61. E.W. Dijkstra Archive: On One Of Cayley's Theorems (EWD 677) On one of Cayley s theorems. A (finite) graph consists of a (finite) set of This is ascertained by one of Cayley s famous theorems, of which this note http://www.cs.utexas.edu/users/EWD/transcriptions/EWD06xx/EWD677.html

On one of Cayley's theorems A (finite) graph consists of a (finite) set of nodes with at most one edge between any two nodes. The so-called "multiplicity" of a node is defined as the number of edges of which that node is an endpoint. A connected graph without cycles is called a "tree"; a tree with N nodes has N-1 edges. In a tree the nodes with a multiplicity =1 are called its "leaves". A tree with at least 2 nodes has at least 2 leaves. In the following "a labeled tree" is a tree in which each node is labeled with a distinct integer. From a labeled tree with at least 2 nodes we can remove the leaf with the lowest number, together with the edge connecting it to the rest of the tree; the remaining graph is again a labeled tree. Hence this action can be repeated until the tree has been reduced to a single node; that remaining node is the one with the maximum number. For instance, the tree would give rise to the following sequence - in the order from left to right - of edge removals; each time we have written the number of the leaf being removed in the upper line: We remark that the right-most value of the bottom line is always the highest node number that the top line is always a permutation of the remaining node numbers that the number of times that a value occurs in such a scheme equals the multiplicity of the corresponding node.

62. FOM: Steve's Barber; Lou On Faltings' Theorem It s exciting that Lou has promised to explicate the general intellectual interestof Faltings and other famous theorems of number theory. http://www.cs.nyu.edu/pipermail/fom/1997-October/000116.html

FOM: Steve's barber; Lou on Faltings' theorem Stephen G Simpson simpson at math.psu.edu Wed Oct 29 12:43:13 EST 1997 Previous message: FOM: accessibility of abstract notion of algorithm Next message: FOM: accessibility of "algorithm" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Previous message: FOM: accessibility of abstract notion of algorithm Next message: FOM: accessibility of "algorithm" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... More information about the FOM mailing list

63. Encyclopedia: Mathematics 2.7 famous theorems and conjectures 2.8 Important theorems and conjectures 2.9Foundations and methods 2.10 History and the world of mathematicians http://www.nationmaster.com/encyclopedia/Mathematics

Supporter Benefits Signup Login Sources ... Pies Related Articles People who viewed "Mathematics" also viewed: Philosophy of mathematics Mathematical practice Foundations of mathematics Foundations problem in mathematics ... Ian Stewart (mathematician) What's new? Our next offering Latest newsletter Student area Lesson plans Recent Updates List of largest wikis List of internet slang List of hooded figures List of historical novels ... More Recent Articles Top Graphs Richest Most Murderous Most Taxed Most Populous ... More Stats Updated 19 hours 29 minutes ago. Other descriptions of Mathematics Mathematics Mathematics is the investigation of axiomatically defined abstract structures using symbolic logic and mathematical notation . It is commonly defined as the study of patterns of structure, change , and space ; even more informally, one might say it is the study of "figures and numbers". Because it is not empirical, it is not a science Mathematical knowledge is constantly growing, through research and application. Mathematics is usually regarded as a tool for science, even though the development of mathematics is not necessarily done with science in mind. The specific structures that are investigated by mathematicians sometimes do have their origin in natural and social sciences, including

64. Encyclopedia: Pierre De Fermat Pierre de Fermat Fermats last theorem (sometimes abbreviated as FLT and alsocalled Fermats great theorem) is one of the most famous theorems in the history http://www.nationmaster.com/encyclopedia/Pierre-de-Fermat

Supporter Benefits Signup Login Sources ... Pies Related Articles People who viewed "Pierre de Fermat" also viewed: List of mathematical theorems Fermat's last theorem Fermat prime Fermat's little theorem ... 1665 in science What's new? Our next offering Latest newsletter Student area Lesson plans Recent Updates List of largest wikis List of internet slang List of hooded figures List of historical novels ... More Recent Articles Top Graphs Richest Most Murderous Most Taxed Most Populous ... More Stats Encyclopedia: Pierre de Fermat Updated 6 days 5 hours 58 minutes ago. Other descriptions of Pierre de Fermat Pierre de Fermat Pierre de Fermat August 20 January 12 ) was a French lawyer at the Parlement of Toulouse , southern France , and a mathematician who is given credit for the development of modern calculus . In particular, he is the precursor of differential calculus with his method of finding the greatest and the smallest ordinates of curved lines, analogous to that of the then unknown differential calculus . Perhaps even more important, his brilliant researches in the theory of numbers entitle him to the rank of the founder of the modern theory. He also made notable contributions to

65. Mathematics - Simple English Wikipedia famous theorems and conjectures. These theorems have interested mathematiciansand nonmathematicians. Pythagorean theorem – Fermat s last theorem http://simple.wikipedia.org/wiki/Mathematics

Mathematics From Wikipedia, a free encyclopedia written in simple English for easy reading. Mathematics is the study of quantity structure (how things are organized), space (where things are) and change Because mathematics looks at such general topics, it uses abstract (simpler, less specific) descriptions of objects to talk about things. The simplest example of this is numbers . In the real world two apples plus two apples makes four apples, two bricks plus two bricks makes four bricks so in mathematics this becomes the general statement "two plus two equals four". This is arithmetic Another very simple example comes from set theory . If all blackbirds are black and one bird is not black it is not a blackbird. If all snow is white and another thing is not white it is not snow. In math we make this idea abstract by saying: if A is a subset of B then "not B" is a subset of "not A". For example, if you have something that is not black, it cannot be a blackbird. By finding a general way to say something mathematics solves many problems at the same time. The examples of snow and blackbirds are easy to understand without math, but harder situations can be much easier to understand with math. Sometimes mathematics studies rules or ideas which have not yet been found in the real world. Often if the rules are chosen because they are simple, later on they are found in the real world and studying the rules helps us understand the world better.

66. Theorem mathematics for a list of famous theorems and conjectures. list of theorems Gödel s incompleteness theorem CategoryTheorems http://copernicus.subdomain.de/theorem

Suche: Main Page Theorem A '''theorem''' is a statement which can be proven true within some logical framework. Proving theorems is a central activity of mathematics . Note that "theorem" is distinct from " theory condition In general mathematics a statement must be interesting or important in some way to be called a theorem. Less important statements are called: lemma ''''': a statement that forms part of the proof of a larger theorem. Of course, the distinction between theorems and lemmas is rather arbitrary, since one mathematician's major result is another's minor claim. Gauss' lemma and Zorn's lemma , for example, are interesting enough '' per se '' for some authors to stop at the nominal lemma without going on to use that result in any "major" theorem. '''''corollary''''': a statement which follows immediately or very simply from a theorem. A proposition ''A'' is a corollary of a proposition or theorem ''B'' if ''A'' can be deduced quickly and easily from ''B''. '''''proposition''''': a result not associated with any particular theorem. '''''claim''''': a very minor, but necessary or interesting result, which may be part of the proof of another statement. Despite the name, claims are proven.

67. Theorem In propositional logic, any proven statement is called a theorem. See also.mathematics for a list of famous theorems and conjectures. http://www.fact-index.com/t/th/theorem.html

Main Page See live article Alphabetical index Theorem A theorem is a statement which can be proven true within some logical framework. Proving theorems is a central activity of mathematics . Note that 'theorem' is distinct from ' theory A theorem generally has a set-up - a number of conditions, which may be listed in the theorem or described beforehand. Then it has a conclusion - a mathematical statement which is true under the given set up. The proof, though necessary to the statement's classification as a theorem is not considered part of the theorem. In general mathematics a statement must be interesting or important in some way to be called a theorem. Less important statements are called: lemma : a statement that forms part of the proof of a larger theorem. Of course, the distinction between theorems and lemmas is rather arbitrary, since one mathematician's major result is another's minor claim. Gauss' Lemma and Zorn's Lemma , for example, are interesting enough per se for some authors to stop at the nominal lemma without going on to use that result in any "major" theorem. corollary : a statement which follows immediately or very simply from a theorem. A proposition

68. Article About "Fermat's Last Theorem" In The English Wikipedia On 24-Apr-2004 Fermat s last theorem. Fermat s last theorem (also called Fermat s great theorem)is one of the most famous theorems in the history of mathematics. http://fixedreference.org/en/20040424/wikipedia/Fermat's_last_theorem

The Fermat's last theorem reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference : snapshots of Wikipedia from wikipedia.org) Fermat's last theorem Fermat's last theorem (also called Fermat's great theorem ) is one of the most famous theorems in the history of mathematics . It states that: There are no positive natural numbers a b , and c such that in which n is a natural number greater than 2. The 17th-century mathematician Pierre de Fermat wrote about this in in his copy of Claude-Gaspar Bachet 's translation of famous Diophantus Arithmetica , "I have discovered a truly remarkable proof but this margin is too small to contain it". The reason why this statement is so significant is that all the other theorems proposed by Fermat were settled either by proofs he supplied, or by more rigorous proofs supplied afterwards. Mathematicians long were baffled by this statement, for they were unable either to prove or to disprove it. The theorem was not the last Fermat conjectured, but the last to be proved. The theorem has the credit of the largest number of wrong proofs. For various special exponents n , the theorem had been proved over the years, but the general case remained elusive. In

69. 1. Introduction ABSTRACT As is known, G.Cantor formulated his famous Continuum Hypothesis generalized and proved the famous theorems of Hilbert, Lagrange, Wieferich, http://members.tripod.com/vismath1/zen/zen1.htm

setAdGroup('67.18.104.18'); var cm_role = "live" var cm_host = "tripod.lycos.com" var cm_taxid = "/memberembedded" Search: Lycos Tripod Star Wars Share This Page Report Abuse Edit your Site ... Next 1. INTRODUCTION. COGNITIVE VISUALIZATION OF NUMBER-THEORETICAL ABSTRACTIONS ]: "... Continuum Hypothesis is a rather dramatic example of what can be called (from our today's point of view) an absolutely undecidable assertion, ..." (p.13). The complete absence of any progress in the Continuum Hypothesis proof (or dispoof) on the way of modern meta-mathematics during last decades confirms the validity of Cohen's pessimism. So, it is obviously that new ways are necessary here. One of such new ways - a NON-meta-mathematical and NON-mathematical-logic way based on a so-called scientific cognitive computer visualization technique - to a new comprehension of the Continuum Problem itself is offered below. ]. For example, in classical Number Theory (NT) such the main feature giving rise to many famous NT-problems (such as Fermat's, Goldbach's, Waring's problems) is, by B.N.Delone and A.Ya.Hintchin, a hard comprehended connection between two main properties of natural numbers - their additivity and multiplicativity. Nevertheless, by means of CV-approach, we visualized this twice abstract connection in the form of color-musical 2D-images (so-called pythograms) of abstract NT-objects, and obtained really a lot of new NT-results. In particularly, we generalized well-known Classical Waring's Problem, generalized and proved the famous theorems of Hilbert, Lagrange, Wieferich, Balasubramanian, Desouillers, and Dress, discovered a new type of NT-objects, a new universal additive property of the natural numbers and a new method, - the so-called Super-Induction method, - for the rigorous proving of general mathematical statements of the form

70. Australian Mathematics Trust relation. Fermat s Last Theorem. Pierre de Fermat (16011665) posed one of themost famous theorems in Mathematics, stating that the equation http://www.amt.canberra.edu.au/euler.html

Leonhard Euler (1707-1783) Leonhard Euler was the most published mathematician of all time. There is probably not a single branch of mathematics known during his lifetime which he did not influence. If a difficult problem arose, Euler was generally consulted, and could often solve it. Euler was born near Basel, Switzerland and raised in the village of Riehen. His father was a Protestant Minister, and his mother was also from a clerical family. He was expected to follow his father into the clergy. He was an able student, mastering languages and mathematics and a memory for matters of detail. He entered the University of Basel at the age of 14. A Professor of mathematics there was Johann Bernoulli (1667-1748), arguably the world's greatest active mathemtician. Euler became a good friend of Bernoulli, who became his mentor. Both men appeared to have inspired each other greatly during their regular meetings. He obtained a Bachelor of Arts and Master of Philosophy Degree from Basel University. He did afterwards enter divinity school but found the call of mathematics to be greater. Bernoulli's son Daniel (1700-1782) moved to Russia in 1725 to take up a position at the newly formed St Petersburg Academy. In the following year Euler was invited to join him and he arrived in the year 1727. Living at the same home as Daniel Bernoulli Euler was able to discuss and collaborate with him extensively.

71. Gödel's Theorems And Truth His two famous theorems changed mathematics, logic, and even the way we look atour universe. This article explains what Gödel proved and why it matters to http://www.rae.org/godel.html

Gödel's Theorems and Truth Author: Dan Graves Subject: Date: Essays by Author Essays by Subject Essays by Date Shopping Cart Summary Famed mathematician Kurt Gödel proved two extraordinary theorems. Accepted by all mathematicians, they have revolutionized mathematics, showing that mathematical truth is more than logic and computation. Does Gödel's work imply that someone or something transcends the universe? Truth and Provability Kurt Gödel has been called the most important logician since Aristotle.(1) Such praise is evidence of how seriously Gödel's ideas are taken by mathematicians. His two famous theorems changed mathematics, logic, and even the way we look at our universe. This article explains what Gödel proved and why it matters to Christians. But first we must set the stage. A very simple formal system cannot support number theory but such a system is easily proven to be self-consistent. All we have to do is to show that it can't make a silly proof such as A=Non-A, which would be like saying 2=17. To handle number theory a complex formal system is needed. But as systems get more complex, they are harder to prove consistent. One result is that we don't know if our number theories are sound or if there are contradictions hidden in them. Gödel worked with such problems. He especially studied undecidable statements. An undecidable statement is one which can neither be proven true nor false in a formal system. Gödel proved that any formal system deep enough to support number theory has at least one undecidable statement.(2) Even if we know that the statement is true, the system cannot prove it. This means the system is incomplete. For this reason, Gödel's first proof is called "the incompleteness theorem".

72. Citations Graph Minors XX. Wagner S Conjecture - Robertson In a paper of 1950, Dilworth 9 showed one of the most famous theorems forpartially ordered sets, or posets. That is, for any poset, the maximum size of http://citeseer.ist.psu.edu/context/66859/0

73. Foundations.Cognition: Re: Descartes And The Mind There are some famous theorems like Fermat s Last Theorem whose truth wedid not know until very recently (and it s still not sure that the Fermat http://www.ecs.soton.ac.uk/~harnad/Hypermail/Foundations.Cognition/0008.html

Re: Descartes and the Mind From: Harnad, Stevan ( harnad@cogsci.soton.ac.uk Date: Thu Oct 19 1995 - 23:57:58 BST Next message: Parker, Chris: "Re: Descartes and the Mind" Previous message: Baden, Denise: "Re: Descartes and the Mind" Maybe in reply to: Harnad, Stevan: "Descartes and the Mind" Next in thread: Parker, Chris: "Re: Descartes and the Mind" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] > From: "Baden, Denise" < DB193@psy.soton.ac.uk Not quite. It's not all just definition. Remember that a lot of mathematics has to do with axioms and theorems that "follow" from them. Now the axioms we don't prove. We simply suppose they are true: "If these axioms were true, what would FOLLOW from them?" Then we start proving theorems. And the proofs (if you take them apart) turn out to be of the form: If you try to suppose that the axioms are true and the theorem is false then that leads to a contradiction. That's not just definition any more. (When I define something, I know what I've said, but when I define a set of axioms, I don't know which

74. CHOICE Magazine | About Choice Magazine either true or false, the famous theorems of Godel and Cohen have forced the Their book culminates with complete expositions of Godel s theorems, http://archive.ala.org/acrl/choice/35-3912.html

Site Map Contact Us About Choice Magazine About ALA ... Sample Reviews Mathematics CIP Adamowicz, Zofia. Logic of mathematics: a modern course of classical logic, by Zofia Adamowicz and Pawel Zbierski. Wiley, 1997. 260p bibl index afp ISBN 0-471-06026-7, $49.95 MARC Smullyan, Raymond. Knopf, 1997. 224p ISBN 0-679-44634-6, $22.00 CHOICE is a publication of the a division of American Library Association. For questions or comments, contact the Website editor. American Library Association. Site designed and developed by Fyrsta.com ChoiceReviews.online Reviews since 1988

75. Automated Development Of Fundamental Mathematical Theories By Art Quaife Let us prove the famous theorem that has this and the clause {is_a_man(Socrates)}as its The culminating results are two famous theorems of Euclid. http://psyche.cs.monash.edu.au/v2/psyche-2-28-fearnley_sande.html

This book may be purchased from Amazon.Com Automated Theorem Proving and Its Prospects Review of Automated Development of Fundamental Mathematical Theories by Art Quaife Desmond Fearnley-Sander Department of Mathematics University of Tasmania HOBART TAS 7001 AUSTRALIA dfs@hilbert.maths.utas.edu.au Received: December 28, 1995; Accepted: March 17, 1996; Published: June 19, 1996 PSYCHE, 2(28), June 1996 http://psyche.cs.monash.edu.au/v2/psyche-2-28-fearnley_sande.html KEYWORDS: automated theorem proving, OTTER, Quaife, resolution, understanding. REVIEW OF: Automated Development of Fundamental Mathematical Theories by Art Quaife (1992). Kluwer Academic Publishers. 271pp. $US123 hbk. ISBN: 0-7923-2021-2. 1. Introduction In his recent books one of the main pre-occupations of Roger Penrose has been to show (to prove!) that computers are intrinsically limited, compared to humans, when it comes to the doing of mathematics. Even those who think that such things can be proved may be interested in the empirical question: what mathematics can computers do? Art Quaife's book addresses the aspect of doing mathematics that is reported in mathematical journals: proof of theorems. It shows us the extent to which this high exercise of rationality can currently be automated, and the extent to which it cannot. To me this book was fascinating. After reading it, I found myself turning over in my mind the question of the role played in theorem proving by understanding, a quality that Penrose regards as a prime desideratum of consciousness. I'll come back to that at the end of this review.

76. Geometry, Table Of Contents 2.6 Some famous theorems of Geometry Chapter 3 Lines and Angles 3.1 NumberOperations and Equality 3.2 The Ruler and Distance http://www.fun-books.com/books/toc/3612-TOC.htm

For a lifetime of learning fun! If you are unable to see the blue navigation buttons below, go to Contents for text links or use our Search function. Geometry - Seeing, Doing, Understanding Table of Contents All Chapters include: Chapter Review and Algebra Review Introduction Inductive Reasoning Chapter 1: An Introduction to Geometry 1.1 Lines in Designing a City 1.2 Angles in Measuring the Earth 1.3 Polygons and Polyhedra: Pyramid Architecture 1.4 Constructions: Telling Time with Shadows 1.5 We Can't Go On Like This Chapter 2: The Nature of Deductive Reasoning 2.1 Conditional Statements 2.2 Definitions 2.3 Direct Proof 2.4 Indirect Proof 2.5 A Deductive System 2.6 Some Famous Theorems of Geometry Chapter 3: Lines and Angles 3.1 Number Operations and Equality 3.2 The Ruler and Distance 3.3 The Protractor and Angle Measure 3.4 Bisection 3.5 Complementary and Supplementary Angles 3.6 Linear Pairs and Vertical Angles 3.7 Perpendicular and Parallel Lines

77. Sites To Use To Practice Skills Needed On The Algebra II End Of Course Assessmen you will look at a few proofs and several applications of one of the mostfamous theorems in mathematics; Proof of the theorem is demonstrated through a http://www.internet4classrooms.com/eoc_algebra2.htm

Daily Dose of the Web Links for K-12 Teachers On-Line Practice Modules End-of-Course - Algebra II sites to help students practice skills needed for the Algebra II exam Standards Algebra Geometry Measurement Data Analysis and Probability Links verified 4/17/05 = a PowerPoint show Standard 1 - Number and Operations Students will recognize, represent, model, and apply real numbers and operations and will demonstrate an understanding of properties and operations of the complex number system. Level 1 order a given set of real numbers Rational and Irrational number review identify the reciprocal of a real number Rational Number and its Reciprocal - a step-by-step examination of a problem by Dr. Math How do I find the reciprocal of a decimal mixed number? - an answer to a student's question multiply two polynomials with each factor having no more than two terms. Interactive Algebra Review - Multiplying Algebraic Expressions Multiplying Polynomials - a step-by-step explanation from PurpleMath Interactive Algebra Review - Solving Polynomial Equations (see also part B of this tutorial Factoring Trinomials Self Quiz - Check your knowledge about factoring polynomials.

78. Nut03s but the course is much more than just the proof of two famous theorems. Basic notions and theorems about divisibility, primes and congruences, http://www.renyi.hu/~dezso/budsem/04spring/nut2_04s.html

Topics in Number Theory NUT2 Instructor: Dr. Antal BALOG Text: Selected chapters of Harold Davenport, Multiplicative Number Theory. In case of departing from the book, and also for gaining some preliminary routine, supplements will be handed out. Prerequisite: General mathematical experience of the undergraduate level is expected. This includes elementary algebra (Abelian groups, vector spaces, systems of linear equations) and calculus (limits, derivatives, integration, infinite series). A first course of number theory (divisibility, congruences, Chinese Remainder Theorem, primitive roots and power residues) and a course of complex function theory (analytic functions, continuation, power series, complex line integrals, calculation of residues) are essential , although the basic concepts of the applied theories and theorems will always be explained. Taking CLX parallel to this course is enough. Course description: Our aim is to provide a classical introduction to analytic number theory, focused on the connection between the zeros of the Riemann z (s) -function and prime numbers. We will follow the history and development of a beautiful discipline, rich in problems, methods and ideas. The highlights are a proof of Dirichlet's Theorem about the distribution of primes in arithmetic progressions, and of the Prime Number Theorem, but the course is much more than just the proof of two famous theorems.

79. Home Contains information on famous mathematicians, theorems, and topics. http://www.mathsisgoodforyou.com/

This site is being developed and will have all the pages currently listed at the moment functional by the end of the summer 2005. It already has lots of resources, so explore! About Maths is Good for You... See updates or more about courses mathematicians timeline topics in the history of mathematics ... Click here to find out why is maths good for you! Key Stage 3 Year 7 Year 8 Year 9 Key Stage 4 introduction to maths through the history of maths artefacts from the history of maths some mathematical concepts in context famous mathematicians ... famous places These materials have been written by Dr. Snezana Lawrence as part of a Gatsby Teacher Fellowship, made possible by funding from Gatsby Technical Education projects (GTEP). Responsibility for the materials lies with the author, and GTEP cannot accept any liability for the content of these materials or any consequences arising from their use. You can learn more about Gatsby Teacher Fellowships and how to apply for grants from their web site. e-mail

80. Mathematics And Statistics Easily the most famous of all mathematical theorems—the one that nearly everyoneknows. This is probably the second most famous mathematical theorem. http://www.geocities.com/~mikemclaughlin/MS.html

The World of Numbers This page, like the companion Science page , contains items and links that appeal to me for one reason or another. Here you will find various topics related to mathematics and statistics , including links to reference material, tutorials, Macintosh freeware, and miscellaneous tidbits as well as to other sites like this one. The list is not meant to be comprehensive and the selection is based, as always, on my personal perspective . Thus, this page will always be under construction! Mathematics It would be difficult to find any other subject, of comparable relevance and importance, about which so many are so proud of their ignorance. Even the most fastidious individuals, who would never tolerate the unforgivable solecism of a split infinitive, will not hesitate to proclaim, to one and all, their utter inadequacy when it comes to anything pertaining to numbers. This includes activities even as mundane as rescaling a recipe, reading a weather map, or computing the total cost of a mortgage. Tell people that the Sun is far away and they will nod their head and try to look intelligent. Say that it is 93,000,000 miles away and their eyes will start to glaze over as they search for some means of escape. If you are reading this page, then you are likely in that happy minority who know better. Therefore, you might find things here that you will appreciate.

Page 4     61-80 of 96    Back | 1  | 2  | 3  | 4  | 5  | Next 20