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Prolongable Morphism
In mathematics and computer science, a morphic word or substitutive word is an infinite sequence of symbols which is constructed from a particular class of endomorphism of a free monoid. Every automatic sequence is morphic. Definition Let ''f'' be an endomorphism of the free monoid ''A''∗ on an alphabet ''A'' with the property that there is a letter ''a'' such that ''f''(''a'') = ''as'' for a non-empty string ''s'': we say that ''f'' is prolongable at ''a''. The word : a s f(s) f(f(s)) \cdots f^(s) \cdots \ is a pure morphic or pure substitutive word. Note that it is the limit of the sequence ''a'', ''f''(''a''), ''f''(''f''(''a'')), ''f''(''f''(''f''(''a''))), ... It is clearly a fixed point of the endomorphism ''f'': the unique such sequence beginning with the letter ''a''.Lothaire (2011) p. 10Honkala (2010) p.505 In general, a morphic word is the image of a pure morphic word under a coding, that is, a morphism that maps letter to letter. If a morphic word is c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set ''S'' to itself. In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms a monoid, the full transformation monoid, and denoted (or to emphasize the category ). Automorphisms An invertible endomorphism of is called an automorphism. The set of all automorphisms is a subset of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication: Endomorphism rings Any two endomorphisms of an abelian group, , can be added toge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturmian Word
In mathematics, a Sturmian word (Sturmian sequence or billiard sequence), named after Jacques Charles François Sturm, is a certain kind of infinitely long sequence of characters. Such a sequence can be generated by considering a game of English billiards on a square table. The struck ball will successively hit the vertical and horizontal edges labelled 0 and 1 generating a sequence of letters. This sequence is a Sturmian word. Definition Sturmian sequences can be defined strictly in terms of their combinatoric properties or geometrically as cutting sequences for lines of irrational slope or codings for irrational rotations. They are traditionally taken to be infinite sequences on the alphabet of the two symbols 0 and 1. Combinatorial definitions Sequences of low complexity For an infinite sequence of symbols ''w'', let ''σ''(''n'') be the complexity function of ''w''; i.e., ''σ''(''n'') = the number of distinct contiguous subwords (factors) in ''w'' o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup Theory
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
European Journal Of Combinatorics
European, or Europeans, or Europeneans, may refer to: In general * ''European'', an adjective referring to something of, from, or related to Europe ** Ethnic groups in Europe ** Demographics of Europe ** European cuisine, the cuisines of Europe and other Western countries * ''European'', an adjective referring to something of, from, or related to the European Union ** Citizenship of the European Union ** Demographics of the European Union In publishing * ''The European'' (1953 magazine), a far-right cultural and political magazine published 1953–1959 * ''The European'' (newspaper), a British weekly newspaper published 1990–1998 * ''The European'' (2009 magazine), a German magazine first published in September 2009 *''The European Magazine'', a magazine published in London 1782–1826 *''The New European'', a British weekly pop-up newspaper first published in July 2016 Other uses * * Europeans (band), a British post-punk group, from Bristol See also * * * Europe (disam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valérie Berthé
Valérie Berthé (born 16 December 1968) is a French mathematician who works as a director of research for the Centre national de la recherche scientifique (CNRS) at the Institut de Recherche en Informatique Fondamentale (IRIF), a joint project between CNRS and Paris Diderot University. Her research involves symbolic dynamics, combinatorics on words, discrete geometry, numeral systems, tessellations, and fractals. Education Berthé completed her baccalauréat at age 16, and studied at the École Normale Supérieure from 1988 to 1993. She earned a licentiate and master's degree in pure mathematics from Pierre and Marie Curie University in 1989, a Diplôme d'études approfondies from University of Paris-Sud in 1991, completed her agrégation in 1992, and was recruited by CNRS in 1993. Continuing her graduate studies, she defended a doctoral thesis in 1994 at the University of Bordeaux 1. Her dissertation, ''Fonctions de Carlitz et automates: Entropies conditionnelles'' was supervis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wojciech Szpankowski
Wojciech Szpankowski (born February 18, 1952 in Wapno) is the Saul Rosen Professor of Computer Science at Purdue University. He is known for his work in analytic combinatorics, analysis of algorithms and analytic information theory. He is the director of the NSF Science and Technology Center for Science of Information. Biography Szpankowski received his MS and PhD in Electrical Engineering and Computer Science from the Technical University of Gdańsk in 1970 and 1980 respectively. Awards and honors * Fellow of IEEE * The Erskine Fellow * Flajolet Lecture Prize The Philippe Flajolet Lecture Prize is awarded to for contributions to analytic combinatorics and analysis of algorithms, in the fields of theoretical computer science. This prize is named in memory of Philippe Flajolet. History The Flajolet Le ... References {{DEFAULTSORT:Szpankowski, Wojciech Living people Polish computer scientists Polish mathematicians Theoretical computer scientists Information theoris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Waterman
Michael Spencer Waterman (born June 28, 1942) is a Professor of Biology, Mathematics and Computer Science at the University of Southern California (USC), where he holds an Endowed Associates Chair in Biological Sciences, Mathematics and Computer Science. He previously held positions at Los Alamos National Laboratory and Idaho State University. Education and early life Waterman grew up near Bandon, Oregon, and earned a bachelor's degree in Mathematics from Oregon State University, followed by a PhD in statistics and probability from Michigan State University in 1969. Research and career Waterman is one of the founders and current leaders in the area of computational biology. He focuses on applying mathematics, statistics, and computer science techniques to various problems in molecular biology. His work has contributed to some of the most widely used tools in the field. In particular, the Smith-Waterman algorithm (developed with Temple F. Smith) is the basis for many sequence al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sophie Schbath
Sophie Schbath (born 1969) is a French statistician whose research concerns the statistics of pattern matching in strings and formal languages, particularly as applied to genomics. She is a director of research for the French National Institute for Research in Agriculture, Food, and Environment (INRAE), and a former president of the French BioInformatics Society. Education and career Schbath was born on 19 December 1969 in Nantes. She earned a master's degree in stochastic modeling and statistics in 1992 from Paris-Sud University, and completed a Ph.D. in 1995 at Paris Descartes University. Her dissertation was ''Étude asymptotique du nombre d'occurrences d'un mot dans une chaîne de Markov et application à la recherche de mots de fréquence exceptionnelle dans les séquences d'ADN''. She earned a habilitation in 2003 at the University of Évry Val d'Essonne. After postdoctoral research in 1996 at the University of Southern California, Schbath became a researcher for INRAE. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gesine Reinert
Gesine Reinert is a University Professor in Statistics at the University of Oxford. She is a Fellow of Keble College, Oxford, a Fellow of the Alan Turing Institute, and a Fellow of the Institute of Mathematical Statistics. Her research concerns the probability theory and statistics of biological sequences and biological networks. Reinert has also been associated with the M. Lothaire pseudonymous mathematical collaboration on combinatorics on words. Education Reinert earned a diploma in mathematics from the University of Göttingen in 1989. She went on to graduate study in applied mathematics at the University of Zurich, completing her Ph.D. in 1994. Her dissertation, in probability theory, was ''A Weak Law of Large Numbers for Empirical Measures via Stein's Method, and Applications'', and was supervised by Andrew Barbour. Career Reinert worked as a lecturer at the University of Southern California from 1994 to 1996 and the University of California, Los Angeles from 1996 to 1998 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mehryar Mohri
Mehryar Mohri is a Professor and theoretical computer scientist at the Courant Institute of Mathematical Sciences. He is also a Research Director at Google Research where he heads the Learning Theory team. Career Prior to joining the Courant Institute, Mohri was a Research Department Head and later Technology Leader at AT&T Bell Labs, where he was a Member of the Technical Staff for about ten years. Mohri has also taught as an Assistant Professor at the University of Paris 7 (1992-1993) and Ecole Polytechnique (1992-1994). Research Mohri's main area of research is machine learning, in particular learning theory. He is also an expert in automata theory and algorithms. He is the author of several core algorithms that have served as the foundation for the design of many deployed speech recognition and natural language processing systems. Publications Mohri is the author of the reference book Foundations of Machine Learning used as a textbook in many graduate-level machine learni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxime Crochemore
Maxime Crochemore (born 1947) is a French computer scientist known for his numerous contributions to algorithms on strings. He is currently a professor at King's College London. Biography Crochemore earned his doctorate (PhD) in 1978 and his Doctorat d'état (DSc) in 1983 from the University of Rouen. He was a professor at Paris 13 University in 1985–1989, and moved to a professorship at Paris Diderot University in 1989. In 2002–2007, Crochemore was a senior research fellow at King's College London, where he is a professor since 2007. Since 2007, he is also a professor emeritus at the University of Marne-la-Vallée. Crochemore holds an honorary doctorate (2014) from the University of Helsinki. A festschrift in his honour was published in 2009 as a special issue of Theoretical Computer Science. Research contributions Crochemore published over 100 journal papers on string algorithms. He in particular introduced new algorithms for pattern matching, string indexing and text com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Berstel
Jean Berstel (born 1941) is a French mathematician and theoretical computer scientist known for his contributions to combinatorics on words and formal language theory. He is a currently a professor emeritus at the University of Marne-la-Vallée. Biography Berstel earned his doctorate (doctorat d'État) at Paris Diderot University in 1973. In 1973–1995 he was a professor at Pierre and Marie Curie University, and in 1995–2005 a professor at the University of Marne-la-Vallée, where he has been a professor emeritus since 2005. In 2006, Berstel was awarded an honorary doctorate from the University of Turku, Finland. A festschrift in his honour was published in 2003 as a special issue of ''Theoretical Computer Science''. Research contributions Berstel has been a member of the Lothaire group of mathematicians that developed the foundations of combinatorics of words. He has published several scientific monographs, including ''Transductions and Context-free Languages'' (1979), ''The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
