Geometric Measure Theory: A Beginner's Guide This third edition of Geometric Measure Theory A Beginner's Guide presents, forthe first time in print, the proofs of the double bubble conjecture and the http://www.uni-protokolle.de/buecher/isbn/0125068514/
Extractions: ISBN: 0125068514 Kurzbeschreibung Geometric measure theory is the mathematical framework for the study of crystal growth, clusters of soap bubbles, and similar structures involving minimization of energy. It has contributed over the past thirty years to major advances in geometry and analysis, including, for example, the original proof of the positive mass conjecture in cosmology. The subject is receiving more and more attention. This third edition of Geometric Measure Theory: A Beginner's Guide presents, for the first time in print, the proofs of the Double Bubble Conjecture and the Hexagonal Honeycomb Conjecture. Within four new chapters, readers are also led through treatments of the Weaire-Phelan counterexample to Kelvin's conjecture, Almgren's optimal isoperimetric inequality, and immiscible fluids and crystals. The abundant illustrations, examples, exercises, and solutions in this book will enhance its reputation as the most accessible introduction to the subject. Synopsis Geometric measure theory is the mathematical framework for the study of crystal growth, clusters of soap bubbles, and similar structures involving minimization of energy. It has contributed to major advances in geometry and analysis, including, for example, the original proof of the positive mass conjecture in cosmology. This third edition of "Geometric Measure Theory: A Beginner's Guide" presents the proofs of the Double Bubble Conjecture and the Hexagonal Honeycomb Conjecture. Within four new...
Note That If You See A Reference To The Heart Of Math Book, Then General double bubble conjecture in R^3 Solved, Focus The Newsletter of the MathematicalAssociation of America, May/June 2000, Volume 20, Number 5, p. 45. http://www.cs.appstate.edu/~sjg/class/1010/mathematician/mathematicianreferences
Extractions: One of Fuller's Calculations A History of Computers the history of computer speed MAD page http://www.math.buffalo.edu/mad/special/fuller_thomas_1710-1790.html African Slave and Calculating Prodigy: Bicentenary of the Death of Thomas Fuller, by Fauvel and Gerdes, Historia Mathematics 17 (1990), 141-151. The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies, Past and Present, by Steven B. Smith, 1983, On Mathematics in the History of Sub-Saharan Africa, by Paulus Gerdes, Historia Mathematica 21 (1994), 345-376, p. 345, 361-2, 366, 373. Maria Gaetana Agnesi The Living Witch of Agnesi http://www.astr.ua.edu/4000ws/witch-of-agnesi.html Why bother to learn Calculus - http://www.karlscalculus.org/why.html Definition of Calculus Women of mathematics : a biobibliographic sourcebook edited by Louise S. Grinstein p. 1-5. The Witch of Agnesi: A Lasting Contribution from the First Surviving Mathematical Work Written by a Woman - A commemoritive on the 200th anniversary of her death, by S. I. B. Gray and Tagui Malakyan, The College Mathematics Journal, Vol 30, No 4, September 1999, p. 258-268.
Dr. Joel Foisy's Home Page The trial of the semester. double bubble conjecture Proved! To see the pageof my graduate advisor, John Harer. My high school alma mater, SLC. http://www2.potsdam.edu/MATH/foisyjs/
Extractions: Students should feel welcome to drop by at any time. I am thrilled that SUNY Potsdam had REU (Research Experience for Undergraduates) for the summers in 1997, 1999-2002 and we will have another in 2003. REU information and links. Calculus Resources and links. SUNY Potsdam Mathematics Departement homepage. Links of interest to me: The best band in the region. The trial of the semester. Double Bubble Conjecture Proved! To see the page of my graduate advisor, John Harer. ... Would you like to purchase a Klein Bottle?!.
ADM Seminar for Primality Testing. Sept. 19, none, Sept. 26, none, Frank Morgan Proof of the double bubble conjecture at Furman University. Oct.3, http://www.math.clemson.edu/~gmatthe/admF02.html
Extractions: Aug. 29 Robert E. Jamison Clemson University Rank Tolerance Graph Classes Sept. 5 Gretchen L. Matthews Clemson University Semigroups of Triples of Points on Hermitian Curves Sept. 12 Kevin James Clemson University A New Algorithm for Primality Testing Sept. 19 none Sept. 26 none Frank Morgan "Proof of the Double Bubble Conjecture" at Furman University Oct. 3 Kevin James Neil Calkin Clemson University REU in Computational Number Theory and Combinatorics Oct. 10 Clemson Miniconference Oct. 17 Jirapha Limbupasiriporn Clemson University Small Sets of Even Type in Finite Projective Planes of Even Order Oct. 25 Gary Salazar* Trinity University An Examination of Affine Variety Codes with an Improved Minimum Distance Bound Nov. 1 Shuhong Gao* Clemson University Multivariate polynomial interpolation and decoding of algebraic geometric codes Nov. 7 Doug Shier Stochastic Minimum Spanning Trees Nov. 14 Wayne Goddard Computer Science,
Notes From The Lab FOURDIMENSIONAL PROOF. Yvonne Lai, a junior in mathematics, has helped extend arecent mathematical proof of the double bubble conjecture to four dimensions. http://web.mit.edu/newsoffice/tt/2000/apr26/labnotes.html
Extractions: April 26 Tech Talk Search MIT News ... MIT WEDNESDAY, APRIL 26, 2000 FOUR-DIMENSIONAL PROOF Yvonne Lai, a junior in mathematics, has helped extend a recent mathematical proof of the "double bubble conjecture" to four dimensions. In a March address to the Undergraduate Mathematics Conference at the Rose-Hulman Institute of Technology, mathematicians from Williams College, Stanford University and the University of Granada announced their proof that the familiar double soap bubble is indeed the optimal shape for enclosing and separating two chambers of air. In a postscript, a group of undergraduates from Stanford, Williams and MIT including Ms. Lai extended the theorem to four-dimensional bubbles. Working last summer at Williams, they found a way to extend the proof to 4-space and certain cases in 5-space and above. Their work was part of the Research Experiences for Undergraduates sponsored by the National Science Foundation and Williams College. The group's paper on their work is awaiting publication. METALS FOUND IN BOSTON HARBOR Caroline Tuit, a graduate student in the MIT/Woods Hole Oceanographic Institution Joint Program, is co-author of a study that reveals high levels of platinum and palladium in Boston Harbor surface sediments. The researchers say the most likely source of these metals is the use of catalytic converters in cars, as well as industrial waste entering the harbor through the sewage system.
CIM Bulletin #9 17, 2000 Science is a piece by Barry Cipra Why Double Bubbles Form the Way TheyDo, and reporting on the recent solution of the double bubble conjecture. http://at.yorku.ca/i/a/a/h/13.htm
Extractions: Topology Atlas Document # iaah-13 from CIM Bulletin #9 Ivars Peterson reports in the December 2, 2000 Science on recent progress towards the resolution of this 150-year-old conjecture. Catalan noted that 8 = 2 and 9 = 3 are consecutive integers and conjectured that they were the only set of consecutive whole powers. This translates to the Fermat-like statement that the equation xp - yq = 1 has no whole-number solutions other than 3 A new Federal encryption algorithm was reported in the October 20, 2000 Chronicle of Higher Education. The article, by Florence Olsen, relates how the Commerce Department, after a 4-year search, has declared the new federal standard for protecting sensitive information to be Rijndael, an algorithm named after its inventors Vincent Rijmen and Joan Daemen. The two Belgians beat out 20 other entries, including teams from IBM and RSA. The new encryption algorithm, of which no mathematical details were given, can be made stronger as more powerful computer processors are developed. This was an entry requirement for the competition. According to Raymond G. Kammer of NIST, which managed the selection process, it should be good for about 30 years, "that is, if quantum computing doesn't manifest itself in five or six years." Squeeze in a few more?
Archived Front Pages 21/4/2000 Onwards 28/4/00. . .The double bubble conjecture.. On February 25, 2000 itwas announced that the double bubble conjecture had been proved. http://www.madras.fife.sch.uk/maths/ArchHomePages9.html
Extractions: Welcome to our P7 visitors today! Today we see the start of a week of visits from our future S1 students. They will work through a timetable of sample lessons given by the various subject departments. At Mathematics some may even be reading this sentence at this precise moment and are about to be shown other SENTENCES created by our present S1 students. Their homework will be to create their own sentences, send them to us and if they stretch our minds into complicated enough logical knots then we will publish them here. May at the Nrich Site Try some excellent sets of problems for and . They have a deadline of 22nd May for sending in solutions. In this month's Interact Magazine our S3 students: Sam Larg, Dave Stewart, Richard Mason, Joe Neilson, Matthew Broadbent and Ross Craig have had their work published on the problem Never Prime Sue Liu in S5 had an excellent month with solutions published to By the quad - quick solve Shape and territory and Napoleon's Hat Congratulations to all these students for their excellent effort and results.
Nonius Translate this page Millennium Prize Problems Goldbach's Conjecture $1,000,000 challengedouble bubble conjecture Proved. A semana de chamada de atenção http://www.mat.uc.pt/~jaimecs/
12 It makes you think of the double bubble conjecture, doesn't it? Well, this isn'tapure double, but it is close enough to view the 120 degree meeting. http://www.allie.com/171.htm
Extractions: 12/11/2002 Will it pop? While doing my dishes today, I noticed a double bubble formation. It makes you think of the double bubble conjecture, doesn't it? Well, this isn't a pure double, but it is close enough to view the 120 degree meeting. The little bubbles of soap might affect the stability of the two larger ones. Also, it is hard to take a picture of something transparent.
EXN.ca | Discovery Now, mathematicians at Williams College have proved the double bubble conjecture. Using a relatively simple mathematical formula, they showed this is indeed http://www.exn.ca/Stories/2000/03/20/54.asp
Extractions: On Monday, a new take on the causes of global climate change emerged from the Proceedings of the National Academy of Sciences. Researchers studying ice and sedimentary core records say they've identified a new climate cycle - one that repeats every 1,800 years. They say it's related to the lunar tides. It's thought that tidal forces cause cooling at the sea surface by vertically mixing the oceans. If the idea is correct, it would mean that global warming predicted over the next few decades would be the result of a natural warming trend that began at the end of the last Ice Age. The warming trend would continue into the 32nd century. And, according to the research, it would produce warmer conditions than the Earth has seen in the past millennium. Bubbling with answers A lot toil and trouble has gone into understanding the double bubble and now there are some answers. For years, mathematicians have wondered why ordinary soap bubbles act the way they do. When two round bubbles come together, they form a double bubble, with the surface between them bowing a bit into the larger bubble. The angle between the two surfaces is always 120 degrees. The assumption has been that this is the optimum shape for the bubbles to take. Now, mathematicians at Williams College have proved the "double bubble conjecture." Using a relatively simple mathematical formula, they showed this is indeed the most efficient shape for enclosing two chambers of air. They also proved why other bubble formations don't appear in nature.
Math Coffees Frank Morgan of Williams College talk on The Proof of the double bubble conjectureand The Soap Bubble Geometry Contest at 4 pm and 730 pm, respectively, on http://www.davidson.edu/math/frontpage/Math_Coffees.htm
Extractions: Faculty Courses Programs Math Center ... Student Job Opportunities Math Coffees Bernard Review Bernard Lecture [ Math Coffees ] Problem Contest Math Coffees Math Coffees are weekly meetings of students and faculty to hear a local or visiting speaker. Check here frequently for updates! Rob McLean , DC'03 4 p.m.5 p.m., Thursday, March 20, Chambers 125 In the Euclidean plane there are only two curves of constant curvature, the line and the circle. However, in the hyperbolic plane the curves of constant curvature are more abundant. I will introduce hyperbolic geometry and summarize two methods that classify the curves of constant curvature in the hyperbolic plane. In one method I will introduce a model for hyperbolic geometry and give an argument that classifies the curves of constant curvature based solely on the structure of the model. In the other method I will introduce the notion of an orbit of a point and use a more algebraic argument to classify the curves of constant curvature. Spring 2003 Math Coffee Schedule (click on an image at the left to view the Math Coffee announcement) January 23
HRUMC'00 - Session III Details Exciting New Minimal Surfaces! Joshua White, Williams College; Thedouble bubble conjecture - Frank Morgan, Williams College. Room 301. http://www.skidmore.edu/academics/mcs/00sess3.htm
Extractions: First talk: 3:30 PM Second talk: 3:50 PM Third talk: 4:10 PM Fourth talk: 4:30 PM (An asterisk (*) indicates a level-II talk) The title of each session appears on a button. Press the button to see the abstracts of the talks from that session. All talks are in Rockefeller Hall Room: 101 Chair: Gary Krahn, United States Military Academy AN Awesome Sequence I - Michele Kelley, Colgate University AN Awesome Sequence II - Nathan Bailey, Colgate University Enumeration of Kirkman Triple Systems of Order 21 - Lee Ann Ives, University of Vermont Investigation of the Proof by N. G. de Bruijn of the Number of de Bruijn Cycles - Phillip M. LaCasse, United States Military Academy Room: 201 Chair: Phil Beaver, United States Military Academy Java Servlets for Data Access Solutions Across the Internet - Jason Miele and Paul Evans, Siena College A Web Interface for MIDI Transformations - Aileen Ang, St. Lawrence University TCP/IP Subnetting - Doug Chimento, St. Lawrence University Chaos and Complex Adaptive Systems - Mary Oliastro, United States Military Academy
GoSeekIt And MosaicFX Directory - Multilingual Directory Of The 5. Proof of the double bubble conjecture proof that the standard double bubble providesthe leastarea way to enclose and separate two regions of prescribed http://uk.dir.mosaicfx.com/serf.pl?data=9293
Page014 March 2001 The double bubble conjecture is a true statement in dimension 4 Thiswas proved by the undergraduate students Ben Reichardt, Cory Heilmann, Yvonne http://www.math.utoledo.edu/~jevard/Page014.htm
Extractions: Albany, New York June 24 - 28, 2000 The Program Saturday, June 24, Lecture Center 2 Welcome. Christopher D'Elia , Vice Present for Research, University at Albany Nat Friedman , Director, ISAMA Ivars Peterson Moebius Fantasies and Other Excursions Into Mathematical Art George Hart Sculpture Based on Propellorized Polyhedra Douglas Dunham A Unified Classification For Repeating Patterns Lunch Thom O'Connor Geometry and Light Stephanie Strickland Enumeration, Constraint, and Other Mathematical/Literary Delights Robert Krawczyk The Art of Spirolateral Reversals Eva Knoll and Simon Morgan Decomposing Deltahedra Douglas Peden Gridfield Form and Patterning Sunday, June 25, Lecture Center 2 Larry Kagan New Sculptural Form in Steel and Light, and a Speculative Meditation on the Margins of Dimensionality Robert Longhurst Poetry in Wood Thomas Sakoulas Antistrophe Lunch Edward Mayer Architectural Sculpture Charles Ginnever Rashomon Allen Linder Divisions of Space Susan Happersett Mathematical Meditations Hanna Bizek Non-Cubical Designs Made From Rubik's Cubes Monday, June 26, Lecture Center 2
Extractions: November 13, 2002 SUMMARY ABSTRACT n cities, a Voronoi diagram is generated. Voronoi edge lengths are used to identify city poor regions of the graph and place bubble centers. Cities are associated with a minimum of one bubble and a maximum of three in compliance with Plateau's laws, and bubble arcs are flattened to lines giving a graph G (V,E) with all vertices lying at the intersection of a minimum of two edges. A start city is defined and a Hamiltonian cycle traced in a clockwise direction. The algorithm correctly predicts the location of bubbles and the cities bound to each bubble. The predicted bubbles are verifiable on a physical soap bubble array as the bubble configuration needed to solve a given graph, and the algorithm attempts to extract the shortest path from the bubble configuration. The algorithm is guaranteed to always produce a Hamiltonian cycle for a subset of G in an amount of time bounded by a fourth order polynomial and the runtime of the algorithm does not appear to be influenced by the arrangement of the cities on the graph. Due to weak edge detection methods, the algorithm excludes fringe cities from the Hamiltonian cycle for graphs of large n during the tracing phase. The number of cities included in the Hamiltonian cycle is based on the arrangement of cities in the graph. The author is working to implement more robust edge detection methods capable of including fringe cities.
Extractions: Abstract: Machine interval arithmetic has become an important tool in computer assisted proofs in analysis. Usually, an interval arithmetic computation is just one of many ingredients in such a proof. The purpose of this contribution is to highlight and to summarize the role of interval arithmetic in some outstanding results obtained in computer assisted analysis. 'Outstanding' is defined through the observation that the importance of a mathematical result is at least to some extent indicated by the fact ... (Update)
Guardian | Blowing Out The Bubble Reputation Hutchins of Stanford university, and Manuel Ritori and Antonio Ros of the universityof Granada, in Spain, have, they say, proved the doublebubble conjecture. http://www.guardian.co.uk/Print/0,3858,3977398,00.html
Extractions: The Guardian Last Saturday, at a mathematics meeting in Indiana, four mathematicians announced that they had solved a long-standing conundrum about the shape of soap bubbles. Frank Morgan, of Williams college, Michael Hutchins of Stanford university, and Manuel Ritori and Antonio Ros of the university of Granada, in Spain, have, they say, proved the double-bubble conjecture. This says that the two-chambered soap bubble you can see in your bath is the one that has the least surface area compared with any other double-bubble combination enclosing the same two volumes of air. (A computer-generated picture of this double-bubble is shown in the illustration. The conjecture describes its shape precisely.) The conjecture goes back to work of the French mathematician Plateau in 1873, although it was not formulated precisely until a student at Williams College in Massachusetts - Joel Foisy - wrote an undergraduate thesis about soap bubbles in 1991. Since thin soap films adopt the shape that uses the minimum surface area to enclose a given volume of air, the question about the possible shapes of soap bubbles is of interest to both mathematicians and physicists. The first mathematician to think about the geometry of soap bubbles was Archimedes - he of the famous bath story. He claimed that the sphere was the most efficient way to enclose a given volume, but he did not have the mathematics to prove this. That had to wait until a German mathematician called Schwarz found a proof in 1884.
Bubbles For 2 areas, a standard double bubble is shortest (early 90s). Will we live longenough to see the proof of the planar triple bubble conjecture? , Time. http://www.math.uiuc.edu/~wichiram/maths/bubbles.html
Math Department Colloquia Series The double bubble Theorem, proved in 2000, says that the double soap bubble, formedwhen They proved this conjecture for the case when the image group is http://www.math.unt.edu/colloquiaarchive.htm
Extractions: Math Department Colloquia Series Archive September, 18, 2002 Nicholas Alikakos, University of North Texas Title: "The Normalized Mean Curvature Flow for a Small Bubble in a Riemannian Manifold" April 12, 2002 Frank Morgan, Williams College Title: "2000 Proof of the Double Bubble Theorem" Abstract: A round soap bubble is the most efficient, least area shape for enclosing a given volume of air. The Double Bubble Theorem, proved in 2000, says that the double soap bubble, formed when two bubbles come together, provides the least area shape for enclosing and separating two given volumes of air. Undergraduate students contributed to the proof of this theorem. June 6, 2002 Miklós Laczkovich, Eötvös Loránd University, Budapest, Hungary Title: "Squaring the Circle" Abstract: In 1925, A.Tarski asked whether or not the disc can be decomposed into finitely many pieces and the pieces can be moved to get a decomposition of a square. We outline the history and the solution of Tarskis problem, and give a review of some recent developments and of some open questions.
