Academic literature on the topic 'Lemmas'

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Journal articles on the topic "Lemmas"

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Schecter, Stephen. "Exchange lemmas 1: Deng's lemma." Journal of Differential Equations 245, no. 2 (July 2008): 392–410. http://dx.doi.org/10.1016/j.jde.2007.08.011.

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Schecter, Stephen. "Exchange lemmas 2: General Exchange Lemma." Journal of Differential Equations 245, no. 2 (July 2008): 411–41. http://dx.doi.org/10.1016/j.jde.2007.10.021.

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Sivaraman, Aishwarya, Alex Sanchez-Stern, Bretton Chen, Sorin Lerner, and Todd Millstein. "Data-driven lemma synthesis for interactive proofs." Proceedings of the ACM on Programming Languages 6, OOPSLA2 (October 31, 2022): 505–31. http://dx.doi.org/10.1145/3563306.

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Interactive proofs of theorems often require auxiliary helper lemmas to prove the desired theorem. Existing approaches for automatically synthesizing helper lemmas fall into two broad categories. Some approaches are goal-directed, producing lemmas specifically to help a user make progress from a given proof state, but they have limited expressiveness in terms of the lemmas that can be produced. Other approaches are highly expressive, able to generate arbitrary lemmas from a given grammar, but they are completely undirected and hence not amenable to interactive usage. In this paper, we develop an approach to lemma synthesis that is both goal-directed and expressive. The key novelty is a technique for reducing lemma synthesis to a data-driven program synthesis problem, whereby examples for synthesis are generated from the current proof state. We also describe a technique to systematically introduce new variables for lemma synthesis, as well as techniques for filtering and ranking candidate lemmas for presentation to the user. We implement these ideas in a tool called lfind, which can be run as a Coq tactic. In an evaluation on four benchmark suites, lfind produces useful lemmas in 68% of the cases where a human prover used a lemma to make progress. In these cases lfind synthesizes a lemma that either enables a fully automated proof of the original goal or that matches the human-provided lemma.
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Wei, Longxing. "The Bilingual Mental Lexicon and Lemmatic Transfer in Second Language Learning." English Language Teaching and Linguistics Studies 2, no. 3 (August 31, 2020): p43. http://dx.doi.org/10.22158/eltls.v2n3p43.

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There have been numerous studies of first Language (L1) transfer in second Language (L2) learning. Various models have been proposed to explore the sources of language transfer and have also caused many controversies over the nature of language transfer and its effects on interlanguage. Different from most previous studies remaining at a surface level of observation, this study proposes an abstract approach, which is abstract because it goes beyond any superficial observation and description by exploring the nature and activity of the bilingual mental lexicon in L2 learning. This approach adopts the Bilingual Lemma Activation Model (BLAM) (Wei, 2006a, 2006b) and tests its crucial assumptions and claims: The bilingual mental lexicon does not simply contain lexemes but abstract entries, called “lemmas”, about them; lemmas in the bilingual mental lexicon are language-specific; language-specific lemmas in the bilingual mental lexicon are in contact in L2 learning, lemmas underlying L1 abstract lexical structure may replace those underlying L2 abstract lexical structure. Lemmas in the bilingual mental lexicon are about three levels of abstract lexical structure: lexical-conceptual structure, predicate-argument structure, and morphological realization patterns. The typical instances of L1 lemma transfer in L2 learning are discussed and explained in support of the BLAM.
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Anosov, D. V., and E. V. Zhuzhoma. "Closing lemmas." Differential Equations 48, no. 13 (December 2012): 1653–99. http://dx.doi.org/10.1134/s0012266112130010.

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Szemerédi, Endre. "Arithmetic Progressions, Different Regularity Lemmas and Removal Lemmas." Communications in Mathematics and Statistics 3, no. 3 (September 2015): 315–28. http://dx.doi.org/10.1007/s40304-015-0062-1.

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Lungu, Nicolaie, and Sorina Anamaria Ciplea. "Optimal Gronwall lemmas." Fixed Point Theory 18, no. 1 (March 1, 2017): 293–304. http://dx.doi.org/10.24193/fpt-ro.2017.1.23.

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Carbery, Anthony. "Covering lemmas revisited." Proceedings of the Edinburgh Mathematical Society 31, no. 1 (February 1988): 145–50. http://dx.doi.org/10.1017/s0013091500006647.

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In this note we intend to discuss the method of A. Córdoba and R. Fefferman of using covering lemmas to control maximal functions, and make some simplifications which allow us to obtain alternative proofs of some of their results.
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Gasser, I., P. A. Markowich, and B. Perthame. "Dispersion Lemmas Revisited." VLSI Design 9, no. 4 (January 1, 1999): 365–75. http://dx.doi.org/10.1155/1999/81341.

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We investigate regularizing dispersive effects for various classical equations, e.g., the Schrödinger and Dirac equations. After Wigner transform, these dispersive estimates are reduced to moment lemmas for kinetic equations. They yield new regularization results for the Schrödinger equation (valid up to the semiclassical limit) and the Dirac equation.
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Sun, Shu-Hao. "On separation lemmas." Journal of Pure and Applied Algebra 78, no. 3 (April 1992): 301–10. http://dx.doi.org/10.1016/0022-4049(92)90112-s.

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Dissertations / Theses on the topic "Lemmas"

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JACHELLI, KEILLA LOPES CASTILHO. "SPERNER S LEMMAS AND APPLICATIONS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2017. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=33127@1.

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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE MESTRADO PROFISSIONAL EM MATEMÁTICA EM REDE NACIONAL
Esse trabalho visa demonstrar os lemas de Sperner e aplicá-los nasdemonstrações do teorema de Monsky em Q2 e do teorema do ponto fixo deBrouwer em R2. Além disso, relatamos como esses lemas foram abordados com alunos da educação básica tendo como ferramenta educacional jogos de tabuleiro.
This work aims to prove the Sperner s Lemmas and to apply them in proving the Monsky s Theorem in Q2 and the Brouwer fixed point Theorem in R2. Moreover, we report how these lemmas were addressed with students in basic education using board games as educational tools.
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Pfeiffer, Markus Johannes. "Adventures in applying iteration lemmas." Thesis, University of St Andrews, 2013. http://hdl.handle.net/10023/3671.

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The word problem of a finitely generated group is commonly defined to be a formal language over a finite generating set. The class of finite groups has been characterised as the class of finitely generated groups that have word problem decidable by a finite state automaton. We give a natural generalisation of the notion of word problem from finitely generated groups to finitely generated semigroups by considering relations of strings. We characterise the class of finite semigroups by the class of finitely generated semigroups whose word problem is decidable by finite state automata. We then examine the class of semigroups with word problem decidable by asynchronous two tape finite state automata. Algebraic properties of semigroups in this class are considered, towards an algebraic characterisation. We take the next natural step to further extend the classes of semigroups under consideration to semigroups that have word problem decidable by a finite collection of asynchronous automata working independently. A central tool used in the derivation of structural results are so-called iteration lemmas. We define a hierarchy of the considered classes of semigroups and connect our original results with previous research.
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Johansson, Moa. "Automated discovery of inductive lemmas." Thesis, University of Edinburgh, 2009. http://hdl.handle.net/1842/9807.

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The discovery of unknown lemmas, case-splits and other so called eureka steps are challenging problems for automated theorem proving and have generally been assumed to require user intervention. This thesis is mainly concerned with the automated discovery of inductive lemmas. We have explored two approaches based on failure recovery and theory formation, with the aim of improving automation of firstand higher-order inductive proofs in the IsaPlanner system. We have implemented a lemma speculation critic which attempts to find a missing lemma using information from a failed proof-attempt. However, we found few proofs for which this critic was applicable and successful. We have also developed a program for inductive theory formation, which we call IsaCoSy. IsaCoSy was evaluated on different inductive theories about natural numbers, lists and binary trees, and found to successfully produce many relevant theorems and lemmas. Using a background theory produced by IsaCoSy, it was possible for IsaPlanner to automatically prove more new theorems than with lemma speculation. In addition to the lemma discovery techniques, we also implemented an automated technique for case-analysis. This allows IsaPlanner to deal with proofs involving conditionals, expressed as if- or case-statements.
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Lovász, László Miklós. "Regularity and removal lemmas and their applications." Thesis, Massachusetts Institute of Technology, 2017. http://hdl.handle.net/1721.1/112899.

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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 123-127).
In this thesis, we analyze the regularity method pioneered by Szemerédi, and also discuss one of its prevalent applications, the removal lemma. First, we prove a new lower bound on the number of parts required in a version of Szemerédi's regularity lemma, determining the order of the tower height in that version up to a constant factor. This addresses a question of Gowers. Next, we turn to algorithms. We give a fast algorithmic Frieze-Kannan (weak) regularity lemma that improves on previous running times. We use this to give a substantially faster deterministic approximation algorithm for counting subgraphs. Previously, only an exponential dependence of the running time on the error parameter was known; we improve it to a polynomial dependence. We also revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for some co-NP-complete problems. We show how to use the Frieze-Kannan regularity lemma to approximate the regularity of a pair of vertex sets. We also show how to quickly find, for each [epsilon]' > [epsilon], an [epsilon]'-regular partition with k parts if there exists an [epsilon]-regular partition with k parts. After studying algorithms, we turn to the arithmetic setting. Green proved an arithmetic regularity lemma, and used it to prove an arithmetic removal lemma. The bounds obtained, however, were tower-type, and Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. The previous best known bound was tower-type with a logarithmic tower height. We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in Fn/p. Finally, we give a new proof of a regularity lemma for permutations, improving the previous tower-type bound on the number of parts to an exponential bound.
by László Miklós Lovász.
Ph. D.
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Michael, Ifeanyi Friday. "On a unified categorical setting for homological diagram lemmas." Thesis, Stellenbosch : Stellenbosch University, 2011. http://hdl.handle.net/10019.1/18085.

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Thesis (MSc)--Stellenbosch University, 2011.
ENGLISH ABSTRACT: Some of the diagram lemmas of Homological Algebra, classically known for abelian categories, are not characteristic of the abelian context; this naturally leads to investigations of those non-abelian categories in which these diagram lemmas may hold. In this Thesis we attempt to bring together two different directions of such investigations; in particular, we unify the five lemma from the context of homological categories due to F. Borceux and D. Bourn, and the five lemma from the context of modular semi-exact categories in the sense of M. Grandis.
AFRIKAANSE OPSOMMING: Verskeie diagram lemmata van Homologiese Algebra is aanvanklik ontwikkel in die konteks van abelse kategorieë, maar geld meer algemeen as dit behoorlik geformuleer word. Dit lei op ’n natuurlike wyse na ’n ondersoek van ander kategorieë waar hierdie lemmas ook geld. In hierdie tesis bring ons twee moontlike rigtings van ondersoek saam. Dit maak dit vir ons moontlik om die vyf-lemma in die konteks van homologiese kategoieë, deur F. Borceux en D. Bourn, en vyflemma in die konteks van semi-eksakte kategorieë, in die sin van M. Grandis, te verenig.
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Nunes, Alexmay Soares. "As permutaÃÃes caÃticas, o problema de Lucas e a teoria dos permanentes." Universidade Federal do CearÃ, 2015. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=15598.

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In this work we cover some counting techniques used to solve some classic problems in Combinatorics. We also show a link between the so called ârencontre problemâ, the âmÃnage problemâ and the permanent of a square matrix.
Neste trabalho abordamos algumas tÃcnicas de contagem utilizadas para solucionar alguns problemas clÃssicos da AnÃlise CombinatÃria. Mostramos tambÃm uma relaÃÃo entre o problema das cartas mal endereÃadas, o problema de Lucas e os permanentes de uma matriz quadrada.
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Edmundo, Mario Jorge. "O-minimal expansions of groups." Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.312447.

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Aimino, Romain. "Vitesse de mélange et théorèmes limites pour les systèmes dynamiques aléatoires et non-autonomes." Thesis, Toulon, 2014. http://www.theses.fr/2014TOUL0005/document.

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Dans cette thèse, nous nous intéressons aux propriétés statistiques des systèmes dynamiques aléatoires et non-autonomes. Dans le premier chapitre, consacré aux systèmes aléatoires, nous établissons un cadre fonctionnel abstrait, couvrant une large classe de systèmes dilatants en dimension 1 et supérieure, permettant de démontrer de nombreux théorèmes limites annealed. Nous donnons aussi une condition nécessaire et suffisante pour que la version quenched du théorème de la limite centrale soit valide en dimension 1. Dans le chapitre deux, après avoir introduit la notion de système non-autonome, nous étudions un système composé d'applications en dimension 1 ayant un point fixe neutre commun, et nous montrons que celui-ci admet une vitesse de perte de mémoire polynomiale. Le chapitre trois est consacré aux inégalités de concentration. Nous établissons de telles inégalités pour des systèmes dynamiques aléatoires et non-autonomes, et nous étudions diverses applications. Dans le chapitre quatre, nous nous intéressons aux lemmes dynamiques de Borel-Cantelli pour l'induction de Rauzy-Veech-Zorich, et présentons quelques résultats liés aux statistiques de récurrence pour cette application
The first chapter, devoted to random systems, we establish an abstract functional framework, including a large class of expanding systems in dimension 1 and higher, under which we can prove annealed limit theorems. We also give a necessary and sufficient condition for the quenched central limit theorem to hold in dimension 1. In chapter 2, after an introduction to the notion of non-autonomous system, we study an example consisting of a family of maps of the unit interval with a common neutral fixed point, and we show that this system admits a polynomial loss of memory. The chapter 3 is devoted to concentration inequalities. We establish such inequalities for random and non-autonomous dynamical systems in dimension 1, and we study some of their applications. In chapter 4, we study dynamical Borel-Cantelli lemmas for the Rauzy-Veech-Zorich induction, and we present some results concerning statistics of recurrence for this map
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Cobra, Thiago Taglialatela [UNESP]. "Sobre coincidências e pontos fixos de aplicações." Universidade Estadual Paulista (UNESP), 2010. http://hdl.handle.net/11449/94372.

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Made available in DSpace on 2014-06-11T19:27:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-12-09Bitstream added on 2014-06-13T20:47:43Z : No. of bitstreams: 1 cobra_tt_me_rcla.pdf: 485593 bytes, checksum: 107d36859b5a9c932411b3a54094c4ac (MD5)
O principal objetivo deste trabalho é apresentar conceitos básicos sobre coincidências e pontos fixos de aplicações contínuas usando como ferramentas os Lemas Combinatórios de Sperner e grau de aplicações. Apresentamos também um cálculo do número de Lefschetz de f; g : T2 ¡! T3, onde Th denota uma superfície de genus h, através da fórmula dada por Gonçalves e Oliveira em [3]
The main goal of this work is present basic concepts on coincidences and fixed points of continuous maps with Sperner’s Combinatorial Lemmas, and degree maps approaches. We also present a calculation of the Lefschetz number of f; g : T2 ¡! T3, where Th denotes surface of genus h, by using the formula given by Gonçalves and Oliveira in [3]
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Cobra, Thiago Taglialatela. "Sobre coincidências e pontos fixos de aplicações /." Rio Claro : [s.n.], 2010. http://hdl.handle.net/11449/94372.

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Orientador: Alice Kimie Miwa Libardi
Banca: Edson de Oliveira
Banca: Thiago de Melo
Resumo: O principal objetivo deste trabalho é apresentar conceitos básicos sobre coincidências e pontos fixos de aplicações contínuas usando como ferramentas os Lemas Combinatórios de Sperner e grau de aplicações. Apresentamos também um cálculo do número de Lefschetz de f; g : T2 ¡! T3, onde Th denota uma superfície de genus h, através da fórmula dada por Gonçalves e Oliveira em [3]
Abstract: The main goal of this work is present basic concepts on coincidences and fixed points of continuous maps with Sperner's Combinatorial Lemmas, and degree maps approaches. We also present a calculation of the Lefschetz number of f; g : T2 ¡! T3, where Th denotes surface of genus h, by using the formula given by Gonçalves and Oliveira in [3]
Mestre
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Books on the topic "Lemmas"

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Dragomir, Sever Silvestru. The Gronwall type lemmas and applications. Timișoara: Tipografia Universității din Timișoara, 1987.

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Rossi, Richard J. Theorems, Corollaries, Lemmas, and Methods of Proof. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2006. http://dx.doi.org/10.1002/9781118031575.

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Lalitte, Scott, ed. Lemons! lemons! lemons! New York, NY: Viking, 1994.

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The Schwarz lemma. Oxford: Clarendon Press, 1989.

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Dineen, Seán. The Schwarz lemma. Mineola, New York: Dover Publications, 2016.

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Tziatzi, Maria, Margarethe Billerbeck, Franco Montanari, and Kyriakos Tsantsanoglou, eds. Lemmata. Berlin, München, Boston: DE GRUYTER, 2015. http://dx.doi.org/10.1515/9783110354348.

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Idone, Christopher. Lemons. San Francisco: CollinsPublishers San Francisco, 1993.

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Savage, Melissa. Lemons. New York: Crown Children's Books, 2017.

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service), SpringerLink (Online, ed. The Borel-Cantelli Lemma. India: Springer India, 2012.

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Chandra, Tapas Kumar. The Borel-Cantelli Lemma. India: Springer India, 2012. http://dx.doi.org/10.1007/978-81-322-0677-4.

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Book chapters on the topic "Lemmas"

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Pfanzagl, Johann. "Lemmas." In Asymptotic Expansions for General Statistical Models, 451–86. New York, NY: Springer New York, 1985. http://dx.doi.org/10.1007/978-1-4615-6479-9_14.

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Łojasiewicz, Stanisław. "Fundamental Lemmas." In Introduction to Complex Analytic Geometry, 178–202. Basel: Birkhäuser Basel, 1991. http://dx.doi.org/10.1007/978-3-0348-7617-9_6.

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Simpson, Carlos. "Finiteness lemmas." In Asymptotic Behavior of Monodromy, 60–67. Berlin, Heidelberg: Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/bfb0094558.

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Herzog, Bernd. "Basic lemmas." In Kodaira-Spencer Maps in Local Algebra, 18–29. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/bfb0074028.

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Gorenstein, Daniel, Richard Lyons, and Ronald Solomon. "General lemmas." In Mathematical Surveys and Monographs, 1–18. Providence, Rhode Island: American Mathematical Society, 1999. http://dx.doi.org/10.1090/surv/040.4/01.

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Gorenstein, Daniel, Richard Lyons, and Ronald Solomon. "General lemmas." In The Classification of the Finite Simple Groups, Number 6, 13–59. Providence, Rhode Island: American Mathematical Society, 2004. http://dx.doi.org/10.1090/surv/040.6/02.

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Feldmeier, Achim. "Arithmetic Lemmas." In Introduction to Arnold's Proof of the Kolmogorov–Arnold–Moser Theorem, 175–84. Boca Raton: CRC Press, 2022. http://dx.doi.org/10.1201/9781003287803-8.

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Feldmeier, Achim. "Analytic Lemmas." In Introduction to Arnold's Proof of the Kolmogorov–Arnold–Moser Theorem, 115–26. Boca Raton: CRC Press, 2022. http://dx.doi.org/10.1201/9781003287803-5.

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Feldmeier, Achim. "Convergence Lemmas." In Introduction to Arnold's Proof of the Kolmogorov–Arnold–Moser Theorem, 161–74. Boca Raton: CRC Press, 2022. http://dx.doi.org/10.1201/9781003287803-7.

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Feldmeier, Achim. "Geometric Lemmas." In Introduction to Arnold's Proof of the Kolmogorov–Arnold–Moser Theorem, 127–60. Boca Raton: CRC Press, 2022. http://dx.doi.org/10.1201/9781003287803-6.

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Conference papers on the topic "Lemmas"

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McLaughlin, Craig, James McKinna, and Ian Stark. "Triangulating context lemmas." In CPP '18: Certified Proofs and Programs. New York, NY, USA: ACM, 2018. http://dx.doi.org/10.1145/3167081.

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McLaughlin, Craig, James McKinna, and Ian Stark. "Triangulating context lemmas." In the 7th ACM SIGPLAN International Conference. New York, New York, USA: ACM Press, 2018. http://dx.doi.org/10.1145/3176245.3167081.

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Verdu, Sergio. "Non-asymptotic covering lemmas." In 2015 IEEE Information Theory Workshop (ITW). IEEE, 2015. http://dx.doi.org/10.1109/itw.2015.7133173.

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Johnsonbaugh, Richard, and David P. Miller. "Converses of pumping lemmas." In the twenty-first SIGCSE technical symposium. New York, New York, USA: ACM Press, 1990. http://dx.doi.org/10.1145/323410.319073.

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Preiner, Mathias, Aina Niemetz, and Armin Biere. "Better lemmas with lambda extraction." In 2015 Formal Methods in Computer-Aided Design (FMCAD). IEEE, 2015. http://dx.doi.org/10.1109/fmcad.2015.7542262.

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Gishboliner, Lior, and Asaf Shapira. "Removal lemmas with polynomial bounds." In STOC '17: Symposium on Theory of Computing. New York, NY, USA: ACM, 2017. http://dx.doi.org/10.1145/3055399.3055404.

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Smith, Therese, and Robert McCartney. "Mathematization in teaching pumping lemmas." In 2013 IEEE Frontiers in Education Conference (FIE). IEEE, 2013. http://dx.doi.org/10.1109/fie.2013.6685122.

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Bansal, Nikhil, and Ryan Williams. "Regularity Lemmas and Combinatorial Algorithms." In 2009 IEEE 50th Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2009. http://dx.doi.org/10.1109/focs.2009.76.

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Kato, Ryo, Naohisa Nishida, Ryo Hirano, Tatusmi Oba, Yuji Unagami, Shota Yamada, Tadanori Teruya, Nuttapong Attrapadung, Takahiro Matsuda, and Goichiro Hanaoka. "Embedding Lemmas for Functional Encryption." In 2018 International Symposium on Information Theory and Its Applications (ISITA). IEEE, 2018. http://dx.doi.org/10.23919/isita.2018.8664231.

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Chattopadhyay, Eshan, Pooya Hatami, Kaave Hosseini, Shachar Lovett, and David Zuckerman. "XOR lemmas for resilient functions against polynomials." In STOC '20: 52nd Annual ACM SIGACT Symposium on Theory of Computing. New York, NY, USA: ACM, 2020. http://dx.doi.org/10.1145/3357713.3384242.

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Reports on the topic "Lemmas"

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Sowers, R. Stein's Lemma - A Large Deviations Approach. Fort Belvoir, VA: Defense Technical Information Center, March 1989. http://dx.doi.org/10.21236/ada209090.

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Block, Henry W., and Zhaoben Fang. A Multivariate Extension of Hoeffding's Lemma. Fort Belvoir, VA: Defense Technical Information Center, November 1985. http://dx.doi.org/10.21236/ada170170.

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Hoover, Donald R. An Improved First Borel-Cantelli Lemma. Fort Belvoir, VA: Defense Technical Information Center, July 1991. http://dx.doi.org/10.21236/ada239489.

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Gibbons, Robert, and Lawrence Katz. Layoffs and Lemons. Cambridge, MA: National Bureau of Economic Research, May 1989. http://dx.doi.org/10.3386/w2968.

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House, Christopher, and Jing Zhang. Layoffs, Lemons and Temps. Cambridge, MA: National Bureau of Economic Research, March 2012. http://dx.doi.org/10.3386/w17962.

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Stewart, John, Ron Halbgewachs, Adrian Chavez, Rhett Smith, and David Teumim. Lemnos Interoperable Security Program. Office of Scientific and Technical Information (OSTI), January 2012. http://dx.doi.org/10.2172/1051509.

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Hu, Luojia, and Christopher Taber. Layoffs, Lemons, Race, and Gender. Cambridge, MA: National Bureau of Economic Research, July 2005. http://dx.doi.org/10.3386/w11481.

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Tapia, R. A., and M. W. Trosset. Extending the Farkas Lemma Approach to Necessity Conditions to Infinite Programming. Fort Belvoir, VA: Defense Technical Information Center, May 1993. http://dx.doi.org/10.21236/ada452706.

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CIFOR. Studi komparatif global tentang REDD+: lembar fakta. Center for International Forestry Research (CIFOR), 2014. http://dx.doi.org/10.17528/cifor/004510.

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Bebchuk, Lucian Arye, and Marcel Kahan. The 'Lemons Effect' in Corporate Freeze-Outs. Cambridge, MA: National Bureau of Economic Research, February 1999. http://dx.doi.org/10.3386/w6938.

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