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# Topic: Lemma (mathematics)

 lemma.ca   (Site not responding. Last check: 2007-10-16) "In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than an independent statement, in and of itself. A good stepping stone leads to many others, so some of the most powerful results in mathematics are known as lemmata, such as Zorn's lemma, Bézout's lemma, Gauss's lemma, Fatou's lemma, and Nakayama lemma. The Greek word "lemma" itself means "anything which is received, such as a gift, profit, or a bribe." According to [1], the plural "lemmas" is commonly used. lemma.ca   (152 words)

 Mathematics   (Site not responding. Last check: 2007-10-16) Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'. Mathematics might be seen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships. Although mathematics itself is not usually considered a natural science, the specific structures that are investigated by mathematicians often have their origin in the natural sciences, most commonly in physics. uncover.us /en/wikipedia/m/ma/mathematics.html   (2162 words)

 PlanetMath: lemma A lemma is a proven statement, typically named a lemma to distinguish it as a truth used as a stepping stone to a larger result rather than an important statement in and of itself. The greek word ``lemma'' itself means ``anything which is received, such as a gift, profit, or a bribe.'' According to [1], the plural 'Lemmas' is commonly used. This is version 11 of lemma, born on 2003-07-22, modified 2005-03-18. planetmath.org /encyclopedia/Lemma.html   (338 words)

 Five lemma In mathematics, especially homological algebra and other applications of Abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma can be thought of as a combination of two other theorems, the four lemmas, which are dual[?] to each other. The five lemma states that, if the rows are exact, m and p are isomorphisms, l is an epimorphism, and q is a monomorphism, then n is also an isomorphism. www.ebroadcast.com.au /lookup/encyclopedia/fi/Five_lemma.html   (626 words)

 Mathematics - Simple English Wikipedia Mathematics (math or maths for short) is the study of quantity (how many things there are), structure (how things are organized), space (where things are) and change. Mathematical prediction is the basis of hard science, which almost always uses equations to predict events in physics. The philosophy of mathematics explores these issues, and others regarding how mathematics fits in with philosophy, ethics and real life. simple.wikipedia.org /wiki/Mathematics   (832 words)

 Talk:Lemma - Wikipedia, the free encyclopedia The article lemma was moved to lemma (mathematics), with the former being made into a disamibig. It is true that mathematics as a field is larger and older than linguistics, and thus the term is overall more commonly used for mathematics. Note about the use of lemma in linguistics: The term lemma is not used all that often in theoretical linguistics because the thing it describes has only a practical use, not a theoretical one, because a lemma is just a form chosen for convention to represent a lexeme. en.wikipedia.org /wiki/Talk:Lemma   (1607 words)

 Lemma - Wikipedia, the free encyclopedia Lemma (linguistics), the canonical form of a word Shadowing lemma, an animal which exists in only 2 dimensions and eats mathematicians in Terry Pratchett's fictional Discworld series This is a disambiguation page—a list of articles associated with the same title. en.wikipedia.org /wiki/Lemma   (100 words)

 Mathematics   (Site not responding. Last check: 2007-10-16) The specific structures that are investigated by mathematicians often have their origin in the natural sciences, most commonly in physics, but mathematicians also define and investigate structures for reasons purely internal to mathematics, because the structures may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common calculations. Mathematics is often abbreviated to math in North America and maths in other English-speaking countries. An alphabetical list of mathematical topics is available; together with the "Watch links" feature, this list is useful to track changes in mathematics articles. www.icyclopedia.com /encyclopedia/m/ma/mathematics.html   (1692 words)

 Gauss lemma Category:Modular arithmetic\nIn mathematics, there is more than one Gauss lemma; all are named after Carl Friedrich Gauss. In the theory of polynomials, the Gauss lemma relates the highest common factor of a product of two polynomials with integer coefficients to the corresponding hcfs of the factors. The Gauss lemma in number theory is involved in some proofs of quadratic reciprocity. encyclopedia.codeboy.net /wikipedia/g/ga/gauss_lemma.html   (365 words)

 Sets and Models of Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Lemma 4.3 With notation as in Lemma 4.2, for all This is a uniform relativization of part 2 of Lemma 3.16. www.math.psu.edu /simpson/papers/pizowkl   (1962 words)

 The Role of Reverse Mathematics Reverse Mathematics is a highly developed research program whose purpose is to investigate the role of strong set existence axioms in ordinary mathematics. Reverse Mathematics is a technique which frequently yields precise answers to special cases of this question. This might be called ``forward mathematics.'' But in order to establish that the axioms are needed for a proof of the theorem, one must reverse the process and deduce the axioms from the theorem. www.math.psu.edu /simpson/papers/hilbert/node5.html   (729 words)

 [No title] Mathematics is classified with both the humanities and the sciences. Departments in a variety of fields which use mathematics, including some in the social and biological sciences as well as in engineering and the physical sciences, are interested in attracting math majors into their graduate programs. Students with only minimum and intermediate mathematical competence are strongly advised to remove this deficiency by independent study through the UW-Extension or by enrolling for the summer session preceding the freshman year. www.wisc.edu /pubs/ug/10lettsci/depts/math.html   (5050 words)

 Pythagorean Pages M&IQ - Summer 2002   (Site not responding. Last check: 2007-10-16) Lemma 3 shows that if (a,b) is a solution of (1), then so is (b,nb - a); and b > a, nb - a = (n - 1)b + (b - a) > b, so (b,nb - a) is a strictly greater solution than (a,b). This is just Lemma 3 with a and b interchanged, and is again a direct consequence of (2). If we start with any positive solution (a,b) of (1) for a particular value of n, Lemma 4 provides a method of obtaining a sequence of strictly decreasing positive solutions; this is the exact reverse of the previous method of obtaining a strictly increasing sequence. www.mualphatheta.org /Mathematical_Log/Issues/Summer_02/MAO_Mathematical_Log_MIQ_Summer_02.htm   (571 words)

 [No title] Majors are advised to complete Mathematics 13 or 18 and Mathematics 46 by the end of their sophomore year. Mathematics 5 is a one-term calculus course and is not adequate preparation for Mathematics 12. Prerequisites: Mathematics 13 or 18, and 46, or consent. ase.tufts.edu /bulletin/mathematics.html   (2215 words)

 Albion College Math/CS - Colloquium Schedule The simplest version of this lemma says if your cat is inside when you go to bed and outside when you wake up in the morning, it must have gone through its little cat door an odd number of times. A degree in mathematics or computer science is excellent preparation for graduate school in areas such as mathematics, statistics, computer science, engineering, finance, and law. Mathematical models involving ordinary and partial differential equations and corresponding convergent numerical algorithms for computing their solutions can be used to create simulation packages to study physical phenomena. www.albion.edu /MathCS/Colloquium.asp   (1665 words)

 Lemma - TheBestLinks.com - Czech language, Discworld, Inflected language, Linguistics, ...   (Site not responding. Last check: 2007-10-16) Lemma - TheBestLinks.com - Czech language, Discworld, Inflected language, Linguistics,... Lemma, Czech language, Discworld, Inflected language, Linguistics, Mathematics... In mathematics, a lemma expresses a minor theorem, of interest primarily because it serves in the proof of a major result. www.thebestlinks.com /Lemma.html   (131 words)

 Convex Structures Convexity was the organizing principle of mathematical economics - particularly as practiced by those associated with the Cowles Commission (Koopmans, Debreu, Arrow, etc.) throughout the 1950s. Adoption and extension of these tools were relatively swift and revolutionized mathematical economics and drew it away from the "Paretian" use of differential calculus and into axiomatic reasoning. The Minkowski-Farkas Lemma is merely the reverse of this. cepa.newschool.edu /het/essays/math/convex.htm   (1799 words)

 Schwarz   (Site not responding. Last check: 2007-10-16) In 1869 he was appointed as professor of mathematics at the Eidgenössische Technische Hochschule in Zurich then, in 1875, he accepted appointment to the chair of mathematics at Göttingen University. We shall describe some of his very fine mathematical achievements in a moment, but first we note that he had several interests outside mathematics, although his marriage was a mathematical one since he married Kummer's daughter. Outside mathematics he was the captain of the local Voluntary Fire Brigade and, more surprisingly, he assisted the stationmaster at the local railway station by closing the doors of the trains. www-history.mcs.st-and.ac.uk /history/Mathematicians/Schwarz.html   (1074 words)

 Origins of some arithmetic terms #2 In the study of graphs in discrete mathematics the term diameter is often used for the greatest path length necessary to move between two nodes of a connected graph. Lemma Mathematicians use the word lemma to describe a proof which is a preliminary step to a more important or more complete result. Mathematics The origin of the word mathematics is in the Greek word manthanein, to learn. www.pballew.net /arithme2.html   (6166 words)

 Clay Mathematics Institute Gérard Laumon and Bao-Châu Ngô were recognized for their proof of the Fundamental Lemma for unitary groups. The lemma is a conjectured identity between orbital integrals for two groups, e.g., the unitary groups U(n) and U(p)xU(q), where p+q = n. The proof for the unitary case, which is significant for applications, requires many new ideas, including Laumon's deformation strategy and Ngô's purity result which is based on a geometric interpretation of the endoscopy theory of Langlands and Kottwitz in terms of the Hitchin fibration. www.claymath.org /research_award/Laumon-Ngo   (304 words)

 Seminars at Weizmann Institute of Science Faculty of Mathematical Sciences: Sun May 29 03:00:01 2005 Iterative application of this lemma yields a self-contained combinatorial proof for the PCP Theorem. The amplification lemma relies on a new notion of ``graph powering" that can be applied to systems of constraints, for example to 3SAT systems. We also apply the amplification lemma to construct *short* PCPs and locally-testable codes whose length is linear up to a polylog factor, and whose correctness can be probabilistically verified by making a constant number of queries. www.weizmann.ac.il /usersfiles/math/html/sem.2005:05:29:11:00:9:17897.shtml   (161 words)

 Practical Foundations of Mathematics In Set, image factorisation is stable: this is necessary in Lemma 5.8.6 to make relational composition associative, and in Theorem 9.3.11 for the existential quantifier to be invariant under substitution. Y, let q:X\twoheadrightarrow Q be the coequaliser of the kernel pair K\rightrightarrows X of f; by Lemma 5.6.6(b) this is also the kernel pair of q. From Lemma 5.7.6(a) it follows that all epis are regular. www.geocities.com /yury_bendersky/b/f/s57.html   (1337 words)

 Mathematics in Movies This is a collection of movie clips in which Mathematics appears. A lecture on the Snake lemma in homological algebra in with a student pretends to know things better. P.S. A flboard in Columbia University with mathematical formulas written by astrophysisist Peter, the ex-husband of the main character Louise www.math.harvard.edu /~knill/mathmovies   (365 words)

 Interactive Mathematics Miscellany and Puzzles But a Lemma and a Theorem and a very real proof that exploits well known properties of parallel lines are still available. At times entire Mathematics Departments were split between proponents of two different solutions. is a collection of diverse mathematical facts many of which I hope will be of interest to an average visitor. www.cut-the-knot.org /front.shtml   (1122 words)

 Amazon.com: The Joy of Sets : Fundamentals of Contemporary Set Theory (Undergraduate Texts in Mathematics): Books   (Site not responding. Last check: 2007-10-16) The author is a well-known mathematician and the editor of the "Computers in Mathematics" column in the AMS Notices and of FOCUS, the magazine published by the MAA. Keith Devlin is one of those rare research mathematicians who is able to make recent advances in mathematics understandable and interesting to those whose mathematical education is obsolete or incomplete. There are a number of other books on topics in pure mathematics about which I feel the same way, but they are a tiny minority among the deluge of texts that will never be read by anyone who doesn't have to. www.amazon.com /exec/obidos/tg/detail/-/0387940944?v=glance   (1232 words)

 Math/CS, Emory Univ.   (Site not responding. Last check: 2007-10-16) Introduction to Differential Geometry II Prerequisites: Mathematics 545 or permission of the instructor. Prerequisites: Mathematics 531 and 532; Mathematics 535-536 or permission of the instructor. Prerequisites: Mathematics 531 and 532; Mathematics 500 or permission of the instructor. www.mathcs.emory.edu /Graduate/Course/Catalog-1999-2000.html   (2025 words)

 CEU, Department of Mathematics and its Applications   (Site not responding. Last check: 2007-10-16) Polynomials (Gauss Lemma, cyclotomic polynomials, polynomials in several variables, homogeneous polynomials, symmetric polynomials, formal power series, Newton's Formulas, twisted polynomials, polynomials in non-commuting variables). Partially ordered sets and lattices (Hasse-diagram, chain conditions, Zorn Lemma, lattices as posets and as algebraic structures, modular and distributive lattices, modularity of the lattice of normal subgroups, Boolean algebras, Stone Representation Theorem). This course is a first introduction to the subject for an audience with diverse mathematical interest. www.ceu.hu /math/Courses/fall0405.html   (2845 words)

 Set Theory:Zorn's Lemma and the Axiom of Choice - Wikibooks, collection of open-content textbooks These notions are phrased in terms of Zorn's Lemma, and the Axiom of Choice and though they sound very different, they are equivalent to one another. That is, given Zorn's Lemma, one can derive the Axiom of Choice and vice versa. There are literally hundreds of mathematical statements that are known to be logically equivalent to the Axiom of Choice. en.wikibooks.org /wiki/Set_Theory:Zorn's_Lemma_and_the_Axiom_of_Choice   (783 words)

 Albion College Math/CS - Calendar of Events The MATH Challenge is a team-oriented mathematics competition for undergraduates. The Mathematical Contest in Modeling is a contest where teams of undergraduates use mathematical modeling to present their solutions to real world problems. The goal of Mathematics Awareness Month, held in April each year, is to increase public understanding of and appreciation for mathematics. www.albion.edu /mathcs/Calendar.asp   (3225 words)

 Mathematics Department 2000-01 Colloquia He also proved that a formal system that includes a significant amount of mathematics cannot be shown to be consistent (free from contradictions) without using methods more powerful than what is included in the system itself. In the 2001 Mathematical Contest in Modeling, we were asked to model the evacuation of the coastal population of South Carolina in the face of an oncoming hurricane. In the 2001 Mathematical Contest in Modeling, we were asked to provide a table demonstrating the conditions for which the ideal rear wheel of a racing bicycle is a spoked wheel and those for which it is solid. www.calpoly.edu /~math/colloquia0001.html   (4521 words)

 Amazon.com: Introduction to Topological Manifolds (Graduate Texts in Mathematics): Books   (Site not responding. Last check: 2007-10-16) A course on manifolds differs from most other introductory mathematics graduate courses in that the subject matter is often completely unfamiliar. It is even possible to get through an entire undergraduate mathematics education without ever hearing the word "manifold." Yet manifolds are part of the basic vocabulary of modern mathematics, and students need to know them as intimately as they know the integers, the real numbers, Euclidean spaces, groups, rings, and fields. Contains the essential topological ideas that are needed to further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields. www.amazon.com /exec/obidos/tg/detail/-/0387950265?v=glance   (1140 words)

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