 A Proof of the Full ShimuraTaniyamaWeil Conjecture  Henri Darmon
Download an article in PDFformat on the completion of the proof by Wiles, Breuil, Conrad, Diamond and Taylor.
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 A Proof of the Pythagorean Theorem by Liu Hui (third century A.D.)  Donald B. Wagner
Web version of an article which appeared in Historia mathematica, 1985, 12, pp. 713. Includes Chinese characters, which were not in the published version. Some statements here are no longer up to date, but the article has not been significantly modified.
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 Proofs Without Words  Cut the Knot!, Alexander Bogomolny
Martin Gardner wrote, "There is no more effective aid in understanding certain algebraic identities than a good diagram. One should, of course, know how to manipulate algebraic symbols to obtain proofs, but in many cases a dull proof can be supplemented
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 Prophet of Chaos  Ivars Peterson (MathTrek)
"'...it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.'
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 The Pros and Cons of Technology in the Classroom  Roy Pea, Larry Cuban, Bay Area School Reform Collaborative, TAPPED IN
Nationally recognized experts on the subject, Dr. Roy Pea and Dr. Larry Cuban offer their insights in this transcript of speeches each gave to a gathering of organizations that fund school reform projects. A transcript of questions and answers from the
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 Providing HandsOn, MindsOn, and Authentic Learning Experiences in Mathematics (NCREL)  Cathy Cook; Pathways to School Improvement
A linked essay giving overview, background, and issues around constructivist theory and its implementation. "When educators begin to see learning as knowledge construction, they change their thinking about curriculum, instruction, and assessment, developing
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 Psychosemiotic Theory and Educational Practice  Alice Leonard
Walkerdine and Corran's (1979) paper showed how psychosemiology may account for developmental data as a potentially fruitful alternative to the better known stage theories, like that of Piaget. They argued that learning mathematics is an ideal activity
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 P.T. Chang
Articles by this math professor from the University of Alaska Anchorage's MatSu College include "The Ostentatious World of Algebra," "InService Training for the Math Teacher of the 21st Century," and "Retesting in a College Remedial Mathematics Course."
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 Publications  Douglas N. Arnold; Institute for Mathematics and its Applications
Most of Arnold's papers written since the late 1980s are available in PostScript, TeX DVI or plaintext file format. The Ph.D. theses of students he has supervised are also online. Primary research interests center around numerical analysis, with an emphasis
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 Publications  J. F. Jardine; Mathematics Dept., Univ. of Western Ontario, Canada
On this page there are: a link to a full list of publications; a book subdirectory (Simplicial Homotopy Theory, by P. G. Goerss and J. F. Jardine); and abstracts of papers: A^1local symmetric spectra; Presheaves of symmetric spectra; A rigidity theorem
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 Publications of François Morain
Links to papers on elliptic curves: Primality proving; Complex multiplication; Pseudoprimes; Elliptic curves over finite fields; Discrete logarithm over algebraic curves; Character sums; Miscellaneous. Also in French.
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 Public Understanding of Mathematics  Key Issues in Mathematics at the Math Forum
Links to organizations and articles that explore perceptions and assumptions nonmathematicians hold about mathematics and mathematicians.
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 Publishing or Programming? A Framework for the Development and Presentation of Work by Children on the World Wide Web  Judy Keiner
A paper to be presented at SIGGRAPH97, Los Angeles, August 8th 1997, involving process writing approaches, and including three criteria for evaluating Web pages. Also links to example sites.
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 Punctured Polyhedra  Ivars Peterson (MathTrek)
A tetrahedron has four triangular faces, four vertices, and six edges. When a vertex of one tetrahedron pierces the face of a second tetrahedron it forms a new, more complicated polyhedron: one triangular face has a triangular "hole" where the face was
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 Puzzling Lines  Ivars Peterson (MathTrek)
Sol LeWitt has often featured geometric and combinatorial themes in his sculptures, paintings, and drawings. In 1973 he composed "Straight Lines in Four Directions and All Their Possible Combinations," a grid of 15 squares, each inscribed with one or
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 The Puzzling World of Barry R. Clarke  Barry R. Clarke
Brain teasers from Clarke's column in The Daily Telegraph (UK) and from his various puzzle books: liar problems, logic teasers, number conundrums, matchstick puzzles, lateral mindbenders, and more. With links to other puzzles sites and an article on
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 Pythagoras Plays Ball  Ivars Peterson (MathLand)
Spring, baseball, statistics, math. In many ways, baseball is a numbers game: Strikeouts, home runs, batting averages, innings pitched, complete games, earned run averages, number of players left on base in scoring position in the bottom of the ninth
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 Pythagoras' Theorem: Elaboration on the Similarity Argument  Interactive Mathematics Miscellany and Puzzles, Alexander Bogomolny
Euclid included two proofs of the Pythagorean theorem in his Elements. A better known Proposition I.47 (Bride's chair) and VI.31 which is consistently overlooked in the introductory geometry courses. The latter,
however, renders the theorem virtually
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 Pythagoras' Theorem in Babylonian Mathematics  MacTutor Math History Archives
Linked essay examining four Babylonian tablets that all have some connection with the Pythagorean theorem. With references and other related web sites.
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 The Pythagorean Theorem and the Maya Long Count  Charles Wm. Johnson; Science in Ancient Artwork 
Various essays dealing with the possible extension of the Pythagorean Theorem to the third power in relation to the mathematical and geometrical appearance of the Maya long count fractals within progressive series of right triangles. It would appear that
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 The Pythagorean Theorem  Jim Loy
A proof of the theorem, courtesy Legendre, and other proofs and miscellaneous information.
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 Pythagorean Theorem  Over 20 Proofs  Interactive Mathematics Miscellany and Puzzles, Alexander Bogomolny
A general exposition of the Pythagorean theorem, with proofs from Euclid to Garfield  and more. Also links to other sites on the Web with proofs.
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 Quadratic, Cubic, and Quartic Equations  MacTutor Math History Archives
Linked article describing the progress made on these equations from the Babylonians through Leibniz in 1673, with another Web site and 12 references.
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 Quadratic Formula  Jim Loy
The normal derivation of the quadratic formula (completing the square), and another using a pair of factors, finding all possible factors with whole numbered terms...
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 The Quadray Papers  Kirby Urner, 4D Solutions
An Introduction to Quadrays. The game of quadrays starts with a regular tetrahedron and four rays pointing from its center to the four corners; these or any ray built from these rays are labelled 4tuples  i.e. four numbers separated by commas and enclosed
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 The Quake Project  The Department of Civil and Environmental Engineering and the School of Computer Science at Carnegie Mellon University; and the University of Southern California's Southern California Earthquake Center (SCEC)
An NSF and DARPAfunded project developing the capability for predicting, by computer simulation, the ground motion of large basins during strong earthquakes, and using this capability to study the seismic response of the Greater Los Angeles Basin. Research
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 Quantitative and Qualitative Research Supporting Prentice Hall Mathematics Programs  Pearson Education, Inc.
Download PDFs: the Research Summary of Prentice Hall Mathematics programs; Quantitative Research on 1999 Course 1, Algebra LVR, 2000 PreAlgebra Research Summary, and 1999 Middle School Math; Qualitative Research on Unique Progress and on Reading and
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 Quantitative Reasoning  Bill Briggs; Dept. of Mathematics, Univ. of Colorado at Denver
Ideas and information about teaching Quantitative Reasoning courses (also called Mathematics for Liberal Arts Students), links to many Internet sites that support QR courses, and some information about a new QR book. Includes an article by William L.
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 Quantitative Reasoning for College Graduates: A Complement to the Standards  MAA Online
What quantitative literacy requirements should be established for all students who receive a bachelor's degree? Over the years, the Mathematical Association of America has approached this question in various ways, most recently by establishing, in 1989,
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 The Quantum Age Begins  MacTutor Math History Archives
Linked essay describing the beginnings of quantum theory, from Kirchoff's blackbody radiation work in 1859 to Bohr, Einstein, and von Neumann in the 1930's, with 33 references (books/articles).
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 Quantum Games  Ivars Peterson  Science News Online
Scientists studying quantum computation offer new perspectives on coin tossing, chess, and game theory. Theorists intend to apply the peculiarities of quantum behavior to game theory, two previously unrelated fields. These researchers propose that the
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 Quantum Information Page  John Aaron Smolin; UCLA Particle Beam Physics Laboratory (PBPL)
Papers on quantum computation, cryptography, etc.; quantum information on the Net.
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 Quantum Lie Algebras  Gustav Delius
A comprehensive, continuously updated bibliography of literature on Quantum Lie Algebras. Mathematica notebooks containing calculations and animations of quantum root systems are included.
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 QuantumQuick Queries  Ivars Peterson  Science News Online
Computer scientist Lov K. Grover of AT&T Bell Laboratories has found an an algorithm that relies on quantum mechanical principles that significantly speeds up the process of identifying a particular item in an unsorted list. Whereas the best possible
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 Quantum Theory  Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to Quantum Theory, the study of solutions of the Schroedinger (differential) equation. Also includes a good deal of Lie group theory and quantum group theory, theory of distributions and topics from
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 QUdata.com  Icestone Trustee Limited
Online tools and reference library resources for data mining and knowledge discovery. Collection of free online statistical and scientific calculators, 2D and 3D function plotters.
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 Quirky Video Poker  Ivars Peterson (MathTrek)
The lure of easy money brings gullible bettors back again and again to the game of video poker  an immensely popular pastime in casinos and other gambling venues throughout the United States. Most players are bound to lose money, says Todd D. Mateer,
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 Rachel W. Hall's Research Page  Rachel Wells Hall
Rachel W. Hall in the Department of Math and Computer Science, Saint Joseph's University, Philadelphia, PA, offers links to her articles on mathematics and music in PDF format, including: Geometrical Music Theory; A Course in Multicultural Mathematics;
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 Radical Constructivism  Alex Riegler, Ed.
People, organizations, and online papers that deal with radical, epistemological, psychological and educational constructivism and second order cybernetics. Includes general introductions, online books and papers, links, conferences, and a URL submission
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 Radical Math
A site for integrating issues of social justice into the classroom. Contribute your own resources, search them by keyword, or browse by math topic, title, or types such as lesson plans, article, charts, graphs, data sets, maps, books, websites, and more.
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 RAND Education  RAND Corporation
A division of the RAND Corporation, RAND Education is a nonprofit institution that helps improve policy and decisionmaking through research and analysis. In particular, see the RAND Mathematics Study Panel (RSMP), part of its Achievement for All research
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 Random Walk on the Speiser Graph of a Riemann Surface [PDF]  Peter G. Doyle
A paper that appeared in the Bulletin of the American Mathematical society, which considers the problem of determining the conformal type of a covering surface of the Riemann sphere with punctures. To such a surface there corresponds a Speiser graph of
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 Real Functions  Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to real functions studied in calculus classes; the focus here is on their derivatives and integrals, and general inequalities. This category includes familiar functions such as rational
functions. History;
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 A Really Friendly Guide to Wavelets  C. Valens
A tutorial in wavelet transforms and their applications to signal theory and image compression, aimed at engineers. The tutorial may be read online, or downloaded as a PDF or zipped PostScript file.
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 The Real Y2K Problem (Math Chat)  Frank Morgan, MAA Online
Solution to the challenge: Every year in the millennium from 2000 to 2999 can be represented as the sum of consecutive integers, except one year. Which year cannot be represented as the sum of consecutive integers? New challenge: Where do you think you
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 Reasons for Studying (Learning and Mathematics)  Susan Bobbitt Nolen; Math Forum
In this 1988 article, "Reasons for studying: Motivational orientations and study strategies," Susan Bobbitt Nolen, of the University of Washington, argues that in order to foster meaningful learning and effective study strategies, teachers should reduce
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 Reassessing U.S. International Mathematics Performance: New Findings from the 2003 TIMSS and PISA  American Institutes for Research
"Despite a widely held belief that U.S. students do well in mathematics in grade school but decline precipitously in high school, a new study comparing the math skills of students in industrialized nations finds that U.S. students in 4th and
8th grade
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 Recent Papers in Category Theory  Prof. George F. Nassopoulos
Recent papers in Category theory and its applications to functional analysis, global analysis, and abstract differential geometry.
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 Recognizing Numerical Constants (Organic Mathematics Proceedings)  David H. Bailey, Simon Plouffe
Article and presentation. Abstract: The advent of inexpensive, highperformance computers and new efficient algorithms have made possible the automatic recognition of numerically computed constants. In other words, techniques now exist for determining,
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 Recycling Topology  Ivars Peterson (MathLand)
It's hard to miss the triangle of three bent arrows that signifies recycling. It appears in newspapers and magazines and on bottles, envelopes, cardboard cartons, and other containers. But have you noticed that there are two versions of this ubiquitous
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