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Topic: Dense mathematics

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	Dense graph - Wikipedia, the free encyclopedia
		In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges.
		The distinction between sparse and dense graphs is rather vague.
		One possibility is to choose a number k with 1 < k < 2 and to define sparse graph to be a graph with
	en.wikipedia.org /wiki/Dense_graph (205 words)

	Dense set - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be "well-approximated" by points in A.
		Formally, A is dense in X if for any point x in X, any neighborhood of x contains at least a point from A.
		An alternative definition in case of the metric spaces is the following: A set A in a metric space X is dense if every x in X is a limit of a sequence of elements in A.
	en.wikipedia.org /wiki/Dense_set (206 words)

	Dense
		As an example, the set of rational numbers is a dense subset of the real numbers.
		Note that the first notion of "dense" depends on the surrounding space, while the second notion is completely internal to the ordered set.
		The rationals in [0,1] for instance are dense as an ordered set and they are dense in the space [0,1] but they are not dense in the real numbers.
	www.ebroadcast.com.au /lookup/encyclopedia/de/Dense.html (198 words)

	Density (disambiguation) - Biocrawler (Site not responding. Last check: 2007-10-31)
		A partial order on a set S is said to be dense (or dense-in-itself) if, for all x and y in S for which x < y, there is a z in S such that x < z < y.
		In case the order is a linear order, then B is dense in A in the present sense if and only if B is dense in the order topology on A.
		In forcing, a subset D of a forcing notion (P, ≤) is called dense in P if for any p in P there is d in D with d≤p.
	www.biocrawler.com /encyclopedia/Dense (451 words)

	Dense And Nowhere Dense in ZhurnalWiki
		Rational numbers are "dense" because no matter how tiny a neighborhood you pick around any number, there are still an infinite number of fractions that lie in that neighborhood.
		Simultaneously, the rationals are "nowhere dense" because however tiny a zone you pick around a rational number, there are infinitely many non-rational numbers (irrationals, numbers not expressible as fractions) in that zone too.
		A set E is said to be dense in a set X if the closure of E is X. For example, the set of rational numbers is dense in the set of real numbers.
	zhurnal.net /ww/zw?DenseAndNowhereDense (435 words)

	Math Forum - Searching for New Mathematics
		To most outsiders, modern mathematics represents unknown territory; its borders are protected by dense thickets of technical terms, its landscapes strewn with indecipherable equations and impenetrable concepts.
		And if mathematics is more than just a private game, if it is a consumer good, then mathematicians must take some responsibility for communicating their ideas in ways that convey the meaning of their work to broader audiences.
		It is encouraging to an outsider today that mathematics as a whole seems to show a renewed emphasis on applications, a return to concrete images, and an increasing and more explicit role for mathematical experiments.
	mathforum.org /social/articles/ivars.html (2743 words)

	Patchen Barss - John A. Adam: Mathematics in Nature
		Furthermore, one advantage of such a math-filled book (I belabour the point because so many books purported to be about mathematics for a lay audience practically trumpet the dearth of actual math contained within) is the precision with which it describes the natural world.
		Mathematical modelling, the basis of this book, is a curious field.
		This book is too advanced to serve as an introduction to the mathematics of nature, and it is by no means the last word on the subject.
	www3.sympatico.ca /patchen/adam.html (848 words)

	sciforums.com - New Article: The Relativity Program
		This article MATHEMATICALLY proves that the physical basis of the Special Theory of Relativity is contradicted by Einstein's own mathematics, thus that the THEORY of relativity is proved false by the very same experiments that confirm the validity of Lorentz's equations themselves.
		The point is to use the mathematics to open it up for a more realistic, causal and visualizable theory such as Sorce Theory at www.anpheon.org.
		The pattern is merely represented, in the standard model, as a limited, abstract, mathematical algorithm from the 18th century known as Bode’s Law.
	www.sciforums.com /showthread.php?t=36036 (2039 words)

	Physics of warm dense matter in the high energy density regime
		In a recent report, the US national academy of sciences identified the so called warm dense matter regime as one of their key areas for development in the 21st century (Frontiers in High Energy Density Physics: The X-Games of Contemporary Science, National Academies Press2003).
		The use of X-rays to determine the structure of crystals is well known and it is well understood that the long range order in a crystal leads to sharp diffraction features.
		Likewise, by looking at the spectral content of the scattered X-rays, we can look at the dynamical structure, that is to say the fluctuations that occur on very short timescales within the microscopic structure of the sample.
	www.qub.ac.uk /schools/SchoolofMathematicsandPhysics/PostgraduateOpportunities/SelectionProcedures/ListofPhDProjectsfor2006/Physicsofwarmdensematterinthehighenergydensityregime (628 words)

	Chris Fenwick's mathematics education (self-organised reading-to-learn) research and teaching page
		When reading a mathematics text students are then confronted by a type of discourse which might be seen to be highly rarefied and as a result difficult to understand and learn from.
		Beyond the terms/phrases themselves, the text as a whole may also be described as dense in the sense that their is a high concentration of these within the organisation of sentences, paragraphs, sections, etc...
		Consequently, their are very few, if any, terms of the texts which may be ignored or deleted, as well as very few, if any, terms of the text which may be replaced (in the sense of synonyms) by other terms or phrases carrying precisely the same meaning in precisely the same context.
	www.ucl.ac.uk /~uczlcfe/documents/research/writings/RtoL_intro_p3.html (517 words)

	Elegant Mathematics
		Elegant Mathematics Ltd can solve a linear system with this matrix within 5 hours on AMD 2.0 Optheron with only 16 Gb main memory.
		Elegant Mathematics Ltd can compute the solution of this linear system with 125,000,000 unknowns within 3 hours on AMD 2.0 Optheron with only 16 Gb main memory.
		Dense and Sparse data sets, the input could be the data stored in a file, or the channel that provides data upon the request.
	www.elegant-mathematics.com (924 words)

	Caltech - Applied & Computational Mathematics
		The aim is to cover the interactions existing between applied mathematics, namely applied and computational harmonic analysis, approximation theory, etc., and statistics and signal processing.
		The mathematical theory of ordinary and partial differential equations, including a discussion of elliptic regularity, maximal principles, solubility of equations.
		The aim of the course is to develop the basic mathematical tools, and to demonstrate these through concrete applications in engineering, finance and physics.
	www.acm.caltech.edu /courses.shtml (1113 words)

	Bank Street Corner: Mathematics (Site not responding. Last check: 2007-10-31)
		One of the central convictions of the mathematics reform movement is that children be introduced to the full range of mathematics (which of course includes arithmetic) starting at kindergarten.
		In one sense, it is reasonable to say that informal mathematics lives in a child’s stomach, whereas formal mathematics resides in the brain.
		Constructivist learning, where children make active sense of the formal mathematics they are adding to their repertoires, decreases the stress and increases the depth of thought, the joy, and the creativity in mathematics study.
	www.bankstreetcorner.com /develop_practices.shtml (1274 words)

	A Diagnostic Analysis of a Proportional Reasoning Test Item
		In that study, eighth graders from 20 countries were administered a variety of mathematics tests.
		Cognitive diagnosis is necessary for the prescription of effective instruction in mathematics (Glaser, 198 1; Linn, 1986; Bart and Orton, 1991).
		Continued progress in cognitive diagnostic testing in mathematics education would be an important measure in overall plans to correct the mediocre status of typical student achievement in mathematics.
	education.umn.edu /rationalnumberproject/94_1.html (4275 words)

	Read This: Briefly Noted, February 2005
		It is dense with mathematics, but there is also motivation and discussion.
		Annalisa Crannell's primary research is in topological dynamical systems, but she is also active in developing curricular materials for courses on "Mathematics and Art" as well as materials for writing across the curriculum.
		July, 2003: mathematical circles, the Scottish Café, and the zeta function.
	www.maa.org /reviews/brief_feb05.html (2168 words)

	Computing Sciences: Applied Mathematics
		Mathematics Department member Alexandre Chorin, also a University Professor at the University of California, was awarded the Norbert Wiener Prize in Applied Mathematics for his seminal work in computational fluid dynamics, statistical mechanics, and turbulence, as well as the National Academy of Sciences award in applied mathematics and numerical analysis.
		Mathematics department member Grigory Barenblatt is a winner of the Lagrange Medal, the Modesto Panetti Prize and Medal, the G. Taylor Medal, the J. Maxwell Medal and Prize, and the Timoshenko Medal.
		Applied mathematics research is also carried out by the Scientific Computing Group, which researches numerical linear algebra techniques such as dense and sparse linear and eigen system solvers.
	www.lbl.gov /cs/html/applied-math.html (782 words)

	Poincare Conjecture and Grigori Perelman
		Those around them might be saying that studying mathematics could lead to a career as an accountant (which is an important profession) or perhaps to becoming a teacher (an even more important job), but these opportunities are not necessarily going to inspire every mathematically inclined teenager.
		John Ball, president of the International Mathematical Union, said that he had urged Perelman to accept the medal, but Perelman said he felt isolated from the mathematics community and "does not want to be seen as its figurehead." Ball offered no further details of the conversation.
		His refusal won as much public attention as his groundbreaking mathematical work, possibly because the refusal is easier to describe, although it is not necessarily easier for the media to understand.
	agutie.homestead.com /files/world_news_map/poincare_conjecture.htm (4333 words)

	Mathematics Magazine: October 2003
		Practically every student of mathematics is acquainted with the arithmetic and geometric means of two given positive numbers, a and b.
		This mathematics allows us to see that a uniform distribution of numbers in the DLA simulation models this process more accurately than a normal distribution.
		The set of rational numbers that are abundancy indices is dense in the closed interval from one to infinity.
	www.maa.org /pubs/mag_oct03_toc.html (950 words)

	Topological Preliminaries
		Topology is one of (quite a few) mathematical theories that permeate other branches of Mathematics connecting them into one coherent whole.
		A set is nowhere dense if its closure has no internal points.
		The set Q of all rational numbers is dense in R, thick, neither open nor closed, and without internal points.
	www.cut-the-knot.org /do_you_know/topology.shtml (759 words)

	Amazon.com: Encyclopedia of Mathematics Education: Books: L. Grinstein (Site not responding. Last check: 2007-10-31)
		This single-volume reference is designed for readers and researchers investigating national and international aspects of mathematics education at the elementary, secondary, and post-secondary levels.
		The scope is comprehensive, encompassing all major areas of mathematics education, including assessment, content and instructional procedures, curriculum, enrichment, international comparisons, and psychology of learning and instruction.
		This is an excellent graduation present for anyone planning to teach in mathematics or the sciences, at any level.
	www.amazon.com /exec/obidos/tg/detail/-/081531647X?v=glance (983 words)

	Dense
		By the definition you gave, (1, 2) is not dense: for example, the interval (7, 96) does not contain a point of (1, 2).
		If you have two sets which are dense, then their intersection would also be dense if some part of one set is a subset of the other.
		Given any set not dense in the reals, I can find two sets that are dense in the reals, and have that set as their intersection.
	www.physicsforums.com /showthread.php?t=60708 (1481 words)

	Vanishing act has math pros re-solving riddle
		Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose among them.
		In the early 1980s, Richard Hamilton of Columbia suggested a new technique, called the Ricci flow, borrowed from the kind of mathematics that underlies Einstein's general theory of relativity and string theory, to investigate the shapes of spaces.
		Applied mathematically to this metric, the Ricci flow acts like heat, flowing through the space in question, smoothing and straightening all its bumps and curves to reveal its essential shape, the way a hair dryer shrink-wraps plastic.
	www.sfgate.com /cgi-bin/article.cgi?file=/c/a/2006/08/20/MNG71KJJPK1.DTL (2088 words)

	[No title]
		For p = 2, we show that is dense in an index 2 subgroup of S2.
		Gorbounov, Mahowald, and Symonds [9] studied dense subgroups of Sl0n(Sl in their notation), and we prove Theorem 0.2 using their methods.
		In Section 2, we recall the homological criterion that is employed in [9] to detect dense subgroups of Sl02for p > 2.
	hopf.math.purdue.edu /Behrens-Lawson/dense.txt (3679 words)

	Art
		Mathematics isn't classed as art by the general public, yet neither is engineering, nor computer programming.
		All three have in common with all other forms of art that there is a central theme to them that is also clearly at the heart of more commonly accepted fields like painting, sculpture or music.
		When something mathematical becomes part of the general body of knowledge it is because the element is like an image: something you continue to remember.
	www.cs.ucla.edu /~klinger/art.html (589 words)

	The extraordinary story of Fermat's Last Theorem
		The most important mathematical problem from the 17th century was coupled to the most significant problem of the 20th.
		Then, in August, he received a query which forced him to think seriously about one of the logical links in his argument: "For a while it seemed to be as trivial as the other problems, but by September I began to realise that this wasn't just a minor difficulty but a fundamental flaw.
		The proof is a masterpiece of modern mathematics, which leads to the conclusion that Wiles's proof of the Last Theorem is not the same as Fermat's.
	www.cs.uleth.ca /~kaminski/esferm03.html (5332 words)

	Staff Info
		My research interests centre around dense plasmas ; sometimes called warm dense matter.
		In particular I have been carrying out experiments into X-ray scattering from dense plasmas, with the objective of gaining information about their static and dynamic structures.
		This has potential benefits in understanding, for example, planetary interiors which are at pressures of millions of atmospheres and temperatures of thousands of degrees.
	www.qub.ac.uk /schools/SchoolofMathematicsandPhysics/StaffList/StaffInfo/?staffid=56 (85 words)

	Annals of Mathematics, Volume 157, no. 1
		The Annals of Mathematics is published bimonthly with the cooperation of Princeton University and the Institute for Advanced Study.
		Founded in 1884 by Ormond Stone of the University of Virginia, the journal was transferred in 1899 to Harvard University, and in 1911 to Princeton University.
		Axiom A maps are dense in the space of unimodal maps in the $C^k$ topology
	projecteuclid.org /Dienst/UI/1.0/Journal?authority=euclid.annm&issue=1044909491 (123 words)

	Mathematics (Site not responding. Last check: 2007-10-31)
		Partition the points on a circle into 2 sets such that, by rotating one of the sets by some amount, a hole appears in the circle's circumference and yet the 2 sets remain disjoint.
		Another interesting math-related site is the Mathematics Geneaology Project, which is a "family tree" of mathematicians.
		Biographies of the famous mathematicians can be found at the MacTutor History of Mathematics archive.
	www.cs.cornell.edu /~djm/math (457 words)

	Elusive Proof, Elusive Prover: A New Mathematical Mystery \| The Agonist
		The other Fields medalists this year are Andrei Okounkov, a professor of mathematics at Princeton; Terence Tao, a professor of mathematics at the University of California, Los Angeles; and Wendelin Werner, a professor of mathematics at the University of Paris-South in Orsay, France.
		And Dr. Perelman was resolute in saying no. “The reasons center around his feeling of isolation from the mathematical community,” Sir John said, “and in consequence his not wanting to be a figurehead for it or wanting to represent it.” He added: “I don’t think he meant it as an insult.
		The names of the nominees for the Fields Prize (the equivalent of the Nobel Prize for mathematics) were announced August 22 during the ceremony opening the International Congress of Mathematicians in Madrid.
	agonist.org /node/33402 (1218 words)

	GAMS : Other Sources of Math Software Information
		A directory of software and other materials useful in the teaching of mathematics at the college level maintained by the Mathematics Department at the University of Tennessee at Knoxville.
		A collection of C++ classes representing basic mathematical objects such as vectors, linear and nonlinear operators, and functions.
		A mathematical assistant for secondary level or high school students and teachers.
	gams.nist.gov /OtherSources.html (4369 words)

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