 Phase State Diagrams  Harold Brochmann
Phase diagrams reveal much information about physical phenomena. Brochmann illustrates with a variety of graphs created by computer programs. Phase state diagrams of unstable orbits reveal 'strange attractors', useful in a variety of fields including
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 Philippe Flajolet
Philippe Flajolet researches the analysis of algorithms, analytic combinatorics, computer algebra, asymptotic analysis, special functions, random structures, and natural languages. An extensive collection of his articles, and the first chapters of
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 Phi: That Golden Number  Mark Freitag, Univ. of Georgia
Phi can be found in many geometric shapes as an approximation of the Golden Ratio. A geometric figure commonly associated with Phi is the Golden Rectangle, which has sides A and B that are in proportion to the Golden Ratio. It has been said that the Golden
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 PhotoMath  Martha Jean Barrett; Kodak
A project where students created their own mathematical board games for use with younger or less advanced students, for grades 48. Working in teams, advanced students covered seven math categories: whole numbers, decimals, fractions, measurement, geometry,
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 Phyllotaxis  Pau Atela, Chris Golé; Smith College
An interactive site for the mathematical study of plant pattern formation. Sections of the site include: Introduction to Phyllotaxis (classification, Fibonacci/golden angle, microscope, mathematical model, and lattices); history; glossary; references;
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 Physics (MathPages)  Kevin Brown
More than 60 "informal notes" by Kevin Brown on mathematical physics: Doppler for sound and light, optical rays in heterogeneous media, radial orbitgs in Schwarzschild geometry, the formation and growth of black holes, can Schrodinger's cat factor numbers?,
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 The Physics of Baseball  Alan Nathan
Read about the benefits (or not) of corked bats, the differences between aluminum and wood bats, the issues involved with characterizing the performance of bats from a physics point of view, and a description of the collision between a wooden bat and
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 Pi: A 2000Year Search Changes Direction  Adamchik, Wagon
A New Formula for Pi: David Bailey of NASA/Ames and Peter Borwein and Simon Plouffe of the Centre for Experimental and Computational Mathematics at Simon Fraser University (BBP) have discovered a remarkable and remarkably simple new formula for Pi. Sum
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 PianoMath Home Page: Improving Math Skills through Music  Peter Taussig, E. Paul Goldenberg
Essays and activities on links between music and mathematics. PianoMath is
a crossdisciplinary curriculum of musical classroom activities relating to
arithmetic, geometry, graphing, logic foundation, and prealgebra. It is
now being tested at an elementary
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 Pick a Digit, Any Digit  Ivars Peterson (MathTrek)
One of the most amazing mathematical results of the last few years was the discovery of a surprisingly simple formula for computing digits of the number pi. Unlike previously known methods, this one allows you to calculate isolated digits  without computing
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 Picking Winners  Ivars Peterson (MathTrek)
One of the attractive features of spectator sports is the uncertainty of the outcome. Even when one team is overwhelmingly favored to win, the underdog may still come through with a surprising victory. Nonetheless, the ability to pick winners can be of
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 Pick's Theorem  Cut the Knot!, Alexander Bogomolny
A description with proof of Pick's theorem, including links to further pages including exercises with an applet for the geoboard, Farey Series, and SternBrocot Tree. An interactive column for MAA Online that uses a Java applet to simulate a puzzle or
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 Pi  Jim Loy
An elementary introduction to pi.
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 Pinpointing Prey  Ivars Peterson (MathTrek)
The sand scorpion uses two types of sensors on the tips of its legs to detect mechanical vibrations transmitted over relatively short distances across the surface. One sensor, called the basitarsal slit sensillum, detects surface waves, or ripples. The
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 PiSearch
Search the first four billion binary digits of Pi for a string. Enter the string in hex (4 bit) or character (5 bit) format. Includes some statistics (probability of occurrence of characters, digits). Related links include articles and papers.
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 Pitfalls of Data Analysis  Clay Helberg
Subtitled "... or How to Avoid Lies and Damned Lies," this paper discusses what people "often overlook in their data analysis, and ways people sometimes 'bend the rules' of statistics to support their viewpoint." Tips to ensure the clarity and accuracy
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 Pi Through the Ages  MacTutor Math History Archives
A history of pi: the Rhind papyrus (Egypt), Ptolemy, Tsu Ch'ung Chi, al'Khwarizmi, Al'Kashi, Viet, Romanus, Van Ceulen, Gregory, Shanks, Lambert, Euler, Buffon, and many more, with other Web sites and 30 references (books/articles).
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 pLab: Random Number Generation  Peter Hellekalek; Mathematics Dept., University of Salzburg
Results and links on this fundamental topic in stochastic simulation. RNGs are deterministic algorithms that produce numbers with certain distribution properties. Roughly speaking, these numbers should behave similar to realizations of independent, identically
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 Place Value  Cut the Knot!, Alexander Bogomolny
Dictionary definitions of place value and what they lack; the emphasis on place value in the NCTM Standards; its importance in understanding our numeration system; how it arises from one of several possible representations of grouping in the counting
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 Plane Isometries  Interactive Mathematics Miscellany and Puzzles, Alexander Bogomolny
A collection of basic facts and partial explanations on plane isometries. With accompanying Java applets for translation, reflection, rotation and glide reflection.
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 Plane Patterns  Ivars Peterson (MathTrek)
Exploring the link between mathematical symmetry and artistic effort. One and twodimensional repeating patterns appear ubiquitously on surfaces, from quilts and colored fabrics to pottery and ceramic tiles... Analyzing the symmetries that underlie such
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 Planes of Budapest  Ivars Peterson (MathTrek)
Nearly every Sunday during the winter of 1933 in Budapest, a small group of students would meet somewhere in the city at a park or cafe to discuss mathematics. The gathering typically included Paul Erdös (19131996), György (George) Szekeres,
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 Planet Math
Designed to be a central repository for mathematical knowledge on the web, with a pedagogical slant. All content is usercontributed, either to the Encyclopedia or to the reviewed listings of papers, books, and lecture notes. Polls, discussions, and feedback
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 Plato: Acceptance of Astronomy as a Science  Michael Tirabassi
Translations from Plato's Dialogues (Loeb edition): The Myth of Er (a multihemispere model of the sky  Republic, X, 616B617). The True Astronomer Must be the Wisest of Men (Epinomis 990a); Pursue Astronomy as you Would Geometry (Republic 530b); Realise
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 Platonic Solids  Paul Bourke
Definitions, formulas, and illustrations of the threedimensional regular polyhedra: tetrahedron, octahedron, hexahedron (cube), icosahedron, dodecahedron, including the solids as drawn in Kepler's Mysterium Cosmographicum, and as represented in stone
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 Plato's Molecule  Ivars Peterson (MathTrek)
In his dialogue Timaeus, the Greek philosopher Plato (427347 B.C.) carefully laid out his reasoning for ascribing certain geometric shapes to the minuscule particles that constituted the four elements of matter. In his view, these elements  fire, air,
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 Playing with RuthAaron Pairs  Ivars Peterson (MathLand)
On April 8, 1974, Henry (Hank) Aaron hit his 715th major league home run, surpassing the previous mark of 714 career home runs long held by baseball great Babe Ruth. Understandably, the event received considerable coverage in newspapers and magazines
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 Plus Magazine  Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Plus (formerly PASSMaths) is a triannual online magazine for older pupils and the general public. It publishes articles explaining the diverse applications of mathematics used in solving problems in physics, chemistry, biology, engineering, and economics.
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 Poe's Secret  Ivars Peterson (MathTrek)
Writer Edgar Allan Poe (18091849) is famous for his short stories of the mysterious and the macabre. His popular tale "The GoldBug," published in 1843, is often cited as one of the best works of fiction that turn upon a secret message. Poe had a longstanding
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 Polynomials, Rational Functions  Dave Rusin; The Mathematical Atlas
A page reserved for the analytic study of polynomial functions studied in calculus classes. History; applications and related fields and subfields; textbooks, reference works, and tutorials; software and tables; selected topics; other web sites with this
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 Polytopes and Polyhedra  Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to geometric objects made from straight pieces: polygons, polyhedra, and generalizations. Applications and related fields and subfields; textbooks, reference works, and tutorials; software and tables;
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 Population Index 19861998  Office of Population Research, Princeton University
The primary reference tool to the world's population literature, presenting an annotated bibliography of recently published books, journal articles, working papers, and other material on population topics. It covers all fields of interest to demographers,
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 The Position of the Sun  Dave Rusin; The Mathematical Atlas
A computer program to calculate the position of the sun, with an article describing the calculation of the apparent sun position in local coordinates for an earth observer: notation; solution; removing the effect of one variable and simplifying the solution;
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 Potential Theory  Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to potential theory, the mathematical treatment of the potentialenergy functions used in physics to study gravitation and electromagnetism. If some electrically charged particles are distributed in
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 The Power of Innovative Scheduling  Robert Lynn Canady; Educational Leadership Journal
A November 1995 article. "Alternative schedules may not add hours to the school day, but they can vastly improve the quality of the time students spend at school." Issues facing all schools; scheduling tips for elementary, middle and high schools; references.
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 The Power of One  Robert Matthews, The New Scientist
Benford's law demands that around 30 per cent of the numbers in a given data set will start with a 1, 18 per cent with a 2, right down to just 4.6 per cent starting with a 9. This essay provides a mathematical history behind the counterintuitive law,
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 Powers of 5 and Traveling to the North Pole (Math Chat)  Frank Morgan; Christian Science Monitor
What is the remainder when you divide 5^999,000 by 7?
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 Powersums, Fermat, Waring; Quotes, History...  Nico F. Benschop
Links to papers on the ideas of Fermat, Waring, Goldbach, Cantor, Hensel, HasseMinkowski, and Selmer, and the following mathematical concepts: semigroups, sequential logic synthesis, cubic roots, and state machines and their sequential closure. With
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 Practical Foundations of Mathematics  Paul Taylor
The full text (online, in HTML) of the book (some diagrams may be missing), published by Cambridge University Press in March, 1999 as number 59 in their series Cambridge Studies in Advanced Mathematics. Practical Foundations collects the methods of construction
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 A Practical Guide to Wavelet Analysis  Christopher Torrence and Gilbert P. Compo
Read an introduction to wavelet analysis and Monte Carlo methods, or download it in PDF or PostScript format. Use an interactive wavelet plotting form to investigate sample time series or your own data. Download wavelet software in Fortran, IDL, or
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 The Practice of Logical Frameworks (1996)  Frank Pfenning
A survey article discussing the history of work on logical frameworks and their current open problems. Deductive systems, given via axioms and rules of inference, are a common conceptual tool in mathematical logic and computer science. They are used to
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 Practice Online! The Electronic Complement to INTERFACES (POL)  Institute for Operations Reseach and the Management Sciences (INFORMS)
A practiceoriented electronic information service of INFORMS, jointly sponsored by INTERFACES and the INFORMS Roundtable. Forum, submission guidelines, supplemental materials online, tables of contents and abstracts, Edelman abstracts, practiceoriented
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 Precollege science and math 'lack focus'  Janet Raloff  Science News Online
A National Science Foundation report finds no one at the helm of U.S. science and math education, with the result that students are exposed to smatterings of topics that vary regionally and encourage little indepth learning. The analysis, released in
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 Preconditioned Eigensolvers  Andrew Knyazev; Dept. of Mathematics, Univ. of Colorado at Denver
Investigations into matrixfree iterative methods for partial eigenvalue
problems that take advantage of using preconditioners to accelerate convergence. Papers, an interactive discussion, a bibliography, related conferences, and links to software.
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 Preparing for University Calculus  Mathematics and Statistics Committee of the Atlantic Provinces Council on the Sciences (APICS)
Answers frequently asked questions about university level calculus courses, such as "What math will I need in university?" and "What is calculus?" Also download the booklet, which offers sample questions and hints of the sort often found on calculus placement
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 Preparing Teachers of Mathematics: Asking the Right Questions  Lynn Arthur Steen; St. Olaf College
Lynn Steen writes that the problems of teacher preparation arise from multiple sources and require correspondingly different solutions. About mathematics teachers, prospective mathematics teachers, and curriculum issues.
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 Preservice Teachers' Conceptions of Mathematics and How to Teach It  Annie and John Selden; Mathematical Association of America
"The way teachers explain mathematics depends, to a large extent, on the conceptual grasp they acquire in their college classes. In addition, they often teach as they were taught, modeling themselves after their college mathematics teachers as much as
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 Presidential Election and the Length of the Shortest Piece (Math Chat)  Frank Morgan, MAA Online
Presidential election with 48% of the US popular vote does not come close to the extreme possibility, even with just two candidates, of winning with less than about 22% of the popular vote. See Math Chat of November 8, 1996. Answer to the Challenge: 1.
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 President's Column (MAA)  Ken Ross
MAA President Ken Ross wrote this column for Focus on the need to connect K12 and collegelevel reforms.
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 Pretty Functions  Ivars Peterson (MathTrek)
About graphing calculators, and the software Graphing Calculator, a computer program for quickly visualizing two and threedimensional mathematical objects. Graphs illustrate plotting a given function, then seeing what happens when you
modify the function
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