bner 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

Harmony of Gröbner Bases and the Modern Industrial Society

Author : Takayuki Hibi
Publisher : World Scientific
ISBN 13 : 9814383457
Total Pages : 385 pages
Book Rating : 4.8/5 (143 download)

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Book Synopsis Harmony of Gröbner Bases and the Modern Industrial Society by : Takayuki Hibi

Download or read book Harmony of Gröbner Bases and the Modern Industrial Society written by Takayuki Hibi and published by World Scientific. This book was released on 2012 with total page 385 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume consists of research papers and expository survey articles presented by the invited speakers of the conference on "Harmony of Grobner 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 Grobner bases and will stimulate further development of many research areas surrounding Gr bner bases. Contents: Polyhedral Approach to Statistical Learning Graphical Models; Implementation of a Primary Decomposition Package; Computing Tropical Resultants; Running Markov Chain Without Markov Basis; Incomplete A-Hypergeometric Systems; Degree Bounds for a Minimal Markov Basis for the Three-State Toric Homogeneous Markov Chain Model.

Statistics and Simulation

Author : Jürgen Pilz
Publisher : Springer
ISBN 13 : 3319760351
Total Pages : 417 pages
Book Rating : 4.3/5 (197 download)

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Book Synopsis Statistics and Simulation by : Jürgen Pilz

Download or read book Statistics and Simulation written by Jürgen Pilz and published by Springer. This book was released on 2018-05-17 with total page 417 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume features original contributions and invited review articles on mathematical statistics, statistical simulation and experimental design. The selected peer-reviewed contributions originate from the 8th International Workshop on Simulation held in Vienna in 2015. The book is intended for mathematical statisticians, Ph.D. students and statisticians working in medicine, engineering, pharmacy, psychology, agriculture and other related fields. The International Workshops on Simulation are devoted to statistical techniques in stochastic simulation, data collection, design of scientific experiments and studies representing broad areas of interest. The first 6 workshops took place in St. Petersburg, Russia, in 1994 – 2009 and the 7th workshop was held in Rimini, Italy, in 2013.

Algebraic Statistics

Author : Seth Sullivant
Publisher : American Mathematical Soc.
ISBN 13 : 1470435179
Total Pages : 490 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Algebraic Statistics by : Seth Sullivant

Download or read book Algebraic Statistics written by Seth Sullivant and published by American Mathematical Soc.. This book was released on 2018-11-19 with total page 490 pages. Available in PDF, EPUB and Kindle. Book excerpt: Algebraic statistics uses tools from algebraic geometry, commutative algebra, combinatorics, and their computational sides to address problems in statistics and its applications. The starting point for this connection is the observation that many statistical models are semialgebraic sets. The algebra/statistics connection is now over twenty years old, and this book presents the first broad introductory treatment of the subject. Along with background material in probability, algebra, and statistics, this book covers a range of topics in algebraic statistics including algebraic exponential families, likelihood inference, Fisher's exact test, bounds on entries of contingency tables, design of experiments, identifiability of hidden variable models, phylogenetic models, and model selection. With numerous examples, references, and over 150 exercises, this book is suitable for both classroom use and independent study.

Topics in Statistical Simulation

Author : V.B. Melas
Publisher : Springer
ISBN 13 : 1493921045
Total Pages : 531 pages
Book Rating : 4.4/5 (939 download)

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Book Synopsis Topics in Statistical Simulation by : V.B. Melas

Download or read book Topics in Statistical Simulation written by V.B. Melas and published by Springer. This book was released on 2014-12-05 with total page 531 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Department of Statistical Sciences of the University of Bologna in collaboration with the Department of Management and Engineering of the University of Padova, the Department of Statistical Modelling of Saint Petersburg State University, and INFORMS Simulation Society sponsored the Seventh Workshop on Simulation. This international conference was devoted to statistical techniques in stochastic simulation, data collection, analysis of scientific experiments, and studies representing broad areas of interest. The previous workshops took place in St. Petersburg, Russia in 1994, 1996, 1998, 2001, 2005, and 2009. The Seventh Workshop took place in the Rimini Campus of the University of Bologna, which is in Rimini’s historical center.

Monomial Algebras

Author : Rafael Villarreal
Publisher : CRC Press
ISBN 13 : 148223470X
Total Pages : 704 pages
Book Rating : 4.4/5 (822 download)

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Book Synopsis Monomial Algebras by : Rafael Villarreal

Download or read book Monomial Algebras written by Rafael Villarreal and published by CRC Press. This book was released on 2018-10-08 with total page 704 pages. Available in PDF, EPUB and Kindle. Book excerpt: Monomial Algebras, Second Edition presents algebraic, combinatorial, and computational methods for studying monomial algebras and their ideals, including Stanley–Reisner rings, monomial subrings, Ehrhart rings, and blowup algebras. It emphasizes square-free monomials and the corresponding graphs, clutters, or hypergraphs. New to the Second Edition Four new chapters that focus on the algebraic properties of blowup algebras in combinatorial optimization problems of clutters and hypergraphs Two new chapters that explore the algebraic and combinatorial properties of the edge ideal of clutters and hypergraphs Full revisions of existing chapters to provide an up-to-date account of the subject Bringing together several areas of pure and applied mathematics, this book shows how monomial algebras are related to polyhedral geometry, combinatorial optimization, and combinatorics of hypergraphs. It directly links the algebraic properties of monomial algebras to combinatorial structures (such as simplicial complexes, posets, digraphs, graphs, and clutters) and linear optimization problems.

Algebraic and Geometric Methods in Discrete Mathematics

Author : Heather A. Harrington
Publisher : American Mathematical Soc.
ISBN 13 : 1470423219
Total Pages : 277 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Algebraic and Geometric Methods in Discrete Mathematics by : Heather A. Harrington

Download or read book Algebraic and Geometric Methods in Discrete Mathematics written by Heather A. Harrington and published by American Mathematical Soc.. This book was released on 2017-03-16 with total page 277 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the AMS Special Session on Algebraic and Geometric Methods in Applied Discrete Mathematics, held on January 11, 2015, in San Antonio, Texas. The papers present connections between techniques from “pure” mathematics and various applications amenable to the analysis of discrete models, encompassing applications of combinatorics, topology, algebra, geometry, optimization, and representation theory. Papers not only present novel results, but also survey the current state of knowledge of important topics in applied discrete mathematics. Particular highlights include: a new computational framework, based on geometric combinatorics, for structure prediction from RNA sequences; a new method for approximating the optimal solution of a sum of squares problem; a survey of recent Helly-type geometric theorems; applications of representation theory to voting theory and game theory; a study of fixed points of tensors; and exponential random graph models from the perspective of algebraic statistics with applications to networks. This volume was written for those trained in areas such as algebra, topology, geometry, and combinatorics who are interested in tackling problems in fields such as biology, the social sciences, data analysis, and optimization. It may be useful not only for experts, but also for students who wish to gain an applied or interdisciplinary perspective.

Women in Commutative Algebra

Author : Claudia Miller
Publisher : Springer Nature
ISBN 13 : 3030919862
Total Pages : 439 pages
Book Rating : 4.0/5 (39 download)

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Book Synopsis Women in Commutative Algebra by : Claudia Miller

Download or read book Women in Commutative Algebra written by Claudia Miller and published by Springer Nature. This book was released on 2022-03-18 with total page 439 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume features contributions from the Women in Commutative Algebra (WICA) workshop held at the Banff International Research Station (BIRS) from October 20-25, 2019, run by the Pacific Institute of Mathematical Sciences (PIMS). The purpose of this meeting was for groups of mathematicians to work on joint research projects in the mathematical field of Commutative Algebra and continue these projects together long-distance after its close. The chapters include both direct results and surveys, with contributions from research groups and individual authors. The WICA conference was the first of its kind in the large and vibrant area of Commutative Algebra, and this volume is intended to showcase its important results and to encourage further collaboration among marginalized practitioners in the field. It will be of interest to a wide range of researchers, from PhD students to senior experts.

ISCS 2014: Interdisciplinary Symposium on Complex Systems

Author : Ali Sanayei
Publisher : Springer
ISBN 13 : 3319107593
Total Pages : 362 pages
Book Rating : 4.3/5 (191 download)

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Book Synopsis ISCS 2014: Interdisciplinary Symposium on Complex Systems by : Ali Sanayei

Download or read book ISCS 2014: Interdisciplinary Symposium on Complex Systems written by Ali Sanayei and published by Springer. This book was released on 2014-08-28 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book you hold in your hands is the outcome of the “2014 Interdisciplinary Symposium on Complex Systems” held in the historical city of Florence. The book consists of 37 chapters from 4 areas of Physical Modeling of Complex Systems, Evolutionary Computations, Complex Biological Systems and Complex Networks. All 4 parts contain contributions that give interesting point of view on complexity in different areas in science and technology. The book starts with a comprehensive overview and classification of complexity problems entitled Physics in the world of ideas: Complexity as Energy” , followed by chapters about complexity measures and physical principles, its observation, modeling and its applications, to solving various problems including real-life applications. Further chapters contain recent research about evolution, randomness and complexity, as well as complexity in biological systems and complex networks. All selected papers represent innovative ideas, philosophical overviews and state-of-the-art discussions on aspects of complexity. The book will be useful as an instructional material for senior undergraduate and entry-level graduate students in computer science, physics, applied mathematics and engineering-type work in the area of complexity. The book will also be valuable as a resource of knowledge for practitioners who want to apply complexity to solve real-life problems in their own challenging applications.

Monomial Ideals

Author : Jürgen Herzog
Publisher : Springer Science & Business Media
ISBN 13 : 0857291068
Total Pages : 311 pages
Book Rating : 4.8/5 (572 download)

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Book Synopsis Monomial Ideals by : Jürgen Herzog

Download or read book Monomial Ideals written by Jürgen Herzog and published by Springer Science & Business Media. This book was released on 2010-09-28 with total page 311 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book demonstrates current trends in research on combinatorial and computational commutative algebra with a primary emphasis on topics related to monomial ideals. Providing a useful and quick introduction to areas of research spanning these fields, Monomial Ideals is split into three parts. Part I offers a quick introduction to the modern theory of Gröbner bases as well as the detailed study of generic initial ideals. Part II supplies Hilbert functions and resolutions and some of the combinatorics related to monomial ideals including the Kruskal—Katona theorem and algebraic aspects of Alexander duality. Part III discusses combinatorial applications of monomial ideals, providing a valuable overview of some of the central trends in algebraic combinatorics. Main subjects include edge ideals of finite graphs, powers of ideals, algebraic shifting theory and an introduction to discrete polymatroids. Theory is complemented by a number of examples and exercises throughout, bringing the reader to a deeper understanding of concepts explored within the text. Self-contained and concise, this book will appeal to a wide range of readers, including PhD students on advanced courses, experienced researchers, and combinatorialists and non-specialists with a basic knowledge of commutative algebra. Since their first meeting in 1985, Juergen Herzog (Universität Duisburg-Essen, Germany) and Takayuki Hibi (Osaka University, Japan), have worked together on a number of research projects, of which recent results are presented in this monograph.

Mathematical Concepts and Methods in Modern Biology

Author : Raina Robeva
Publisher : Academic Press
ISBN 13 : 0124157939
Total Pages : 372 pages
Book Rating : 4.1/5 (241 download)

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Book Synopsis Mathematical Concepts and Methods in Modern Biology by : Raina Robeva

Download or read book Mathematical Concepts and Methods in Modern Biology written by Raina Robeva and published by Academic Press. This book was released on 2013-02-26 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical Concepts and Methods in Modern Biology offers a quantitative framework for analyzing, predicting, and modulating the behavior of complex biological systems. The book presents important mathematical concepts, methods and tools in the context of essential questions raised in modern biology. Designed around the principles of project-based learning and problem-solving, the book considers biological topics such as neuronal networks, plant population growth, metabolic pathways, and phylogenetic tree reconstruction. The mathematical modeling tools brought to bear on these topics include Boolean and ordinary differential equations, projection matrices, agent-based modeling and several algebraic approaches. Heavy computation in some of the examples is eased by the use of freely available open-source software. Features self-contained chapters with real biological research examples using freely available computational tools Spans several mathematical techniques at basic to advanced levels Offers broad perspective on the uses of algebraic geometry/polynomial algebra in molecular systems biology

Chromatic Polynomials And Chromaticity Of Graphs

Author : Fengming Dong
Publisher : World Scientific
ISBN 13 : 9814480460
Total Pages : 386 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Chromatic Polynomials And Chromaticity Of Graphs by : Fengming Dong

Download or read book Chromatic Polynomials And Chromaticity Of Graphs written by Fengming Dong and published by World Scientific. This book was released on 2005-06-23 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first book to comprehensively cover chromatic polynomials of graphs. It includes most of the known results and unsolved problems in the area of chromatic polynomials. Dividing the book into three main parts, the authors take readers from the rudiments of chromatic polynomials to more complex topics: the chromatic equivalence classes of graphs and the zeros and inequalities of chromatic polynomials. The early material is well suited to a graduate level course while the latter parts will be an invaluable resource for postgraduate students and researchers in combinatorics and graph theory.

Linear Algebra and Its Applications with R

Author : Ruriko Yoshida
Publisher : CRC Press
ISBN 13 : 1000400263
Total Pages : 290 pages
Book Rating : 4.0/5 (4 download)

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Book Synopsis Linear Algebra and Its Applications with R by : Ruriko Yoshida

Download or read book Linear Algebra and Its Applications with R written by Ruriko Yoshida and published by CRC Press. This book was released on 2021-06-23 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book developed from the need to teach a linear algebra course to students focused on data science and bioinformatics programs. These students tend not to realize the importance of linear algebra in applied sciences since traditional linear algebra courses tend to cover mathematical contexts but not the computational aspect of linear algebra or its applications to data science and bioinformatics. The author presents the topics in a traditional course yet offers lectures as well as lab exercises on simulated and empirical data sets. This textbook provides students a theoretical basis which can then be applied to the practical R and Python problems, providing the tools needed for real-world applications. Each section starts with working examples to demonstrate how tools from linear algebra can help solve problems in applied science. These exercises start from easy computations, such as computing determinants of matrices, to practical applications on simulated and empirical data sets with R so that students learn how to get started with R along with computational examples in each section and then they learn how to apply what they learn to problems in applied sciences. This book is designed from first principles to demonstrate the importance of linear algebra through working computational examples with R and python including tutorials on how to install R in the Appendix. If a student has never seen R, they can get started without any additional help. Since Python is one of the most popular languages in data science, optimization, and computer science, code supplements are available for students who feel more comfortable with Python. R is used primarily for computational examples to develop student’s practical computational skills. Table of Contents Preface List of Figures List of Tables 1. Systems of Linear Equations and Matrices 2. Matrix Arithmetic 3. Deteminants 4. Vector Spaces 5. Inner Product Space 6. Eigen values and Eigen vectors 7. Linear Regression 8. Linear Programming Network Analysis Appendices A) Introduction to RStudio via Amazon Web Service (AWS) B) Introduction to R Bibliography Index Biography Dr. Ruriko Yoshida is an Associate Professor of Operations Research at the Naval Postgraduate School. She received her Ph.D. in Mathematics from the University of California, Davis. Her research topics cover a wide variety of areas: applications of algebraic combinatorics to statistical problems such as statistical learning on non-Euclidean spaces, sensor networks, phylogenetics, and phylogenomics. She teaches courses in statistics, stochastic models, probability, and data science.

Mr Sraffa on Joint Production and Other Essays

Author : Bertram Schefold
Publisher : Routledge
ISBN 13 : 1134998945
Total Pages : 398 pages
Book Rating : 4.1/5 (349 download)

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Book Synopsis Mr Sraffa on Joint Production and Other Essays by : Bertram Schefold

Download or read book Mr Sraffa on Joint Production and Other Essays written by Bertram Schefold and published by Routledge. This book was released on 1989 with total page 398 pages. Available in PDF, EPUB and Kindle. Book excerpt: Interest in the application of the classical theory of prices, in its significance for thecritique of economic theory, and in the special aspect of joint production hasdeveloped slowly but steadily since Sraffa's Production of Commodities by Means ofCommodities (henceforth to be abbreviated as PCMC) was published in 1960. The coreof this book (Part II) consists of my PhD thesis 'Mr Sraffa On Joint Production',which was written in 1969/70 when I was first a Visitor to the Faculty of Economics,then an Advanced Student at King's College, Cambridge. At that time, almostnobody had written about the.

Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes)

Author : Sirakov Boyan
Publisher : World Scientific
ISBN 13 : 9813272899
Total Pages : 5396 pages
Book Rating : 4.8/5 (132 download)

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Book Synopsis Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes) by : Sirakov Boyan

Download or read book Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes) written by Sirakov Boyan and published by World Scientific. This book was released on 2019-02-27 with total page 5396 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.

Algebraic and Discrete Mathematical Methods for Modern Biology

Author : Raina Robeva
Publisher : Academic Press
ISBN 13 : 0128012714
Total Pages : 382 pages
Book Rating : 4.1/5 (28 download)

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Book Synopsis Algebraic and Discrete Mathematical Methods for Modern Biology by : Raina Robeva

Download or read book Algebraic and Discrete Mathematical Methods for Modern Biology written by Raina Robeva and published by Academic Press. This book was released on 2015-05-09 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: Written by experts in both mathematics and biology, Algebraic and Discrete Mathematical Methods for Modern Biology offers a bridge between math and biology, providing a framework for simulating, analyzing, predicting, and modulating the behavior of complex biological systems. Each chapter begins with a question from modern biology, followed by the description of certain mathematical methods and theory appropriate in the search of answers. Every topic provides a fast-track pathway through the problem by presenting the biological foundation, covering the relevant mathematical theory, and highlighting connections between them. Many of the projects and exercises embedded in each chapter utilize specialized software, providing students with much-needed familiarity and experience with computing applications, critical components of the "modern biology" skill set. This book is appropriate for mathematics courses such as finite mathematics, discrete structures, linear algebra, abstract/modern algebra, graph theory, probability, bioinformatics, statistics, biostatistics, and modeling, as well as for biology courses such as genetics, cell and molecular biology, biochemistry, ecology, and evolution. Examines significant questions in modern biology and their mathematical treatments Presents important mathematical concepts and tools in the context of essential biology Features material of interest to students in both mathematics and biology Presents chapters in modular format so coverage need not follow the Table of Contents Introduces projects appropriate for undergraduate research Utilizes freely accessible software for visualization, simulation, and analysis in modern biology Requires no calculus as a prerequisite Provides a complete Solutions Manual Features a companion website with supplementary resources

Computational Mathematics with SageMath

Author : P. Zimmermann
Publisher : SIAM
ISBN 13 : 161197545X
Total Pages : 459 pages
Book Rating : 4.6/5 (119 download)

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Book Synopsis Computational Mathematics with SageMath by : P. Zimmermann

Download or read book Computational Mathematics with SageMath written by P. Zimmermann and published by SIAM. This book was released on 2018-12-10 with total page 459 pages. Available in PDF, EPUB and Kindle. Book excerpt: This fantastic and deep book about how to use Sage for learning and doing mathematics at all levels perfectly complements the existing Sage documentation. It is filled with many carefully thought through examples and exercises, and great care has been taken to put computational functionality into proper mathematical context. Flip to almost any random page in this amazing book, and you will learn how to play with and visualize some beautiful part of mathematics. --- William A. Stein, CEO, SageMath, and professor of mathematics, University of Washington SageMath, or Sage for short, is an open-source mathematical software system based on the Python language and developed by an international community comprising hundreds of teachers and researchers, whose aim is to provide an alternative to the commercial products Magma, Maple, Mathematica, and MATLAB. To achieve this, Sage relies on many open-source programs, including GAP, Maxima, PARI, and various scientific libraries for Python, to which thousands of new functions have been added. Sage is freely available and is supported by all modern operating systems. Sage provides a wonderful scientific and graphical calculator for high school students, and it efficiently supports undergraduates in their computations in analysis, linear algebra, calculus, etc. For graduate students, researchers, and engineers in various mathematical specialties, Sage provides the most recent algorithms and tools, which is why several universities around the world already use Sage at the undergraduate level.