Merge lemma logic date December 2011 In mathematics , a lemma plural lemmata or lemmas ref name mathhandbook cite book last Higham first Nicholas J. title Handbook of Writing for the Mathematical Sciences publisher Society for Industrial and Applied Mathematics year 1998 isbn 0898714206 pages 16 ref from the Greek language Greek wikt en Ancient Greek lemma , anything which is received, such as a gift, profit, or a bribe is a proven Proposition mathematics proposition which is used as a stepping stone to a larger result rather than as a statement in and of itself. There is no formal distinction between a lemma and a theorem , only one of usage and convention see Theorem Terminology . A good stepping stone leads to many others, so some of the most powerful results in mathematics are known as lemmas, such as B zout s lemma , Urysohn s lemma , Dehn s lemma , Fatou s lemma , Gauss s lemma disambiguation Gauss s lemma , Nakayama s lemma , Poincar s lemma , Riesz s lemma , Schwarz s lemma , It s lemma and Zorn s lemma . See also wiktionary lemma Corollary Fundamental lemma List of lemmas References reflist External links Doron Zeilberger , http www.math.rutgers.edu zeilberg Opinion82.html Opinion 82 A Good Lemma is Worth a Thousand Theorems planetmath id 4492 title Lemma maths stub Category Mathematical terminology Category Lemmas Category Greek loanwords ar bg ca Lema matem tiques cs Lemma matematika de Hilfssatz es Lema matem ticas eo Lemo fr Lemme math matiques hi id Lema matematika it Lemma matematica he lt Lema hu Lemma mk nl Hulpstelling pl Lemat pt Lema matem tica ro Lem ru simple Lemmamathematics sk Lemma fi Lemma sv Lemma tt th tg uk zh ... more details
Lemma may refer to wiktionary Lemmamathematics , a proven statement used as a stepping stone toward the proof of another statement Lemma morphology , the canonical form or citation form of a word Lemma psycholinguistics , an intermediate form a word about to be uttered takes during speech production Headword , in lexicons Lemma logic , which is simultaneously a premise for a contention above it and a contention for premises below it Lemma botany , one of the specialised bracts enclosing a floret in a grass inflorescence Misspellings Analemma , the curve traced out by a celestial body over the course of a year on the celestial sphere of another body a phenomenon that may be used to determine the time of year Morris Iemma Note spelling with a capital i , former premier of the Australian state of New South Wales disambig cs Lemma da Lemma de Lemma el fr Lemme io Lemo id Lemma it Lemma lb Lemma nl Lemma simple Lemma ... more details
uses see Mathematics disambiguation and Math disambiguation . File Euclid.jpg thumb Euclid , Greek ... . ref Mathematics from Greek language Greek m th ma knowledge, study, learning is the study ..., then mathematical reasoning often provides insight or predictions. Through the use of abstraction mathematics abstraction and logic al reasoning , mathematics developed from counting , calculation , measurement .... Practical mathematics has been a human activity for as far back as History of Mathematics written records exist. Logic Rigorous arguments first appeared in Greek mathematics , most notably in Euclid Euclid s Euclid s Elements Elements . Mathematics developed at a relatively slow pace until the Renaissance ... History of Mathematics 1. Newton and Leibniz , BBC Radio 4 , 27 09 2010. ref Carl Friedrich Gauss 1777 1855 referred to mathematics as the Queen of the Sciences . ref Waltershausen ref Benjamin Peirce 1809 1880 called mathematics the science that draws necessary conclusions . ref Peirce, p. 97. ref David Hilbert said of mathematics We are not speaking here of arbitrariness in any sense. Mathematics ..., Birkh user 1992 . ref Albert Einstein 1879 1955 stated that as far as the laws of mathematics ... . ref name certain Mathematics is used throughout the world as an essential tool in many fields, including natural science , engineering , medicine , and the social sciences . Applied mathematics , the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires ... mathematics , or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. ref Peterson ref Etymology The word mathematics comes from the ancient ... mean to learn . The word mathematics in Greek came to have the narrower and more technical meaning ... until around 1700, the term mathematics more commonly meant astrology or sometimes astronomy ... more details
Gauss s lemma can mean any of several Lemmamathematics lemmas named after Carl Friedrich Gauss Gauss s lemma polynomial Gauss s lemma number theory Gauss s lemma Riemannian geometry See also List of topics named after Carl Friedrich Gauss mathdab Category Lemmas eo Ga sa lemo fr Lemme de Gauss it Lemma di Gauss ... more details
In mathematics, Tukey s lemma , named after John Tukey , states that every nonempty collection of finite character has a maximal element with respect to inclusion. It is equivalent to the Axiom of Choice . References Brillinger, David R. John Wilder Tukey http www.ams.org notices 200202 fea tukey.pdf Category set families Category order theory Category axiom of choice Category lemmas settheory stub de Lemma von Teichm ller Tukey hu Teichm ller Tukey lemma nl Lemma van Teichm ller Tukey ... more details
In mathematics , Lindel f s lemma is a simple but useful Lemmamathematicslemma in topology on the real line , named for the Finland Finnish mathematician Ernst Leonard Lindel f . Statement of the lemma Let the real line have its standard topology. Then every Open set open subset of the real line is a Countable set countable Union set theory union of open Interval mathematics interval s. Generalization Lindel f s lemma is also known as the statement that every open cover in a second countable space has a countable cover topology subcover Kelley 1955 49 This means that every second countable space is also a Lindel f space . References J.L. Kelley 1955 , General Toplogy , van Nostrand. Category Covering lemmas Category Lemmas Category Topology topology stub ... more details
merge lemmamathematics date December 2011 Other uses Lemma disambiguation Unreferenced date November 2006 In informal logic and argument map ping, a lemma is simultaneously a main contention contention for premise s below it and a premise for a contention above it. See also Co premise Objection argument Objection Inference objection Category Concepts in logic Philo stub pt Lema filosofia ... more details
Unreferenced date December 2009 For Lebesgue s lemma for open covers of compact spaces in topology see Lebesgue s number lemma In mathematics , Lebesgue s lemma is an important statement in approximation theory . It provides a bound for the projection error. Statement Let V , be a normed vector space , U be a subspace of V and let math P math be a projection linear algebra linear projector on math U math . Then, for each v in V math v Pv leq 1 P inf u in U v u . math See also Lebesgue constant interpolation DEFAULTSORT Lebesgue s Lemma Category Lemmas Category Approximation theory eo Lebega lemo fr Lemme de Lebesgue ... more details
In mathematics , the nine lemma is a statement about commutative diagram s and exact sequence s valid in any abelian category , as well as in the category of group mathematics group s. It states if image nine lemma.png is a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well. Likewise, if all columns as well as the two top rows are exact, then the bottom row is exact as well. The nine lemma can be proved by direct diagram chasing , or by applying the snake lemma to the two bottom rows in the first case, and to the two top rows in the second case . Linderholm p. 201 offers a satirical view of the nine lemma Draw a tic tac toe noughts and crosses board... Do not fill it in with noughts and crosses... Instead, use curved arrows... Wave your hands about in complicated patterns over this board. Make some noughts, but not in the squares put them at both ends of the horizontal and vertical lines. Make faces. You have now proved a the Nine Lemma b the Sixteen Lemma c the Twenty five Lemma... References cite book first Carl last Linderholm year 1971 title Mathematics Made Difficult publisher Wolfe isbn 0 7234 0415 1 Category Homological algebra Category Lemmas de Neunerlemma zh ... more details
In the theory of formal language s in computability theory , a pumping lemma or pumping argument states that, for a particular language to be a member of a language class, any sufficiently long string in the language contains a section, or sections, that can be removed, or repeated any number of times, with the resulting string remaining in that language. The proofs of these lemmas typically require counting argument s such as the pigeonhole principle . The two most important examples are the pumping lemma for regular languages and the pumping lemma for context free languages . Ogden s lemma is a second, stronger pumping lemma for context free language s. These lemmamathematicslemma s can be used to determine if a particular language is not in a given language class. However, they cannot be used to determine if a language is in a given class, since satisfying the pumping lemma is a necessary and sufficient necessary , but not sufficient, condition for class membership. References cite book author Michael Sipser year 1997 title Introduction to the Theory of Computation publisher PWS Publishing isbn 0 534 94728 X Section 1.4 Nonregular Languages, pp. 77&ndash 83. Section 2.3 Non context free Languages, pp. 115&ndash 119. cite book author Thomas A. Sudkamp year 2006 title Languages and Machines, Third edition publisher Adison Wesley isbn 0 321 32221 5 Chapter 6 Properties of Regular Languages pp. 205 210 Category Formal languages Category Lemmas bs Osobina napuhavanja cs Lemma o vkl d n de Pumping Lemma es Lema del bombeo fr Lemme d it ration ko hr Svojstvo napuhavanja it Pumping lemma nl Pompstelling ja pt Lema do bombeamento ro Lema de pompare ru sr uk zh ... more details
Image Butterfly lemma.svg thumb 300px right Hasse diagram of the Zassenhaus butterfly lemma smaller subgroups are towards the top of the diagram In mathematics , the butterfly lemma or Zassenhaus lemma , named after Hans Julius Zassenhaus , is a technical result on the lattice of subgroups of a group mathematics group or the lattice of submodules of a module, or more generally for any modular lattice . ref See Pierce, p. 27, exercise 1. ref Lemma Suppose math G, Omega math is a group with operators and math A math and math C math are subgroup s. Suppose math B triangleleft A math and math D triangleleft C math are stable subgroup s. Then, math A cap C B A cap D B math is isomorphism isomorphic to math A cap C D B cap C D. math Zassenhaus proved this lemma specifically to give the smoothest proof of the Schreier refinement theorem . The butterfly becomes apparent when trying to draw the Hasse diagram of the various groups involved. Notes references References citation title Associative algebras first1 R. S. last1 Pierce publisher Springer pages 27 isbn 0387906932 . citation title An introduction to noncommutative noetherian rings first1 K. R. last1 Goodearl first2 Robert B. last2 Warfield publisher Cambridge University Press year 1989 isbn 9780521369251 pages 51, 62 . citation first Serge last Lang title Algebra pages 20 21 edition Revised 3rd series Graduate Texts in Mathematics publisher Springer Verlag isbn 9780387953854 . Carl Clifton Faith, Nguyen Viet Dung, Barbara Osofsky. Rings, Modules and Representations . p. 6. AMS Bookstore, 2009. ISBN 0821843702 External links Zassenhaus Lemma and proof at http www.artofproblemsolving.com Wiki index.php Zassenhaus 27s Lemma DEFAULTSORT Zassenhaus Lemma Category Group theory Category Lemmas Category Isomorphism theorems Category Theorems in algebra fr Lemme de Zassenhaus pl Lemat Zassenhausa ... more details
In mathematics , especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemmamathematicslemma about commutative diagram s. The five lemma is valid not only for abelian categories but also works in the category of groups , for example. The five lemma can be thought of as a combination of two other theorems, the four lemmas , which are duality category theory dual to each other. Statements Consider the following commutative diagram in any abelian category such as the category of abelian group s or the category of vector space s over a given field algebra field or in the category of group mathematics group s. image FiveLemma.png The five lemma states that, if the rows are exact sequence exact , m and p are isomorphism s, l is an epimorphism , and q is a monomorphism , then n is also an isomorphism. The two four lemmas state br 1 If the rows in the commutative diagram image FourLemma01.png are exact and m and p are epimorphisms and q is a monomorphism, then n is an epimorphism. 2 If the rows in the commutative diagram .... We shall prove the five lemma by individually proving each of the 2 four lemmas. To perform diagram chasing, we assume that we are in a category of module mathematics modules over some ring mathematics ... of the diagram as function mathematics function s in fact, homomorphism s acting on those elements ... five lemma. Applications The five lemma is often applied to long exact sequence s when computing homology mathematics homology or cohomology of a given object, one typically employs a simpler subobject ... lemma can then be used to determine the unknown homology groups. See also Short five lemma , a special case of the five lemma for short exact sequence s Snake lemma , another lemma proved by diagram chasing Nine lemma Notes references References W. R. Scott Group Theory , Prentice Hall, 1964. Citation ... topology edition 3rd volume 127 series Graduate texts in mathematics publisher Springer ... more details
In mathematics , and more specifically, in the fractal dimension theory of fractal dimensions , Frostman s lemma provides a convenient tool for estimating the Hausdorff dimension of sets. Lemma Let A be a Borel measurable Borel subset of R sup n sup , and let s 0. Then the following are equivalent H sup s sup A 0, where H sup s sup denotes the s dimensional Hausdorff measure . There is an unsigned Borel measure &mu satisfying &mu A 0, and such that math mu B x,r le r s math holds for all x &isin R sup n sup and r 0. Otto Frostman proved this lemma for closed sets A as part of his PhD dissertation at Lund University in 1935. The generalization to Borel sets is more involved, and requires the theory of Suslin set s. A useful corollary of Frostman s lemma requires the notions of the s capacity of a Borel set A &sub R sup n sup , which is defined by math C s A sup Bigl Bigl int A times A frac d mu x ,d mu y x y s Bigr 1 mu text is a Borel measure and mu A 1 Bigr . math Here, we take inf &empty &infin and Frac 1 &infin 0. As before, the measure math mu math is unsigned. It follows from Frostman s lemma that for Borel A &sub R sup n sup math mathrm dim H A sup s ge 0 C s A 0 . math References Citation last1 Mattila first1 Pertti title Geometry of sets and measures in Euclidean spaces publisher Cambridge University Press isbn 978 0 521 65595 8 year 1995 mr 1333890 series Cambridge Studies in Advanced Mathematics volume 44 Category Dimension theory Category Fractals Category Metric geometry mathanalysis stub fi Frostmanin lemma ... more details
dablink Abhyankar s lemma is not directly related to Abhyankar s conjecture . In mathematics , Abhyankar s lemma named after Shreeram Shankar Abhyankar allows one to kill tame ramification by taking an extension of a base field. More precisely, Abhyankar s lemma states that if A , B , C are local field s such that A and B are finite extension s of C , with ramification index ramification indices a and b , and B is tamely ramified over C and b divides a , then the compositum AB is an unramified extension of A . References Gary Cornell http links.jstor.org sici?sici 0002 9947 28198206 29271 3A2 3C501 3AOTCORG 3E2.0.CO 3B2 T On the Construction of Relative Genus Fields Theorem 3, page 504. Transactions of the American Mathematical Society, Vol. 271, No. 2. Jun., 1982 , pp. 501 511. Gold, Robert Madan, M. L. Some applications of Abhyankar s lemma. Math. Nachr. 82 1978 , 115 119. A. Grothendieck http www.arxiv.org abs math.AG 0206203 S minaire de G om trie Alg briques du Bois Marie 1960 61. Lecture Notes in Math. 224. Springer Verlag 1971, http modular.fas.harvard.edu sga sga 1 1t 279.html page 279 . Category Algebraic geometry Category Lemmas Category Algebraic number theory algebra stub ... more details
In theoretical computer science and mathematics , especially in the area of combinatorics , the Levi lemma states that, for all string computer science strings u , v , x and y , if uv xy , then there exists a string w such that either uw x and v wy or u xw and wv y That is, there is a string w that is in the middle , and can be grouped to one side or the other. ref Mathematical Foundations of Computer Science 2004 Ji Fiala , V clav Koubek , Jan Kratochv l ISBN 3540228233, 9783540228233 ref The above is known as the Levi lemma for strings the lemma can occur in a more general form in graph theory and in monoid theory for example, there is a more general Levi lemma for trace monoid traces . ref name Messner1997 Citation title Pattern matching in trace monoids url http www.springerlink.com index d17g454526765k88.pdf year 1997 author Messner, J. journal Lecture Notes in Computer Science pages 571 582 accessdate 2009 05 11 ref See also String operations String functions programming Transfinite strings Notes reflist Category Formal languages Category Semigroup theory Category Lemmas combin stub ... more details
In mathematics , Spijker s lemma is a result in the theory of rational mapping s of the Riemann sphere . It states that the image mathematics image of a circle under a complex rational map with numerator and denominator having degree of a polynomial degree at most n has length at most 2 n . See also Buffon s needle External links MathWorld title Spijker s Lemma urlname SpijkersLemma References cite journal last Wegert first Elias coauthors Trefethen, Lloyd N. title From the Buffon Needle Problem to the Kreiss Matrix Theorem journal The American Mathematical Monthly volume 101 issue 2 pages 132 139 month February year 1994 doi 10.2307 2324361 jstor 2324361 Category Complex analysis Category Lemmas ... more details
Unreferenced date December 2009 Shephard s lemma is a major result in microeconomics having applications in the theory of the firm and in consumer choice. ref Microeconomic Analysis Third Edition, Hal Varian 1992 ref The lemmamathematicslemma states that if indifference curves of the expenditure or cost function are convex function convex , then the cost minimizing point of a given good math i math with price math p i math is unique. The idea is that a consumer will buy a unique ideal amount of each item to minimize the price for obtaining a certain level of utility given the price of goods in the market . The lemma is named after Ronald Shephard who gave a Mathematical proof proof using the distance formula in his book Theory of Cost and Production Functions Princeton University Press, 1953 . The equivalent result in the context of consumer theory was first derived by Lionel W. McKenzie in 1957. It states that the partial derivatives of the expenditure function with respect the prices of goods equal the Hicksian demand functions for the relevant goods. Similar results had already been derived by John Hicks 1939 and Paul Samuelson 1947 . Definition In consumer theory, Shephard s lemma states that the demand for a particular good i for a given level of utility u and given prices ... , the lemma gives a similar formulation for the Conditional factor demands conditional factor ... s lemma use the envelope theorem . Proof for the Differentiable Case The proof is stated for the two ... i.e. the Hicksian demand function for good 1 . This completes the proof. Application Shephard s lemma gives a relationship between expenditure or cost functions and Hicksian demand. The lemma can be re ... Marshallian demand function . See also Hotelling s lemma Convex preferences References reflist DEFAULTSORT Shephard s Lemma Category Underlying principles of microeconomic behavior Category Lemmas az epard lemmas de Shephards Lemma it Lemma di Shephard vi B Shephard ... more details
lemma prevails in Poland and Russia. Portal Mathematics Equivalent forms of Zorn s lemma Zorn s lemma ... how Zorn s lemma can be seen as a powerful tool, especially in the sense of unified mathematics Clarify ...For the film by Hollis Frampton Zorns Lemma film Zorn s lemma , also known as the Kuratowski Zorn lemma ... . An equivalent formulation of the lemma is therefore blockquote Suppose a non empty partially ordered ... Zorn s lemma often involve taking a union of some sort to produce an upper bound. The case of an empty chain, hence empty union is a boundary case that is easily overlooked. Zorn s lemma is equivalent ... algebra that every nonzero ring algebra ring has a maximal ideal and that every field mathematics ... s lemma the proof that every nontrivial ring R with Unital ring unity contains a maximal ideal ... maximal ideals by definition are not equal to R . We want to apply Zorn s lemma, and so we take a non ... R from P . The condition of Zorn s lemma has been checked, and we thus get a maximal element in P , in other ... of the proof of Zorn s lemma from the axiom of choice A sketch of the proof of Zorn s lemma follows. Suppose the lemma is false. Then there exists a partially ordered set, or poset, P such that every ... bound has a bigger element. To actually define the function mathematics function b , we need to employ .... This proof shows that actually a slightly stronger version of Zorn s lemma is true Quote If P ... s lemma. Kazimierz Kuratowski K. Kuratowski proved in 1922 ref cite journal first Casimir last ... tresc.php?wyd 1&tom 3 ref a version of the lemma close to its modern formulation it applied to sets ... equivalence with the axiom of choice in another paper, which never appeared. The name Zorn s lemma ... Axiom of choice Well ordering theorem . Moreover, Zorn s lemma or one of its equivalent forms implies ... 0 cite journal last Campbell first Paul J. year 1978 month February title The Origin of Zorn s Lemma ... External links http www.apronus.com provenmath choice.htm Zorn s Lemma at ProvenMath contains a formal ... more details
A Shadowing lemma is also a fictional creature in the Discworld. See Flora and fauna of the Discworld Shadowing Lemma Shadowing lemma . In the Dynamical systems theory theory of dynamical systems , the shadowing lemma is a lemmamathematicslemma describing the behaviour of pseudo orbits near a Hyperbolic set hyperbolic invariant set . Informally, it says that every pseudo orbit which one can think as of a numerically computed trajectory with rounding errors on every step ref MathWorld title Shadowing Theorem urlname ShadowingTheorem ref stays uniformly close to some true trajectory with slightly altered initial position in other words, a pseudo trajectory is shadowed by a true one. Formal statement Given a map f X &rarr X of a metric space X , d to itself, define a &epsilon pseudo orbit or &epsilon orbit as a sequence math x n math of points such that math x n 1 math belongs to a &epsilon neighborhood of math f x n math . Then, near a hyperbolic invariant set, the following statement holds ref A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems, Theorem 18.1.2. ref Let &Lambda be a hyperbolic invariant set of a diffeomorphism f. There exists a neighborhood U of &Lambda with the following property for any &delta 0 there exists &epsilon 0, such that any finite or infinite &epsilon pseudo orbit that stays in U also stays in a &delta neighborhood of some true orbit. math forall x n , , x n in U, , d x n 1 ,f x n varepsilon quad exists y n , , , y n 1 f y n , quad text such that , , forall n , , x n in U delta y n . math References ref list Scholarpedia article http mathworld.wolfram.com ShadowingTheorem.html Shadowing Theorem Category Dynamical systems Category Lemmas mathanalysis stub ... more details
In topology , Urysohn s lemma is a lemmamathematicslemma that states that a topological space is normal space normal if and only if any two disjoint sets disjoint closed set closed subsets can be separated by a function . Urysohn s lemma is sometimes called the first non trivial fact of point set topology and is commonly used to construct continuous functions with various properties on normal spaces. It is widely applicable since all metric space s and all compact space compact Hausdorff space s are normal. The lemma is generalized by and usually used in the proof of the Tietze extension theorem . The lemma is named after the mathematician Pavel Samuilovich Urysohn . Formal statement Two disjoint sets disjoint closed set closed subsets A and B of a topological space X are said to be separated by neighbourhoods if there are neighbourhood topology neighbourhood s U of A and V of B that are also disjoint. A and B are said to be separated by a function if there exists a continuous function topology continuous function f from X into the unit interval nowiki 0,1 nowiki such that nowrap begin f a 0 nowrap end for all a in A and nowrap begin f b 1 nowrap end for all b in B . Any such function ... disjoint closed sets can be separated by neighbourhoods. Urysohn s lemma states that a topological ... normal space s. Urysohn s lemma has led to the formulation of other topological properties such as the Tychonoff property and completely Hausdorff spaces . For example, a corollary of the lemma ... has completely formalized and automatically checked a proof of Urysohn s lemma in the http www.mizar.org ... id 3597 title proof of Urysohn s lemma Category Lemmas Category Topology Category Articles containing proofs Category Separation axioms de Lemma von Urysohn fr Lemme d Urysohn it Lemma di Urysohn he nl Lemma van Urysohn pl Lemat Urysohna pt Lema de Urysohn fi Urysonin lemma ru sv Urysohns lemma zh ... more details
In set theory , a branch of mathematics, the condensation lemma is a result about sets in the constructible universe . It states that if math L alpha math is a level of the constructible hierarchy, X is an elementary submodel of the model math L alpha, in math then math X, in math is isomorphic to some math L beta, in math , math beta leq alpha math . The lemma was formulated and proved by Kurt G del in his proof that the axiom of constructibility implies Continuum hypothesis The generalized continuum hypothesis GCH . References cite book last Devlin first Keith authorlink Keith Devlin year 1984 title Constructibility publisher Springer id ISBN 3 540 13258 9 Category Set theory Category Lemmas Category Article Feedback 5 settheory stub ... more details
In mathematics , Tucker s lemma , named after ?????? Tucker , is a combinatorics combinatorial analog of the Borsuk&ndash Ulam theorem . Let T be a triangulation of the closed n dimensional ball math mathbb B n math . Assume T is antipodally symmetric on the boundary math mathbb S n 1 math . That means that the subset of simplices of T which are in math mathbb S n 1 math provides a triangulation of math mathbb S n 1 math where if is a simplex then so is . Let math L V T to 1, 1, 2, 2,..., n, n math be a labelling of the vertices of T which satisfies L v L v for all vertices v in math mathbb S n 1 math . Then Tucker s Lemma states that there exists a 1 simplex in T whose vertices are labelled by the same number but with opposite signs. File TuckerLemmaDiagram.png In the above example, where n 2, the red 1 simplex has vertices which are labelled by the same number with opposite signs. Tucker s lemma states that for such a triangulation at least one such 1 simplex must exist. See also Brouwer fixed point theorem Borsuk&ndash Ulam theorem Topological combinatorics References cite book author Ji Matou ek mathematician Ji Matou ek title Using the Borsuk&ndash Ulam Theorem publisher Springer Verlag date 2003 isbn 3 540 00362 2 pages 34 Category Combinatorics Category Topology Category Lemmas combin stub ... more details
In mathematics , Borel s lemma is an important result about partial differential equation s named after mile Borel . Suppose math U math is an open set in the Euclidean space R sup n sup , and suppose that math f 0, f 1, ... math is a sequence of smooth function smooth , complex number complex valued function mathematics functions on math U math . Then there exists a smooth function math F F t,x math defined on R × math U math with complex values, such that math left frac partial k partial t k F right 0,x f k x , math for all math k 0,1,... math , and math x math in math U. math A constructive proof of this result is given in Golubitsky and Guillemin 1974 . References M. Golubitsky, V. Guillemin 1974 . Stable mappings and their singularities . Springer Verlag, Graduate texts in Mathematics Vol. 14. ISBN 0 387 90072 1. planetmath title Borel lemma id 6185 Category Partial differential equations Category Lemmas Category Theorems in analysis eo Borela lemo fr Th or me de Borel ... more details
In mathematics , particularly in set theory , Fodor s lemma states the following If math kappa math is a Regular cardinal regular , uncountable Cardinal number cardinal , math S math is a stationary set stationary subset of math kappa math , and math f S rightarrow kappa math is regressive that is, math f alpha alpha math for any math alpha in S math , math alpha neq 0 math then there is some math gamma math and some stationary math S 0 subseteq S math such that math f alpha gamma math for any math alpha in S 0 math . In modern parlance, the nonstationary ideal is normal . Proof We can assume that math 0 notin S math by removing 0, if necessary . If Fodor s lemma is false, for every math alpha kappa math there is some club set math C alpha math such that math C alpha cap f 1 alpha emptyset math . Let math C Delta alpha kappa C alpha math . The club sets are closed under diagonal intersection , so math C math is also club and therefore there is some math alpha in S cap C math . Then math alpha in C beta math for each math beta alpha math , and so there can be no math beta alpha math such that math alpha in f 1 beta math , so math f alpha geq alpha math , a contradiction . The lemma was first proved by the Hungarian set theorist, G za Fodor mathematician G za Fodor in 1956. It is sometimes also called The Pressing Down Lemma . Fodor s lemma also holds for Thomas Jech s notion of stationary sets as well as for the Stationary set Generalized notion general notion of stationary set. References G. Fodor, Eine Bemerkung zur Theorie der regressiven Funktionen, Acta Scientiarum Mathematicarum Acta Sci. Math. Szeged , 17 1956 , 139 142. Karel Hrbacek & Thomas Jech, Introduction to Set Theory , 3rd edition, Chapter 11, Section 3. Mark Howard, Applications of Fodor s Lemma to Vaught s Conjecture . Ann. Pure and Appl. Logic 42 1 1 19 1989 . Simon Thomas, The Automorphism Tower Problem . PostScript file at http www.math.rutgers.edu sthomas book.ps planetmath id 3232 title Fodor s lemma ... more details
In stability theory and nonlinear control , Massera s lemma , named after Jos Luis Massera , deals with the construction of the Lyapunov function to prove the stability of a dynamical system . ref name Khalil2001 Citation author Khalil, H.K. year 2001 title Nonlinear Systems publisher Prentice Hall isbn 0130673897 ref The lemma appears in harv Massera 1949 p 716 as the first lemma in section 12, and in more general form in harv Massera 1956 p 195 as lemma 2. In 2004, Massera s original lemma for single variable functions was extended to the multivariable case, and the resulting lemma was used to prove the stability of switched dynamical systems, where a common Lyapunov function describes the stability of multiple modes and switching signals. Massera s original lemma Massera s lemma is used in the construction of a converse Lyapunov function of the following form also known as the integral construction math V zeta int 0 infty G varphi t, zeta dt math for an asymptotically stable dynamical system whose stable trajectory starting from math zeta text is varphi t, zeta math The lemma states blockquote Let math g 0, infty rightarrow R math be a positive, continuous, strictly decreasing function with math g t rightarrow 0 math as math t rightarrow infty math . Let math h 0, infty rightarrow R math be a positive, continuous, nondecreasing function. Then there exists a function math G 0 .... math blockquote Extension to multivariable functions Massera s lemma for single variable functions ... id MathSciNet id 0035354 year 1949 journal Annals of Mathematics Annals of Mathematics. Second Series volume 50 pages 705 721 issue 3 publisher Annals of Mathematics jstor 1969558 Citation last1 Massera ... 1969955 id MathSciNet id 0079179 year 1956 journal Annals of Mathematics Annals of Mathematics. Second Series volume 64 pages 182 206 issue 1 publisher Annals of Mathematics jstor 1969955 Citation ... Pure and Applied Mathematics, Vol. 21 id MathSciNet id 0212324 year 1966 Footnotes reflist Category ... more details
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