General Mathematics - Abstract Algebra - Analysis - Numerical Analysis - Calculus - Differential Equations - Geometry - Graph Theory - Linear Algebra & Matrices - Number Theory - Operations Research - Physics - Statistics - Videos - Miscellaneous
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GENERAL MATHEMATICSThe Book "A=B" by Marko Petkovsek, Herbert Wilf and Doron Zeilberger."A=B" is about identities in general, and hypergeometric identities in particular, with emphasis on computer methods of discovery and proof. The book describes a number of algorithms for doing these tasks, and we intend to maintain the latest versions of the programs that carry out these algorithms on this page." Basic Concepts of Mathematics by Elias Zakon. "This book helps the student complete the transition from purely manipulative to rigorous mathematics. The clear exposition covers many topics that are assumed by later courses but are often not covered with any depth or organization: basic set theory, induction, quantifiers, functions and relations, equivalence relations, properties of the real numbers (including consequences of the completeness axiom), fields, and basic properties of n-dimensional Euclidean spaces... In particular, it gives students the skills they need to succeed in the first courses in Real Analysis and Abstract Algebra/Modern Algebra." Design of Comparative Experiments by R. A. Bailey. Math Alive by Ingrid Daubechies, Princeton University. "How is life different from 25 or even 10 years ago? Mathematics has profoundly changed our world, from banking & computers to listening to music. This course is designed for those who haven't had college mathematics but would like to understand some of the mathematical concepts behind important modern applications. It will consist of largely independent 2-week units:
You can navigate through the units using the navigation bar on the left. Each unit is divided into two parts. For each part you can download Lecture notes in PDF or PS format. Each part has its problem set and a corresponding on-line Lab. You can find the Lecture Notes, On-Line Labs, and Problem Sets through corresponding links on the left. You can also find them (for each unit) through the corresponding unit links. The On-Line Labs review some of the material seen in class, and give you interactive windows to try out various things. The problem sets contain questions and assignments for you to answer or complete. To answer some of these questions you'll need the interactive pages from the On-Line Labs. " ALGEBRAAlgebra and Analysis for Computer Science, by Jean Gallier. Book in progress, approx. 254 pages. Covers linear algebra, determinants, Gaussian elimination, LU- and Cholesky-factoring, affine geometry, polynomials, PID's, UFD's, basic topology, and differential calculus.Abstract Algebra with GAP, Authored by By J. G. Rainbolt and J. A. Gallian. "This GAP Manual accompanies Contemporary Abstract Algebra, Fifth Edition, and provides instruction and exercises for students who use the GAP software program. You may choose either to download each chapter separately or download the entire manual at once. The subroutines and art figures needed for this GAP manual are saved in separate files. For your convenience, you will find a helpful Student Solutions Manual that provides the odd-numbered answers to the exercises found in this GAP Manual." A Course In Algebraic Number
Theory by Robert B. Ash. "We study fundamental
algebraic structures, namely groups, rings, fields and modules, and
maps between these structures. The techniques are used in many areas of
mathematics, and there are applications to physics, engineering and
computer science as well. In addition, I have attempted to communicate
the intrinsic beauty of the subject. Mathematical Methods of Engineering Analysis, by Erhan Çinlar and Robert J. Vanderbei. ANALYSISAnalysis WebNotes "Analysis is one of the large divisions of modern mathematics. (Algebra, applied mathematics, topology, and discrete mathematics are some others.) Analysis is the study of limits. Anything in mathematics which has limits in it uses ideas of analysis. Some of the examples which will be important in this course are sequences, infinite series, derivatives of functions, and integrals." A Companion to Analysis by Tom Körner, Trinity Hall, Cambridge. "This book is intended for those students who might find rigorous analysis a treat. It aims to provide a foundation for later courses in functional analysis, differential geometry and measure theory." A Companion to Analysis (Answers) by Tom Körner, Trinity Hall, Cambridge. Advanced Calculus and Analysis by Ian Craw. Complex Analysis by George Cain. "This is a textbook for an introductory course in complex analysis. It has been used for our undergraduate complex analysis course here at Georgia Tech and at a few other places that I know of. " Complex Analysis by Douglas N. Arnold. Functional Analysis by Douglas N. Arnold. Graphics for Complex Analysis by Douglas N. Arnold. "This is a collection of graphical demonstrations of concepts in complex analysis which I developed for a course I gave on that subject. The most common method of visualizing a complex map is to show the image under the map of a set of curves, e.g., a set of line segments of constant real and/or imaginary part (a Cartesian grid), or a set of concentric circles and spokes (a polar grid). A weakness of this approach is that it can be difficult or impossible to infer which points of the original curves are mapped to which points of the final images. The graphics on this page use two techniques to overcome this problem. First, most are animated so that the original curves are continuously deformed into the image curves, and the eye can follow which points move where. Second, I use colors to distinguish different points and curves." Interactive Real Analysis "Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in one real variable. It deals with sets, sequences, series, continuity, differentiability, integrability (Riemann and Lebesgue), topology, and more." Numerical Analysis and MethodsNumerical
Recipes in C, The Art of Scientific Computing, 2nd Ed.,
William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian
P. Flannery, Cambridge University Press. "The scope of Numerical
Recipes is supposed to be 'everything up to, but not including, partial
differential equations.' We honor this in the breach: First, we do have
one introductory chapter on methods for partial differential equations
(Chapter 19). Second, we obviously cannot include everything else. All
the so-called “standard” topics of a numerical
analysis course
have been included in this book: linear equations (Chapter 2),
interpolation and extrapolation (Chaper 3), integration (Chaper 4),
nonlinear root-finding (Chapter 9), eigensystems (Chapter
11), and ordinary differential equations (Chapter 16). Most of these
topics have
been taken beyond their standard treatments into some advanced material
which we
have felt to be particularly important or useful. Numerical Recipes in Fortran 77, The Art of Scientific Computing, 2nd Ed (Volume 1 of Numerical Recipes in Fortran), William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, Cambridge University Press. Numerical Recipes in Fortran 90, The Art of Scientific Computing, 2nd Ed (Volume 2 of Numerical Recipes in Fortran), William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, Cambridge University Press. Numerical Computing with MATLAB, Cleve Moler (Free electronic edition published by The MathWorks): "Numerical Computing with MATLAB is a textbook for an introductory course in numerical methods, MATLAB, and technical computing. It emphasizes the informed use of mathematical software. Topics include matrix computation, interpolation and zero finding, differential equations, random numbers, and Fourier analysis. Based on MATLAB, the textbook provides more than 70 M-files. Many of the more than 200 exercises involve modifying and extending these programs. The book also makes extensive use of computer graphics, including interactive graphical expositions of numerical algorithms." CALCULUSCalculus
without Limits (Lecture Notes for Applied Calculus) by
Karl Heinz Dovermann, (302 pp). "We introduce diffrentiability as a
local property without using limits. The philosophy behind this idea is
that limits are the a big stumbling A Summary of Calculus by Karl Heinz Dovermann, (164 pp). The Calculus of Functions of Several Variables by Dan Sloughter. Difference Equations to Differential Equations: An Introduction to Calculus by Dan Sloughter. Graphics for the Calculus by Douglas N. Arnold. "These are excerpts from a collection of graphical demonstrations I developed for first year calculus." Multivariable Calculus by George Cain & James Herod. "This is a textbook for a course in multivariable calculus. It has been used for the past few years here at Georgia Tech. The notes are available as Adobe Acrobat documents." Steven Shreve: Stochastic Calculus and Finance prepared by Prasad Chalasani and Somesh Jha, CMU. Visual Calculus by Lawrence S. Husch. "Originally, this collection was designed for instructors to give some ideas how technology, in particular, computers, can be used in the teaching of calculus. Detailed instructions on implementing these ideas with various public domain, shareware and commercial software packages are provided. The collection has been expanded to include tutorials, interactive modules (LiveMath, Java, and Javascript) which can be used by either students or faculty and includes detailed instructions for TI-85 and TI-86 graphing calculators. As an additional aid for students, modules containing quizzes and drill problems have also been added. " GEOMETRYClassical Geometry by Danny Calegari. "An introduction to spherical, Euclidean and hyperbolic geometry in two and three dimensions, with an emphasis on the similarities and differences between these flavors of geometry. The most important tool in analyzing these geometries will be a study of their symmetries; we will see how this leads naturally to basic notions in group theory and topology. Topics to be covered might include classical tessellations, the Gauss-Bonnet theorem, scissors congruence, orbifolds, and fibered geometries." Classical Geometry and Low-Dimensional Topology by Danny Calegari. "A continuation of the study of spherical, Euclidean and especially hyperbolic geometry in two and three dimensions begun in Mathematics 138. The emphasis will be on the relationship with topology, and the existence of metrics of constant curvature on a vast class of two and three dimensional manifolds. We will concentrate mainly on a detailed study of examples, and we will try to be as explicit and as elementary as possible. Topics to be covered might include: uniformization for surfaces, shapes and volumes of hyperbolic polyhedra, circle packing and Andreev's theorem, and hyperbolic structures on knot complements." GRAPH THEORYGraph Theory, the complete electronic version of Reinhard Diestel's book on graph theory. "Here is a complete electronic edition of the book, which you are welcome to view or download to your computer for offline use. It comes as a hyperlinked pdf file." Introduction
to Social Network Methods by Robert A.
Hanneman. "An on-line textbook supporting Sociology 157, an
undergraduate introductory course on social network analysis. Robert A.
Hanneman of the Department of Sociology teaches the course at the
University of California, Riverside. Social Network Analysis by Steve Borgatti, Dept. of Organization Studies, Boston College. "There has been a dramatic rise in the use of social network analysis over the last decade. The availability of standard texts and robust software has undoubtedly contributed to this increase. Social network analysis focuses on the relationships between actors and acknowledges that an individual's behaviour is influenced by those around them. Actors and their actions are viewed as interdependent rather than independent units. This view means that the unit of analysis is not the individual, but an entity consisting of the individuals and the linkages connecting them. This will be a practical course focused around the collection and analysis of network data. The course is centred on the software packages UCINET and NetDraw. We shall look at the description and visualisation of network data and consider issues of validity and representation. We will then focus on uncovering structural properties of individual actors and the detection and description of groups. Finally we will consider how to test network hypothesis." LINEAR ALGEBRAIntroduction to Matrix Algebra by Autar K Kaw, Professor, University of South Florida. "This book is written primarily for students who are at freshman level or do not take a full course in Linear/Matrix Algebra, or wanting a contemporary and applied approach to Matrix Algebra. I am making this book available FREE of charge at this time. I would like your comments if you use this book so that future versions would be made better and expanded. " Elements of Abstract and Linear Algebra by Edwin H. Connell. "This is a foundational textbook on abstract algebra with emphasis on linear algebra. You may download parts of the book or the entire textbook. Elementary Linear Algebra. by Keith Matthews. "This book is an introduction to linear algebra, based on lectures given by me over 17 years, in the (now defunct) first year course MP103 at the University of Queensland. The section on subspaces is meant to be a gentle introduction to the second course, where abstract vector spaces are met in detail. Things of substance are met here, including the rank of a matrix. The section on three dimensional geometry makes use of the earlier sections on linear equations, matrices and determinants and some of the proofs are more algebraic (even pedantic) than some readers would like." Solutions to Elementary Linear Algebra prepared by Keith Matthews. Linear Algebra and Applications Textbook by Thomas S. Shores. "In order to enable prospective users to preview my text easily and conveniently, I'm putting a copy of it on the web for your perusal. Why this text? I'm committed to a balanced blend of theory, application and computation. Mathematicians are beginning to see their discipline as more of an experimental science, with computer software as the "laboratory" for mathematical experimentation. I believe that the teaching of linear algebra should incorporate this new perspective." Visual Linear Algebra for Maple and Mathematica: "Visual Linear Algebra is a new kind of textbook. With its thorough integration of technology, it offers a unique environment for teaching and learning linear algebra. Its goals, however remain quite traditional. Foremost among these it to provide a rich context in which students can achieve robust understanding of the core topics of linear algebra and real competence in using them. Students who work in this dynamic environment, we have found, become quite actively engaged in learning linear algebra." Ordinary Differential Equations (ODE)Differential Equations With Boundary Value Problems: Prentice Hall Companion Web site. Differential Equations With Boundary Value Problems: by the author Selwyn Hollis. Partial Differential Equations (PDE)Elementary PDEs and Applications: by Björn Birnir, UCSB. "This book is designed for students who will eventually be solving partial differential equations (PDEs) numerically. The aim is to teach them enough appreciation for the properties of the solutions to be able to judge with confidence, whether their numerical solutions make sense and how to interpret them. The examples, applications and exercises included in this book should give students a good intuition for the distinct qualities of waves, heat diffusion and fields." NUMBER THEORYElementary Number Theory, by W. Edwin Clark: "At first blush one might think that of all areas of mathematics certainly arithmetic should be the simplest, but it is a surprisingly deep subject. We assume that students have some familiarity with basic set theory, and calculus. But very little of this nature will be needed. To a great extent the book is self-contained. It requires only a certain amount of mathematical maturity. And, hopefully, the student’s level of mathematical maturity will increase as the course progresses. Before the course is over students will be introduced to the symbolic programming language Maple which is an excellent tool for exploring number theoretic questions." An Introduction to the Theory of Numbers, by Leo Moser: "This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties, greatest common divisor, etc.), is an expanded version of a series of lectures for graduate student on elementary number theory. Topics include: Compositions and Partitions; Arithmetic Functions; Distribution of Primes; Irrational Numbers; Congruences; Diophantine Equations; Combinatorial Number Theory; and Geometry of Numbers. Three sections of problems (which include exercises as well as unsolved problems) complete the test." OPERATIONS RESEARCHLinear Programming: Foundations and Extensions by Robert Vanderbei, Princeton University. "This book is about constrained optimization. It begins with a thorough treatment of linear programming and proceeds to convex analysis, network flows, integer programming, quadratic programming, and convex optimization. Along the way, dynamic programming and the linear complementarity problem are touched on as well. The book aims to be a first introduction to the subject. Specific examples and concrete algorithms precede more abstract topics. Nevertheless, topics covered are developed in some depth, a large number of numerical examples are worked out in detail, and many recent topics are included, most notably interior-point methods. The exercises at the end of each chapter both illustrate the theory and, in some cases, extend it." PHYSICSEssential Physics 1, by Frank W. K. Firk, Yale University. "Essential Physics 1 is an intensive introduction to classical and special relativity, Newtonian dynamics and gravitation, Einsteinian dynamics and gravitation, and wave motion. Mathematical methods are discussed, as needed; they include: elements of differential geometry, linear operators and matrices, ordinary differential equations, calculus of variations, orthogonal functions and Fourier series, and non-linear equations for chaotic systems. The contents of this book can be taught in one semester. It is a book for first-year college students who have an interest in pursuing a career in Physics or a closely related field." Introduction to Groups, Invariants & Particles, by Frank W. K. Firk, Yale University. "Introduction to Groups, Invariants & Particles is a book for Seniors and advanced Juniors who are majoring in the Physical Sciences or Mathematics. The book places the subject matter in its historical context with discussions of Galois groups, algebraic invariants, Lie groups and differential equations, presented at a level that is not the standard fare for students majoring in the Physical Sciences. A sound mathematical basis is thereby provided for the study of special unitary groups and their applications to Particle Physics." The Age of Einstein, by Frank W. K. Firk, Yale University. "The Age of Einstein is a brief introduction to Einstein's Theories of Special and General Relativity. It is a book for the inquisitive general reader who wishes to gain an understanding of the key ideas put forward by the greatest scientist of the 20th-century. No more than a modest grasp of High School Mathematics is required to follow the arguments." Motion Mountain: A hike beyond space and time along the concepts of modern physics, by Christoph Schiller. "How does one empty a bottle as rapidly as possible? How does one connect water pipes to a turning wheel? What are the dangers of a can of beans? What is the one single page of unsolved problems in fundamental physics? Is the universe a set? This physics textbook with 1000 pages, on the undergraduate level, is written to be entertaining, surprising and challenging on every page. It provides a structured introduction to classical physics, relativity, quantum theory, and the present unification attempts. At the same time it provides the best known physical quizzes and physical curiosities. Over 500 solved challenges, over 200 figures and over 70 tables are included. " Classical Electrodynamics, by Bo Thidé. "Intended for the advanced undergraduate or graduate student, Electromagnetic Field Theory is a textbook on the theory of electrodynamics, at roughly the same level as the well-known textbooks by Jackson and Panofsky&Phillips. The book is written mainly from a classical field theoretical point of view, emphasising fundamental and subtle properties of the EM field and includes a comprehensive appendix on the mathematical methods used. It treats relativistic covariance and the Lagrangian/Hamiltionan formulation of electromagnetic field theory, with an eye on modern ideas of duality and unification of theories, and includes a rigorous, comprehensive and detailed treatment of EM radiation phenomena. " Light and Matter series: Introductory physics textbooks. "These books are meant for a one-year algebra- or calculus-based course. Applications of calculus are treated in optional sections at the end of certain chapters.
Discover Physics. "A conceptual physics textbook intended for students in a nonmathematical one-semester general-education course." Simple Nature. "A physics textbook intended for students in a three-semester introductory calculus-based course." Fields (1999, 2nd edition 2002), by Warren Siegel. "The first free comprehensive textbook on quantum (and classical) field theory. The approach is pragmatic, rather than traditional or artistic. It covers many recent topics at an introductory yet nontrivial level, such as supersymmetry, general relativity, supergravity, and strings. It introduces many topics not appearing in other textbooks, including: 1/N expansion (color ordering) in QCD, including relation to random worldsheets spacecone (spinor helicity), including explicit calculations of 4- and 5-point S-matrices in Yang-Mills many useful gauges, such as Gervais-Neveu, Nielsen-Kallosh, unitary lightcone, and even string gauges in gravity finite N=1 supersymmetric theories " Introduction to String Field Theory, by Warren Siegel. "Includes an introduction to string theory in general, as well as relativistic first-quantization and other topics prerequisite for string theory." Superspace: One Thousand and One Lessons in Supersymmetry, by Warren Siegel. Jim Gates, Marc Grisaru, and Martin Rocek."The first and best comprehensive textbook on supersymmetry, by some of the developers of superspace methods." STATISTICSElectronic Statistics Textbook:
"This Electronic Statistics Textbook offers training in the
understanding and application of statistics. The material was developed
at the StatSoft R&D department based on many years of teaching
undergraduate and graduate statistics courses and covers a wide variety
of applications, including laboratory research (biomedical,
agricultural, etc.), business statistics and forecasting, social
science statistics and survey research, data mining, engineering and
quality control applications, and many others. VIDEOSDovermann's Derive Videos: Over 200 MB of free video files (.avi) about the computer algebra system (CAS) Derive. The videos can be viewed online or downloaded on your hard drive. MISCELLANEOUSFibonacci Numbers and the Golden Section.
"This is the Home page for Ron Knott's Surrey University multimedia web
site on the Fibonacci numbers, the Golden section and the Golden
string. There is a large amount of information at this site (more than
200 pages if it was printed), so if all you want is a quick
introduction then the first link takes you to an introductory page on
the Fibonacci numbers and where they appear in Nature. The rest of this
page is a brief introduction to all the web pages at this site on Teaching with Maple: "These articles present and summarize some of the issues involved in using mathematical software - like Maple - for teaching purposes." The MacTutor History of Mathematics archive:
by the School of Mathematics and Statistics OPEN SOURCE & FREE SOFTWAREAxiom "Axiom is a general purpose Computer Algebra system. It is useful for doing mathematics by computer and for research and development of mathematical algorithms. It defines a strongly typed, mathematically correct type hierarchy. It has a programming language and a built-in compiler." CMAT:
"CMAT is a matrix calculator program, written in C. GAP: "GAP (Groups, Algorithms and Programming) is a system for computational discrete algebra with particular emphasis on, but not restricted to computational group theory. GAP was developed at Lehrstuhl D für Mathematik (LDFM), RWTH Aachen, Germany from 1986 to 1997. After the retirement of J. Neubüser from the chair of LDFM, the development and maintenance of GAP is coordinated by the School of Mathematical and Computational Sciences at the University of St. Andrews, Scotland. Several users have contributed to the system via share packages which can be used in the same form as the main library." KeyPlayer: " KeyPlayer is a program for identifying an optimal set of nodes in a network for one of two basic purposes: (a) crippling the network by removing key nodes, and (b) selecting which nodes to either keep under surveillance or to try to influence via some kind of intervention. The two purposes are different and require different procedures. KeyPlayer provides two approaches for the first goal, and one approach for the second." KrackPlot: "KrackPlot is a program for network visualization designed for social network analysts. It runs on Dos systems, but there is now an experimental web interface using forms." Maxima: "Maxima is a system for the manipulation of symbolic and numerical expressions, including differentiation, integration, Taylor series, Laplace transforms, ordinary differential equations, systems of linear equations, and vectors, matrices, and tensors. Maxima produces high precision results by using exact fractions and arbitrarily long floating point representations, and can plot functions and data in two and three dimensions." NetDraw: "NetDraw is a program for drawing networks. It uses (or will use) several different algorithms for laying out nodes in 2-dimensional space (3D will come later). Netdraw reads UCINET system files, UCINET DL text files, and Pajek text files (.net, .clu and .vec). It can save data to Pajek and to Mage. It can save diagrams as EMF, WMF, BMP and JPG files. It can also print directly from the program at high resolution (much better than printing document containing embedded graphics)." Octave: "GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with Matlab. It may also be used as a batch-oriented language. Octave has extensive tools for solving common numerical linear algebra problems, finding the roots of nonlinear equations, integrating ordinary functions, manipulating polynomials, and integrating ordinary differential and differential-algebraic equations. It is easily extensible and customizable via user-defined functions written in Octave's own language, or using dynamically loaded modules written in C++, C, Fortran, or other languages." Yacas: "Yacas is a general purpose easy to use Computer Algebra System (a CAS is a program that can be used to do symbolic manipulation of mathematical expressions). It is built on top of its own programming language designed for this purpose, in which new algorithms can easily be implemented. In addition, it comes with extensive documentation on the functionality implemented and methods used to implement them." COMMERCIAL SOFTWAREDerive: "For school and university level mathematics, Derive 6 offers an easy to use, powerful CAS teaching tool with maximum computational power and 3-D graphing capability at an affordable price. As a teaching tool, it provides the freedom and ease-of-use necessary to explore and document different approaches to solving problems. As a research tool, it provides an efficient and reliable environment for symbolically solving a wide range of mathematical problems. Derive can solve both symbolic and numeric problems, and then plot the results as 2D graphs or 3D surfaces." Limpdep: "An integrated program for estimation and analysis of linear and nonlinear models, with cross section, time series and panel data. The main feature of the package is a suite of more than 100 built-in estimators for all forms of the linear regression model, and stochastic frontier, discrete choice and limited dependent variable models, including models for binary, censored, truncated, survival, count, discrete and continuous variables and a variety of sample selection models. No other program offers a wider range of single and multiple equation linear and nonlinear models." Maple: "Maple is a complete mathematical tool for professors, industry professionals, and students. Maple lets you explore and visualize mathematical concepts, develop technical applications, and share information with the Web, Microsoft® Excel, MATLAB® and your programs. You can create professional reports, presentations, and interactive technical documents for teaching using its unique worksheet environment." Mathematica: "From simple calculator operations to large-scale programming and interactive document preparation, Mathematica is the tool of choice at the frontiers of scientific research, in engineering analysis and modeling, in technical education from high school to graduate school, and wherever quantitative methods are used." Matlab: "MATLAB and companion toolboxes provide engineers, scientists, mathematicians, and educators with an environment for technical computing applications. These products serve a broad range of tasks across a variety of industries from automotive and electronics to industrial equipment and telecommunications." UCINET: "UCINET is a comprehensive program for the analysis of social networks and other proximity data. The program contains dozens of network analytic routines (e.g., centrality measures, dyadic cohesion measures, positional analysis algorithms, clique finders, etc.), stochastic dyad models (P1), network hypothesis testing procedures (including QAP matrix correlation/regression and categorical and continuous attribute autocorrelation tests), plus general statistical and multivariate analysis tools such as multidimensional scaling, correspondence analysis, factor analysis, cluster analysis, multiple regression, etc. In addition, UCINET provides a host of data management and transformation tools ranging from graph-theoretic procedures to a full-featured matrix algebra language." |
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