Like the Rubik's Cube, protein folding is a combinatorial optimization problem.
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"Sentence Order" consists of combinatorial, second-person flash fictions presented in five discrete fragments.
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Regulators within Europe have already expressed concern about this combinatorial approach to user tracking.
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Combinatorial creativity is interesting, [an AI] could learn patterns and apply them to new areas.
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By contrast, combinatorial objects like graphs and matroids are purely discrete objects—assemblages of dots and sticks.
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Here's James Vincent: Regulators within Europe have already expressed concern about this combinatorial approach to user tracking.
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There are these two traditions, to take combinatorial objects and either make them geometric or make them algebraic.
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I think we found a really beautiful connection between the geometric and the algebraic structure of combinatorial objects.
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He and his colleagues call their method sci-RNA-seq (short for single-cell combinatorial indexing RNA sequencing).
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Catalog's combinatorial approach does mean that more DNA is needed per byte stored than other DNA-based methods require.
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She and her team set out hunting for combinatorial constraints on viral assembly pathways, this time using graph theory.
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There has been a "combinatorial explosion" according to Andrus, particularly for engineering teams that have chosen a microservices architecture.
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The app's quantitative approach made cooking a simple, combinatorial thing, an equation with variables waiting to be filled in.
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But Babbitt's music, despite its use of concepts with names like superarray and all-combinatorial hexachord, sparkles with a hip lucidity.
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It's that special affinity, the theory now goes, along with the combinatorial nature of olfactory reactions, that accounts for unique scents.
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The innovation will happen when this computer-generated data can be shared safely and securely across the city to foster new combinatorial innovations.
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Some of these new settings have been combinatorial, which encouraged Huh to wonder whether relationships from Hodge theory might underlie these log concave patterns.
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Their work provides a fully combinatorial vision of Hodge theory, which in turn provides a whole new way to approach open problems in combinatorics.
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And finally, "Dream," when the concert hall fuses all this together to generate a sort of combinatorial fantasia — to "hallucinate," as Mr. Anadol put it.
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And the paper you'll be presenting at the Joint Mathematics Meetings connects two different ways of understanding combinatorial structures, through the lenses of geometry and algebra.
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More important, despite the tendency to dot his book with such daunting phrases as "combinatorial game theory" and "stochastic equations," he tells a surprisingly captivating story.
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The only problem here is that the amount of data you need is a combinatorial function of task complexity, so even slightly complex tasks can become prohibitively expensive.
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With a street light here, a sensor there — when solutions are not clearly connected and without critical mass — you don't get the combinatorial benefit of different systems working together.
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They call this "combinatorial future," where you just combine little pieces of the future that you see today, and it can turn into a new story about what's possible.
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There's combinatorial creativity, where you're taking two ideas that have nothing to do with each other to see how associations in one can help stimulate new ideas in the other.
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All that's needed are better algorithms to prove that complicated questions—such as protein folding, efficient marketplaces, and combinatorial analyses—are merely variations of simpler problems that supercomputers are already able to solve.
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That problem was posed 56 years ago by the Italian mathematician Gian-Carlo Rota, and it deals with combinatorial objects—Tinkertoy-like constructions, like graphs, which are "combinations" of points and line segments glued together.
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Combinatorial chemistry, which emerged in the 1980s and was adopted by the pharmaceutical industry in the 1990s, enabled chemists to rapidly generate immense libraries of potentially novel drugs by mixing and matching their molecular building blocks.
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While some were veterans, many were newcomers to Marvel's ever-broadening paracosm—a narrative web that already crammed more than 60 heroes into May's Avengers: Endgame, and will continue its combinatorial creep over at least the next three years.
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And the data is so complex and combinatorial — 2,000 transcription factors, in terms of regulation of those genes, they then interact in network to do the protein-protein interactions, you've got epigenetic aspects of that, you could even start adding cell microbiome effects to that later — so you've got a lot of factors that could influence the phenotype of the cell that's coming out the other end.
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The answer, in short, is more politics: a political system that is fluid and competitive; A system that leverages diversity and creates opportunity for experimentation and change; A political system that expands, not limits, the combinatorial possibilities of political innovation and deal-making; A political system that helps citizens to aggregate and realize their interests in the most efficacious ways, rather than simultaneously expecting them to be super-engaged and expert while giving them few meaningful choices.
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Topics include algebraic combinatorics, combinatorial geometry, combinatorial number theory, combinatorial optimization, designs and configurations, enumerative combinatorics, extremal combinatorics, graph theory, ordered sets, random methods, and topological combinatorics.
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Simple combinatorial problems are the ones that can be solved by applying just one combinatorial operation (variations, permutations, combinations, …). These problems can be classified into three different models, called implicit combinatorial models.
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A subset of a combinatorial cube is a smaller combinatorial cube if it can be obtained by a composition in this way.
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ACS Combinatorial Science, (usually abbreviated as ACS Comb. Sci.), formerly Journal of Combinatorial Chemistry (1999-2010), is a peer-reviewed scientific journal, published since 1999 by the American Chemical Society. ACS Combinatorial Science publishes articles, reviews, perspectives, accounts and reports in the field of Combinatorial Chemistry. JCS is currently indexed in: Chemical Abstracts Service (CAS), SCOPUS, EBSCOhost, PubMed, and Web of Science.
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The connection of the Bernoulli number to various kinds of combinatorial numbers is based on the classical theory of finite differences and on the combinatorial interpretation of the Bernoulli numbers as an instance of a fundamental combinatorial principle, the inclusion–exclusion principle.
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Combinatorial aspects of dynamical systems is another emerging field. Here dynamical systems can be defined on combinatorial objects. See for example graph dynamical system.
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A combinatorial map is a combinatorial object modelling topological structures with subdivided objects. Historically, the concept was introduced informally by J. Edmonds for polyhedral surfaces Edmonds J., A Combinatorial Representation for Polyhedral Surfaces, Notices Amer. Math. Soc., vol. 7, 1960 which are planar graphs.
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Nonnegative rank has important applications in Combinatorial optimization:Mihalis Yannakakis. Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci., 43(3):441–466, 1991.
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One application of ADS is software testing, particularly combinatorial testing. A framework has been proposed based on ADS for concurrent combinatorial testing using AR and TA.
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In combinatorial number theory, the barycentric-sum problems are questions that can be answered using combinatorial techniques. The context of barycentric-sum problems are the barycentric sequences.
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Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games. Examples include chess and go. Games that involve imperfect information may also have a strong combinatorial character, for instance backgammon. There is no unified theory addressing combinatorial elements in games.
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In statistics, combinatorial data analysis (CDA) is the study of data sets where the order in which objects are arranged is important. CDA can be used either to determine how well a given combinatorial construct reflects the observed data, or to search for a suitable combinatorial construct that does fit the data.
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Combinatorial auctions were first proposed by Rassenti, Smith, and Bulfin (1982), for the allocation of airport landing slots. Their paper introduced many key ideas on combinatorial auctions, including the mathematical programming formulation of the auctioneer’s problem, the connection between the winner determination problem and the set-packing problem, the issue of computational complexity, the use of techniques from experimental economics for testing combinatorial auctions, and consideration of issues of incentive compatibility and demand revelation in combinatorial auctions.
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The Journal of Combinatorial Theory, Series AJournal of Combinatorial Theory, Series A - Elsevier and Series B,Journal of Combinatorial Theory, Series B - Elsevier are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. Series A is concerned primarily with structures, designs, and applications of combinatorics. Series B is concerned primarily with graph and matroid theory.
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Combinatorial Chemistry & High Throughput Screening is a peer-reviewed scientific journal that covers combinatorial chemistry. It was established in 1998 and is published by Bentham Science Publishers. The editor-in-chief is Gerald H. Lushington (LiS Consulting, Lawrence, KS, USA). The journal has 5 sections: Combinatorial/ Medicinal Chemistry, Chemo/Bio Informatics, High Throughput Screening, Pharmacognosy, and Laboratory Automation.
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Combinatorial modelling is the process which lets us identify a suitable mathematical model to reformulate a problem. These combinatorial models will provide, through the combinatorics theory, the operations needed to solve the problem.
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It has been applied on both numerical and combinatorial optimization problems.
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The methods used in combinatorial chemistry are applied outside chemistry, too.
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The presentation in this article borrows somewhat from Joyal's combinatorial species.
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In mathematics, a generalized map is a topological model which allows one to represent and to handle subdivided objects. This model was defined starting from combinatorial maps in order to represent non-orientable and open subdivisions, which is not possible with combinatorial maps. The main advantage of generalized map is the homogeneity of one-to-one mappings in any dimensions, which simplifies definitions and algorithms comparing to combinatorial maps. For this reason, generalized maps are sometimes used instead of combinatorial maps, even to represent orientable closed partitions.
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Lee is the author of A First Course in Combinatorial Optimization (Cambridge University Press, 2004) and A First Course in Linear Optimization (Reex Press, 2013). He is co-editor of: Trends in Optimization (American Mathematical Society, 2004), Mixed Integer Nonlinear Programming (Springer, 2012), Integer Programming and Combinatorial Optimization (Lecture Notes in Computer Science, Vol. 8494; Springer, 2014), Special Issue: Integer Programming and Combinatorial Optimization, 2014 (Mathematical Programming, Series B. Issue 1-2, December 2015), and Combinatorial Optimization (Lecture Notes in Computer Science, Vol. 10856; Springer, 2018).
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Combinatorial and Artificial Intelligence Methods in Materials Science II, MRS Proceedings Volume 804, Fall 2004 The application of appropriate informatics tools is critical to handle, administer, and store the vast volumes of data produced.QSAR and Combinatorial Science, 24, Number 1 (February 2005) New types of Design of experiments methods have also been developed to efficiently address the large experimental spaces that can be tackled using combinatorial methods.J. N. Cawse, Ed., Experimental Design for Combinatorial and High Throughput Materials Development, John Wiley and Sons, 2002.
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Additionally, jointly with Wiktor Marek, Lipski published a monograph on Combinatorial analysis.
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Many combinatorial identities arise from double counting methods or the method of distinguished element. Generating functions and recurrence relations are powerful tools that can be used to manipulate sequences, and can describe if not resolve many combinatorial situations.
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Specifically, it is NP-hard, meaning that it is conjectured that there does not exist a polynomial-time algorithm which finds the optimal allocation. The combinatorial auction problem can be modeled as a set packing problem. Therefore, many algorithms have been proposed to find approximated solutions for combinatorial auction problem. For example, Hsieh (2010) proposed a Lagrangian relaxation approach for combinatorial reverse auction problems.
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The concept was later extended to represent higher-dimensional orientable subdivided objects. Combinatorial maps are used as efficient data structures in image representation and processing, in geometrical modeling. This model is related to simplicial complexes and to combinatorial topology. Note that combinatorial maps were extended to generalized maps that allow also to represent non-orientable objects like the Möbius strip and the Klein bottle.
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The fifth Aussois Workshop on Combinatorial Optimization in 2001 was dedicated to him.
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Various combinatorial problems have been reduced to the Chinese Postman Problem, including finding a maximum cut in a planar graph and a minimum-mean length circuit in an undirected graph.A. Schrijver, Combinatorial Optimization, Polyhedra and Efficiency, Volume A, Springer. (2002).
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Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae.
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A collection of circles and the corresponding unit disk graph Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.
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The criss-cross algorithm is a simply stated algorithm for linear programming. It was the second fully combinatorial algorithm for linear programming. The partially combinatorial simplex algorithm of Bland cycles on some (nonrealizable) oriented matroids. The first fully combinatorial algorithm was published by Todd, and it is also like the simplex algorithm in that it preserves feasibility after the first feasible basis is generated; however, Todd's rule is complicated.
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Topics in combinatorial group theory. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993.
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A combinatorial map is a boundary representation model; it represents object by its boundaries.
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Dynamical systems can be defined on combinatorial objects; see for example graph dynamical system.
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Some applications of combinatorial game theory to chess endgames were found by Elkies (1996).
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Combinatorial physics or physical combinatorics is the area of interaction between physics and combinatorics.
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Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory.
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They characterize combinatorial depth n as a condition on the number of conjugates of H intersecting in G thereby showing that combinatorial depth is finite. In more detail, one defines an ascending chain of sets of subgroups of H starting with the zero'th stage singleton set of H, the first stage intersecting H by all its conjugate subgroups, and the n'th stage is to intersect all subgroups of H in the (n-1)'st stage by all conjugates of H. Then the combinatorial depth of H in G is 2n if the n'th stage subset is equal to the (n-1)'st stage subset. For example, H is a normal subgroup of G if and only if H has combinatorial depth two in G. The minimum combinatorial depth follows from taking n to be minimum, and a technical definition of odd combinatorial depth. For example, d_c(H,G)=1 if and only if G=H C_G(H) (i.e.
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This was the first resolution of one of the Hilbert Problems. Dehn's interests later turned to topology and combinatorial group theory. In 1907 he wrote with Poul Heegaard the first book on the foundations of combinatorial topology, then known as analysis situs.
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Mei-Chu Chang is a mathematician who works in algebraic geometry and combinatorial number theory.
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Section 3.3.3 of Handbook of Discrete and Combinatorial Mathematics. Kenneth H. Rosen, ed. CRC Press. .
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The planted motif search is another motif discovery method that is based on combinatorial approach.
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Its mathematical topics include probability and statistics, graph theory and combinatorial geometry, and number theory.
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Mosheiov research field is mainly Combinatorial Optimization, and focus on various types of scheduling problems.
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The theory of combinatorial species and its extension to analytic combinatorics provide a language for describing many important combinatorial classes, constructing new classes from combinations of previously defined ones, and automatically deriving their counting sequences. For example, two combinatorial classes may be combined by disjoint union, or by a Cartesian product construction in which the objects are ordered pairs of one object from each of two classes, and the size function is the sum of the sizes of each object in the pair. These operations respectively form the addition and multiplication operations of a semiring on the family of (isomorphism equivalence classes of) combinatorial classes, in which the zero object is the empty combinatorial class, and the unit is the class whose only object is the empty set..
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Selection and identification of specific desired molecular traits (e.g. antigen response, antimicrobial response) required a selection algorithm and process. In 1991, he and his colleagues published one of the major papers in combinatorial biology—the paper described a method to generate peptides capturable to contemporary protein microarrays through the creation of synthetic peptide combinatorial libraries (SPCL). Houghten continued his work in combinatorial biology with an article in Methods, the journals section of Methods in Enzymology.
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Young diagram of a partition (5,4,1). Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. Algebraic combinatorics is continuously expanding its scope, in both topics and techniques, and can be seen as the area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant.
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Combinatorial optimization is the study of optimization on discrete and combinatorial objects. It started as a part of combinatorics and graph theory, but is now viewed as a branch of applied mathematics and computer science, related to operations research, algorithm theory and computational complexity theory.
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Hochgürtel, M.; Lehn, J.-M. Dynamic combinatorial diversity in drug discovery. In Fragment-based approaches in drug discovery; Jahnke, W., Erlanson, D. A., Ed.; Wiley-VCH: Weinheim, 2006; Chapter 16, pp 341–364. Scheme illustrating the theory of protein-directed dynamic combinatorial chemistry (DCC).
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Computational geometry applies computer algorithms to representations of geometrical objects. Discrete geometry and combinatorial geometry are about combinatorial properties of discrete collections of geometrical objects. A long-standing topic in discrete geometry is tiling of the plane. Computational geometry applies algorithms to geometrical problems.
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In mathematics, a Moufang set is a particular kind of combinatorial system named after Ruth Moufang.
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He is also one of seven honorary members of the Egerváry Research Group on Combinatorial Optimization.
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On two combinatorial problems arising from automata theory. Annals of Discrete Math., 17, 535-548, 1983.
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She is best known for her works in group theory, algebraic graph theory and combinatorial designs.
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S. Shrikhande is a professor of combinatorial mathematics at Central Michigan University in Mt. Pleasant, Michigan.
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Ralph P. Grimaldi (2000). "Nonhomogeneous Recurrence Relations". Section 3.3.3 of Handbook of Discrete and Combinatorial Mathematics.
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As above mentioned, combinatorial chemistry was a key technology enabling the efficient generation of large screening libraries for the needs of high-throughput screening. However, now, after two decades of combinatorial chemistry, it has been pointed out that despite the increased efficiency in chemical synthesis, no increase in lead or drug candidates has been reached. This has led to analysis of chemical characteristics of combinatorial chemistry products, compared to existing drugs or natural products. The chemoinformatics concept chemical diversity, depicted as distribution of compounds in the chemical space based on their physicochemical characteristics, is often used to describe the difference between the combinatorial chemistry libraries and natural products.
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Like combinatorial maps, generalized maps are used as efficient data structure in image representation and processing, in geometrical modeling, they are related to simplicial set and to combinatorial topology, and this is a boundary representation model (B-rep or BREP), i.e. it represents object by its boundaries.
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254 534-538. Combinatorial auctions are smart markets in which goods are indivisible,Pekec, Aleksandar and Michael H. Rothkopf (2003), Combinatorial Auction Design, MANAGEMENT SCIENCE, Vol. 49, No. 11, November 2003, pp. 1485-1503. but some smart markets allocate divisible goods such as electricity and natural gas.
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John Francis Riordan (April 22, 1903 - August 27, 1988)John F. Riordan, 85, Ex-Bell Labs Engineer, The New York Times obituary, August 31, 1988 was an American mathematician and the author of major early works in combinatorics, particularly Introduction to Combinatorial Analysis and Combinatorial Identities.
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Combinatorial chemistry comprises chemical synthetic methods that make it possible to prepare a large number (tens to thousands or even millions) of compounds in a single process. These compound libraries can be made as mixtures, sets of individual compounds or chemical structures generated by computer software. Combinatorial chemistry can be used for the synthesis of small molecules and for peptides. Strategies that allow identification of useful components of the libraries are also part of combinatorial chemistry.
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Even though combinatorial chemistry has been an essential part of early drug discovery for more than two decades, so far only one de novo combinatorial chemistry-synthesized chemical has been approved for clinical use by FDA (sorafenib, a multikinase inhibitor indicated for advanced renal cancer).D. Newman and G. Cragg "Natural Products as Sources of New Drugs over the Last 25 Years" J Nat Prod 70 (2007) 461 The analysis of the poor success rate of the approach has been suggested to connect with the rather limited chemical space covered by products of combinatorial chemistry.M. Feher and J. M. Schmidt "Property Distributions: Differences between Drugs, Natural Products, and Molecules from Combinatorial Chemistry" J. Chem. Inf. Comput. Sci., 43 (2003) 218 When comparing the properties of compounds in combinatorial chemistry libraries to those of approved drugs and natural products, Feher and Schmidt noted that combinatorial chemistry libraries suffer particularly from the lack of chirality, as well as structure rigidity, both of which are widely regarded as drug-like properties.
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162-163 An alternate term is combinatorial logic. Clive Maxfield. "FPGAs: World Class Designs". p. 70. 2009.
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Reidemeister's interests were mainly in combinatorial group theory, combinatorial topology, geometric group theory, and the foundations of geometry. His books include Knoten und Gruppen (1926), Einführung in die kombinatorische Topologie (1932), and Knotentheorie (1932). He co-edited the journal Mathematische Annalen from 1947 until 1963.Title page of vol.
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In mathematics, a digital manifold is a special kind of combinatorial manifold which is defined in digital space i.e. grid cell space. A combinatorial manifold is a kind of manifold which is a discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes.
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There are, however, mathematical tools that can solve particular problems and answer general questions. Games of perfect information have been studied in combinatorial game theory, which has developed novel representations, e.g. surreal numbers, as well as combinatorial and algebraic (and sometimes non- constructive) proof methods to solve games of certain types, including "loopy" games that may result in infinitely long sequences of moves. These methods address games with higher combinatorial complexity than those usually considered in traditional (or "economic") game theory.
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It is the first book in the history of mathematics to devote a whole chapter to combinatorial problems.
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Vocal grooming (the production of pleasing sounds lacking syntax or combinatorial semantics) then evolved somehow into syntactical speech.
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In particular this implies the Euler characteristic of the combinatorial boundary of the polyhedron is 2. The combinatorial manifold model of solidity also guarantees the boundary of a solid separates space into exactly two components as a consequence of the Jordan-Brouwer theorem, thus eliminating sets with non-manifold neighborhoods that are deemed impossible to manufacture. The point-set and combinatorial models of solids are entirely consistent with each other, can be used interchangeably, relying on continuum or combinatorial properties as needed, and can be extended to n dimensions. The key property that facilitates this consistency is that the class of closed regular subsets of ℝn coincides precisely with homogeneously n-dimensional topological polyhedra.
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Splitting a necklace with two cuts. Combinatorial analogs of concepts and methods in topology are used to study graph coloring, fair division, partitions, partially ordered sets, decision trees, necklace problems and discrete Morse theory. It should not be confused with combinatorial topology which is an older name for algebraic topology.
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The Fano matroid, derived from the Fano plane. Matroids are one of many areas studied in algebraic combinatorics. Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
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This is in contrast to combinatorial rules or Slater-Kirkwood equation applied for development of the classical force fields. The combinatorial rules state that interaction energy of two dissimilar atoms (e.g., C…N) is an average of the interaction energies of corresponding identical atom pairs (i.e., C…C and N…N).
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In DAC, pp. 526 - 529, 1997. like Capo. Quadratic placement later outperformed combinatorial solutions in both quality and stability.
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The approximated solution of the combinatorial optimization problem is a string z that is close to maximizing C(z) .
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Aviezri Siegmund Fraenkel () (born June 7, 1929) is an Israeli mathematician who has made contributions to combinatorial game theory.
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Combinatorial mutagenesis is a site-directed protein engineering technique whereby multiple mutants of a protein can be simultaneously engineered based on analysis of the effects of additive individual mutations. It provides a useful method to assess the combinatorial effect of a large number of mutations on protein function. Large numbers of mutants may be screened for a particular characteristic by combinatorial analysis. In this technique, multiple positions or short sequences along a DNA strand may be exhaustively modified to obtain a comprehensive library of mutant proteins.
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Gould has published over 200 papers, which have appeared in about 20 countries. His research has been in combinatorial analysis, number theory, special functions of mathematical physics, and the history of mathematics and astronomy.(with Jocelyn Quaintance) Floor and Roof Function Analogs of the Bell Numbers, Vol 7 Combinatorial Number Theory ISSN 1553-1732Bell Numbers and Variant Sequences Derived from a General Functional Differential Equation (pp. 581–589) Vol9 Combinatorial Number Theory ISSN 1553-1732 Gould served as mathematics consultant to the 'Dear Abby' newspaper column.
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Other books on the combinatorics of experimental design include Statistical Design and Analysis of Experiments (John, 1971), Constructions and Combinatorial Problems in Design of Experiments (Rao, 1971), Design Theory (Beth, Jungnickel, and Lenz, 1985), and Combinatorial Theory and Statistical Design (Constantine, 1987). Compared to these, Combinatorics of Experimental Design makes the combinatorial aspects of the subjects more accessible to statisticians, and its last two chapters contain material not covered by the other books. However, it omits several other topics that were included in Rao's more comprehensive text.
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More generally, combinatorial algorithms researchers have defined a Gray code for a set of combinatorial objects to be an ordering for the objects in which each two consecutive objects differ in the minimal possible way. In this generalized sense, the Steinhaus–Johnson–Trotter algorithm generates a Gray code for the permutations themselves.
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Combinatorial physics can be characterized by the use of algebraic concepts to interpret and solve physical problems involving combinatorics. It gives rise to a particularly harmonious collaboration between mathematicians and physicists. Among the significant physical results of combinatorial physics, we may mention the reinterpretation of renormalization as a Riemann–Hilbert problem,A.
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Open problems in combinatorial group theory It is known that the Zeeman conjecture on collapsibility implies the Andrews–Curtis conjecture.
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Eric Charles Milner, FRSC (May 17, 1928 - July 20, 1997) was a mathematician who worked mainly in combinatorial set theory.
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A hexachord that is self-complementing for all the canonic operations—inversion, retrograde, and retrograde inversion—is called all- combinatorial.
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In the field of mathematics called combinatorial optimization, the method of symmetry-breaking constraints can be used to take advantage of symmetries in many constraint satisfaction and optimization problems, by adding constraints that eliminate symmetries and reduce the search space size. Symmetries in a combinatorial problem increase the size of the search space and therefore, time is wasted in visiting new solutions which are symmetric to the already visited solutions. The solution time of a combinatorial problem can be reduced by adding new constraints, referred as symmetry breaking constraints, such that some of the symmetric solutions are eliminated from the search space while preserving the existence of at least one solution. Symmetry is common in many real-life combinatorial problems.
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András Sebő (born 24 April 1954) is a Hungarian-French mathematician working in the areas of combinatorial optimization and discrete mathematics. Sebő is a French National Centre for Scientific Research (CNRS) Director of Research and the head of the Combinatorial Optimization. group in Laboratory G-SCOP, affiliated with the University of Grenoble and the CNRS.
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He has been one of three editors-in-chief of the Journal of Combinatorial Designs since 1992.Journal of Combinatorial Designs web site, Wiley Periodicals, Inc., retrieved 2011-03-26. In 2004, the Institute of Combinatorics and its Applications named Colbourn as that year's winner of their Euler Medal for lifetime achievements in combinatorics.
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Analytic Combinatorics is a book on the mathematics of combinatorial enumeration, using generating functions and complex analysis to understand the growth rates of the numbers of combinatorial objects. It was written by Philippe Flajolet and Robert Sedgewick, and published by the Cambridge University Press in 2009. It won the Leroy P. Steele Prize in 2019.
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The topological spaces so generated may be multiply connected (as contrasted with simply connected). The relationship to Noyes' bit-string physics is explained. Subsequently McGoveran developed a combinatorial and phenomenological argument for computing the fine structure constant from the combinatorial hierarchy, accurate to four decimal places. While suggestive, the argument was not considered convincing.
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Since he structures the components are unknown deconvolution methods need to be used in screening. One of the most important features of combinatorial libraries is that the whole mixture can be screened in a single process. This makes these libraries very useful in pharmaceutical research. Partial libraries of full combinatorial libraries can also be synthesized.
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Pregeometry, and in full combinatorial pregeometry, are essentially synonyms for "matroid". They were introduced by Gian-Carlo Rota with the intention of providing a less "ineffably cacophonous" alternative term. Also, the term combinatorial geometry, sometimes abbreviated to geometry, was intended to replace "simple matroid". These terms are now infrequently used in the study of matroids.
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An example of the combinatorial game Toads And FrogsThe combinatorial game Toads and Frogs is a partisan game invented by Richard Guy. This mathematical game was used as an introductory game in the book Winning Ways for your Mathematical Plays. Known for its simplicity and the elegance of its rules, Toads-and-Frogs is useful to illustrate the main concepts of combinatorial game theory. In particular, it is not difficult to evaluate simple games involving only one toad and one frog, by constructing the game tree of the starting position.
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346, Beginning in 1969 Klarner made significant contributions to the theory of combinatorial enumeration, especially focusing on polyominoesAnother Fine Math You've Got Me Into. . ., By Ian Stewart, Dover Publications (January 15, 2004), p. 21, and box-packing.Packing a rectangle with congruent n-ominoes Journal of Combinatorial Theory, Vol. 7, Issue 2, September 1969, Pages 107-115Klarner systems and tiling boxes with polyominoes by Michael Reid, Journal of Combinatorial Theory, Series A, Vol. 111, Issue 1, July 2005, Pages 89-105 Working with Ronald L. Rivest he found upper bounds on the number of n-ominoes.
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Go also contributed to development of combinatorial game theory (with Go Infinitesimals being a specific example of its use in Go).
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In mathematics, an algebraic matroid is a matroid, a combinatorial structure, that expresses an abstraction of the relation of algebraic independence.
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Operations with species are supported by SageMathSage documentation on combinatorial species. and, using a special package, also by Haskell.Haskell package species.
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J. Statist. Plan. Infer. 144, 3–18., Multivariate analysis and Combinatorial mathematics. Srivastava was a Fellow of Institute of Mathematical Statistics.
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These include dynamic programming, reinforcement learning and combinatorial optimization. Languages used to describe planning and scheduling are often called action languages.
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Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers. Algebraic or analytic methods are powerful in this field.
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In the context of combinatorial mathematics, stars and bars is a graphical aid for deriving certain combinatorial theorems. It was popularized by William Feller in his classic book on probability. It can be used to solve many simple counting problems, such as how many ways there are to put n indistinguishable balls into k distinguishable bins.
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The ideas of combinatorial conformal geometry that underlie Cannon's proof of the "combinatorial Riemann mapping theorem", were applied by Cannon, Floyd and Parry (2000) to the study of large-scale growth patterns of biological organisms.J. W. Cannon, W. Floyd and W. Parry. Crystal growth, biological cell growth and geometry. Pattern Formation in Biology, Vision and Dynamics, pp. 65–82.
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A fairly precise date can be supplied in the internal notes of the Bourbaki group. While topology was still combinatorial in 1942, it had become algebraic by 1944. gives documentation (translated into English from French originals). Azriel Rosenfeld (1973) proposed digital topology for a type of image processing that can be considered as a new development of combinatorial topology.
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But now consider the position 11333233, which must be filled with either a cross or a nought. If it is filled with a cross, then the combinatorial line 11xxx2xx is filled entirely with crosses, contradicting our hypothesis. If instead it is filled with a nought, then the combinatorial line 11xxx233 is filled entirely with noughts, again contradicting our hypothesis.
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Markov random fields were a generalization over the Ising model and have, since then, been used widely in combinatorial optimizations and networks.
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Another equivalent term, used more in a combinatorial context, is transversal. The notions of hitting set and set cover are equivalent too.
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1916, 92, 129-168.,Souge, J. L. Analytic solutions to Smoluchowski’s coagulation equation: a combinatorial inter-pretation. J. Phys. A.: Math. Gen.
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The above binomial coefficient, due to its combinatorial interpretation, gives precisely the number of all these sequences of length k + r − 1\.
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Brugha earned a BSc, an MSc (Mathematical Science – UCD), a PhD (Combinatorial Optimization - UCD) and MBA (TCD) Prize winner for his dissertation.
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In 1902 Bouton published a solution of the game Nim. This result is today viewed as the birth of combinatorial game theory.
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Ng computed the linearized contact homology in this case, providing an entirely combinatorial model for it which is a powerful knot invariant.
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From the Introduction by Marc Kac to the Special Issue of the JCTA in honor of John Riordan: : Foremost among the keepers of the barely flickering combinatorial flame was John Riordan. John’s work in Combinatorial Theory (or Combinatorial Analysis as he prefers to call it) is uncompromisingly classical in spirit and appearance. Though largely tolerant of modernity he does not let anyone forget that Combinatorial Analysis is the art and science of counting (enumerating is the word he prefers) and that a generating function by any other name or definition is still a generating function. From an interview with Neil Sloane published by Bell Labs: : Even at the end of my first year as a graduate student at Cornell, in 1962, I managed to arrange a summer job at Bell Labs in Holmdel.
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Thomas Lengauer (born November 12, 1952) is a German computer scientist, working in the fields of computational biology, computational chemistry and combinatorial optimization.
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Cynthia A. Phillips is a researcher at the Center for Computing Research of Sandia National Laboratories, known for her work in combinatorial optimization.
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He continued as one of the organizers until at least 1991, at which point it was the largest combinatorial meeting in the world.
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Douglas B. West: Combinatorial Mathematics. Cambridge University Press, 2020, p. 61Steven Vadja: Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications.
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If “finite vector spaces” is replaced by “finite sets”, one gets the notion of combinatorial species (to be precise, those of polynomial nature).
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Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. These concepts are not made precise so that a wide range of objects can be thought of as being under the same umbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other times it could involve the spatial arrangement of entries in an array as in sudoku grids. Combinatorial design theory can be applied to the area of design of experiments.
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A combinatorial auction is a type of smart market in which participants can place bids on combinations of discrete heterogeneous items, or “packages”, rather than individual items or continuous quantities. These packages can be also called lots and the whole auction a multi-lot auction. Combinatorial auctions are applicable when bidders have superadditive valuations on bundles of items, that is, they value combinations of items more than the sum of their valuations of individual elements of the combination. Simple combinatorial auctions have been used for many years in estate auctions, where a common procedure is to accept bids for packages of items.
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Another area of medicinal research in which the Janda laboratory has made contributions encompasses techniques to create molecular diversity, uncover active components from complex mixtures and the separation of synthetic targets by phase tagging. He has published methodologies that allow implementation of what has been termed "encoded combinatorial libraries", providing a means whereby the alternating parallel synthesis of peptides and oligonucleotides can be performed in a routine manner. His group has also demonstrated a technology termed "recursive deconvolution of combinatorial libraries" and "liquid phase combinatorial synthesis" which showed that reactants, products and by-products can be effectively "tagged" and targeted to different phases,.
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The Group was founded (under the name EU/ME – EUropean chapter on MEtaheuristics) in 2000 by Marc Sevaux, Kenneth Sörensen and Christelle Wynants, following the 2000 EURO Winter Institute on metaheuristics for combinatorial optimization held in Lac Noir, Switzerland.Hertz, Alain, and Marino Widmer. "Guidelines for the use of meta- heuristics in combinatorial optimization." European Journal of Operational Research 151.2 (2003): 247–252.
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Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, open access journal, Combinatorial Theory, was announced in 2020, that aims to be a continuation of JCTA independently from Elsevier. Most of the editorial board of JCTA will resign at the end of 2020 and transition to Combinatorial Theory.
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It also does not specify whether the configuration is purely combinatorial (an abstract incidence pattern of lines and points) or whether the points and lines of the configuration are realizable in the Euclidean plane or another standard geometry. The type (214) is highly ambiguous: there is an unknown but large number of (combinatorial) configurations of this type, 200 of which were listed by .
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Comparison of natural and synthetic selection processes for peptide generation. In biotechnology, combinatorial biology is the creation of a large number of compounds (usually proteins or peptides) through technologies such as phage display. Similar to combinatorial chemistry, compounds are produced by biosynthesis rather than organic chemistry. This process was developed independently by Richard A. Houghten and H. Mario Geysen in the 1980s.
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Parallel-Synthesis was developed by Mario Geysen and his colleagues and is not a true type of combinatorial synthesis, but can be incorporated into a combinatorial synthesis.H. M. Geysen, R. H. Meloen, S. J. Barteling Proc. Natl. Acad. Sci. USA 1984, 81, 3998. This group synthesized 96 peptides on plastic pins coated with a solid support for the solid phase peptide synthesis.
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A minimum spanning tree of a weighted planar graph. Finding a minimum spanning tree is a common problem involving combinatorial optimization. Combinatorial optimization is a subfield of mathematical optimization that is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, applied mathematics and theoretical computer science.
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Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Thus the combinatorial topics may be enumerative in nature or involve matroids, polytopes, partially ordered sets, or finite geometries. On the algebraic side, besides group and representation theory, lattice theory and commutative algebra are common.
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Maria Chudnovsky (born January 6, 1977) is an Israeli-American mathematician working on graph theory and combinatorial optimization. She is a 2012 MacArthur Fellow.
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So by controlling the order and proceeding with calculations with the smallest product of sizes, there will be less calculation and less combinatorial explosion.
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Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer-Verlag, New York, 2001. "Classics in Mathematics" series, reprint of the 1977 edition.
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It expands the concept of data flow oriented domains by the possibility to activate or deactivate parts of the model according to combinatorial logic.
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Ricerca Sci., pp. 129–156, CNR, Rome, Italy, 1981. reflected in the name of the combinatorial structure called by some the Lascoux–Schützenberger tree.
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One possible scenario is that differential phosphorylation of coactivators may direct their combinatorial recruitment into different transcriptional complexes at distinct promoters in specific cells.
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Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975). Congr. Numer. 15 (1976) 355–359. # (with J. A. Bondy) ‘Pancyclic graphs II’.
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Subsequently, provided a proof that for all , the maximum rotation distance is exactly . Pournin's proof is combinatorial, and avoids the use of hyperbolic geometry.
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Frontiers of Economic Research. Princeton University Press, Princeton, NJ, 1998. xii+272 pp. Zimmermann, U. Linear and combinatorial optimization in ordered algebraic structures. Ann.
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The NK model has found use in many fields, including in the study of spin glasses, epistasis and pleiotropy in evolutionary biology, and combinatorial optimisation.
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Employment of scavenger resins has become increasingly popular in solution-phase combinatorial chemistry. Used primarily in the synthesis of medicinal drugs, solution-phase combinatorial chemistry allows for the creation of large libraries of structurally related compounds. When purifying a solution, many approaches can be taken. In general chemical synthesis laboratories, a number of traditional techniques for purification are used as opposed to the employment of scavenger resins.
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The mathematical disciplines of combinatorics and dynamical systems interact in a number of ways. The ergodic theory of dynamical systems has recently been used to prove combinatorial theorems about number theory which has given rise to the field of arithmetic combinatorics. Also dynamical systems theory is heavily involved in the relatively recent field of combinatorics on words. Also combinatorial aspects of dynamical systems are studied.
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His research deals with combinatorial analysis, multidimensional complex analysis, and algorithms of integral representation and calculation of combinatorial sums and their applications in various fields of mathematics and science. In particular, his research has applied the Egorychev method to the basis of tensor calculus and to the theory of matrix functions, including permanents and determinants over various algebraic systems. He has published over 80 articles.
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Molecular Informatics is a peer-reviewed scientific journal published by Wiley VCH. It covers research in cheminformatics, quantitative structure–activity relationships, and combinatorial chemistry. It was established in 1981 as Quantitative Structure-Activity Relationships and renamed to QSAR & Combinatorial Science in 2003, before obtaining its present name in 2010. According to the Journal Citation Reports, the journal has a 2012 impact factor of 2.338.
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Together with J.P. Grossman and Richard Nowakowski, Albert invented the game Clobber.Getting Clobbered article at Science News Albert has also contributed to the Combinatorial Game Suite game analysis software, and is a coauthor of Lessons in Play: An Introduction to Combinatorial Game Theory. Another significant topic of his research has been permutation patterns. Albert is a keen bridge player, and has won tournaments internationally.
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In its modern form, combinatorial chemistry has probably had its biggest impact in the pharmaceutical industry. Researchers attempting to optimize the activity profile of a compound create a 'library' of many different but related compounds. Advances in robotics have led to an industrial approach to combinatorial synthesis, enabling companies to routinely produce over 100,000 new and unique compounds per year.Jeffrey W. Noonan et al.
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He held positions at the Massachusetts Institute of Technology and the University of Pennsylvania before moving to Haverford. Greene has written highly cited research papers on Sperner families,. . Young tableaux,. . . and combinatorial equivalences between hyperplane arrangements, zonotopes, and graph orientations.. With Daniel Kleitman, he has also written a highly cited survey paper on combinatorial proof techniques.. In 2012 he became a fellow of the American Mathematical Society.
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In computer science and artificial intelligence, combinatorial search studies search algorithms for solving instances of problems that are believed to be hard in general, by efficiently exploring the usually large solution space of these instances. Combinatorial search algorithms achieve this efficiency by reducing the effective size of the search space or employing heuristics. Some algorithms are guaranteed to find the optimal solution, while others may only return the best solution found in the part of the state space that was explored. Classic combinatorial search problems include solving the eight queens puzzle or evaluating moves in games with a large game tree, such as reversi or chess.
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A diagram depicting a use of dynamic combinatorial library to select an optimal receptor. In contrast to traditional combinatorial synthesis where a library of catalysts were first generated and later screened (as in the two above approaches), dynamic combinatorial library approach utilizes a mixture of multicomponent building blocks that reversibly form library of catalysts. With out a template, the library consists of a roughly equal mixture of different combination of building blocks. In the presence of a template which is either a starting material or a TSA, the combination that provides the best binding to the template is thermodynamically favorable and thus that combination is more prevalent than other library members.
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Many of these aspects of combinatorial auctions, including some real-world examples, are also discussed in the comprehensive book edited by Cramton, Shoham and Steinberg (2006).
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The combinatorial Riemann mapping theorem. Acta Mathematica 173 (1994), no. 2, pp. 155–234. that was motivated by the classic Riemann mapping theorem in complex analysis.
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A less combinatorial example is the operad of little intervals: The space A(n) consists of all embeddings of n disjoint intervals into the unit interval.
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In partition calculus, part of combinatorial set theory, a branch of mathematics, the Erdős–Rado theorem is a basic result extending Ramsey's theorem to uncountable sets.
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The counting sequence of a combinatorial class is the sequence of the numbers of elements of size i for i = 0, 1, 2, ...; it may also be described as a generating function that has these numbers as its coefficients. The counting sequences of combinatorial classes are the main subject of study of enumerative combinatorics. Two combinatorial classes are said to be isomorphic if they have the same numbers of objects of each size, or equivalently, if their counting sequences are the same.. Frequently, once two combinatorial classes are known to be isomorphic, a bijective proof of this equivalence is sought; such a proof may be interpreted as showing that the objects in the two isomorphic classes are cryptomorphic to each other. For instance, the triangulations of regular polygons (with size given by the number of sides of the polygon, and a fixed choice of polygon to triangulate for each size) and the set of unrooted binary plane trees (up to graph isomorphism, with a fixed ordering of the leaves, and with size given by the number of leaves) are both counted by the Catalan numbers, so they form isomorphic combinatorial classes.
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The abstract definition allows some more general combinatorial structures than traditional definitions of a polytope, thus allowing many new objects that have no counterpart in traditional theory.
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The method has been applied to several Combinatorial Optimization Problems including the Job-Shop Scheduling Problems, Flow-Shop Problems, Vehicle Routing Problems as well as many others.
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They have been used recently for truckload transportation, bus routes, industrial procurement, and in the allocation of radio spectrum for wireless communications. In recent years, procurement teams have applied reverse combinatorial auctions in the procurement of goods and services. This application is often referred to as sourcing optimization. Although they allow bidders to be more expressive, combinatorial auctions present both computational and game-theoretic challenges compared to traditional auctions.
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LOTH hinges on the belief that the mind works like a computer, always in computational processes. The theory believes that mental representation has both a combinatorial syntax and compositional semantics. The claim is that mental representations possess combinatorial syntax and compositional semantic—that is, mental representations are sentences in a mental language. Alan Turing's work on physical machines implementation of causal processes that require formal procedures was modeled after these beliefs.
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Five binary trees on three vertices, an example of Catalan numbers. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of certain combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. Fibonacci numbers is the basic example of a problem in enumerative combinatorics.
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Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties. Block designs are combinatorial designs of a special type. This area is one of the oldest parts of combinatorics, such as in Kirkman's schoolgirl problem proposed in 1850. The solution of the problem is a special case of a Steiner system, which systems play an important role in the classification of finite simple groups.
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Cramton is a major advocate for the use of auctions, rather than the grandfathering, of emission permits. He has helped to design many electricity auctions that are conducted throughout the world. In the area of telecommunications, together with Larry Ausubel and Paul Milgrom, he invented the combinatorial clock auction. This combinatorial auction design has been used for more than ten major spectrum auctions in Europe, Canada and Australia.
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Structure-based combinatorial protein engineering (SCOPE) is a synthetic biology technique for creating gene libraries (lineages) of defined composition designed from structural and probabilistic constraints of the encoded proteins. The development of this technique was driven by fundamental questions about protein structure, function, and evolution, although the technique is generally applicable for the creation of engineered proteins with commercially desirable properties. Combinatorial travel through sequence spacetime is the goal of SCOPE.
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See for a detailed history of combinatorial group theory. A proto-form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron. The foundations of combinatorial group theory were laid by Walther von Dyck, student of Felix Klein, in the early 1880s, who gave the first systematic study of groups by generators and relations.
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In 1980s the focus of Cannon's work shifted to the study of 3-manifolds, hyperbolic geometry and Kleinian groups and he is considered one of the key figures in the birth of geometric group theory as a distinct subject in late 1980s and early 1990s. Cannon's 1984 paper "The combinatorial structure of cocompact discrete hyperbolic groups"J. W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups. Geometriae Dedicata, vol.
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Noyes has cited McGoveran's calculation of the Sommerfeld-Dirac formula and corrections to both the combinatorial hierarchy computation of the fine structure and gravitational constants as convincing him that the evolving combinatorial hierarchy construction could be the starting point for a new physics and physical cosmology.Noyes, H. P., et al. (2001). "Bit-String Physics: A Finite and Discrete Approach to Natural Philosophy". p.343. Series on Knots and Everything. Vol. 27.
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In the mathematical study of combinatorics on words, a parameter word is a string over a given alphabet having some number of wildcard characters. The set of strings matching a given parameter word is called a parameter set or combinatorial cube. Parameter words can be composed, to produce smaller subcubes of a given combinatorial cube. They have applications in Ramsey theory and in computer science in the detection of duplicate code.
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List of combinatorial computational geometry topics enumerates the topics of computational geometry that states problems in terms of geometric objects as discrete entities and hence the methods of their solution are mostly theories and algorithms of combinatorial character. See List of numerical computational geometry topics for another flavor of computational geometry that deals with geometric objects as continuous entities and applies methods and algorithms of nature characteristic to numerical analysis.
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The fact that systematicity and productivity depend on the compositional structure of language means that language has a combinatorial semantics. If thought also has such a combinatorial semantics, then there must be a language of thought. The second argument that Fodor provides in favour of representational realism involves the processes of thought. This argument touches on the relation between the representational theory of mind and models of its architecture.
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Auctions with more than one winner are called multi-winner auctions. Multiunit auction, Combinatorial auction, Generalized first-price auction and Generalized second-price auction are multi-winner auctions.
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In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety X is a combinatorial invariant of importance to the birational geometry of X.
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The first two examples of this equation are :, :. Thus the Bernoulli numbers vanish at odd index because some non-obvious combinatorial identities are embodied in the Bernoulli numbers.
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V. M. Mirsky, V. Kulikov, Q. Hao, O. S. Wolfbeis. Multiparameter High Throughput Characterization of Combinatorial Chemical Microarrays of Chemosensitive Polymers. Macromolec. Rap. Comm., 2004, 25, 253-258H.
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Silvia Heubach is a German-American mathematician specializing in enumerative combinatorics, combinatorial game theory, and bioinformatics. She is a professor of mathematics at California State University, Los Angeles.
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The synthetic, combinatorial library compounds seem to cover only a limited and quite uniform chemical space, whereas existing drugs and particularly natural products, exhibit much greater chemical diversity, distributing more evenly to the chemical space. The most prominent differences between natural products and compounds in combinatorial chemistry libraries is the number of chiral centers (much higher in natural compounds), structure rigidity (higher in natural compounds) and number of aromatic moieties (higher in combinatorial chemistry libraries). Other chemical differences between these two groups include the nature of heteroatoms (O and N enriched in natural products, and S and halogen atoms more often present in synthetic compounds), as well as level of non-aromatic unsaturation (higher in natural products). As both structure rigidity and chirality are well-established factors in medicinal chemistry known to enhance compounds specificity and efficacy as a drug, it has been suggested that natural products compare favourably to today's combinatorial chemistry libraries as potential lead molecules.
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Combinatorial designs date to antiquity, with the Lo Shu Square being an early magic square. One of the earliest datable application of combinatorial design is found in India in the book Brhat Samhita by Varahamihira, written around 587 AD, for the purpose of making perfumes using 4 substances selected from 16 different substances using a magic square. Combinatorial designs developed along with the general growth of combinatorics from the 18th century, for example with Latin squares in the 18th century and Steiner systems in the 19th century. Designs have also been popular in recreational mathematics, such as Kirkman's schoolgirl problem (1850), and in practical problems, such as the scheduling of round-robin tournaments (solution published 1880s).
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Combinatorial optimization is a good strategy to solve MSA problems. The idea of combinatorial optimization strategy is to transform the multiple sequence alignment into pair sequence alignment to solve this problem. Depending on its transformation strategy, the combinatorial optimization strategy can be divided into the tree alignment algorithm and the star alignment algorithm. For a given multi-sequence set S ={s_1,...,s_n }, find an evolutionary tree which has n leaf nodes and establishing one to one relationship between this evolutionary tree and the set S. By assigning the sequence to the internal nodes of the evolutionary tree, we calculate the total score of each edge, and the sum of all edges' scores is the score of the evolutionary tree.
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He received the IBM Faculty award, and was awarded funding from the DFG and Yahoo!. Landau co-chaired the International Symposium on Combinatorial Pattern Matching in both 2001 and 2008. He serves on the editorial board of Journal of Discrete Algorithms, and served as a guest editor for TCS and Discrete Applied Mathematics. He has served on numerous program committees for international conferences, most recently, International Conference on Language and Automata Theory and Applications (LATA), International Symposium on String Processing and Information Retrieval (SPIRE), International Symposium on Algorithms and Computation (ISAAC), Annual Symposium on Combinatorial Pattern Matching (CPM), Workshop on Algorithms in Bioinformatics (WABI), International Workshop on Combinatorial Algorithms (IWOCA), and Brazilian Symposium on Bioinformatics (BSB).
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For m=1, this is just two times the ordinary Catalan numbers, and for m=n, the numbers have an easy combinatorial description. However, other combinatorial descriptions are only known for m=2, 3 and 4, and it is an open problem to find a general combinatorial interpretation. Sergey Fomin and Nathan Reading have given a generalized Catalan number associated to any finite crystallographic Coxeter group, namely the number of fully commutative elements of the group; in terms of the associated root system, it is the number of anti-chains (or order ideals) in the poset of positive roots. The classical Catalan number C_n corresponds to the root system of type A_n.
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The Interdependence category contains five task types that Steiner describes as combinatorial strategies illustrating how the individual contributions of members of a group can be combined in different ways.
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In combinatorial mathematics, a picture is a bijection between skew diagrams satisfying certain properties, introduced by in a generalization of the Robinson–Schensted correspondence and the Littlewood–Richardson rule.
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These two reactions are generally quite reliable, lending themselves to combinatorial chemistry. The antiviral drug zidovudine (AZT) contains an azido group. Some azides are valuable as bioorthogonal chemical reporters.
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Peptide aptamers can also be selected from combinatorial peptide libraries constructed by phage display and other surface display technologies such as mRNA display, ribosome display, bacterial display and yeast display. These experimental procedures are also known as biopannings. Among peptides obtained from biopannings, mimotopes can be considered as a kind of peptide aptamers. All the peptides panned from combinatorial peptide libraries have been stored in a special database with the name MimoDB.
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The new combinatorial topology formally treated topological classes as abelian groups. Homology groups are finitely generated abelian groups, and homology classes are elements of these groups. The Betti numbers of the manifold are the rank of the free part of the homology group, and the non-orientable cycles are described by the torsion part. The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "algebraic topology".
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In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used. The rule of sum, rule of product, and inclusion–exclusion principle are often used for enumerative purposes. Bijective proofs are utilized to demonstrate that two sets have the same number of elements. The pigeonhole principle often ascertains the existence of something or is used to determine the minimum or maximum number of something in a discrete context.
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The cross-entropy (CE) method is a Monte Carlo method for importance sampling and optimization. It is applicable to both combinatorial and continuous problems, with either a static or noisy objective. The method approximates the optimal importance sampling estimator by repeating two phases:Rubinstein, R.Y. and Kroese, D.P. (2004), The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning, Springer-Verlag, New York . #Draw a sample from a probability distribution.
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The terminology used in the field of Dynamic Combinatorial Chemistry (DCC) and Constitutional Dynamic Chemistry (CDC). Dynamic combinatorial chemistry (DCC); also known as constitutional dynamic chemistry (CDC) is a method to the generation of new molecules formed by reversible reaction of simple building blocks under thermodynamic control.Schaufelberger, F.; Timmer, B. J. J.; Ramström, O. Principles of Dynamic Covalent Chemistry. In Dynamic Covalent Chemistry: Principles, Reactions, and Applications; Zhang, W.; Jin, Y., Eds.
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Although the work of Busch helped to establish the template method as a powerful synthetic route to stable macrocyclic structures, this approach remained exclusively within the domain of inorganic chemistry until the early 1990s, when Sanders et al. first proposed the concept of dynamic combinatorial chemistry. Their work combined thermodynamic templation in tandem with combinatorial chemistry, to generate an ensemble complex porphyrin and imine macrocycles using a modest selection of simple building blocks.
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Sanders then developed this early manifestation of dynamic combinatorial chemistry as a strategy for organic synthesis; the first example being the thermodynamically-controlled macrolactonisation of oligocholates to assemble cyclic steroid-derived macrocycles capable of interconversion via component exchange. Early work by Sanders et al. employed transesterification to generate dynamic combinatorial libraries. In retrospect, it was unfortunate that esters were selected for mediating component exchange, as transesterification processes are inherently slow and require vigorous anhydrous conditions.
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The Teiresias algorithm is a combinatorial algorithm for the discovery of rigid patterns (motifs) in biological sequences. It is named after the Greek prophet Teiresias and was created in 1997 by Isidore Rigoutsos and Aris Floratos.Rigoutsos, I, Floratos, A (1998) Combinatorial pattern discovery in biological sequences: The TEIRESIAS algorithm. Bioinformatics 14: 55-67 The problem of finding sequence similarities in the primary structure of related proteins or genes arises in the analysis of biological sequences.
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He is known for his pioneering research in molecular self- assembly. Molecular self-assembly is the assembly of molecules without guidance or management from an outside source. His main field of expertise and research interests are Host Guest Chemistry, Molecular Recognition, Liquid Crystals/Organic Gelators, Sugar Sensing/Sugar-Based Combinatorial Chemistry, Boronic-acids, Polysaccharide-Polynucleotide Interactions, Sol-Gel Transcription and Inorganic Combinatorial Chemistry. His most recent research is related to chiral discrimination using AIE.
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In the same year of 1928, two Indian scientists C. V. Raman and K. S. Krishnan were looking for "Compton component" of scattered light in liquids and vapors. They found the same combinatorial scattering of light. Raman stated that "The line spectrum of the new radiation was first seen on 28 February 1928." Thus, combinatorial scattering of light was discovered by Mandelstam and Landsberg a week earlier than by Raman and Krishnan.
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After his Ph.D., Ryser spent a year at Princeton's Institute for Advanced Study, then joined the faculty of Ohio State University. In 1962 he took a professorship at Syracuse University, and in 1967 moved to Caltech. His doctoral students include Richard A. Brualdi, Clement W. H. Lam, and Marion Tinsley. Ryser contributed to the theory of combinatorial designs, finite set systems, the permanent, combinatorial functions, and to many other topics in combinatorics.
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Bernhard H. Korte (born November 3, 1938 in Bottrop, Germany) is a German mathematician and computer scientist, a professor at the University of Bonn, and an expert in combinatorial optimization.
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The concepts of orientability, orientation and degree of a mapping make sense for pseudomanifolds and moreover, within the combinatorial approach, pseudomanifolds form the natural domain of definition for these concepts.
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Daniel Mier Gusfield is an American computer scientist, Distinguished Professor of Computer Science at the University of California, Davis. Gusfield is known for his research in combinatorial optimization and computational biology.
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Schilling was included in the 2019 class of fellows of the American Mathematical Society "for contributions to algebraic combinatorics, combinatorial representation theory, and mathematical physics and for service to the profession".
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In mathematics, and particularly in axiomatic set theory, ♣S (clubsuit) is a family of combinatorial principles that are a weaker version of the corresponding ◊S; it was introduced in 1975.
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Bruce Lee Rothschild (born August 26, 1941) is an American mathematician and educator, specializing in combinatorial mathematics. He is a professor emeritus of mathematics at the University of California, Los Angeles.
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Therefore, every n-dimensional solid may be unambiguously represented by its boundary and the boundary has the combinatorial structure of an n−1-dimensional polyhedron having homogeneously n−1-dimensional neighborhoods.
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He joined the Georgia Tech faculty in 1994. Johnson's research interests in logistics include crew scheduling and real-time repair, fleet assignment and routing, distribution planning, network problems, and combinatorial optimization.
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A finite-state machine with only one state is called a "combinatorial FSM". It only allows actions upon transition into a state. This concept is useful in cases where a number of finite-state machines are required to work together, and when it is convenient to consider a purely combinatorial part as a form of FSM to suit the design tools.Brutscheck, M., Berger, S., Franke, M., Schwarzbacher, A., Becker, S.: Structural Division Procedure for Efficient IC Analysis.
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Combinatorial properties of Kazhdan–Lusztig polynomials and their generalizations are a topic of active current research. Given their significance in representation theory and algebraic geometry, attempts have been undertaken to develop the theory of Kazhdan–Lusztig polynomials in purely combinatorial fashion, relying to some extent on geometry, but without reference to intersection cohomology and other advanced techniques. This has led to exciting developments in algebraic combinatorics, such as pattern-avoidance phenomenon. Some references are given in the textbook of .
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MallbaE. Alba, F. Almeida, M. J. Blesa, J. Cabeza, C. Cotta, M. Diaz, I. Dorta, J. Gabarro, C. Leon, J. Luna, L. M. Moreno, C. Pablos, J. Petit, A. Rojas, and F. Xhafa. "Mallba: A library of skeletons for combinatorial optimisation (research note)." In Euro-Par '02: Proceedings of the 8th International Euro-Par Conference on Parallel Processing, pages 927–932, London, UK, 2002. Springer-Verlag. is a library for combinatorial optimizations supporting exact, heuristic and hybrid search strategies.
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Palo Alto to be home of expanded biopharma research effort In 1988 he co-founded Affymax, specializing in combinatorial chemistry to reduce the cost and time of identifying new medicines. In 1991 he founded Affymetrix, specializing in using genetics for developing new medicine. He was also involved in the creation of Perlegen Sciences, an Affymetrix spin-off which works on finding genetic causes of disease. In 1994, he founded Symyx Technologies, a company dedicated to utilizing combinatorial chemistry technologies.
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Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of Combinatorial Principles is a book on reverse mathematics in combinatorics, the study of the axioms needed to prove combinatorial theorems. It was written by Denis R. Hirschfeldt, based on a course given by Hirschfeldt at the National University of Singapore in 2010, and published in 2014 by World Scientific, as volume 28 of the Lecture Notes Series of the Institute for Mathematical Sciences, National University of Singapore.
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Next the products from this cleavage are ligated together, resulting in the insertion of the gene into the target plasmid. An alternative form of cassette mutagenesis called combinatorial cassette mutagenesis is used to identify the functions of individual amino acid residues in the protein of interest. Recursive ensemble mutagenesis then utilizes information from previous combinatorial cassette mutagenesis. Codon cassette mutagenesis allows you to insert or replace a single codon at a particular site in double stranded DNA.
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Koinuma et al. "Combinatorial solid state materials science and technology" Sci. Technol. Adv. Mater. 1 (2000) 1 free downloadAndrei Ionut Mardare et al. "Combinatorial solid state materials science and technology" Sci. Technol. Adv. Mater. 9 (2008) 035009 free download as well as companies with large research and development programs (Symyx Technologies, GE, Dow Chemical etc.). The technique has been used extensively for catalysis,Applied Catalysis A, Volume 254, Issue 1, Pages 1-170 (10 November 2003) coatings,J.
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Some alternative procedures describe unspecific covalent binding and adhesive immobilization. However, lithographic methods can be used to overcome the problem of excessive number of coupling cycles. Combinatorial synthesis of peptide arrays onto a microchip by laser printing has been described, where a modified colour laser printer is used in combination with conventional solid- phase peptide synthesis chemistry. Amino acids are immobilized within toner particles, and the peptides are printed onto the chip surface in consecutive, combinatorial layers.
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In combinatorics, especially in analytic combinatorics, the symbolic method is a technique for counting combinatorial objects. It uses the internal structure of the objects to derive formulas for their generating functions. The method is mostly associated with Philippe Flajolet and is detailed in Part A of his book with Robert Sedgewick, Analytic Combinatorics. Similar languages for specifying combinatorial classes and their generating functions are found in work by Bender and Goldman, Foata and Schützenberger, and Joyal.
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Czarnik joined the Bio-organic Chemistry Department at the Ohio State University as assistant professor in 1983. He later was promoted to associate professor. Czarnik worked at Ohio State University until 1993, when he was offered a position as director of the bio-organic chemistry group at Parke-Davis Research Laboratory in Ann Arbor, Michigan. Czarnik was the founding editor of ACS Combinatorial Science (formerly Journal of Combinatorial Chemistry), an academic journal published by the American Chemical Society.
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36, Cambridge University Press, Cambridge-New York, 1979; and the books of Allen Hatcher, Gilbert Baumslag,Gilbert Baumslag. Topics in combinatorial group theory. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993.
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Unlike comparable methods for pyridine synthesis, the Kröhkne synthesis benefits from being a high-yielding one pot synthesis, which ultimately allows for abbreviation of synthetic pathways and further simplifies combinatorial library cataloging.
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Friedmann (1990), p. 104. The only more redundant hexachord is 6-35. It is also Ernő Lendvai's "1:3 Model" scale and one of Milton Babbitt's six all-combinatorial hexachord "source sets".
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Tamas Kalman developed a combinatorial invariant for loops of Legendrian knots, with which he detected differences between the fundamental groups of the space of smooth knots and of the space of Legendrian knots.
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This possibly represents the first instance that a combinatorial problem involving permutations was attempted. Xenocrates also supported the idea of "indivisible lines" (and magnitudes) in order to counter Zeno's paradoxes.Simplicius, in Arist. Phys.
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The Brunn-Minkowski inequality continues to be relevant to modern geometry and algebra. For instance, there are connections to algebraic geometry, and combinatorial versions about counting sets of points inside the integer lattice.
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In combinatorial mathematics, cyclic sieving is a phenomenon by which evaluating a generating function for a finite set at roots of unity counts symmetry classes of objects acted on by a cyclic group.
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Other more combinatorial objects associated to this group (in particular the curve complex) have also been related to Teichmüller space, and this is a very active subject of research in geometric group theory.
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She was the first woman Reader in Combinatorial Mathematics in Australia. she had supervised 30 doctorates and had 71 academic descendants. Her notable students have included Peter Eades, Mirka Miller, and Deborah Street.
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Materials science has applied the techniques of combinatorial chemistry to the discovery of new materials. This work was pioneered by P.G. Schultz et al. in the mid-nineties X. -D. Xiang et al.
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Techniques involved in this interdisciplinary research include proteomics, X-ray crystallography, nuclear magnetic resonance (NMR) spectroscopy, biological mass spectrometry, molecular biology, enzyme kinetics, protein-directed dynamic combinatorial chemistry and organic synthesis/medicinal chemistry.
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The biggest technical change after 1950 has been the development of sieve methods, particularly in multiplicative problems. These are combinatorial in nature, and quite varied. The extremal branch of combinatorial theory has in return been greatly influenced by the value placed in analytic number theory on quantitative upper and lower bounds. Another recent development is probabilistic number theory, which uses methods from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has.
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Edmonds 1965 paper “Paths, Trees and Flowers” was a preeminent paper in initially suggesting the possibility of establishing a mathematical theory of efficient combinatorial algorithms. One of his earliest and notable contributions is the blossom algorithm for constructing maximum matchings on graphs, discovered in 1961 and published in 1965. This was the first polynomial-time algorithm for maximum matching in graphs. Its generalization to weighted graphs was a conceptual breakthrough in the use of linear programming ideas in combinatorial optimization.
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The biased ratio of the desired catalyst to other combinatorial products could then be frozen by terminating the reversibility of the equilibrium by means such as change in temperature, pH, or radiation to yield the optimal catalyst. For example, Lehn et al. used this method to create a dynamic combinatorial library of imine inhibitor from a set of amines and a set of aldehydes. After some time, the equilibrium was terminated by an addition of NaBH3CN to afford the desired catalyst.
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Blockbusting is a solved combinatorial game introduced in 1987 by Elwyn Berlekamp illustrating a generalisation of overheating. The analysis of Blockbusting may be used as the basis of a strategy for the combinatorial game of Domineering. Blockbusting is a partisan game for two players known as Red and Blue (or Right and Left) played on an n \times 1 strip of squares called "parcels". Each player, in turn, claims and colors one previously unclaimed parcel until all parcels have been claimed.
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There are several approaches to prove Schur positivity of a given symmetric function F. If F is described in a combinatorial manner, a direct approach is to produce a bijection with semi-standard Young tableaux. The Edelman–Green correspondence and the Robinson–Schensted–Knuth correspondence are examples of such bijections. A bijection with more structure is a proof using so called crystals. This method can be described as defining a certain graph structure described with local rules on the underlying combinatorial objects.
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While the use of solid-supported reagents greatly simplifies the synthesis of compounds, many combinatorial syntheses require multiple steps, each of which still requires some form of purification. Armstrong, et al. describe a one-pot method for generating combinatorial libraries, called multiple-component condensations (MCCs). In this scheme, three or more reagents react such that each reagent is incorporated into the final product in a single step, eliminating the need for a multi-step synthesis that involves many purification steps.
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He served as editor of the journals Journal of Combinatorial Theory, Linear and Multilinear Algebra, and Journal of Algebra. Ryser's estate funded an endowment creating undergraduate mathematics scholarships at Caltech known as the H. J. Ryser Scholarships.California Tech, "Mathematics Awards", May 17, 1991, page 1. (Scan of that page on-line via on-line archives at the newspaper.) The Journal of Combinatorial Theory, Series A denoted two issues after Ryser's passing as the "Herbert J. Ryser Memorial Issue", parts 1 and 2.
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Convex hulls have wide applications in many fields. Within mathematics, convex hulls are used to study polynomials, matrix eigenvalues, and unitary elements, and several theorems in discrete geometry involve convex hulls. They are used in robust statistics as the outermost contour of Tukey depth, are part of the bagplot visualization of two-dimensional data, and define risk sets of randomized decision rules. Convex hulls of indicator vectors of solutions to combinatorial problems are central to combinatorial optimization and polyhedral combinatorics.
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In combinatorial optimization and polyhedral combinatorics, central objects of study are the convex hulls of indicator vectors of solutions to a combinatorial problem. If the facets of these polytopes can be found, describing the polytopes as intersections of halfspaces, then algorithms based on linear programming can be used to find optimal solutions.; see especially remarks following Theorem 2.9. In multi- objective optimization, a different type of convex hull is also used, the convex hull of the weight vectors of solutions.
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The use of randomization to improve the time bounds for low dimensional linear programming and related problems was pioneered by Clarkson and by . The definition of LP-type problems in terms of functions satisfying the axioms of locality and monotonicity is from , but other authors in the same timeframe formulated alternative combinatorial generalizations of linear programs. For instance, in a framework developed by , the function is replaced by a total ordering on the subsets of . It is possible to break the ties in an LP-type problem to create a total order, but only at the expense of an increase in the combinatorial dimension.. Additionally, as in LP-type problems, Gärtner defines certain primitives for performing computations on subsets of elements; however, his formalization does not have an analogue of the combinatorial dimension.
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John Horton Conway has a purely combinatorial proof which consequently also holds for points and lines over the complex numbers, quaternions and octonions.Stasys Jukna, Extremal Combinatorics, Second edition, Springer Verlag, 2011, pages 167 - 168.
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Additionally, the reaction does not require anhydrous or inert conditions. As a mild, selective synthesis, the Petasis reaction is useful in generating α-amino acids, and is utilized in combinatorial chemistry and drug discovery.
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Oberwolfach. József Solymosi is a Hungarian-Canadian mathematician and a professor of mathematics at the University of British Columbia. His main research interests are arithmetic combinatorics, discrete geometry, graph theory, and combinatorial number theory.
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He proved a conjecture of Ron Graham in combinatorial number theory jointly with Ramachandran Balasubramanian. He made important contributions in settling the arithmetic Quantum Unique Ergodicity conjecture for Maass wave forms and modular forms.
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Many combinatorial and computational approaches (e.g. Skilling, 1975) treat a vertex figure as the ordered (or partially ordered) set of points of all the neighboring (connected via an edge) vertices to the given vertex.
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András Frank (born 3 June 1949) is a Hungarian mathematician, working in combinatorics, especially in graph theory, and combinatorial optimisation. He is director of the Institute of Mathematics of the Eötvös Loránd University, Budapest.
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When the utilities are not single-parametric (e.g. in combinatorial auctions), the mechanism design problem is much more complicated. The VCG mechanism is one of the only mechanisms that works for such general valuations.
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However, the phenomenon became known as the Raman effect because Raman published his results earlier than Landsberg and Mandelstam did. Nonetheless, in the Russian-language literature it is traditionally called "combinatorial scattering of light".
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This is the case for vegetal trees and vascular, pulmonary, urinary systems. The nervous system may be seen as a system of exchanges between emitting and receiving binary arborizations, offering a huge combinatorial range.
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He completed his PhD, entitled Discrete Isoperimetric Inequalities and Other Combinatorial Results, in 1989, supervised by Béla Bollobás. Godson of mathematical philosopher Imre Lakatos, he is currently a fellow of Trinity College, University of Cambridge.
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Techniques created by Torrey Pines Institute include individual compounds arrays, mixture-based synthetic combinatorial libraries, positional scanning deconvolution, biometrical analysis, libraries from libraries, small molecule and heterocyclic compounds, and direct in-vivo testing of mixtures.
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Bruce Chandler and Wilhelm Magnus. The history of combinatorial group theory. A case study in the history of ideas. Studies in the History of Mathematics and Physical Sciences, vo. 9. Springer-Verlag, New York, 1982.
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The graph application is for manipulating directed and undirected graphs. Some the standard graph functions exist (like for adjacency and cliques) together with combinatorial functions like computing the lattice represented by a directed acyclic graph.
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At early time, placement of integrated circuits is handled by combinatorial approaches. When IC design was of thousand-gate scale, simulated annealingS. Kirkpatrick, C. D. G. Jr., and M. P. Vecchi. Optimization by Simulated Annealing.
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Rockafellar 1984 and 1998. Similarly, matroid theory has influenced the development of combinatorial algorithms, particularly the greedy algorithm.Lawler. Rockafellar 1984 and 1998. More generally, a greedoid is useful for studying the finite termination of algorithms.
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In combinatorial data analysis, seriation is the process of finding an arrangement of all objects in a set, in a linear order, given a loss function. The main goal is exploratory, to reveal structural information.
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Citation on 114. A complement in this context is half of a combinatorial pitch class set and most generally it is the "other half" of any pair including pitch class sets, textures, or pitch range.
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N. Cawse et. al, Progress in Organic Coatings, Volume 47, Issue 2, August 2003, Pages 128-135 electronics,Combinatorial Methods for High-Throughput Materials Science, MRS Proceedings Volume 1024E, Fall 2007 and many other fields.
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In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative integer, such that there are finitely many objects of each size...
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György Elekes ( – ) was a Hungarian mathematician and computer scientist who specialized in Combinatorial geometry and Combinatorial set theory. He may be best known for his work in the field that would eventually be called Additive Combinatorics. Particularly notable was his "ingenious" application of the Szemerédi–Trotter theorem to improve the best known lower bound for the sum- product problem. He also proved that any polynomial-time algorithm approximating the volume of convex bodies must have a multiplicative error, and the error grows exponentially on the dimension.
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Furthermore, lowering the acoustic power required for HIFU yields a safer treatment for the patient, as well as diminished treatment time. Though the treatment itself shows potential, a combinatorial treatment is speculated to be required for a complete treatment. Ultrasound and MB treatment without additional drugs impeded the growth of small tumors but required a combinatorial drug treatment to affect medium- sized tumor growth. With their immune stimulating mechanism, ultrasound and MBs offer a unique ability to prime or enhance immunotherapies for more effective cancer treatment.
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Using an approach similar to the one he used for the case k = 3, Roth gave a second proof for this in 1972. The general case was settled in 1975, also by Szemerédi, who developed an ingenious and complicated extension of his previous combinatorial argument for k = 4 (called "a masterpiece of combinatorial reasoning" by Erdős). Several other proofs are now known, the most important being those by Hillel Furstenberg. in 1977, using ergodic theory, and by Timothy Gowers in 2001, using both Fourier analysis and combinatorics.
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List of numerical computational geometry topics enumerates the topics of computational geometry that deals with geometric objects as continuous entities and applies methods and algorithms of nature characteristic to numerical analysis. This area is also called "machine geometry", computer- aided geometric design, and geometric modelling. See List of combinatorial computational geometry topics for another flavor of computational geometry that states problems in terms of geometric objects as discrete entities and hence the methods of their solution are mostly theories and algorithms of combinatorial character.
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Much of folk psychology involves the attribution of intentional states (or more specifically as a subclass, propositional attitudes). Eliminativists point out that these states are generally ascribed syntactic and semantic properties. An example of this is the language of thought hypothesis, which attributes a discrete, combinatorial syntax and other linguistic properties to these mental phenomena. Eliminativists argue that such discrete and combinatorial characteristics have no place in the neurosciences, which speak of action potentials, spiking frequencies, and other effects which are continuous and distributed in nature.
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A projective spherical variety is a Mori dream space. Spherical embeddings are classified by so-called colored fans, a generalization of fans for toric varieties; this is known as Luna-Vust Theory. In his seminal paper, develops a framework to classify complex spherical subgroups of reductive groups; he reduces the classification of spherical subgroups to wonderful subgroups. He works out completely the case of groups of type A and conjectures that the combinatorial objects (homogeneous spherical data) he introduces indeed provide a combinatorial classification of spherical subgroups.
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GeneDecks is a novel analysis tool to identify similar or partner genes, which provides a similarity metric by highlighting shared descriptors between genes, based on GeneCards’ unique wealth of combinatorial annotations of human genes. # Annotation combinatory: Using GeneDecks, one can get a set of similar genes for a particular gene with a selected combinatorial annotation. The summary table result in ranking the different level of similarity between the identified genes and the probe gene. # Annotation unification: Different data sources often offer annotations with heterogeneous naming system.
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Václav (Vašek) Chvátal ( is a Professor Emeritus in the Department of Computer Science and Software Engineering at Concordia University in Montreal, Quebec, Canada. He has published extensively on topics in graph theory, combinatorics, and combinatorial optimization.
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A standard text book by Adámek and Rosický appeared in 1994.Adamek/Rosický 1994 Accessible categories also have applications in homotopy theory.J. Rosický "On combinatorial model categories", arXiv, 16 August 2007. Retrieved on 19 January 2008.
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3, 461–477. In modern terms, the Kurosh subgroup theorem is a straightforward corollary of the basic structural results of Bass–Serre theory about groups acting on trees.Daniel E. Cohen. Combinatorial group theory: a topological approach.
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Our exposition regarding Whitehead's algorithm mostly follows Ch.I.4 in the book of Lyndon and Schupp,Roger Lyndon and Paul Schupp, Combinatorial group theory. Reprint of the 1977 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.
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A somewhat easier (more symmetrical) puzzle, the 8×8 rectangle with a 2×2 hole in the center, was solved by Dana Scott as far back as 1958.Dana S. Scott (1958). "Programming a combinatorial puzzle".
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Yoccoz's worked on the theory of dynamical systems, his contributions include advances to KAM theory, and the introduction of the method of Yoccoz puzzles, a combinatorial technique which proved useful to the study of Julia sets.
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An additive-combinatorial proof based on formal power product expansions was given by Giedrius Alkauskas. This proof uses neither the Euclidean algorithm nor the binomial theorem, but rather it employs formal power series with rational coefficients.
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However, note that the estimates for the items obtained in the second experiment have errors that correlate with each other. Many problems of the design of experiments involve combinatorial designs, as in this example and others.
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Lawler was an expert on combinatorial optimization and a founder of the field, the author of the widely used textbook Combinatorial Optimization: Networks and Matroids and coauthor of The Traveling Salesman Problem: a guided tour of combinatorial optimization. He played a central role in rescuing the ellipsoid method for linear programming from obscurity in the West.. He also wrote (with D. E. Wood) a heavily cited 1966 survey on branch and bound algorithms,. selected as a citation classic in 1987, and another influential early paper on dynamic programming with J. M. Moore.. Lawler was also the first to observe that matroid intersection can be solved in polynomial time.. The NP- completeness proofs for two of Karp's 21 NP-complete problems, directed Hamiltonian cycle and 3-dimensional matching, were credited by Karp to Lawler. The NP-completeness of 3-dimensional matching is an example of one of Lawler's favorite observations, the "mystical power of twoness": for many combinatorial optimization problems that can be parametrized by an integer, the problem can be solved in polynomial time when the parameter is two but becomes NP-complete when the parameter is three.
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A combinatorial problem, Michigan Math. J. 1 (1952), 81–88. Sagan gave a shifted hook walk proof for the hook length formula for shifted Young tableaux in 1980.Sagan, B. On selecting a random shifted Young tableau.
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In 2015 the Association for Computing Machinery listed her as a Distinguished Member. She became a Fellow of the Society for Industrial and Applied Mathematics in 2016 "for contributions to the theory and applications of combinatorial optimization".
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Dmitry Feichtner-Kozlov Dmitry Feichtner-Kozlov (born December 16, 1972 in Tomsk, Russia) is a Russian-German mathematician. He works in the field of Applied and Combinatorial Topology, where he publishes under the name Dmitry N. Kozlov.
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The original Capelli identity is a statement about determinants. Later, analogous identities were found for permanents, immanants and traces. Based on the combinatorial approach paper by S.G. Williamson was one of the first results in this direction.
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Manfred Wilhelm Padberg (October 10, 1941 in Bottrop, Germany\- May 12, 2014) is a German mathematician who worked with linear and combinatorial optimization. He and Ellis L. Johnson won the John von Neumann Theory Prize in 2000.
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In the 1930s and 1940s, Eric Temple Bell tried unsuccessfully to make this kind of argument logically rigorous. The combinatorialist John Riordan in his book Combinatorial Identities published in the 1960s, used techniques of this sort extensively.
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Kathrin Klamroth (born 1968) is a German mathematician and computer scientist whose research topics include combinatorial optimization and facility location. She is a professor in the department of mathematics and computer science at the University of Wuppertal.
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It categorifies the Alexander polynomial. Knot Floer homology was defined by and independently by . It is known to detect knot genus. Using grid diagrams for the Heegaard splittings, knot Floer homology was given a combinatorial construction by .
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Although this method is polynomial time, it is not used in practice because the lattice has high dimension and huge entries, which makes the computation slow. The exponential complexity in the algorithm of Zassenhaus comes from a combinatorial problem: how to select the right subsets of f_1(x),...,f_r(x). State of the art factoring implementations work in a manner similar to Zassenhaus, except that the combinatorial problem is translated to a lattice problem that is then solved by LLL.M. van Hoeij: Factoring polynomials and the knapsack problem.
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The initial proposed problem for this project, now called Polymath1 by the Polymath community, was to find a new combinatorial proof to the density version of the Hales–Jewett theorem. As the project took form, two main threads of discourse emerged. The first thread, which was carried out in the comments of Gowers's blog, would continue with the original goal of finding a combinatorial proof. The second thread, which was carried out in the comments of Terence Tao's blog, focused on calculating bounds on density of Hales-Jewett numbers and Moser numbers for low dimensions.
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Martin Grötschel is one of the most internationally renowned experts in the field of combinatorial optimization. Martin Grötschel's main mathematical research fields are graph theory, linear and mixed-integer optimization and operations research. Already in his doctoral thesis, Grötschel achieved significant progress in the development of solution methods of the Traveling Salesman Problem, in particular, he contributed significantly to understanding the cutting-plane method. His publications together with L. Lovász and A. Schrijver on the ellipsoid method and its application in the combinatorial and convex optimization gained worldwide recognition.
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In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a natural and geometric way such a presentation. A very closely related topic is geometric group theory, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides. It also comprises a number of algorithmically insoluble problems, most notably the word problem for groups; and the classical Burnside problem.
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Studies to demonstrate the existence of language have been difficult due to the range of possible interpretations. For instance, some have argued that in order for a communication system to count as a language it must be "combinatorial", having an open ended set of grammar- compliant sentences made from a finite vocabulary. Research on parrots by Irene Pepperberg is claimed to demonstrate the innate ability for grammatical structures, including the existence of concepts such as nouns, adjectives and verbs. In the wild, Black-capped Chickadees innate vocalizations have been rigorously shown to have combinatorial language.
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A figure illustrating the vehicle rescheduling problem The vehicle rescheduling problem (VRSP) is a combinatorial optimization and integer programming problem seeking to service customers on a trip after change of schedule such as vehicle break down or major delay. Proposed by Li, Mirchandani and Borenstein in 2007, the VRSP is an important problem in the fields of transportation and logistics. Determining the optimal solution is an NP-complete problem in combinatorial optimization, so in practice heuristic and deterministic methods are used to find acceptably good solutions for the VRSP.
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Both reactants can be mixtures and in this case the procedure would be even more efficient. For practical reasons however, it is advisable to use the split-mix method in which one of two mixtures is replaced by single building blocks (BBs). The mixtures are so important that there are no combinatorial libraries without using mixture in the synthesis, and if a mixture is used in a process inevitably combinatorial library forms. The split-mix synthesis is usually realized using solid support but it is possible to apply it in solution, too.
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He has served two terms as the CMSA's President (2007–09 and 2014). He is an editor in chief of the Electronic Journal of Combinatorics and is on the editorial board of several other journals including the Journal of Combinatorial Designs. Wanless represented Australia at the International Mathematical Olympiad in Cuba in 1987. Wanless is the coauthor (with Colbourn and Dinitz) of the chapter on Latin squares in the CRC Handbook of Combinatorial Designs and the author of the chapter on matrix permanents in the CRC Handbook of Linear Algebra.
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In 1994, his paper Molecular Computation of Solutions To Combinatorial Problems described the experimental use of DNA as a computational system. In it, he solved a seven-node instance of the Hamiltonian Graph problem, an NP- complete problem similar to the travelling salesman problem. While the solution to a seven-node instance is trivial, this paper is the first known instance of the successful use of DNA to compute an algorithm. DNA computing has been shown to have potential as a means to solve several other large-scale combinatorial search problems.
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In modern sources, the Adian–Rabin theorem is usually stated as follows:Roger Lyndon and Paul Schupp, Combinatorial group theory. Reprint of the 1977 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. ; Ch. IV, Theorem 4.1, p. 192G. Baumslag.
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Transactions of the American Mathematical Society, vol. 361 (2009), no. 2, pp. 715-734 Floyd also wrote a number of joint papers with James W. Cannon and Walter R. Parry exploring a combinatorial approach to the Cannon conjectureJ.
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Susan Marie Hermiller is an American mathematician specializing in the computational, combinatorial, and geometric theory of groups. She is a Willa Cather Professor of Mathematics and the current Graduate Chair for Mathematics at the University of Nebraska–Lincoln.
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The multinomial coefficients have a direct combinatorial interpretation, as the number of ways of depositing n distinct objects into m distinct bins, with k1 objects in the first bin, k2 objects in the second bin, and so on.
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Jarvis works at the School of Mathematics and Physics, at the University of Tasmania. His main focus is on algebraic structures in mathematical physics and their applications, especially combinatorial Hopf algebras in integrable systems and quantum field theory.
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Instead, nodes are generated as they are explored, and typically discarded thereafter. A solution to a combinatorial search instance may consist of the goal state itself, or of a path from some initial state to the goal state.
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While investigating dimers on a square lattice (essentially a domino tiling), he independently discovered combinatorial Fisher-Kasteleyn- Temperley algorithm. In a series of papers with C. M. Fortuin he developed random cluster model and obtained the FKG inequality.
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The notion of quasi-randomness have been extended to many other combinatorial structures, such as sequences, tournaments, hypergraphs and graph limits. In general, the theory of quasi- randomness gives a rigorous approach to 'random-like' or 'pseudorandom' alternatives.
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In combinatorial mathematics, the q-difference polynomials or q-harmonic polynomials are a polynomial sequence defined in terms of the q-derivative. They are a generalized type of Brenke polynomial, and generalize the Appell polynomials. See also Sheffer sequence.
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The tilings by these marked tiles are necessarily aperiodic.C. Goodman-Strauss, Matching Rules and Substitution Tilings, Annals Math., 147 (1998), 181-223.Th. Fernique and N. Ollinger, Combinatorial substitutions and sofic tilings, Journees Automates Cellulaires 2010, J. Kari ed.
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In 1953, Roth partially resolved the initial conjecture by proving they must contain an arithmetic progression of length 3 using Fourier analytic methods. Eventually, in 1975, Szemerédi proved Szemerédi's theorem using combinatorial techniques, resolving the original conjecture in full.
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Currently micro-threading is implemented on the Cell Broadband Engine. Three to fivefold performance improvement could be achieved. Currently it is proven for regular and combinatorial algorithms. Some other efforts are trying to prove its viability for scientific algorithms.
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In combinatorial mathematics, Stanley's reciprocity theorem, named after MIT mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the generating function of any rational cone (defined below) and the generating function of the cone's interior.
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In the 8th edition of the International Patent Classification (IPC), which entered into force on January 1, 2006, a special subclass has been created for patent applications and patents related to inventions in the domain of combinatorial chemistry: "C40B".
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The quasi- isometry properties of the history graph can be studied using subdivision rules. For instance, the history graph is quasi-isometric to hyperbolic space exactly when the subdivision rule is conformal, as described in the combinatorial Riemann mapping theorem.
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Micha Sharir (; born 8 June 1950 in Tel Aviv, Israel) is an Israeli mathematician and computer scientist. He is a professor at Tel Aviv University, notable for his contributions to computational geometry and combinatorial geometry, having authored hundreds of papers.
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In 2000, Lehmann and Wegener introduced Dependency Rules with their incarnation of the CTE, the CTE XL (eXtended Logics). Further features include the automated generation of test suites using combinatorial test design (e.g. all- pairs testing). Development was performed by DaimlerChrysler.
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Stanton's main areas of research were in statistics and applied statistics; algebra; mathematical biology; combinatorial design theory, including pair-wise balanced designs, difference sets, covering and packing designs, and room squares; graph theory, including graph models of networks; and algorithms.
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The History of Combinatorial Group Theory: A Case Study in the History of Ideas. Studies in the History of Mathematics and Physical Sciences 9. Springer-Verlag, New York, 1982. and because he spoke German so perfectly, he remained in Berlin.
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In some settings, such as the one described above, a zone diagram can be interpreted as a certain equilibrium between mutually hostile kingdoms,. In a discrete setting it can be interpreted as a stable configuration in a certain combinatorial game.
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If f is linear then it always passes the test. Blum, Luby and Rubinfeld showed that if the test passes with probability 1-\varepsilon then f is O(\varepsilon)-close to a Fourier character. Their proof was combinatorial. Bellare et al.
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She has also significantly contributed to the development of compound collections and combinatorial libraries which investigators can use to develop and compare measures of molecular similarity and diversity. Her collection of MAO (monoamine oxidase) inhibitors has been widely used by researchers.
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His breakthrough in pairwise balanced designs, and orthogonal Latin squares built upon the groundwork set before him, by R. C. Bose, E. T. Parker, S. S. Shrikhande, and Haim Hanani is widely referenced in Combinatorial Design Theory and Coding Theory.
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A partition of a complete graph on 8 vertices into 7 colors (perfect matchings), the case r = 2 of Baranyai's theorem In combinatorial mathematics, Baranyai's theorem (proved by and named after Zsolt Baranyai) deals with the decompositions of complete hypergraphs.
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The term appears to have been first used by Rassenti, Smith, and Bulfin in 1982.Rassenti, S.J., V.L. Smith, and R.L. Bluffing (1982). “A Combinatorial Auction Mechanism for Airport time Slot Allocation,” Bell J. of Economics, v.13, pp. 402-417.
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Ivan Vadimovich Loseu (publishing under the name Ivan Losev; Belarusian Іван Вадзімовіч Лосеў, Russian Иван Вадимович Лосев, born 18 October 1981 in Minsk, Belarus) is a Belarusian-American mathematician, specializing in representation theory, symplectic geometry, algebraic geometry, and combinatorial algebra.
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In applied mathematics, branch and price is a method of combinatorial optimization for solving integer linear programming (ILP) and mixed integer linear programming (MILP) problems with many variables. The method is a hybrid of branch and bound and column generation methods.
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In mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in hypergeometric series. These identities occur frequently in solutions to combinatorial problems, and also in the analysis of algorithms. These identities were traditionally found 'by hand'.
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Petra Mutzel is a German computer scientist, a University Professor of computer science at the Technical University of Dortmund.Faculty profile, TU Dortmund, retrieved 2014-07-04. Her research is in the areas of algorithm engineering, graph drawing and combinatorial optimization.
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From 1987 he was Professor (and Distinguished Professor Emeritus) in the Computer Science Department at New Mexico State University in Las Cruces. He was one of the founders of the Journal of Combinatorial Theory and the Journal of Graph Theory.
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Nicole Megow is a German discrete mathematician and theoretical computer scientist whose research topics include combinatorial optimization, approximation algorithms, and online algorithms for scheduling. She is a professor in the faculty of mathematics and computer science at the University of Bremen.
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Complexity of combinatorial market makers. In Proceedings of the 9th ACM Conference on Electronic Commerce, pages 190-199. ACM, New York, 2008. He has also contributed to a study of the behavior of informed traders working with LMSR market makers.
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Historically, questions about extensions first surfaced in combinatorial optimization, where extensions arise naturally from extended formulations. A seminal work by Yannakakis connected extension complexity to various other notions in mathematics, in particular nonnegative rank of nonnegative matrices and communication complexity.
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The different types of edge in a bidirected graph In the mathematical domain of graph theory, a bidirected graph (introduced by ). Reprinted in Combinatorial Optimization — Eureka, You Shrink!, Springer-Verlag, Lecture Notes in Computer Science 2570, 2003, pp. 27–30, .
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Modern versions of the ping-pong lemma can be found in many books such as Lyndon&Schupp;,Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer-Verlag, New York, 2001. "Classics in Mathematics" series, reprint of the 1977 edition.
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The higher order directed homotopy theory can be developed through cylinder functor and path functor, all constructions and properties being expressed in the setting of categorical algebra^{[1]}. This approach emphasizes the combinatorial role of cubical sets in directed algebraic topology.
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For a polynomial-time list-decoding algorithm to exist, we need the combinatorial guarantee that any Hamming ball of radius pn around a received word r (where p is the fraction of errors in terms of the block length n) has a small number of codewords. This is because the list size itself is clearly a lower bound on the running time of the algorithm. Hence, we require the list size to be a polynomial in the block length n of the code. A combinatorial consequence of this requirement is that it imposes an upper bound on the rate of a code.
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The main emphasis of his work was broadly considered theory of oscillations, which included optics and quantum mechanics. He was a co-discoverer of inelastic combinatorial scattering of light used now in Raman spectroscopy (see below). This paradigm-altering discovery (together with G. S. Landsberg) had occurred at the Moscow State University just one week earlier than a parallel discovery of the same phenomena by C. V. Raman and K. S. Krishnan. In Russian literature it is called "combinatorial scattering of light" (from combination of frequencies of photons and molecular vibrations) but in English it is named after Raman.
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Most notably, these papers demonstrated how a good characterization of the polyhedron associated with a combinatorial optimization problem could lead, via the duality theory of linear programming, to the construction of an efficient algorithm for the solution of that problem. Additional landmark work of Edmonds is in the area of matroids. He found a polyhedral description for all spanning trees of a graph, and more generally for all independent sets of a matroid. Building on this, as a novel application of linear programming to discrete mathematics, he proved the matroid intersection theorem, a very general combinatorial min-max theorem.
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Professor Dordick currently leads the Jonathan S. Dordick Research Group, which works in part at the Center for Biotechnology and Interdisciplinary Studies working on biomedical technologies.The Jonathan S. Dordick Research Group Present and past research has included studies of Biocatalysis in Nonaqueous Media, Combinatorial Biocatalysis, Nanobiotechnology, enzyme technology, molecular bioprocessing. More specifically, they work on the development of enzymatic catalysis under extreme conditions (e.g. high salt concentrations), enzymes in the synthesis and modification of polymeric materials, combinatorial biocatalysis for drug discovery and polymer synthesis, and the generation of biocatalysts and biomimetics with unique activities and selectivities.
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An embedded graph uniquely defines cyclic orders of edges incident to the same vertex. The set of all these cyclic orders is called a rotation system. Embeddings with the same rotation system are considered to be equivalent and the corresponding equivalence class of embeddings is called combinatorial embedding (as opposed to the term topological embedding, which refers to the previous definition in terms of points and curves). Sometimes, the rotation system itself is called a "combinatorial embedding".... An embedded graph also defines natural cyclic orders of edges which constitutes the boundaries of the faces of the embedding.
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He went on to tackle the problem of large sets of disjoint Steiner triple systems. Zhu Lie (), a professor of mathematics at Soochow University working also in combinatorial mathematics, realized the importance of his work and suggested that he submit it to the international journal Journal of Combinatorial Theory, Series A. He wrote to its editorial board that he had essentially solved the problem, and the editors replied to him that if what he said was true, it would be a major achievement. (Many leaders in the field had worked on the problem starting from in 1917.
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Combinatorial biology allows the generation and selection of the large number of ligands for high-throughput screening. Combinatorial biology techniques generally begin with large numbers of peptides, which are generated and screened by physically linking a gene encoding a protein and a copy of said protein. This could involve the protein being fused to the M13 minor coat protein pIII, with the gene encoding this protein being held within the phage particle. Large libraries of phages with different proteins on their surfaces can then be screened through automated selection and amplification for a protein that binds tightly to a particular target.
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One of the key developments within the field of DCC is the use of proteins (or other biological macromolecules, such as nucleic acids) to influence the evolution and generation of components within a DCL.Greaney, M. F.; Bhat, V. T. Protein- directed dynamic combinatorial chemistry. In Dynamic combinatorial chemistry: in drug discovery, bioinorganic chemistry, and materials sciences; Miller, B. L., Ed.; John Wiley & Sons: New Jersey, 2010; Chapter 2, pp 43–82. Protein- directed DCC provides a way to generate, identify and rank novel protein ligands, and therefore have huge potential in the areas of enzyme inhibition and drug discovery.
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In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for analysing discrete structures in terms of generating functions. Examples of discrete structures are (finite) graphs, permutations, trees, and so on; each of these has an associated generating function which counts how many structures there are of a certain size. One goal of species theory is to be able to analyse complicated structures by describing them in terms of transformations and combinations of simpler structures. These operations correspond to equivalent manipulations of generating functions, so producing such functions for complicated structures is much easier than with other methods.
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In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour. The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether,For example L'émergence de la notion de groupe d'homologie, Nicolas Basbois (PDF), note 41, explicitly names Noether as inventing homology groups.
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While the problem can be formulated purely in geometric terms, the methods of the proof drew on commutative algebra techniques. A signature theorem in combinatorial commutative algebra is the characterization of h-vectors of simplicial polytopes conjectured in 1970 by Peter McMullen. Known as the g-theorem, it was proved in 1979 by Stanley (necessity of the conditions, algebraic argument) and by Louis Billera and Carl W. Lee (sufficiency, combinatorial and geometric construction). A major open question was the extension of this characterization from simplicial polytopes to simplicial spheres, the g-conjecture, which was resolved in 2018 by Karim Adiprasito.
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He said that the one combinatorial idea he would like to be remembered for is the correspondence between combinatorial problems and problems of the location of the zeroes of polynomials. He worked on the theory of incidence algebras (which generalize the 19th-century theory of Möbius inversion) and popularized their study among combinatorialists, set the umbral calculus on a rigorous foundation, unified the theory of Sheffer sequences and polynomial sequences of binomial type, and worked on fundamental problems in probability theory. His philosophical work was largely in the phenomenology of Edmund Husserl. Rota founded the Advances in Mathematics journal in 1961.
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Since any two of these positions are part of a combinatorial line, the third element of that line must be occupied by the opposite symbol (since we are assuming that no combinatorial line has all three elements filled with the same symbol). In other words, for each choice of abcdef (which can be thought of as an element of the six-dimensional hypercube W36), there are six (overlapping) possibilities: # abcdef11 and abcdef12 are noughts; abcdef13 is a cross. # abcdef11 and abcdef22 are noughts; abcdef33 is a cross. # abcdef12 and abcdef22 are noughts; abcdef32 is a cross.
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Metaheuristics are used for combinatorial optimization in which an optimal solution is sought over a discrete search-space. An example problem is the travelling salesman problem where the search-space of candidate solutions grows faster than exponentially as the size of the problem increases, which makes an exhaustive search for the optimal solution infeasible. Additionally, multidimensional combinatorial problems, including most design problems in engineeringTomoiagă B, Chindriş M, Sumper A, Sudria-Andreu A, Villafafila-Robles R. Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II. Energies. 2013; 6(3):1439–1455.
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Research Fellows, Rényi Institute He was Research Professor at the Courant Institute of Mathematical Sciences at NYUPersonal website of János Pach, NYU (since 1986), Distinguished Professor of Computer Science at City College, CUNY (1992-2011), and Neilson Professor at Smith College (2008-2009). Between 2008 and 2019, he was Professor of the Chair of Combinatorial Geometry at École Polytechnique Fédérale de Lausanne.János Pach appointed as a full professor of mathematics, EPFL, December 12, 2007.Chair of Combinatorial Geometry, EPFL He was the program chair for the International Symposium on Graph Drawing in 2004 and Symposium on Computational Geometry in 2015.
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Beginning from 1926, Mandelstam and Landsberg initiated experimental studies on vibrational scattering of light in crystals at the Moscow State University. Their intention was to prove the theoretical prediction made by Mandelstam in 1918 regarding the fine structure splitting in Rayleigh scattering due to light scattering on thermal acoustic waves. As a result of this research, Landsberg and Mandelstam discovered the effect of the inelastic combinatorial scattering of light on 21 February 1928 ("combinatorial" – from combination of frequencies of photons and molecular vibrations). They presented this fundamental discovery for the first time at a colloquium on 27 April 1928.
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It is to be hoped that with all of these contributions to the field, as well as those books on graph theory written primarily for an audience of electrical engineers, operations researchers, or social scientists, two developments will become more pronounced: (i) each scholar who finds it convenient to use structural or combinatorial concepts in his own research will not feel obliged to rediscover graph theory for himself, ab initio. (ii) this elegant theory with its applications within mathematics to topology, logic, algebra, and combinatorial analysis will eventually become an undergraduate course at most modern universities.
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In 2003, he won the George B. Dantzig Prize of the Mathematical Programming Society and SIAM for "deep and fundamental research contributions to discrete optimization".Prestigious prizes awarded to Lex Schrijver and Bert Gerards, CWI, retrieved 2012-03-30. In 2006, he was a joint winner of the INFORMS John von Neumann Theory Prize with Grötschel and Lovász for their work in combinatorial optimization, and in particular for their joint work in the book Geometric Algorithms and Combinatorial Optimization showing the polynomial-time equivalence of separation and optimization.INFORMS Awards for Alexander Schrijver , retrieved 2012-03-30.
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Master of Science and PhD in Algorithms and Computation cover very broad areas of algorithms and computation. The scope of this program includes both theoretical and applied research from these fields. The main aim of this program is to promote the exchange of ideas in this active research community. The specific themes targeted for Master of science in Algorithms and Computation are Computational Geometry, Algorithms and Data Structures, Approximation Algorithms, Randomized Algorithms, Graph Drawing and Graph Algorithms, Combinatorial Algorithms, Graphs in Bioinformatics, String Algorithms, Combinatorial Optimization, Computational Biology, Computational Finance, Cryptography, and Parallel and Distributed Algorithms.
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Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, that describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, while an early form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron. Currently combinatorial group theory as an area is largely subsumed by geometric group theory. Moreover, the term "geometric group theory" came to often include studying discrete groups using probabilistic, measure-theoretic, arithmetic, analytic and other approaches that lie outside of the traditional combinatorial group theory arsenal. In the first half of the 20th century, pioneering work of Max Dehn, Jakob Nielsen, Kurt Reidemeister and Otto Schreier, J. H. C. Whitehead, Egbert van Kampen, amongst others, introduced some topological and geometric ideas into the study of discrete groups.
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A combinatorial cube of dimension one is called a combinatorial line. For instance, in the game of tic-tac-toe, the nine cells of a tic-tac-toe board can be specified by strings of length two over the three-symbol alphabet {1,2,3} (the Cartesian coordinates of the cells), and the winning lines of three cells form combinatorial lines. Horizontal lines are obtained by fixing the y-coordinate (the second position of the length-two string) and letting the x-coordinate be chosen freely, and vertical lines are obtained by fixing the x-coordinate and letting the y-coordinate be chosen freely. The two diagonal lines of the tic-tac-toe board can be specified by a parameter word with two wildcard characters that are either constrained to be equal (for the main diagonal) or constrained to be related by a group action that swaps the 1 and 3 characters (for the antidiagonal).
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In particular, for sufficiently well-behaved generating functions, Cauchy's integral formula can be used to recover the power series coefficients (the real object of study) from the generating function, and knowledge of the singularities of the function can be used to derive accurate estimates of the resulting integrals. After an introductory chapter and a chapter giving examples of the possible behaviors of rational functions and meromorphic functions, the remaining chapters of this part discuss the way the singularities of a function can be used to analyze the asymptotic behavior of its power series, apply this method to a large number of combinatorial examples, and study the saddle-point method of contour integration for handling some trickier examples. The final part investigates the behavior of random combinatorial structures, rather than the total number of structures, using the same toolbox. Beyond expected values for combinatorial quantities of interest, it also studies limit theorems and large deviations theory for these quantities.
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Robert Brooks gave a combinatorial scheme to produce pseudocharacters of any free group Fn; this scheme was later shown to yield an infinite-dimensional family of pseudocharacters (see ). Epstein and Fujiwara later extended these results to all non-elementary Gromov-hyperbolic groups.
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Hans Heinrich Wilhelm Magnus known as Wilhelm Magnus (February 5, 1907, Berlin, Germany – October 15, 1990, New Rochelle, NY) was a German-American mathematician. He made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations.
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Michael J. Steele. "Probability theory and combinatorial optimization". SIAM, Philadelphia (1997). . Besides, analogues of Fekete's lemma have been proved for subadditive real maps (with additional assumptions) from finite subsets of an amenable group Theorem 6.1 , and further, of a cancellative left-amenable semigroup.
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Algebraic enumeration is a subfield of enumeration that deals with finding exact formulas for the number of combinatorial objects of a given type, rather than estimating this number asymptotically. Methods of finding these formulas include generating functions and the solution of recurrence relations..
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The optimization objective can be initial engagement, a user action (such as click or install), or a post-install metric (such as purchase, registration, or lifetime value). Optimization of this objective is carried out using some form of discrete or combinatorial optimization.
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10 (2006), pp. 63-99. using finite subdivision rules. This represents one of the few plausible lines of attack of the conjecture.Ilya Kapovich, and Nadia Benakli, in Boundaries of hyperbolic groups, Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp.
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Since 2017 she has been a Visiting Miller Professor at the University of California, Berkeley. Along with colleagues O. Mandelshtam and L. Williams, in 2018 Corteel developed a new characterization of both symmetric and nonsymmetric Macdonald polynomials using the combinatorial exclusion process.
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The PE Applied Biosystems division partnered with Hyseq, Inc., for work on the new DNA chip technology, and also worked with Tecan U.S., Inc., on combinatorial chemistry automation systems, and also with Molecular Informatics, Inc. on genetic data management and analysis automated systems.
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A research monograph on the subject is . , there is no known combinatorial interpretation of all the coefficients of the Kazhdan–Lusztig polynomials (as the cardinalities of some natural sets) even for the symmetric groups, though explicit formulas exist in many special cases.
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Mary Celine Fasenmyer (October 4, 1906, Crown, Pennsylvania – December 27, 1996, Erie, Pennsylvania) was an American mathematician. She is most noted for her work on hypergeometric functions and linear algebra.Rosen, KH and Michaels, JG (2000) Handbook of Discrete and Combinatorial Mathematics, CRC Press.
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Daniel Cremers is a computer scientist, Professor of Informatics and Mathematics and Chair of Computer Vision & Artificial Intelligence at the Technische Universität München. His research foci are computer vision, mathematical image, partial differential equations, convex and combinatorial optimization, machine learning and statistical inference.
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Hermiller became the Willa Cather Professor in 2017. She was included in the 2019 class of fellows of the American Mathematical Society "for contributions to combinatorial and geometric group theory and for service to the profession, particularly in support of underrepresented groups".
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Richard Michael Wilson (23 November 1945) is a mathematician and a professor at the California Institute of Technology. Wilson and his PhD supervisor Dijen K. Ray-Chaudhuri, solved Kirkman's schoolgirl problem in 1968. Wilson is known for his work in combinatorial mathematics.
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In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno's formula.
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Viral received his B.Engg in Computer Engineering from the Padmabhushan Vasantdada Patil Pratishthan's College of Engineering. He went on to study Computer Science at the University of California, Santa Barbara, where he received his Ph.D. in the field of Combinatorial Scientific Computing.
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Fukuda and Terlaky. Compare Ziegler. It has been applied to linear- fractional programming, quadratic-programming problems, and linear complementarity problems. Outside of combinatorial optimization, OM theory also appears in convex minimization in Rockafellar's theory of "monotropic programming" and related notions of "fortified descent".
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His research focuses on topics in algebraic graph theory, particularly the symmetry of graphs and the action of finite groups on combinatorial objects. He is regarded as the founder of the Slovenian school of research in algebraic graph theory and permutation groups.
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Hydrogenography is a combinatorial method based on the observation of optical changes on the metal surface by hydrogen absorption.Determination of thermodynamic properties of gradient films using hydrogenography The method allows the examination of thousands of combinations of alloy samples in a single batch.
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Grigory Samuilovich Landsberg (Russian: Григорий Самуилович Ландсберг; 22 January 1890 – 2 February 1957) was a Soviet physicist who worked in the fields of optics and spectroscopy. Together with Leonid Mandelstam he co- discoverer inelastic combinatorial scattering of light, which known as Raman scattering.
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Positional scanning was introduced independently by Furka et al.Furka Á, Sebestyén F, WC 93/24517, 1993. and Pinilla et al.Pinilla C, Appel JR, Blanc P, Houghten RA (1993) Rapid identification of high affinity peptide ligands using positional scanning synthetic peptide combinatorial libraries.
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In the study of permutation patterns, a combinatorial class of permutation classes, enumerated by permutation length, is called a Wilf class. The study of enumerations of specific permutation classes has turned up unexpected equivalences in counting sequences of seemingly unrelated permutation classes.
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According to the University of Nevada, Reno's official website, Czarnik's research interests include "chemical product improvement using deuterium substitution, combinatorial chemistry as a tool for drug discovery, nucleic acids as targets for small molecule intervention, and fluorescent chemosensors of ion and molecule recognition".
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The Lemke–Howson algorithm is an algorithm that computes a Nash equilibrium of a bimatrix game, named after its inventors, Carlton E. Lemke and J. T. Howson. It is said to be "the best known among the combinatorial algorithms for finding a Nash equilibrium".
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Other precursors of geometric group theory include small cancellation theory and Bass–Serre theory. Small cancellation theory was introduced by Martin Grindlinger in the 1960s and further developed by Roger Lyndon and Paul Schupp.Roger Lyndon and Paul Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin, 1977.
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Coding theory started as a part of design theory with early combinatorial constructions of error-correcting codes. The main idea of the subject is to design efficient and reliable methods of data transmission. It is now a large field of study, part of information theory.
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Vladimir Batagelj (born June 14, 1948 in Idrija, Yugoslavia) is a Slovenian mathematician and an emeritus professor of mathematics at the University of Ljubljana. He is known for his work in discrete mathematics and combinatorial optimization, particularly analysis of social networks and other large networks.
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After several decades, classical AI technologies started to face intractable issues (e.g. combinatorial explosion) when confronted with real-world modeling problems. All approaches to address these issues focus on modeling intelligences situated in an environment. They have become known as the situated approach to AI.
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Savage earned her Ph.D. in 1977 from the University of Illinois at Urbana–Champaign under the supervision of David E. Muller; her thesis concerned parallel graph algorithms. Much of her more recent research has concerned Gray codes and algorithms for efficient generation of combinatorial objects.
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In mathematics, the Kalmanson combinatorial conditions are a set of conditions on the distance matrix used in determining the solvability of the traveling salesman problem. These conditions apply to a special kind of cost matrix, the Kalmanson matrix, and are named after Kenneth Kalmanson.
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Like many puzzles in recreational mathematics, polyominoes raise many combinatorial problems. The most basic is enumerating polyominoes of a given size. No formula has been found except for special classes of polyominoes. A number of estimates are known, and there are algorithms for calculating them.
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In 2016, a proof of the Kelmans–Seymour conjecture was claimed by Xingxing Yu of the Georgia Institute of Technology and his Ph.D. students Dawei He and Yan Wang.; ; ; A sequence four papers proving this conjecture appeared in Journal of Combinatorial Theory Series B.
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A distinct strand of invariant theory, going back to the classical constructive and combinatorial methods of the nineteenth century, has been developed by Gian- Carlo Rota and his school. A prominent example of this circle of ideas is given by the theory of standard monomials.
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An examination of multi-dimensional tic-tac-toe of various numbers of dimensions and board sizes was presented in the article "Hypercube Tic-Tac-Toe" by Golomb and Hales. Another study appears in the book Combinatorial Games: Tic-Tac-Toe Theory by József Beck.
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The abundant water-soluble proteins of the vertebrate lens (by definition, crystallins) are therefore now generally considered to be examples of multifunctional proteins with their precise functions being context- and concentration-dependant, consistent with a combinatorial and quantitative role of gene expression in cell differentiation.
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A set function is called fractionally subadditive (or XOS) if it is the maximum of several additive set functions. This valuation class was defined, and termed XOS, by Noam Nisan, in the context of combinatorial auctions. The term fractionally-subadditive was given by Uriel Feige.
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In mathematics, cyclically reduced word is a concept of combinatorial group theory. Let F(X) be a free group. Then a word w in F(X) is said to be cyclically reduced if and only if every cyclic permutation of the word is reduced.
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Zvezdelina Entcheva Stankova (; born 15 September 1969) is a professor of mathematics at Mills College and a teaching professor at the University of California, Berkeley, the founder of the Berkeley Math Circle, and an expert in the combinatorial enumeration of permutations with forbidden patterns.
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Semidefinite programming has been applied to find approximate solutions to combinatorial optimization problems, such as the solution of the max cut problem with an approximation ratio of 0.87856. SDPs are also used in geometry to determine tensegrity graphs, and arise in control theory as LMIs.
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In addition, infinitely many sites are allowed in the definition (this setting has applications in geometry of numbers and crystallography), but again, in many cases only finitely many sites are considered. In the particular case where the space is a finite-dimensional Euclidean space, each site is a point, there are finitely many points and all of them are different, then the Voronoi cells are convex polytopes and they can be represented in a combinatorial way using their vertices, sides, two- dimensional faces, etc. Sometimes the induced combinatorial structure is referred to as the Voronoi diagram. In general however, the Voronoi cells may not be convex or even connected.
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The combinatorial hierarchy is a mathematical structure of hierarchical sets of bit-strings generated from an algorithm based on "discrimination" (or equivalently XOR). Discovered by Frederick Parker-Rhodes, the hierarchy gives the physical coupling constants from a simple aphysical model. This is a key consequence of bit-string physics, which supposes that reality can be represented by a process of operations on finite strings of dichotomous symbols, or bits (1's and 0's). Bit-string physics has developed from Frederick Parker-Rhodes' 1964 discovery of the combinatorial hierarchy: four numbers produced from a purely mathematical recursive algorithm that correspond to the relative strengths of the four forces.
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Proofs That Really Count: the Art of Combinatorial Proof is an undergraduate- level mathematics book on combinatorial proofs of mathematical identies. That is, it concerns equations between two integer-valued formulas, shown to be equal either by showing that both sides of the equation count the same type of mathematical objects, or by finding a one-to-one correspondence between the different types of object that they count. It was written by Arthur Benjamin and Jennifer Quinn, and published in 2003 by the Mathematical Association of America as volume 27 of their Dolciani Mathematical Expositions series. It won the Beckenbach Book Prize of the Mathematical Association of America.
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In convex geometry, the simplex algorithm for linear programming is interpreted as tracing a path along the vertices of a convex polyhedron. Oriented matroid theory studies the combinatorial invariants that are revealed in the sign patterns of the matrices that appear as pivoting algorithms exchange bases. The development of an axiom system for oriented matroids was initiated by R. Tyrrell Rockafellar to describe the sign patterns of the matrices arising through the pivoting operations of Dantzig's simplex algorithm; Rockafellar was inspired by Albert W. Tucker's studies of such sign patterns in "Tucker tableaux". The theory of oriented matroids has led to breakthroughs in combinatorial optimization.
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The academicians Sergey Mergelyan, Norair Arakelian, Alexandr Talalyan, Raphayel Alexandrian, Rouben V. Ambartzumian and Anry Nersesyan also have greatly influenced the formation of the scientific profile of the Institute and largely contributed to mathematics in general. In particular Rouben V. Ambartzumian is famous for his work in Stochastic Geometry and Integral Geometry, where he created a new branch called Combinatorial Integral Geometry.1982 – R.V. Ambartzumian. Combinatorial Integral Geometry with Applications to Mathematical Stereology, John Wiley, Chichester, NY He has provided solutions to a number of classical problems in particular the solution to the Buffon Sylvester problem as well as the Hilbert's fourth problem in dimensions 2R.
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It served to put the homology theory of the time--the first decade of the twentieth century--on a rigorous basis, since it showed that the topological effect (on homology groups) of continuous mappings could in a given case be expressed in a finitary way. This must be seen against the background of a realisation at the time that continuity was in general compatible with the pathological, in some other areas. This initiated, one could say, the era of combinatorial topology. There is a further simplicial approximation theorem for homotopies, stating that a homotopy between continuous mappings can likewise be approximated by a combinatorial version.
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In combinatorial mathematics, rotation systems (also called combinatorial embeddings) encode embeddings of graphs onto orientable surfaces, by describing the circular ordering of a graph's edges around each vertex. A more formal definition of a rotation system involves pairs of permutations; such a pair is sufficient to determine a multigraph, a surface, and a 2-cell embedding of the multigraph onto the surface. Every rotation scheme defines a unique 2-cell embedding of a connected multigraph on a closed oriented surface (up to orientation preserving topological equivalence). Conversely, any embedding of a connected multigraph G on an oriented closed surface defines a unique rotation system having G as its underlying multigraph.
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Paul Eugene Schupp (born March 12, 1937) is a Professor Emeritus of Mathematics at the University of Illinois at Urbana Champaign. He is known for his contributions to geometric group theory, computational complexity and the theory of computability. He received his Ph.D. from the University of Michigan in 1966 under the direction of Roger Lyndon. Together with Roger Lyndon he is the coauthor of the book "Combinatorial Group Theory" which provided a comprehensive account of the subject of Combinatorial Group Theory, starting with the work of Dehn in the 1910s and to late 1970s and remains a modern standard for the subject of small cancellation theory.
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In the drug discovery process, the synthesis and biological evaluation of small molecules of interest have typically been a long and laborious process. Combinatorial chemistry has emerged in recent decades as an approach to quickly and efficiently synthesize large numbers of potential small molecule drug candidates. In a typical synthesis, only a single target molecule is produced at the end of a synthetic scheme, with each step in a synthesis producing only a single product. In a combinatorial synthesis, when using only single starting material, it is possible to synthesize a large library of molecules using identical reaction conditions that can then be screened for their biological activity.
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The key idea is that of a function which assigns a non-negative integer to each position of a class of combinatorial games, now called impartial games, and which greatly assists in the identification of winning and losing positions, and of the winning moves from the former. The number assigned to a position by this function is called its Grundy value (or Grundy number), and the function itself is called the Sprague–Grundy function, in honour of its co- discoverers.Almost any comprehensive treatment of combinatorial game theory will cover Sprague and Grundy's results in some form. Examples are Berlekamp et al. (1984), Conway (1991), Siegel (2013), and Smith (2015).
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Arithmetic combinatorics is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive combinatorics is the special case when only the operations of addition and subtraction are involved. Ben Green explains arithmetic combinatorics in his review of "Additive Combinatorics" by Tao and Vu.
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Alon has published more than five hundred research papers, mostly in combinatorics and in theoretical computer science, and one book. He has also published under the pseudonym "A. Nilli". Alon is the principal founder of the Combinatorial Nullstellensatz which has many applications in combinatorics and number theory.
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This was at a time when other music theorists were codifying the rules of counterpoint, and writing about other rule-based and combinatorial systems to aid in the composition of music, such as the Arca Musarithmica of Athanasius Kircher. Information about his life and work remains scanty.
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Reprinted in the "Classics in mathematics" series, 2000. It studies van Kampen diagrams, corresponding to finite group presentations, via combinatorial curvature conditions and derives algebraic and algorithmic properties of groups from such analysis. Bass–Serre theory, introduced in the 1977 book of Serre,J.-P. Serre, Trees.
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Sylvie Corteel is a French mathematician at the Centre national de la recherche scientifique and Paris Diderot University who is an editor-in-chief of the Journal of Combinatorial Theory, Series A. Her research concerns the enumerative combinatorics and algebraic combinatorics of permutations, tableaux, and partitions.
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GW501516 was initially discovered during a research collaboration between GSK and Ligand Pharmaceuticals that began in 1992. The discovery of the compound was published in a 2001 issue of PNAS. Oliver et al. reported that they used "combinatorial chemistry and structure-based drug design" to develop it.
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The identification of test relevant aspects usually follows the (functional) specification (e.g. requirements, use cases …) of the system under test. These aspects form the input and output data space of the test object. The second step of test design then follows the principles of combinatorial test design.
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The Topaz application contains all the functions relating to abstract simplicial complexes. Many advance topological calculations over simplicial complexes can be performed like homology groups, orientation, fundamental group. There is also a combinatorial collection of properties that can be computed like a shelling and Hasse diagrams.
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Vladimir Grigorevich Boltyansky (; 26 April 1925 – 16 April 2019), also transliterated as Boltyanski, Boltyanskii, or Boltjansky, was a Soviet and Russian mathematician, educator and author of popular mathematical books and articles. He was best known for his books on topology, combinatorial geometry and Hilbert's third problem.
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Sally Patricia Cockburn (born 1960) is a mathematician whose research ranges from algebraic topology and set theory to geometric graph theory and combinatorial optimization. A Canadian immigrant to the US, she is a professor of mathematics at Hamilton College, and chair of the mathematics department at Hamilton.
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1931, 61, 964–69.. The Passerini reaction This organic reaction was discovered by Mario Passerini in 1921 in Florence, Italy. It is the first isocyanide based multi-component reaction developed, and currently plays a central role in combinatorial chemistry.Dömling, A.; Ugi, I. Angew. Chem. Int. Ed. Engl.
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Since all combinations of subsets and models are exhaustively searched, this method faces the problem of combinatorial complexity. In the current example, noisy ‘smile’ and ‘frown’ patterns are sought. They are shown in Fig.1a without noise, and in Fig.1b with the noise, as actually measured.
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The Fibonacci Association is a mathematical organization that specializes in the Fibonacci number sequence and a wide variety of related subjects, generalizations, and applications, including recurrence relations, combinatorial identities, binomial coefficients, prime numbers, pseudoprimes, continued fractions, the golden ratio, linear algebra, geometry, real analysis, and complex analysis.
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Some ideas underlying the small cancellation theory go back to the work of Max Dehn in the 1910s.Bruce Chandler and Wilhelm Magnus, The history of combinatorial group theory. A case study in the history of ideas. Studies in the History of Mathematics and Physical Sciences, 9.
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Gad Menahem Landau (born 1954) is an Israeli computer scientist noted for his contributions to combinatorial pattern matching and string algorithms and is the founding department chair of the Computer Science Department at the University of Haifa. He has coauthored over 100 peer-reviewed scientific papers.
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Start pages of a 1928 article of Schreier on the Jordan–Hölder theorem Otto Schreier (3 March 1901 in Vienna, Austria – 2 June 1929 in Hamburg, Germany) was a Jewish-Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups.
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Often, "a neuron can receive contacts from up to 10,000 presynaptic neurons, and, in turn, any one neuron can contact up to 10,000 postsynaptic neurons. The combinatorial possibility could give rise to enormously complex neuronal circuits or network topologies, which might be very difficult to understand".
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Clobber is an abstract strategy game invented in 2001 by combinatorial game theorists Michael H. Albert, J.P. Grossman and Richard Nowakowski. It has subsequently been studied by Elwyn Berlekamp and Erik Demaine among others. Since 2005, it has been one of the events in the Computer Olympiad.
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In SBCG, phrasal signs are licensed by correspondence to the mother of some licit construct of the grammar. A construct is a local tree with signs at its nodes. Combinatorial constructions define classes of constructs. Lexical class constructions describe combinatoric and other properties common to a group of lexemes.
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Magliveras does research on combinatorial designs, permutation groups, finite geometries, encryption of data (cryptography), and data security. In 2001 he received the Euler Medal. He is a co-author of the 2007 book Secure group communications over data networks. Magliveras is married since 1961 and has two children.
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If the distance matrix satisfies the Kalmanson combinatorial conditions then Neighbor-net will return the corresponding circular ordering. The method is implemented in the SplitsTree and R/Phangorn packages. Examples of the application of Neighbor-net can be found in virology, horticulture, dinosaur genetics, comparative linguistics, and archaeology.
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Network problems that involve finding an optimal way of doing something are studied under the name of combinatorial optimization. Examples include network flow, shortest path problem, transport problem, transshipment problem, location problem, matching problem, assignment problem, packing problem, routing problem, Critical Path Analysis and PERT (Program Evaluation & Review Technique).
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The theorem has since become well known and repeatedly studied and generalized in graph theory, in part because of its elegant proof using techniques from algebraic graph theory. More strongly, write that all proofs are somehow based on linear algebra: "no combinatorial proof for this result is known".
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Mark Sapir (born February 12, 1957)Mark Sapir's CV, Department of Mathematics, Vanderbilt University. Accessed November 4, 2018 is a U.S. and Russian mathematician working in geometric group theory, semigroup theory and combinatorial algebra. He is a Centennial Professor of Mathematics in the Department of Mathematics at Vanderbilt University.
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A matroid is a mathematical structure that generalizes the notion of linear independence from vector spaces to arbitrary sets. If an optimization problem has the structure of a matroid, then the appropriate greedy algorithm will solve it optimally.Papadimitriou, Christos H., and Kenneth Steiglitz. Combinatorial optimization: algorithms and complexity.
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An early example of pretty-printing was Bill Gosper's "GRINDEF" (i.e. 'grind function') program (c. 1967), which used combinatorial search with pruning to format LISP programs. Early versions operated on the executable (list structure) form of the Lisp program and were oblivious to the special meanings of various functions.
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These combinatorial auctions are cleared as bid, rather than at prices based on dual variables. Only recently have researchers found robust means to obtain dual variables from integer programs.O’Neill, R.P., P.M. Sotkiewicz, B.F. Hobbs, M.H. Rothkopf, W.R. Stewart (2005). Effective market-clearing prices in markets with non-convexities.
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Catherine Greenhill is an Australian mathematician known for her research on random graphs, combinatorial enumeration and Markov chains. She is a professor of mathematics in the School of Mathematics and Statistics at the University of New South Wales, and an editor-in-chief of the Electronic Journal of Combinatorics.
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Chemical biologists used automated synthesis of diverse small molecule libraries in order to perform high-throughput analysis of biological processes. Such experiments may lead to discovery of small molecules with antibiotic or chemotherapeutic properties. These combinatorial chemistry approaches are identical to those employed in the discipline of pharmacology.
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James Anthony Dominic Welsh (born 29 August 1938)Prof Dominic J A Welsh, Debrett's, retrieved 2012-03-11. is an English mathematician, an emeritus professor of Oxford University's Mathematical Institute. He is an expert in matroid theory,. the computational complexity of combinatorial enumeration problems, percolation theory, and cryptography.
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Structads are an approach to the semantics of logic that are based upon generalising the notion of sequent along the lines of Joyal's combinatorial species, allowing the treatment of more drastically nonstandard logics than those described above, where, for example, the ',' of the sequent calculus is not associative.
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In mathematics, Ky Fan's lemma (KFL) is a combinatorial lemma about labellings of triangulations. It is a generalization of Tucker's lemma. It was proved by Ky Fan in 1952. In this example, where n = 2, there is no 2-dimensional alternating simplex (since the labels are only 1,2).
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Two standalone systems for calculations with matroids are Kingan's Oid and Hlineny's Macek. Both of them are open sourced packages. "Oid" is an interactive, extensible software system for experimenting with matroids. "Macek" is a specialized software system with tools and routines for reasonably efficient combinatorial computations with representable matroids.
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The problems of finding a vertex disjoint and edge disjoint cycle covers with minimal number of cycles are NP- complete. The problems are not in complexity class APX. The variants for digraphs are not in APX either.Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties (1999) p.
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While still in Austria, Servatius began working on combinatorial group theory, and her first publication (appearing while she was a graduate student) is in that subject. She switched to the theory of structural rigidity for her doctoral research, and later became the author (with Jack Graver and Herman Servatius) of the book Combinatorial Rigidity (1993). Another well-cited paper of hers in this area characterizes the planar Laman graphs, the minimally rigid graphs that can be embedded without crossings in the plane, as the graphs of pseudotriangulations, partitions of a plane region into subregions with three convex corners studied in computational geometry. Servatius is also the co- editor of a book on matroid theory.
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William Lawrence Kocay is a Canadian professor at the department of computer science at St. Paul's College of the University of Manitoba and a graph theorist. He is known for his work in graph algorithms and the reconstruction conjecture and is affectionately referred to as "Wild Bill" by his students. Bill Kocay is a former managing editor (from Jan 1988 to May 1997) of Ars Combinatoria, a Canadian journal of combinatorial mathematics, is a founding fellow of the Institute of Combinatorics and its Applications. His research interests include algorithms for graphs, the development of mathematical software, the graph reconstruction problem, the graph isomorphism problem, projective geometry, Hamiltonian cycles, planarity, graph embedding algorithms, graphs on surfaces, and combinatorial designs.
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The number of combinatorial types of Hanner polytopes of dimension d is the same as the number of simple series-parallel graphs with d unlabeled edges./ For d = 1, 2, 3, ... it is: :1, 1, 2, 4, 8, 18, 40, 94, 224, 548, ... . A more explicit bijection between the Hanner polytopes of dimension d and the cographs with d vertices is given by . For this bijection, the Hanner polytopes are assumed to be represented geometrically using coordinates in {0,1,−1} rather than as combinatorial equivalence classes; in particular, there are two different geometric forms of a Hanner polytope even in two dimensions, the square with vertex coordinates (±1,±1) and the diamond with vertex coordinates (0,±1) and (±1,0).
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In mathematics, the polynomial method is an algebraic approach to combinatorics problems that involves capturing some combinatorial structure using polynomials and proceeding to argue about their algebraic properties. Recently, the polynomial method has led to the development of remarkably simple solutions to several long-standing open problems. The polynomial method encompasses a wide range of specific techniques for using polynomials and ideas from areas such as algebraic geometry to solve combinatorics problems. While a few techniques that follow the framework of the polynomial method, such as Alon's Combinatorial Nullstellensatz, have been known since the 1990s, it was not until around 2010 that a broader framework for the polynomial method has been developed.
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Combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects.. In many such problems, exhaustive search is not tractable. It operates on the domain of those optimization problems in which the set of feasible solutions is discrete or can be reduced to discrete, and in which the goal is to find the best solution. Typical problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. Some research literature considers discrete optimization to consist of integer programming together with combinatorial optimization (which in turn is composed of optimization problems dealing with graph structures) although all of these topics have closely intertwined research literature.
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Even though natural product drug discovery has not probably been the most fashionable trend in the pharmaceutical industry in recent times, a large proportion of new chemical entities still are nature-derived compounds, and thus, it has been suggested that effectiveness of combinatorial chemistry could be improved by enhancing the chemical diversity of screening libraries.Su QB, Beeler AB, Lobkovsky E, Porco JA, Panek JS "Stereochemical diversity through cyclodimerization: Synthesis of polyketide-like macrodiolides." Org Lett 2003, 5:2149-2152. As chirality and rigidity are the two most important features distinguishing approved drugs and natural products from compounds in combinatorial chemistry libraries, these are the two issues emphasized in so-called diversity oriented libraries, i.e.
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In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete. In his 1972 paper, "Reducibility Among Combinatorial Problems", Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete (also called the Cook-Levin theorem) to show that there is a polynomial time many- one reduction from the boolean satisfiability problem to each of 21 combinatorial and graph theoretical computational problems, thereby showing that they are all NP-complete. This was one of the first demonstrations that many natural computational problems occurring throughout computer science are computationally intractable, and it drove interest in the study of NP- completeness and the P versus NP problem.
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Schrijver was one of the winners of the Delbert Ray Fulkerson Prize of the American Mathematical Society in 1982 for his work with Martin Grötschel and László Lovász on applications of the ellipsoid method to combinatorial optimization; he won the same prize in 2003 for his research on minimization of submodular functions.AMS Awards, retrieved 2012-03-30. He won the INFORMS Frederick W. Lanchester Prize in 1986 for his book Theory of Linear and Integer Programming, and again in 2004 for his book Combinatorial Optimization: Polyhedra and Efficiency. He was an Invited Speaker of the International Congress of Mathematicians (ICM) in 1986 in Berkeley and of the ICM in 1998 in Berlin.
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There are two independent classifications for group-testing problems; every group-testing problem is either adaptive or non-adaptive, and either probabilistic or combinatorial. In probabilistic models, the defective items are assumed to follow some probability distribution and the aim is to minimise the expected number of tests needed to identify the defectiveness of every item. On the other hand, with combinatorial group testing, the goal is to minimise the number of tests needed in a 'worst-case scenario' – that is, create a minmax algorithm – and no knowledge of the distribution of defectives is assumed. The other classification, adaptivity, concerns what information can be used when choosing which items to group into a test.
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Some of the basic theory of combinatorial designs originated in the statistician Ronald Fisher's work on the design of biological experiments. Modern applications are also found in a wide gamut of areas including finite geometry, tournament scheduling, lotteries, mathematical chemistry, mathematical biology, algorithm design and analysis, networking, group testing and cryptography.
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He is both a puzzle historian and a composer of puzzles, and many of his puzzles have been published in newspapers and magazines. In combinatorial number theory, Singmaster's conjecture states that there is an upper bound on the number of times a number other than 1 can appear in Pascal's triangle.
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A type of cone of particular interest to pure mathematicians is the partially ordered set of rational cones. "Rational cones are important objects in toric algebraic geometry, combinatorial commutative algebra, geometric combinatorics, integer programming." . This object arises when we study cones in \R^d together with the lattice \Z^d.
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The overall effect of a well designed Private Electronic Market is what is described as allocative efficiency or in simple terms: a win-win for the seller (who maximizes revenue) and buyers (acquiring exactly what is of highest value to them). PEMs are based on game theory and combinatorial auction theory.
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Combinatorial drug products were proposed for treating diagnosed conditions long before they were proposed for preventive medicine, including "aspolol" (a combination of aspirin and atenolol) for those diagnosed with cardiovascular disease. Fixed-dose combination (FDC) products today are also common for treating other diseases, such as tuberculosis and HIV/AIDS.
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In 2001, Haiman proved that the dimension is indeed n! (see [4]). This breakthrough led to the discovery of many hidden connections and new aspects of symmetric group representation theory, as well as combinatorial objects (e.g., insertion tableaux, Haglund's inversion numbers, and the role of parking functions in representation theory).
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See Otto Schreier published an algebraic proof of this result in 1927, and Kurt Reidemeister included a comprehensive treatment of free groups in his 1932 book on combinatorial topology. Later on in the 1930s, Wilhelm Magnus discovered the connection between the lower central series of free groups and free Lie algebras.
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An average position typically has thirty to forty possible moves, but there may be as few as zero (in the case of checkmate or stalemate) or (in a constructed position) as many as 218. Chess has inspired many combinatorial puzzles, such as the knight's tour and the eight queens puzzle.
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Cannon, Floyd, and Parry first studied finite subdivision rules in an attempt to prove the following conjecture: Cannon's conjecture: Every Gromov hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.James W. Cannon. The combinatorial Riemann mapping theorem. Acta Mathematica 173 (1994), no. 2, pp. 155-234.
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Photolithography can also be used to cross-link cell-seeded photo-polymerizable ECM for three- dimensional studies. Using ECM microarrays to optimize combinatorial effects of collagen, laminin, and fibronectin on stem cells is more advantageous than conventional well plates due to its higher throughput and lower requirement of expensive reagents.
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Plancherel measure appears naturally in combinatorial and probabilistic problems, especially in the study of longest increasing subsequence of a random permutation \sigma. As a result of its importance in that area, in many current research papers the term Plancherel measure almost exclusively refers to the case of the symmetric group S_n.
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Lipski graduated from the Program of Fundamental Problems of Technology, at the Warsaw Technical University. He received Ph.D. in computer science at the Computational Center (later: Institute for Computer Science) of the Polish Academy of Sciences, under supervision of Prof. Wiktor Marek. The dissertation title was: 'Combinatorial Aspects of Information Retrieval'.
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After earning her Ph.D., Fasenmyer published two papers which expanded on her doctorate work. These would be further elaborated by Doron Zeilberger and Herbert Wilf into "WZ theory", which allowed computerized proof of many combinatorial identities. After this, she returned to Mercyhurst to teach and did not engage in further research.
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Rosemary A. Bailey (born 1947) is a British statistician who works in the design of experiments and the analysis of variance and in related areas of combinatorial design, especially in association schemes. She has written books on the design of experiments, on association schemes, and on linear models in statistics.
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Genevac Ltd is a company which was founded in 1990 by Michael Cole. It used to specialize in the manufacture of vacuum pumps and centrifugal evaporators, but has since directed its attention to equipment designed for combinatorial chemistry. Following a series of mergers, it is currently a subsidiary of SP Industries.
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Among symmetric groups, only S6 has a non-trivial outer automorphism, which one can call exceptional (in analogy with exceptional Lie algebras) or exotic. In fact, Out(S6) = C2.Lam, T. Y., & Leep, D. B. (1993). "Combinatorial structure on the automorphism group of S6". Expositiones Mathematicae, 11(4), 289–308.
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Lu Jiaxi (; June 10, 1935 – October 31, 1983) was a self-taught Chinese mathematician who made important contributions in combinatorial design theory. He was a high school physics teacher in a remote city and worked in his spare time on the problem of large sets of disjoint Steiner triple systems.
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That article proposed a combinatorial auction for airplane take-off and landing slots. The U.S. government is now seeking to implement such an auction. The modern electricity market is an important example of a two-sided smart market.,Alvey T., Goodwin D., Xingwang M., Streiffert D. and Sun D. (1998).
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Greenhill was the 2010 winner of the Hall Medal of the Institute of Combinatorics and its Applications. She was president of the Combinatorial Mathematics Society of Australasia for 2011–2013. In 2015 the Australian Academy of Science awarded her their Christopher Heyde Medal for distinguished research in the mathematical sciences.
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This was still on minimal cost networks. During that summer I met another of my heroes, John Riordan, one of the great early workers in combinatorics. His book An Introduction to Combinatorial Analysis is a classic. He was working at Bell Labs in West Street in Manhattan at that time.
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Her graduate research was supported by the Berkeley Fellowship and a National Science Foundation Fellowship. Her PhD thesis Clusters and features from combinatorial stochastic processes looked at clustering and speeding up the analysis of large, streaming data sets. In 2013 she was selected for the Berkeley EECS Rising Stars conference.
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In Normalized Systems separation of concerns is one of the four guiding principles. Adhering to this principle is one of the tools that helps reduce the combinatorial effects that, over time, get introduced in software that is being maintained. In Normalized Systems separation of concerns is actively supported by the tools.
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ACM Names Fellows for Computing Advances that Are Driving Innovation , Association for Computing Machinery, December 8, 2011. In 2014 he was elected as a member of Academia Europaea,. and in 2015 as a fellow of the American Mathematical Society "for contributions to discrete and combinatorial geometry and to convexity and combinatorics.".
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Latin squares and quasigroups are equivalent mathematical objects, although the former has a combinatorial nature while the latter is more algebraic. The listing below will consider the examples of some very small orders, which is the side length of the square, or the number of elements in the equivalent quasigroup.
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240px This example of design experiments is attributed to Harold Hotelling, building on examples from Frank Yates.Herman Chernoff, Sequential Analysis and Optimal Design, SIAM Monograph, 1972. The experiments designed in this example involve combinatorial designs. Weights of eight objects are measured using a pan balance and set of standard weights.
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An association scheme is a collection of binary relations satisfying certain compatibility conditions. Association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.
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In general, finding optimal algorithms for adaptive combinatorial group testing is difficult, and although the computational complexity of group testing has not been determined, it is suspected to be hard in some complexity class. However, an important breakthrough occurred in 1972, with the introduction of the generalised binary-splitting algorithm.
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Diane Margaret Maclagan (born 1974)Birth year from Library of Congress catalog entry, retrieved 2019-03-03. is a professor of mathematics at the University of Warwick. She is a researcher in combinatorial and computational commutative algebra and algebraic geometry, with an emphasis on toric varieties, Hilbert schemes, and tropical geometry.
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INFOCOMP Journal of Computer Science () is an international peer-reviewed quarterly scientific journal. The areas of interest covered are artificial intelligence, combinatorial optimization and meta-heuristics, computer graphics, image processing and virtual reality, databases, graphs, applied mathematics and theory of computation, hypermedia and multimedia, information systems, information technology in education, and software engineering.
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Orbit portraits turn out to be useful combinatorial objects in studying the connection between the dynamics and the parameter spaces of other families of maps as well. In particular, they have been used to study the patterns of all periodic dynamical rays landing on a periodic cycle of a unicritical anti-holomorphic polynomial.
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This usually depends on counting: determining the number of points at stake. Knowledge of counting begins with some simple examples and heuristics. Combinatorial game theory has been implicated in gaining actual proofs rather than practical ways to win positions. Oyose, or large yose, is a term often used in English language literature.
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A core enhancement of copula models are dynamic copulas, introduced by Albanese et al. (2005) and (2007). The "dynamic conditioning" approach models the evolution of multi-factor super-lattices, which correlate the return processes of each entity at each time step. Binomial dynamic copulas apply combinatorial methods to avoid Monte Carlo simulations.
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Checking whether a pattern (or a transaction) supports a given subgraph is an NP-complete problem, since it is an NP-complete instance of the subgraph isomorphism problem. Furthermore, due to combinatorial explosion, according to Lei et al., "mining all frequent subtree patterns becomes infeasible for a large and dense tree database".
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There are several methods for enumerating zeolite frameworks. The method of Earl et al. uses simulated annealing to arrange a fixed number of tetrahedral atoms under periodic symmetry constraints. The method of Treacy and Rivin uses combinatorial enumeration to identify four-valent graphs under fixed symmetry and fixed number of tetrahedral atoms.
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Jacob Fox (born Jacob Licht in 1984) is an American mathematician. He is a professor at Stanford University. His research interests are in Hungarian- style combinatorics, particularly Ramsey theory, extremal graph theory, combinatorial number theory, and probabilistic methods in combinatorics. Fox grew up in West Hartford, Connecticut and attended Hall High School.
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Extremal optimization (EO) is an optimization heuristic inspired by the Bak–Sneppen model of self-organized criticality from the field of statistical physics. This heuristic was designed initially to address combinatorial optimization problems such as the travelling salesman problem and spin glasses, although the technique has been demonstrated to function in optimization domains.
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B. Orthogonal polynomials, Random matrices. Given a weight on a contour, the corresponding orthogonal polynomials can be computed via the solution of a Riemann–Hilbert factorization problem (). Furthermore, the distribution of eigenvalues of random matrices in several classical ensembles is reduced to computations involving orthogonal polynomials (see for example ). C. Combinatorial probability.
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Melody Tung Chan is an American mathematician and violinist who works as Manning Assistant Professor of Mathematics at Brown University. She is a winner of the Alice T. Schafer Prize and of the AWM-Microsoft Research Prize in Algebra and Number Theory. Her research involves combinatorial commutative algebra, graph theory, and tropical geometry.
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Dirac started his studies at St John's College, Cambridge in 1942, but by 1942 the war saw him serving in the aircraft industry. He received his MA in 1949, and moved to the University of London, getting his Ph.D. "On the Colouring of Graphs: Combinatorial topology of Linear Complexes" there under Richard Rado.
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However, this method is complicated and has a high polynomial exponent. More efficient combinatorial algorithms are known for many special cases. For many years the complexity of recognizing Berge graphs and perfect graphs remained open. From the definition of Berge graphs, it follows immediately that their recognition is in co-NP (Lovász 1983).
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To compute χρ(π) where π is a permutation, one can use the combinatorial Murnaghan–Nakayama rule .Richard Stanley, Enumerative Combinatorics, Vol. 2 Note that χρ is constant on conjugacy classes, that is, χρ(π) = χρ(σ−1πσ) for all permutations σ. Over other fields the situation can become much more complicated.
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In 2009, the Polymath Project developed a new proof of the density Hales–Jewett theorem based ideas on from the proof of the corners theorem. Dodos, Kanellopoulos, and Tyros gave a simplified version of the Polymath proof. The Hales–Jewett is generalized by the Graham–Rothschild theorem, on higher-dimensional combinatorial cubes.
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R. A. Brualdi is a professor emeritus of combinatorial mathematics at the University of Wisconsin–Madison. Brualdi received his Ph.D. from Syracuse University in 1964; his advisor was H. J. Ryser. Brualdi is an Editor-in-Chief of the Electronic Journal of Combinatorics. He has over 200 publications in several mathematical journals.
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Deijfen was one of the 2018 recipients of the Paul R. Halmos – Lester R. Ford Award of the Mathematical Association of America for her paper with Alexander E. Holroyd and James B. Martin, "Friendly Frogs, Stable Marriage, and the Magic of Invariance", using combinatorial game theory to analyze the stable marriage problem.
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D. H. Lehmer wrote the article "The Machine Tools of Combinatorics," which is chapter one in the book "Applied Combinatorial Mathematics," by Edwin Beckenbach, 1964. It describes methods for producing permutations, combinations etc. This was a uniquely valuable resource and has only been rivaled recently by Volume 4 of Donald Knuth's series.
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In mathematics, and particularly in axiomatic set theory, the diamond principle is a combinatorial principle introduced by Ronald Jensen in that holds in the constructible universe () and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that the Axiom of constructibility () implies the existence of a Suslin tree.
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Combinatorics of Experimental Design is a textbook on the design of experiments, a subject that connects applications in statistics to the theory of combinatorial mathematics. It was written by mathematician Anne Penfold Street and her daughter, statistician Deborah Street, and published in 1987 by the Oxford University Press under their Clarendon Press imprint.
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A Young tableau (pl.: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900.
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Stanley is known for his two-volume book Enumerative Combinatorics (1986–1999). He is also the author of Combinatorics and Commutative Algebra (1983) and well over 200 research articles in mathematics. He has served as thesis advisor to more than 58 doctoral students, many of whom have had distinguished careers in combinatorial research.
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The Socolar-Taylor tile forms two-dimensional aperiodic tilings, but is defined by combinatorial matching conditions rather than purely by its shape. In higher dimensions, the problem is solved: the Schmitt-Conway-Danzer tile is the prototile of a monohedral aperiodic tiling of three-dimensional Euclidean space, and cannot tile space periodically.
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A CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex).
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In Ramsey theory, parameter words and combinatorial cubes may be used to formulate the Graham–Rothschild theorem, according to which, for every finite alphabet and group action, and every combination of integer values m, k, and r, there exists a sufficiently large number n such that if each combinatorial cube over strings of length n is assigned one of r colors, then there exists a combinatorial cube all of whose subcubes have the same color. This result is a key foundation for structural Ramsey theory, and is used to define Graham's number, an enormous number used to estimate the value of n for a certain combination of values. In computer science, in the problem of searching for duplicate code, the source code for a given routine or module may be transformed into a parameter word by converting it into a sequence of tokens, and for each variable or subroutine name, replacing each copy of the same name with the same wildcard character. If code is duplicated, the resulting parameter words will remain equal even if some of the variables or subroutines have been renamed.
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In combinatorial optimization, the matroid intersection problem is to find a largest common independent set in two matroids over the same ground set. If the elements of the matroid are assigned real weights, the weighted matroid intersection problem is to find a common independent set with the maximum possible weight. These problems generalize many problems in combinatorial optimization including finding maximum matchings and maximum weight matchings in bipartite graphs and finding arborescences in directed graphs. The matroid intersection theorem, due to Jack Edmonds, says that there is always a simple upper bound certificate, consisting of a partitioning of the ground set amongst the two matroids, whose value (sum of respective ranks) equals the size of a maximum common independent set.
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We provide a few examples of the utility of the Redheffer matrices interpreted as a (0,1) matrix whose parity corresponds to inclusion in an increasing sequence of index sets. These examples should serve to freshen up some of the at times dated historical perspective of these matrices, and their being footnote-worthy by virtue of an inherent, and deep, relation of their determinants to the Mertens function and equivalent statements of the Riemann Hypothesis. This interpretation is a great deal more combinatorial in construction than typical treatments of the special Redheffer matrix determinants. Nonetheless, this combinatorial twist on enumerating special sequences of sums has been explored more recently in a number of papers and is a topic of active interest in pre-print archives.
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Chapter 11 connects the low- dimensional faces together into the skeleton of a polytope, and proves the van Kampen–Flores theorem about non-embeddability of skeletons into lower- dimensional spaces. Chapter 12 studies the question of when a skeleton uniquely determines the higher-dimensional combinatorial structure of its polytope. Chapter 13 provides a complete answer to this theorem for three- dimensional convex polytopes via Steinitz's theorem, which characterizes the graphs of convex polyhedra combinatorially and can be used to show that they can only be realized as a convex polyhedron in one way. It also touches on the multisets of face sizes that can be realized as polyhedra (Eberhard's theorem) and on the combinatorial types of polyhedra that can have inscribed spheres or circumscribed spheres.
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The latest volume published by the New York Number Theory seminar is entitled the "Combinatorial and Additive Number Theory III" published in 2020. It concerns the lecture notes of the seminars of the years 2017 and 2018. The only remaining co-founder is Melvyn B. Nathanson who is also the editor of this volume.
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Solving chess means finding an optimal strategy for playing chess, i.e. one by which one of the players (White or Black) can always force a victory, or both can force a draw (see Solved game). It also means more generally solving chess-like games (i.e. combinatorial games of perfect information), such as infinite chess.
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Vertex Pharmaceuticals, Inc. is an American biopharmaceutical company based in Boston, Massachusetts. It was one of the first biotech firms to use an explicit strategy of rational drug design rather than combinatorial chemistry. It maintains headquarters in South Boston, Massachusetts, and three research facilities, in San Diego, California, and Milton Park, near Oxford, England.
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Heinrich was born in Murwillumbah, New South Wales. As an undergraduate at the University of Newcastle in Australia, Heinrich graduated as a University Medalist in 1976. She continued at Newcastle as a graduate student, and completed her doctorate there in 1979. Her dissertation, Some problems on combinatorial arrays, was supervised by Walter D. Wallis.
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In 1850, Kirkman observed that his 1846 solution to Woolhouse's problem had an additional property, which he set out as a puzzle in The Lady's and Gentleman's Diary: This problem became known as Kirkman's schoolgirl problem, subsequently to become Kirkman's most famous result. He published several additional works on combinatorial design theory in later years.
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Software that is used for designing factorial experiments plays an important role in scientific experiments and represents a route to the implementation of design of experiments procedures that derive from statistical and combinatorial theory. In principle, easy-to-use design of experiments (DOE) software should be available to all experimenters to foster use of DOE.
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Thy-1 can be considered as a surrogate marker for various kind of stem cells (e.g. hematopoietic stem cells or HSCs). It is one of the popular combinatorial surface markers for FACS for stem cells in combination with other markers like CD34. In humans, Thy-1 is expressed on neurons and HSCs among others.
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LEMON is an open source graph library written in the C++ language providing implementations of common data structures and algorithms with focus on combinatorial optimization tasks connected mainly with graphs and networks. The library is part of the COIN-OR project. LEMON is an abbreviation of Library for Efficient Modeling and Optimization in Networks.
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Jon Lee (born 1960) is an American mathematician and operations researcher, the G. Lawton and Louise G. Johnson Professor of Engineering at the University of Michigan.Summary of Personnel Actions, Regents of the University of Michigan, October 2011, retrieved 2012-02-25. He is known for his research in nonlinear discrete optimization and combinatorial optimization.
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In fact, both the Aria and the Sonatina are based almost exclusively on all-combinatorial twelve-tone rows with inherent repetitive properties . The Aria features solo bassoon, while the Sonatina has the strongest neoclassical overtones, with a waltz-like accompaniment to the agile solo clarinet. The Finale is an extended lament, dominated by the horn .
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Michael J. Dinneen is an American-New Zealand mathematician and computer scientist working as a senior lecturer at the University of Auckland, New Zealand. He is co-director of the Center for Discrete Mathematics and Theoretical Computer Science. He does research in combinatorial algorithms, distributive programming, experimental graph theory, and experimental algorithmic information theory.
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Mehlhorn's CV..ACM Fellow citation to Mehlhorn for "important contributions in complexity theory and in the design, analysis, and practice of combinatorial and geometric algorithms." He is the 2014 winner of the Erasmus Medal of the Academia Europaea.2014 Erasmus Medal awarded to Professor Dr. Kurt Mehlhorn MAE, Academia Europaea, retrieved 2014-06-21.
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In the 1990s, Geoffrey Sampson worked with William A. Gale of AT&T; to create and implement a simplified and easier-to-use variant of the Good–Turing methodSampson, Geoffrey and Gale, William A. (1995) Good‐turing frequency estimation without tears described below. Various heuristic justifications and a simple combinatorial derivation have been provided.
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In combinatorial game theory, and particularly in the theory of impartial games in misère play, an indistinguishability quotient is a commutative monoid that generalizes and localizes the Sprague–Grundy theorem for a specific game's rule set. In the specific case of misere-play impartial games, such commutative monoids have become known as misere quotients.
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The Combs method is a rule base reduction method of writing fuzzy logic rules described by William E. Combs in 1997. It is designed to prevent combinatorial explosion in fuzzy logic rules. The Combs method takes advantage of the logical equality ((p \land q) \Rightarrow r) \iff ((p \Rightarrow r) \lor (q \Rightarrow r)).
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Combinatorial homophilic interaction between γ-protocadherin multimers greatly expands the molecular diversity of cell adhesion. Proc. Natl. Acad. Sci. USA 107:14893–98Lefebvre JL, Kostadinov D, Chen WV, Maniatis T, Sanes JR. 2012. Protocadherins mediate dendritic self- avoidance in the mammalian nervous system. Nature. Wu W, Ahlsen G, Baker D, Shapiro L, Zipursky SL. 2012.
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The number of computer operations in this example was about 1010. Thus, a problem that was not solvable due to combinatorial complexity becomes solvable using dynamic logic. During an adaptation process, initially fuzzy and uncertain models are associated with structures in the input signals, and fuzzy models become more definite and crisp with successive iterations.
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Affine Lie algebras are a special case of Kac–Moody algebras, which have particular importance in mathematics and theoretical physics, especially conformal field theory and the theory of exactly solvable models. Kac discovered an elegant proof of certain combinatorial identities, Macdonald identities, which is based on the representation theory of affine Kac–Moody algebras.
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MorphoSys has developed a number of antibody technologies that it uses for its proprietary programs as well as partnered programs. MorphoSys’ main technology is HuCAL (Human Combinatorial Antibody Library), which is a collection of more than ten billion fully human antibodies in the form of a phage display bank and a system for their optimization.
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Equitable resolvable coverings. Journal of Combinatorial Designs, 11(2), 113-123. The Oberwolfach problem, of decomposing a complete graph into edge-disjoint copies of a given 2-regular graph, also generalizes Kirkman's schoolgirl problem. Kirkman's problem is the special case of the Oberwolfach problem in which the 2-regular graph consists of five disjoint triangles.
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Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals also belong to a symmetric class.
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Since 2010 he is at the University of the Saarland. His research deals with free probability (with application to random matrices, statistical mechanics and operator algebras) and their combinatorial aspects and with operator algebras. In 2012, Speicher received the Jeffery–Williams Prize. He also received the Research Excellence Award of the President of Ontario.
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The Bottleneck traveling salesman problem (bottleneck TSP) is a problem in discrete or combinatorial optimization. The problem is to find the Hamiltonian cycle in a weighted graph which minimizes the weight of the most weighty edge of the cycle.. It was first formulated by with some additional constraints, and in its full generality by ...
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In the mathematical field of graph theory, an agreement forest for two given (leaf-labeled, irreductible) trees is any (leaf-labeled, irreductible) forest which can, informally speaking, be obtained from both trees by removing a common number of edges. Agreement forests first arose when studying combinatorial problems related to computational phylogenetics, in particular tree rearrangements.
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Collette René Coullard is an American mathematician, industrial engineer, operations researcher, and matroid theorist known for her research on combinatorial optimization problems that combine facility location and stock management. Formerly a professor at Purdue University, the University of Waterloo, Northwestern University, and Lake Superior State University, she has retired to become a professor emeritus.
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Donald Solitar (September 5, 1932 in Brooklyn, New York, United States – April 28, 2008 in Toronto, Canada) was an American and Canadian mathematician, known for his work in combinatorial group theory.. Reprinted as and as . The Baumslag–Solitar groups are named after him and Gilbert Baumslag, after their joint 1962 paper on these groups.
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In graph theory, a partial k-tree is a type of graph, defined either as a subgraph of a k-tree or as a graph with treewidth at most k. Many NP-hard combinatorial problems on graphs are solvable in polynomial time when restricted to the partial k-trees, for bounded values of k.
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Computers can also predict vibrational spectra and vibronic coupling, but also acquire and Fourier transform Infra-red Data into frequency information. The comparison with predicted vibrations supports the predicted shape. ;Molecular modelling: Methods for modelling molecular structures without necessarily referring to quantum mechanics. Examples are molecular docking, protein-protein docking, drug design, combinatorial chemistry.
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In 1930-1960 research on the geometry of numbers was conducted by many number theorists (including Louis Mordell, Harold Davenport and Carl Ludwig Siegel). In recent years, Lenstra, Brion, and Barvinok have developed combinatorial theories that enumerate the lattice points in some convex bodies.Grötschel et alii, Lovász et alii, Lovász, and Beck and Robins.
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In the design and analysis of algorithms for combinatorial optimization, parametric search is a technique invented by for transforming a decision algorithm (does this optimization problem have a solution with quality better than some given threshold?) into an optimization algorithm (find the best solution). It is frequently used for solving optimization problems in computational geometry.
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A skewed grid is a tessellation of parallelograms or parallelepipeds. (If the unit lengths are all equal, it is a tessellation of rhombi or rhombohedra.) A curvilinear grid or structured grid is a grid with the same combinatorial structure as a regular grid, in which the cells are quadrilaterals or [general] cuboids, rather than rectangles or rectangular cuboids.
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He is considered the main host and organiser of the seminars, according to the other hosts in the group. His contributions include organising the annual Workshop on Combinatorial and Additive Number Theory, writing and editing the publications of the seminars. He is now based in the City University and continues to act as the host to the seminar.
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A solved game is a game whose outcome (win, lose or draw) can be correctly predicted from any position, assuming that both players play perfectly. This concept is usually applied to abstract strategy games, and especially to games with full information and no element of chance; solving such a game may use combinatorial game theory and/or computer assistance.
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210, "III. Rule for Eliminating Atomic Formulas". however, their algorithm required trying all ground instances of the given formula. This source of combinatorial explosion was eliminated in 1965 by John Alan Robinson's syntactical unification algorithm, which allowed one to instantiate the formula during the proof "on demand" just as far as needed to keep refutation completeness.
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Thus, we want to describe all the cells of the subdivision, plus all the incidence and adjacency relations between these cells. When all the represented cells are simplexes, a simplicial complex may be used, but when we want to represent any type of cells, we need to use cellular topological models like combinatorial maps or generalized maps.
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Hearn proved that Kōnane is PSPACE-complete with respect to the dimensions of the board, by a reduction from Constraint Logic. There have been some positive results for restricted configurations. Ernst derives Combinatorial-Game-Theoretic values for several interesting positions. Chan and Tsai analyze the 1 × n game, but even this version of the game is not yet solved.
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His area of specialization, included combinatorial optimization, fractional programming, linear programming and network flow problems. He had been in the editorial board of Opsearch, an official journal of ORSI. He was one of the founder members of the "Mathematical Programming Group (MPG)" which was started by retired Prof. R. N. Kaul, Department of Mathematics, University of Delhi in 1972.
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An incomplete game of SOS SOS is paper and pencil game for two or more players. It is similar to tic-tac-toe and dots and boxes, but has greater complexity. SOS is a combinatorial game when played with two players. In terms of game theory, it is a zero-sum, sequential game with perfect information.
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As a special case of the philosophy of the previous paragraph, the geometry of the curve complex is an important tool to link combinatorial and geometric properties of hyperbolic 3-manifolds, and hence it is a useful tool in the study of Kleinian groups. For example, it has been used in the proof of the ending lamination conjecture.
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A simplicial 3-complex. In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.
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Wheels, Life and Other Mathematical Amusements is a book of 22 revised and extended mathematical games columns previously published in Scientific American. It is Gardner's 10th collection of columns, and includes material on Conway's Game of Life, supertasks, nontransitive dice, braided polyhedra, combinatorial game theory, the Collatz conjecture, mathematical card tricks, and Diophantine equations such as Fermat's Last Theorem.
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In 2006 Neven started to explore the application of quantum computing to hard combinatorial problems arising in machine learning. In collaboration with D-Wave Systems he developed the first image recognition system based on quantum algorithms. It was demonstrated at SuperComputing07. At NIPS 2009 his team demonstrated the first binary classifier trained on a quantum processor.
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In computer science, a chart parser is a type of parser suitable for ambiguous grammars (including grammars of natural languages). It uses the dynamic programming approach—partial hypothesized results are stored in a structure called a chart and can be re-used. This eliminates backtracking and prevents a combinatorial explosion. Chart parsing is generally credited to Martin Kay.
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Finite geometry is the study of geometric systems having only a finite number of points. Structures analogous to those found in continuous geometries (Euclidean plane, real projective space, etc.) but defined combinatorially are the main items studied. This area provides a rich source of examples for design theory. It should not be confused with discrete geometry (combinatorial geometry).
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His classic book on group theory was well received when it came out and is still useful today. His book Combinatorial Theory came out in a second edition in 1986, published by John Wiley & Sons. He proposed Hall's conjecture on the differences between perfect squares and perfect cubes, which remains an open problem as of 2015.
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In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ring construction is a basic tool within algebraic combinatorics and combinatorial commutative algebra.Miller & Sturmfels (2005) p.
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Timothy Law Snyder is an American educator, mathematician, academic administrator, and musician. He serves as the 16th President of Loyola Marymount University in Los Angeles, California. Snyder is well known for his academic research, publications and speeches on computational mathematics, data structures, combinatorial optimization, geometric probability, computer music, HIV diagnosis and prevention, and airline flight safety.
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The SFC is employed for combinatorial and high-throughput electrochemical studies. Due to its non- homogenous flow profile distribution, it is currently used for comparative kinetic studies. SFC is predominantly used for coupling of electrochemical measurements with post analytical techniques like UV-Vis, ICP-MS, ICP-OES etc. This makes possible a direct correlation of electrochemical and spectrometric signal.
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The MMs are combined into sub-sets each of which can be identified with a distinct new meaning. The operation is called 'combinate' to emphasise its active character in organising ideas. It is distantly related to the mathematical structure of combinatorial hierarchy and more particularly to the study of multi-term systems introduced by John G. Bennett's systematics. # Integrate.
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In mathematics, the HNN extension is an important construction of combinatorial group theory. Introduced in a 1949 paper Embedding Theorems for Groups by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group G into another group G' , in such a way that two given isomorphic subgroups of G are conjugate (through a given isomorphism) in G' .
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He has given over 200 talks in North America, Europe, Asia, and Australia. These have included keynote addresses at the Conference on Formal Power Series and Algebraic Combinatorics (2006) and the British Combinatorial Conference (2011). He has graduated 15 Ph.D. students. During his time at Michigan State University, he has won two awards for teaching excellence.
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Many classic results of combinatorial group theory, going back to 1949, are now interpreted as saying that a particular group or class of groups is (are) SQ-universal. However the first explicit use of the term seems to be in an address given by Peter Neumann to The London Algebra Colloquium entitled "SQ-universal groups" on 23 May 1968.
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The compound provides real time near-infrared (NIR) fluorescence imaging with an extinction coefficient of 2.8 × 105 M−1 cm−1 and combinatorial phototherapy with dual photothermal and photodynamic therapeutic mechanisms that may be appropriate for adriamycin-resistant tumors. The particles had a hydrodynamic size of 37.66 ± 0.26 nm (polydispersity index = 0.06) and surface charge of −2.76 ± 1.83 mV.
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In game theory, a strictly determined game is a two-player zero-sum game that has at least one Nash equilibrium with both players using pure strategies. The value of a strictly determined game is equal to the value of the equilibrium outcome. Most finite combinatorial games, like tic-tac-toe, chess, draughts, and go, are strictly determined games.
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Although such punctuated-equilibrium behaviour can be "designed" or "hard-coded", it should be stressed that this is an emergent effect of the negative-component- selection principle fundamental to the algorithm. EO has primarily been applied to combinatorial problems such as graph partitioning and the travelling salesman problem, as well as problems from statistical physics such as spin glasses.
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McMullen was invited to speak at the 1974 International Congress of Mathematicians in Vancouver; his contribution there had the title Metrical and combinatorial properties of convex polytopes.ICM 1974 proceedings . He was elected as a foreign member of the Austrian Academy of Sciences in 2006.Awards, Appointments, Elections & Honours, University College London, June 2006, retrieved 2013-11-03.
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In addition to more than a hundred scholarly publications, she has written or edited six books. These include her textbook on Quantitative Drug Design. Editorially, she has worked on Perspectives in Drug Design and Discovery, QSAR: Annual Reports in Computational Chemistry, Journal of Computer-Aided Drug Design and QSAR & Combinatorial Science. She has registered eight patents.
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Although Teirlinck's proof did not follow the outline in the manuscript, it nevertheless made use of the combinatorial structures that Lu had constructed. Lu Jiaxi was awarded posthumously in 1987 the First Class Award of the State Natural Science Award, then the highest honor in science in China, for his work on large sets of disjoint Steiner triple systems.
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Every simplicial complex has an associated face ring, also called its Stanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes.
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Although Jarník's 1921 dissertation, like some of his later publications, was in mathematical analysis, his main area of work was in number theory. He studied the Gauss circle problem and proved a number of results on Diophantine approximation, lattice point problems, and the geometry of numbers. He also made pioneering, but long- neglected, contributions to combinatorial optimization.
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European Journal of Operational Research. 164 269-285. Companies and governments sometimes use smart markets in procurement, as for transportation services. The Chilean government, for example, uses a smart market to choose caterers for school meal programs.Epstein, Rafael, Lysette Henriquez, Jaime Catalán, Gabriel Y. Weintraub, Cristián Martinez, “A Combinatorial Auction Improves School Meals in Chile,” Interfaces, vol.
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Often combinatorial problems arise that make things like computing the partition function of a system difficult. MFT is an approximation method that often makes the original solvable and open to calculation. Sometimes, MFT gives very accurate approximations. In field theory, the Hamiltonian may be expanded in terms of the magnitude of fluctuations around the mean of the field.
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The main types of automation are classified by the type of solid-phase substrates, the methods for adding and removing reagents, and design of reaction chambers. Polymer resins may be used as a substrate for solid-phase.Hardin, J.; Smietana, F., Automating combinatorial chemistry: A primer on benchtop robotic systems. Mol Divers 1996, 1 (4), 270-274.
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A VCG mechanism has to calculate the optimal outcome, based on the agents' reports (step 2 above). In some cases, this calculation is computationally difficult. For example, in combinatorial auctions, calculating the optimal assignment is NP-hard. Sometimes there are approximation algorithms to the optimization problem, but, using such an approximation might make the mechanism non-truthful.
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A turn is legal if it not illegal. An edge-path e1,..., em is said to contain turns ei−1, ei+1 for i = 1,...,m−1\. A combinatorial map f : Γ → Γ is a train-track map if and only if for every edge e of Γ the path f(e) contains no illegal turns.
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One of the earliest presentations of a group by generators and relations was given by the Irish mathematician William Rowan Hamilton in 1856, in his icosian calculus – a presentation of the icosahedral group. The first systematic study was given by Walther von Dyck, student of Felix Klein, in the early 1880s, laying the foundations for combinatorial group theory.
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The molecule talin is thought to be the major initiator of vinculin activation due to its presence in focal complexes. The combinatorial model of vinculin states that either α-actinin or talin can activate vinculin either alone or with the assistance of PIP2 or actin. This activation takes place by separation of the head-tail connection within inactive vinculin.
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The usual use of a clock signal is to synchronize transitions in sequential logic circuits. For most implementations of combinational logic, a clock signal is not even needed. The static/dynamic terminology used to refer to combinatorial circuits should not be confused with how the same adjectives are used to distinguish memory devices, e.g. static RAM from dynamic RAM.
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Geombinatorics is a quarterly scientific journal of mathematics. It was established by editor-in-chief Alexander Soifer in 1991 and is published by the University of Colorado at Colorado Springs. The journal covers problems in discrete, convex, and combinatorial geometry, as well as related areas. The journal is indexed in Zentralblatt MATH, Excellence Research Australia, and MathSciNet.
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In recent 10 years, R&D; investment of NCPC kept occupying over 30% of sales volume annually. NCPC has been transferring from generic imitation into original research in new drug development and forming a development platform of its own. Its original research focuses on the natural and small molecular drugs screening, combinatorial chemical technology and modern biotechnology.
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John Leech (21 July 1926 in Weybridge, Surrey – 28 September 1992 in Scotland) was a British mathematician working in number theory, geometry and combinatorial group theory. He is best known for his discovery of the Leech lattice in 1965. He also discovered Ta(3) in 1957. Leech was married to Jenifer Haselgrove, a British radio scientist.
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MSM is a stochastic volatility model with arbitrarily many frequencies. MSM builds on the convenience of regime-switching models, which were advanced in economics and finance by James D. Hamilton. MSM is closely related to the Multifractal Model of Asset Returns. MSM improves on the MMAR's combinatorial construction by randomizing arrival times, guaranteeing a strictly stationary process.
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New J. Chem., 1998, 22, 493–502 Since the mid-1990s he has been in the forefront (with Jean-Marie Lehn and several other research groups) of developing Dynamic covalent chemistry and the closely related dynamic combinatorial chemistry.Angew. Chemie Intl. Edn., 2002, 41, 898; Chemical Reviews, 2006, 106, 3652; Accounts Chem. Res., 2012, 45, 2211–2221.
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These features also suggest a limited off-target toxicity of CTAG1B-based cancer therapies. The immunisation with CTAG1B could be a successful approach to induce antigen specific immune responses in cancer patients. Up until May 2018, there have been 12 clinical trials registered using a CTAG1B cancer vaccine, 23 using modified T cells, and 13 using combinatorial immunotherapy.
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While this difference between classical and quantum descriptions of systems is fundamental to all of quantum statistics, quantum particles are divided into two further classes on the basis of the symmetry of the system. The spin–statistics theorem binds two particular kinds of combinatorial symmetry with two particular kinds of spin symmetry, namely bosons and fermions.
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In combinatorial mathematics, the Prüfer sequence (also Prüfer code or Prüfer numbers) of a labeled tree is a unique sequence associated with the tree. The sequence for a tree on n vertices has length n − 2, and can be generated by a simple iterative algorithm. Prüfer sequences were first used by Heinz Prüfer to prove Cayley's formula in 1918.
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This is perhaps the simplest known proof, requiring the least mathematical background. It is an attractive example of a combinatorial proof (a proof that involves counting a collection of objects in two different ways). The proof given here is an adaptation of Golomb's proof. To keep things simple, let us assume that is a positive integer.
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Constraint solvers use search, backtracking and constraint propagation techniques to find solutions and determine optimal solutions. They may employ forms of linear and nonlinear programming. They are often used to perform optimization within highly combinatorial problem spaces. For example, they may be used to calculate optimal scheduling, design efficient integrated circuits or maximise productivity in a manufacturing process.
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David Anthony Klarner (October 10, 1940March 20, 1999) was an American mathematician, author, and educator. He is known for his work in combinatorial enumeration, polyominoes,The Tromino Puzzle by Norton Starr and box-packing.A procedure for improving the upper bound for the number of n-ominoes, by D. A. Klarner and R. L. Rivest, Can. J. Math.
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Her dissertation, Application of Combinatorial Structures to Key Predistribution in Sensor Networks and Traitor Tracing, was supervised by Bimal Kumar Roy. After postdoctoral research at Lund University and the University of Ottawa, Ruj returned to India as an assistant professor at the Indian Institute of Technology Indore in 2012, and moved back to the Indian Statistical Institute in 2013.
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Péter Komjáth (born 8 April 1953) is a Hungarian mathematician, working in set theory, especially combinatorial set theory. Komjáth is a professor at the Eötvös Loránd University. He is currently a visiting faculty member at Emory University in the department of Mathematics and Computer Science. Komjáth won a gold medal at the International Mathematical Olympiad in 1971.
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Elwyn Ralph Berlekamp (September 6, 1940 – April 9, 2019) was an American mathematician known for his work in computer science, coding theory and combinatorial game theory. He was a professor emeritus of mathematics and EECS at the University of California, Berkeley.Contributors, IEEE Transactions on Information Theory 42, #3 (May 1996), p. 1048. DOI 10.1109/TIT.1996.490574.
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All these strategies are based on synthesis and testing of partial libraries. The earliest iterative strategy is described in the above mentioned document of Furka notarized in 1982 and.The method was later independently published by Erb et al. under the name „Recursive deconvolution”Erb E, Janda KD, Brenner S (1994) Recursive deconvolution of combinatorial chemical libraries Proc.
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The commission announced that it would now launch an in-depth investigation to validate its findings from the first phase of the investigations. In addition, this phase would study the combinatorial impact from Fitbit, and Google's databases, and also if the interoperability controls with rival wearables with the Android operating system, once the Fitbit acquisition is completed.
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In 1998, she was named Distinguished Professor of Mathematics at the University of California, San Diego. To date, she has over 200 publications to her name. The two best known books are Spectral Graph Theory and Erdős on Graphs. Spectral Graph Theory studies how the spectrum of the Laplacian of a graph is related to its combinatorial properties.
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The matching polynomial of a graph G with n vertices is related to that of its complement by a pair of (equivalent) formulas. One of them is a simple combinatorial identity due to . The other is an integral identity due to . There is a similar relation for a subgraph G of Km,n and its complement in Km,n.
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The 3-CNF Satisfiability instance (x ∨ x ∨ y) ∧ (~x ∨ ~y ∨ ~y) ∧ (~x ∨ y ∨ y) reduced to Clique. The green vertices form a 3-clique and correspond to a satisfying assignment.Adapted from The clique decision problem is NP-complete. It was one of Richard Karp's original 21 problems shown NP-complete in his 1972 paper "Reducibility Among Combinatorial Problems".
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It has been speculated that Pepy's music teacher John Birchensha was influenced by Kircher's combinatorial techniques, as his own "Rules of Composition" bear some similarities. Birchensha, John: Writings on Music] Field and Wardhaugh, editors, 2010, page 42 The Pepysian library does not currently keep their Arca out on display, and there are few photographs available of its contents.
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In combinatorial optimization, Lin–Kernighan is one of the best heuristics for solving the symmetric travelling salesman problem. Briefly, it involves swapping pairs of sub-tours to make a new tour. It is a generalization of 2-opt and 3-opt. 2-opt and 3-opt work by switching two or three edges to make the tour shorter.
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An example of such combinatorial objects are the domino tilings of a given region in the plane. In this case, a flip can be performed when two adjacent dominos cover a square: it consists in rotating these dominos by 90 degrees around the center of the square, resulting in a different domino tiling of the same region.
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In physics, he is remembered partly for his founding contribution to combinatorial physics, based on his elucidation of "combinatorial hierarchy", a mathematical structure of bit-strings generated by an algorithm based on discrimination (exclusive-or between bits). He published some of his ideas in fundamental physics (based on a logical "level below physics") in the book The Theory of Indistinguishables (1981).The Theory of Indistinguishables: A Search for Explanatory Principles below the level of Physics, Springer (1981) It identifies a logic based on adding to equality and inequality a third fundamental relationship which is neither one: indistinguishability-in- principle. For example, even with infinite knowledge, the three dimensions of a completely empty space are totally indistinguishable from one another, but they are still three, not one (contradicting Leibniz's Identity of Indiscernibles).
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Circle packings, as studied in this book, are systems of circles that touch at tangent points but do not overlap, according to a combinatorial pattern of adjacencies specifying which pairs of circles should touch. The circle packing theorem states that a circle packing exists if and only if the pattern of adjacencies forms a planar graph; it was originally proved by Paul Koebe in the 1930s, and popularized by William Thurston, who rediscovered it in the 1970s and connected it with the theory of conformal maps and conformal geometry. As a topic, this should be distinguished from sphere packing, which considers higher dimensions (here, everything is two dimensional) and is more focused on packing density than on combinatorial patterns of tangency. The book is divided into four parts, in progressive levels of difficulty.
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The source submission of dialogues are the graphs of states, system functions on platforms of Windows 95 and Windows NT. The system was created as the tool for fast development and modernising of MMI appendices for open systems of the numerical control. There was developed the method of substitutions of problem solving on the graphs. With the help of the given method it is possible to decide both optimisation problems relating the class of discrete programming problems and combinatorial problems on the graphs with edges without weights. The principle of pair substitutions with combination of the possibility of use the vectors of topology has allowed mathematically to formulate new classes of optimisation and combinatorial problems on the graphs, which could not be solved earlier by known widely classical methods.
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In the mathematics of combinatorial games, the sum or disjunctive sum of two games is a game in which the two games are played in parallel, with each player being allowed to move in just one of the games per turn. The sum game finishes when there are no moves left in either of the two parallel games, at which point (in normal play) the last player to move loses. This operation may be extended to disjunctive sums of any number of games, again by playing the games in parallel and moving in exactly one of the games per turn. It is the fundamental operation that is used in the Sprague–Grundy theorem for impartial games and which led to the field of combinatorial game theory for partisan games.
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In combinatorial mathematics a cycle index is a polynomial in several variables which is structured in such a way that information about how a group of permutations acts on a set can be simply read off from the coefficients and exponents. This compact way of storing information in an algebraic form is frequently used in combinatorial enumeration. Each permutation π of a finite set of objects partitions that set into cycles; the cycle index monomial of π is a monomial in variables a1, a2, … that describes the type of this partition (the cycle type of π): the exponent of ai is the number of cycles of π of size i. The cycle index polynomial of a permutation group is the average of the cycle index monomials of its elements.
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By combining the above equation with px(σ) for a solute x, and adding the σ-independent combinatorial and dispersive contributions, the chemical potential for a solute X in a solvent S results in: In analogy to activity coefficient models used in chemical engineering, such as NRTL, UNIQUAC or UNIFAC, the final chemical potential can be split into a combinatorial and a residual (non ideal) contribution. The interaction energies Eint(σ,σ') of two surface pieces are the crucial part for the final performance of the method and different formulations are used within the various implementations. In addition to the liquid phase terms a chemical potential estimate for the ideal gas phase µgas has been added to COSMO-RS to enable the prediction of vapor pressure, free energy of solvation and related quantities.
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The combinatorial characterization of a set X ⊂ ℝ3 as a solid involves representing X as an orientable cell complex so that the cells provide finite spatial addresses for points in an otherwise innumerable continuum. The class of semi-analytic bounded subsets of Euclidean space is closed under Boolean operations (standard and regularized) and exhibits the additional property that every semi-analytic set can be stratified into a collection of disjoint cells of dimensions 0,1,2,3. A triangulation of a semi-analytic set into a collection of points, line segments, triangular faces, and tetrahedral elements is an example of a stratification that is commonly used. The combinatorial model of solidity is then summarized by saying that in addition to being semi-analytic bounded subsets, solids are three-dimensional topological polyhedra, specifically three-dimensional orientable manifolds with boundary.
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An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones. (the modern standard tool for such construction is the CW complex). In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups, which led to the change of name to algebraic topology.. The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces.. In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism (or more general homotopy) of spaces. This allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure, often making these statement easier to prove.
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In combinatorial number theory, Singmaster's conjecture states that there is a finite upper bound on the number of times a number other than 1 can appear in Pascal's triangle. Paul Erdős suspected that the conjecture is true, but thought it would probably be very difficult to prove. The empirical evidence is consistent with the proposition that the smallest upper bound is 8\.
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Relational frame theory (RFT) is a behavioral theory of human language. It is rooted in functional contextualism and focused on predicting and influencing verbal behavior with precision, scope and depth. Relational framing is relational responding based on arbitrarily applicable relations and arbitrary stimulus functions. The relational responding is subject to mutual entailment, combinatorial mutual entailment and transformation of stimulus functions.
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The Séminaire Lotharingien de Combinatoire (Lotharingian Seminar of Combinatorics) is a peer-reviewed academic journal specialising in combinatorial mathematics, named after Lotharingia. It has existed since 1980 as a regular joint seminar in Combinatorics for the Universities of Bayreuth, Erlangen and Strasbourg. In 1994, it was decided to create a journal under the same name. The regular meetings continue to this day.
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Katherine A. Heinrich (born February 21, 1954) is a mathematician and mathematics educator who became the first female president of the Canadian Mathematical Society. Her research interests include graph theory and the theory of combinatorial designs. Originally from Australia, she moved to Canada where she worked as a professor at Simon Fraser University and as an academic administrator at the University of Regina.
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In the field of selection principles, Tsaban devised the method of omission of intervals for establishing covering properties of sets of real numbers that have certain combinatorial structures. In nonabelian cryptology he devised the algebraic span method that solved a number of computational problems that underlie a number of proposals for nonabelian public-key cryptographic schemes (such as the commutator key exchange).
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He discovered combinatorial cocycles with Shigeyuki Morita for the first and with Nariya Kawazumi for the higher Johnson homomorphisms. Penner has also contributed to theoretical biology in joint work with Jørgen E. Andersen et al. discovering a priori geometric constraints on protein geometry, and with Michael S. Waterman, Piotr Sulkowski, Christian Reidys et al. introducing and solving the matrix model for RNA topology.
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Alpha factoring is based on maximizing the reliability of factors, assuming variables are randomly sampled from a universe of variables. All other methods assume cases to be sampled and variables fixed. Factor regression model is a combinatorial model of factor model and regression model; or alternatively, it can be viewed as the hybrid factor model, whose factors are partially known.
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147:195-208 All peptide sequences obtained from biopanning using combinatorial peptide libraries have been stored in a special freely available database named BDB. This technique is often used for the selection of antibodies too. Biopanning involves 4 major steps for peptide selection.Mandecki W, Chen YC, and Grihalde N. A Mathematical Model for Biopanning (Affinity Selection) Using Peptide Libraries on Filamentous Phage.
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These edges will be bidirectional if both languages borrow from one another. A tree is thus a simple network, however there are many other types of network. A phylogentic network is one where the taxa are represented by nodes and their evolutionary relationships are represented by branches. Another type is that based on splits, and is a combinatorial generalisation of the split tree.
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A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a bijection between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a double counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal.
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A zero-suppressed decision diagram (ZSDD or ZDD) is a particular kind of binary decision diagram (BDD) with fixed variable ordering. This data structure provides a canonically compact representation of sets, particularly suitable for certain combinatorial problems. Recall the OBDD reduction strategy i.e. a node is removed from the decision tree if both out-edges point to the same node.
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A geometrical representation of an abstract simplicial complex that is not a valid simplicial complex. In combinatorics, an abstract simplicial complex (ASC) is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex.
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He is one of the most active mathematicians working on Combinatorial Matrix Theory. He is also noted for his monograph on matrices of sign-solvable linear systems. Besides organizing many workshops, he is a co-principal investigator of Math Teacher Leadership Program, a National Science Foundation project (2009–2014). Shader is special assistant to the Vice-President of Research of University of Wyoming.
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In detailed calculations the combinatorial aspects enter as soon as the cells have to be indexed. Lifted from the Grassmannian, which is a homogeneous space, to the general linear group that acts on it, similar questions are involved in the Bruhat decomposition and classification of parabolic subgroups (by block matrix). Putting Schubert's system on a rigorous footing is Hilbert's fifteenth problem.
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In June 2003 an international group theory conference in honor of Grigorchuk's 50th birthday was held in Gaeta, Italy.International Conference on GROUP THEORY: combinatorial, geometric, and dynamical aspects of infinite groups. Special anniversary issues of the "International Journal of Algebra and Computation" and of the journal "Algebra and Discrete Mathematics" were dedicated to Grigorchuk's 50th birthday.Editorial Statement, Algebra and Discrete Mathematics, (2003), no.
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Jack R. Edmonds (born April 5, 1934) is an American-born and educated computer scientist and mathematician who lived and worked in Canada for much of his life. He has made fundamental contributions to the fields of combinatorial optimization, polyhedral combinatorics, discrete mathematics and the theory of computing. He was the recipient of the 1985 John von Neumann Theory Prize.
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The face ring k[Δ] is a multigraded algebra over k all of whose components with respect to the fine grading have dimension at most 1. Consequently, its homology can be studied by combinatorial and geometric methods. An abstract simplicial complex Δ is called Cohen–Macaulay over k if its face ring is a Cohen–Macaulay ring.Miller & Sturmfels (2005) p.
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New instruments were developed and sold for forensic human identification, protein identification and characterization, metabolite pathway identification, and lead compound identification from combinatorial libraries. On April 27, 1999, the shareholders of Perkin-Elmer Corporation approved the reorganization of Perkin-Elmer into PE Corporation, a pure-play life science company.How can we enable new exploration? a letter from michael hunkapiller, ph.d.
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A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.
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In the acylation step a haloacetic acid, typically bromoacetic acid activated by diisopropylcarbodiimide reacts with the amine of the previous residue. In the displacement step (a classical SN2 reaction), an amine displaces the halide to form the N-substituted glycine residue. The submonomer approach allows the use of any commercially available or synthetically accessible amine with great potential for combinatorial chemistry.
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Dynamic covalent chemistry (DCvC) is a synthetic strategy employed by chemists to make complex supramolecular assemblies from discrete molecular building blocks. DCvC has allowed access to complex assemblies such as covalent organic frameworks, molecular knots, polymers, and novel macrocycles. Not to be confused with dynamic combinatorial chemistry, DCvC concerns only covalent bonding interactions. As such, it only encompasses a subset of supramolecular chemistries.
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Witold Lipski Jr. (July 13, 1949, in Warsaw, Poland – May 30, 1985, in Nantes, France) was a Polish computer scientist (habilitation in computer science), and an author of two books: Combinatorics for Programmers (two editions) and (jointly with Wiktor Marek Combinatorial analysis. Jointly with his PhD student, Tomasz Imieliński, created foundations of the theory of incomplete information in relational databases.
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In combinatorial mathematics, a partial permutation, or sequence without repetition, on a finite set S is a bijection between two specified subsets of S. That is, it is defined by two subsets U and V of equal size, and a one-to- one mapping from U to V. Equivalently, it is a partial function on S that can be extended to a permutation...
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Modifications to the ATAC-seq protocol have been made to accommodate single-cell analysis. Microfluidics can be used to separate single nuclei and perform ATAC-seq reactions individually. With this approach, single cells are captured by either a microfluidic device or a liquid deposition system before tagmentation. An alternative technique that does not require single cell isolation is combinatorial cellular indexing.
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The rule of sum is another basic counting principle. Stated simply, it is the idea that if we have a ways of doing something and b ways of doing another thing and we can not do both at the same time, then there are a + b ways to choose one of the actions.Rosen, Kenneth H., ed. Handbook of discrete and combinatorial mathematics.
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Martin Grötschel (born 10 September 1948) is a German mathematician known for his research on combinatorial optimization, polyhedral combinatorics, and operations research. From 1991 to 2012 he was Vice President of the Zuse Institute Berlin (ZIB) and served from 2012 to 2015 as ZIB's President. Since October 2015 he has been President of the Berlin-Brandenburg Academy of Sciences and Humanities (BBAW).
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The Erdős–Gallai theorem is a result in graph theory, a branch of combinatorial mathematics. It provides one of two known approaches to solving the graph realization problem, i.e. it gives a necessary and sufficient condition for a finite sequence of natural numbers to be the degree sequence of a simple graph. A sequence obeying these conditions is called "graphic".
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223-236 matching, and tree search translate some specific computer vision problems to the more general combinatorial consistent labeling problem and then discuss the theory of the look-ahead operators that speed up the tree search. The most basic of these is called Forward Checking.Increasing Tree Search Efficiency for Constraint Satisfaction Problems, (with G.L. Elliott), Artificial Intelligence, Vol. 14, 1980, pp. 263-313.
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Computational advances have enabled cheaper and faster sequencing. Research has focused on combinatorial chemistry, genomic mining, omic technologies and high throughput screening. As the cost per genetic test decreases, the development of personalized drug therapies will increase. Technology now allows for genetic analysis of hundreds of target genes involved in medication metabolism and response in less than 24 hours for under $1,000.
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The matching problem can be generalized by assigning weights to edges in G and asking for a set M that produces a matching of maximum (minimum) total weight: this is the maximum weight matching problem. This problem can be solved by a combinatorial algorithm that uses the unweighted Edmonds's algorithm as a subroutine. Kolmogorov provides an efficient C++ implementation of this.
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In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in the 18th century. Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second kind. Additionally, Lah numbers are sometimes referred to as Stirling numbers of the third kind.
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Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers.John H. Conway & Richard K. Guy, The Book of Numbers, 1996, p.59-62 It is simply a finite sequence of positive integers separated by rightward arrows, e.g. 2\to3\to4\to5\to6. As with most combinatorial notations, the definition is recursive.
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There are three general strategies for constructing families of expander graphs.see, e.g., The first strategy is algebraic and group-theoretic, the second strategy is analytic and uses additive combinatorics, and the third strategy is combinatorial and uses the zig-zag and related graph products. Noga Alon showed that certain graphs constructed from finite geometries are the sparsest examples of highly expanding graphs.
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Welsh obtained his DPhil from Oxford University under the supervision of John Hammersley. After working as a researcher at Bell Laboratories, he joined the Mathematical Institute in 1963, and became a fellow of Merton College, Oxford in 1966. He chaired the British Combinatorial Committee from 1983 to 1987. Welsh was given a personal chair in 1992, and retired in 2005.
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Although topology is the field of mathematics that formalizes and generalizes the intuitive notion of "continuous deformation" of objects, it gives rise to many discrete topics; this can be attributed in part to the focus on topological invariants, which themselves usually take discrete values. See combinatorial topology, topological graph theory, topological combinatorics, computational topology, discrete topological space, finite topological space, topology (chemistry).
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P. S. Novikov. Pyotr Sergeyevich Novikov (; 15 August 1901, Moscow, Russian Empire – 9 January 1975, Moscow, Soviet Union) was a Soviet mathematician. Novikov is known for his work on combinatorial problems in group theory: the word problem for groups, and Burnside's problem. For proving the undecidability of the word problem in groups he was awarded the Lenin Prize in 1957.
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On 5 November 1923 he was elected a Fellow of St John's. He worked on the foundations of combinatorial topology, and proposed that a notion of equivalence be defined using only three elementary "moves". Newman's definition avoided difficulties that had arisen from previous definitions of the concept. Publishing over twenty papers established his reputation as an "expert in modern topology".
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The idea of a submodular set function has recently emerged as a powerful modeling tool for various summarization problems. Submodular functions naturally model notions of coverage, information, representation and diversity. Moreover, several important combinatorial optimization problems occur as special instances of submodular optimization. For example, the set cover problem is a special case of submodular optimization, since the set cover function is submodular.
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Melvyn Bernard Nathanson (born October 10, 1944, in Philadelphia, Pennsylvania) is an American mathematician, specializing in number theory, and a Professor of Mathematics at Lehman College and The Graduate Center (City University of New York). His principal work is in additive and combinatorial number theory. He is the author of over 150 research papers in mathematics, and author or editor of 20 books.
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In 2014, he became a fellow of the American Mathematical Society "for contributions to discrete geometry and combinatorial optimization as well as for service to the profession, including mentoring and diversity."List of Fellows of the American Mathematical Society, retrieved 2014-12-17 In 2019 he was named a SIAM Fellow "for contributions to discrete geometry and optimization, polynomial algebra, and mathematical software".
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In dynamic covalent chemistry, the most stable accessible product of a mixture is formed using thermodynamically controlled reversible reactions; in dynamic combinatorial chemistry a template is used to direct the synthesis of the molecule that best stabilises the template. In each case unpredictable molecules may be discovered that would not be designed or could not be prepared by conventional chemistry.
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He received private instruction in the combinatorial topology of fiber spaces in Brussels from G. Hirsch of the Agricultural University of Ghent in 1953. During this period he received a stipend from the Dutch-Belgian Cultural Accord. From 1954 to 1957 he taught mathematics at the Delft high school 'Gemeentelijke Hogere Burgerschool HBS'. In 1959 he completed his PhD thesis at Leiden University.
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His early research led to the proof of the theorem devised by Richard M. Wilson on pairwise balance designs. He wrote scholarly papers with Andries Brouwer, Paul Erdős, Alexander Schrijver, and Richard M. Wilson, among others. His papers were published in journals such as Discrete Mathematics, the Journal of Combinatorial Theory, the European Journal of Combinatorics, and the American Mathematical Monthly.
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In combinatorial game theory, a fuzzy game is a game which is incomparable with the zero game: it is not greater than 0, which would be a win for Left; nor less than 0 which would be a win for Right; nor equal to 0 which would be a win for the second player to move. It is therefore a first-player win.
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This conjecture was proved for polytopal spheres by Peter McMullen in 1970McMullen, P. On the upper-bound conjecture for convex polytopes. Journal of Combinatorial Theory, Series B 10 1971 187–200. and by Richard Stanley for general simplicial spheres in 1975. The g-conjecture, formulated by McMullen in 1970, asks for a complete characterization of f-vectors of simplicial d-spheres.
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Combinatorial libraries are special multi- component mixtures of small-molecule chemical compounds that are synthesized in a single stepwise process. They differ from collection of individual compounds as well as from series of compounds prepared by parallel synthesis. It is an important feature that mixtures are used in their synthesis. The use of mixtures ensures the very high efficiency of the process.
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Optimized Markov chain algorithms which use local searching heuristic sub- algorithms can find a route extremely close to the optimal route for 700 to 800 cities. TSP is a touchstone for many general heuristics devised for combinatorial optimization such as genetic algorithms, simulated annealing, tabu search, ant colony optimization, river formation dynamics (see swarm intelligence) and the cross entropy method.
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In a vector space of dimension n, one usually considers only the vectors. According to Hermann Grassmann and others, this presumption misses the complexity of considering the structures of pairs, triples, and general multivectors. Since there are several combinatorial possibilities, the space of multivectors turns out to have 2n dimensions. The abstract formulation of the determinant is the most immediate application.
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The Institute of Combinatorics and its Applications awarded him its Kirkman Medal in 2002 and its Hall Medal in 2008. The Australian Institute of Policy and Science awarded him a Victorian Young Tall Poppy Award in 2008. The Australian Mathematical Society awarded him its medal in 2009. Wanless is a life member of the Combinatorial Mathematics Society of Australasia (CMSA).
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Be2xf1 Nh2-f3+ 19. Ke1-e2 Nf3-d4+ 20. Ke2-d2 Asquiesces the discovered check but other king moves were even worse: Ke2-e1/e3 and Nd4-c2+ wins the queen. : 20. … Nd4-c6+ 21. Qb4-d6 Rd8xd6+ 22. Bf4xd6 Be6xc4 Black reaps the fruit of his combinatorial fireworks: the smoke has cleared and he is two pawns ahead. : 23.
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Aminoshikimic acid is an intriguing alternative to shikimic acid as a starting material for the synthesis of neuraminidase inhibitors such as the antiinfluenza agent oseltamivir. Aminoshikimic acid is also a versatile chiral starting material for the synthesis of new pharmaceuticals. As with shikimic acid, aminoshikimic acid is an attractive candidate for use as the core scaffold for synthesis of combinatorial libraries.
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A row of bowling pins. On their turn, a player may choose to eliminate a single pin, or two adjacent ones. Kayles is a simple impartial game in combinatorial game theory, invented by Henry Dudeney in 1908. Given a row of imagined bowling pins, players take turns to knock out either one pin, or two adjacent pins, until all the pins are gone.
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Note that the central difference will, for odd , have multiplied by non- integers. This is often a problem because it amounts to changing the interval of discretization. The problem may be remedied taking the average of and . Forward differences applied to a sequence are sometimes called the binomial transform of the sequence, and have a number of interesting combinatorial properties.
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That is, a bidder can specify that he or she will pay for items A and B, but only if he or she gets both. In combinatorial auctions, determining the winning bidder(s) can be a complex process where even the bidder with the highest individual bid is not guaranteed to win. For example, in an auction with four items (W, X, Y and Z), if Bidder A offers $50 for items W & Y, Bidder B offers $30 for items W & X, Bidder C offers $5 for items X & Z and Bidder D offers $30 for items Y & Z, the winners will be Bidders B & D while Bidder A misses out because the combined bids of Bidders B & D is higher ($60) than for Bidders A and C ($55). Deferred-acceptance auction is a special case of a combinatorial auction.
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Henry Gould and Jocelyn Quaintance about combinatorial identities involving double factorials He has been a consultant with the National Security Agency, Principal Investigator at West Virginia University with several College of Arts and Sciences grants, and grants from the National Science Foundation on the topic of Combinatorial Identities, and has served as a reviewer for the Mathematical Reviews and the Zentralblatt für Mathematik.Mathematical Reviews Online Between 1974 and 1976, he was a Visiting Lecturer for the Society for Industrial and Applied Mathematics. From 1974 to 1979, Gould was Editor-in-Chief of the Proceedings of the West Virginia Academy of Science .West Virginia Academy of Science In 1976, he was an invited participant to the first Annual Symposium on the History of Mathematics held at the National Museum of Science and Technology, Smithsonian Institution, Washington, D.C., concerned with Cauchy's contributions to analysis.
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The theorem involves sets of strings, all having the same length n, over a finite alphabet, together with a group acting on the alphabet. A combinatorial cube is a subset of strings determined by constraining some positions of the string to contain a fixed letter of the alphabet, and by constraining other pairs of positions to be equal to each other or to be related to each other by the group action. This determination can be specified more formally by means of a labeled parameter word, a string with wildcard characters in the positions that are not constrained to contain a fixed letter and with additional labels describing which wildcard characters must be equal or related by the group action. The dimension of the combinatorial cube is the number of free choices that can be made for these wildcard characters.
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Uchionnye Zapiski Penzenskogo Pedinstituta (Transactions of the Penza Pedagogoical Institute) 4, 75–87 (1956) (in Russian) As he remembers: In 1967, Manuel Blum formulated a set of axioms (now known as Blum axioms) specifying desirable properties of complexity measures on the set of computable functions and proved an important result, the so-called speed-up theorem. The field began to flourish in 1971 when the Stephen Cook and Leonid Levin proved the existence of practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete. In the 1980s, much work was done on the average difficulty of solving NP-complete problems—both exactly and approximately.
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In combinatorial mathematics, the Lubell–Yamamoto–Meshalkin inequality, more commonly known as the LYM inequality, is an inequality on the sizes of sets in a Sperner family, proved by , , , and . It is named for the initials of three of its discoverers. This inequality belongs to the field of combinatorics of sets, and has many applications in combinatorics. In particular, it can be used to prove Sperner's theorem.
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In combinatorial mathematics, the hook length formula is a formula for the number of standard Young tableaux whose shape is a given Young diagram. It has applications in diverse areas such as representation theory, probability, and algorithm analysis; for example, the problem of longest increasing subsequences. A related formula counts the number of semi-standard Young tableaux, which is a specialization of a Schur polynomial.
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In 2016, Complete Genomics contributed over 184 phased human genomes to George Church's Personal Genome Project. In 2019, they published on their new single-tube long fragment read (stLFR) technology, enabling construction of long DNA molecules from short reads using a combinatorial process of DNA barcoding. This enables phasing, SV detection, scaffolding, and cost-effective diploid de novo genome assembly, from second generation sequencing technology.
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Marston Donald Edward Conder (born 9 September 1955) is a New Zealand mathematician, a Distinguished Professor of Mathematics at Auckland University,Staff directory listing entry, Auckland U. Mathematics, retrieved 22 January 2013. and the former co-director of the New Zealand Institute of Mathematics and its Applications. His main research interests are in combinatorial group theory, graph theory, and their connections with each other.
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Herbert Saul Wilf (June 13, 1931 – January 7, 2012) was a mathematician, specializing in combinatorics and graph theory. He was the Thomas A. Scott Professor of Mathematics in Combinatorial Analysis and Computing at the University of Pennsylvania. He wrote numerous books and research papers. Together with Neil Calkin he founded The Electronic Journal of Combinatorics in 1994 and was its editor-in-chief until 2001.
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Hermann Weyl proved a basic result showing that this was equivalent to bounds for exponential sums formed from the sequence. This showed that Diophantine approximation results were closely related to the general problem of cancellation in exponential sums, which occurs throughout analytic number theory in the bounding of error terms. Related to uniform distribution is the topic of irregularities of distribution, which is of a combinatorial nature.
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Websites like eBay provide a potential audience of millions to sellers. Established auction houses, as well as specialist internet auctions, sell everything from antiques and collectibles to holidays, air travel, brand new computers, and household equipment. Private electronic markets use combinatorial auction techniques to continuously sell commodities (coal, iron ore, grain, water...) online to a pre-qualified group of buyers (based on price and non-price factors).
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Simulated Annealing can be used to solve combinatorial problems. Here it is applied to the travelling salesman problem to minimize the length of a route that connects all 125 points. Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem.
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It can be realized as the Voronoi tessellation of the body-centred cubic lattice. Lord Kelvin conjectured that a variant of the bitruncated cubic honeycomb (with curved faces and edges, but the same combinatorial structure) is the optimal soap bubble foam. However, the Weaire–Phelan structure is a less symmetrical, but more efficient, foam of soap bubbles. The honeycomb represents the permutohedron tessellation for 3-space.
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The slime mould Physarum polycephalum is able to solve the Traveling Salesman Problem, a combinatorial test with exponentially increasing complexity, in linear time. Fungi such as Basidiomycetes can also be used to build logical circuits. In a proposed fungal computer, information is represented by spikes of electrical activity, a computation is implemented in a mycelium network, and an interface is realized via fruit bodies..
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El. philos. sect. I de corp 1.1.2. The same idea is found in the work of Leibniz, who had read both Llull and Hobbes, and who argued that logic can be represented through a combinatorial process or calculus. But, like Llull and Hobbes, he failed to develop a detailed or comprehensive system, and his work on this topic was not published until long after his death.
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Jeff Kahn Jeffry Ned Kahn is a professor of mathematics at Rutgers University notable for his work in combinatorics. Kahn received his Ph.D. from The Ohio State University in 1979 after completing his dissertation under his advisor Dijen K. Ray-Chaudhuri. In 1980 he showed the importance of the bundle theorem for ovoidal Möbius planes.Inversive planes satisfying the bundle theorem, Journal Combinatorial Theory, Serie A, Vol.
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Géraud Sénizergues (born March 9, 1957) is a French computer scientist at the University of Bordeaux. He is known for his contributions to automata theory, combinatorial group theory and abstract rewriting systems. He received his Ph.D. (Doctorat d'état en Informatique) from the Université Paris Diderot (Paris 7) in 1987 under the direction of Jean-Michel Autebert. With Yuri Matiyasevich he obtained results about the Post correspondence problem.
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One player places tiles vertically, while the other places them horizontally. (Traditionally, these players are called "Left" and "Right", respectively, or "V" and "H". Both conventions are used in this article.) As in most games in combinatorial game theory, the first player who cannot move loses. Domineering is a partisan game, in that players use different pieces: the impartial version of the game is Cram.
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In applied mathematics, the maximum generalized assignment problem is a problem in combinatorial optimization. This problem is a generalization of the assignment problem in which both tasks and agents have a size. Moreover, the size of each task might vary from one agent to the other. This problem in its most general form is as follows: There are a number of agents and a number of tasks.
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In 1992, her work on proteases led to the identification of the first caspase, caspase-1/Interleukin-1 converting enzyme (ICE). She determined that ICE was the cysteine protease responsible for IL-1β processing in monocytes. Thornberry also developed a novel method for analyzing protease specificities in combinatorial libraries of positional scanning substrates. Her work has led to the broader study of proteases in apoptosis.
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They wrote many articles together and had a major impact on the development of algebraic combinatorics. They succeeded in giving a combinatorial understanding of various algebraic and geometric questions in representation theory. Thus they introduced many new objects related to both fields like Schubert polynomials and Grothendieck polynomials. They were also the first to define the crystal graph structure on Young tableaux (though not under this name).
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Edelsbrunner has over 100 research publicationsDBLP: Herbert Edelsbrunner. and is an ISI highly cited researcher.ISI highly cited researcher: Herbert Edelsbrunner. He has also published four books on computational geometry: Algorithms in Combinatorial Geometry (Springer-Verlag, 1987, ), Geometry and Topology for Mesh Generation (Cambridge University Press, 2001, ), Computational Topology (American Mathematical Society, 2009, 978-0821849255) and A Short Course in Computational Geometry and Topology (Springer-Verlag, 2014, ).
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It uses two restriction enzymes and a combination of the two enzymes for digestion separately. It assumes that complete digestion occurs at each restriction site. The lengths of the DNA fragments are measured and used for ordering of fragments by computation. This approach is easier to handle for the experiment but more difficult to solve when dealing with the combinatorial problem required for mapping.
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In mathematics, the minimum k-cut, is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least k connected components. These edges are referred to as k-cut. The goal is to find the minimum-weight k-cut. This partitioning can have applications in VLSI design, data-mining, finite elements and communication in parallel computing.
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It follows that every monoid (or semigroup) arises as a homomorphic image of a free monoid (or semigroup). The study of semigroups as images of free semigroups is called combinatorial semigroup theory. Free monoids (and monoids in general) are associative, by definition; that is, they are written without any parenthesis to show grouping or order of operation. The non-associative equivalent is the free magma.
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Arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory, and harmonic analysis. It is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive number theory (sometimes also called additive combinatorics) refers to the special case when only the operations of addition and subtraction are involved. One important technique in arithmetic combinatorics is the ergodic theory of dynamical systems.
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Infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. It is a part of set theory, an area of mathematical logic, but uses tools and ideas from both set theory and extremal combinatorics. Gian-Carlo Rota used the name continuous combinatoricsContinuous and profinite combinatorics to describe geometric probability, since there are many analogies between counting and measure.
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Discrete geometry (also called combinatorial geometry) also began as a part of combinatorics, with early results on convex polytopes and kissing numbers. With the emergence of applications of discrete geometry to computational geometry, these two fields partially merged and became a separate field of study. There remain many connections with geometric and topological combinatorics, which themselves can be viewed as outgrowths of the early discrete geometry.
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Ergodic Ramsey theory arose shortly after Endre Szemerédi's proof that a set of positive upper density contains arbitrarily long arithmetic progressions, when Hillel Furstenberg gave a new proof of this theorem using ergodic theory. It has since produced combinatorial results, some of which have yet to be obtained by other means, and has also given a deeper understanding of the structure of measure-preserving dynamical systems.
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In the mathematical subject of group theory, a co-Hopfian group is a group that is not isomorphic to any of its proper subgroups. The notion is dual to that of a Hopfian group, named after Heinz Hopf. Wilhelm Magnus, Abraham Karrass, Donald Solitar, Combinatorial group theory. Presentations of groups in terms of generators and relations, Reprint of the 1976 second edition, Dover Publications, Inc.
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The term "auction algorithm"Dimitri P. Bertsekas. "A distributed algorithm for the assignment problem", original paper, 1979. applies to several variations of a combinatorial optimization algorithm which solves assignment problems, and network optimization problems with linear and convex/nonlinear cost. An auction algorithm has been used in a business setting to determine the best prices on a set of products offered to multiple buyers.
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Some facility location problems which are NP-hard for general graphs, as well as some other graph problems, may be solved in polynomial time for cacti. Since cacti are special cases of outerplanar graphs, a number of combinatorial optimization problems on graphs may be solved for them in polynomial time. Translated from Notices of the BSSR Academy of Sciences, Ser. Phys.-Math. Sci., (1984) no.
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Because generalized permutahedra can be constructed from submodular functions, and every generalized permutahedron has an associated submodular function, we have that there should be a correspondence between generalized permutahedra and polymatroids. In fact every polymatroid is a generalized permutahedron that has been translated to have a vertex in the origin. This result suggests that the combinatorial information of polymatroids is shared with generalized permutahedra.
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Typical instances are relatively easy. This approach to complexity originated in combinatorial group theory, which has a computational tradition going back to the beginning of the last century. The notion of generic complexity was introduced in a 2003 paper,I. Kapovich, A. Myasnikov, P. Schupp and V. Shpilrain, Generic case complexity, decision problems in group theory and random walks, J. Algebra, vol 264 (2003), 665–694.
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Within the polytope application, there are over 230 functions or calculations that can be done with a polytope. These functions range in complexity from simply calculating basic information about a polytope (e.g., number of vertices, number of facets, tests for simplicial polytopes, and converting a vertex description to an inequality description) to combinatorial or algebraic properties (e.g., H-vector, Ehrhart polynomial, Hilbert basis, and Schlegel diagrams).
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In 2005, McCollum designed The Shapes Project, a combinatorial system to produce "a completely unique shape for every person on the planet, without repeating."Princenthal, Nancy, "Shape Shifter", Art in America, February 2007, pp 106-109. The system involves organizing a basic vocabulary of 300 "parts" which can be combined in over 30 billion different ways, created as "vector files" in a computer drawing program.
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He was elected to the Royal Society of Edinburgh in 1969. In 1994, the University of Waterloo gave him an honorary doctorate for his contributions to combinatorics. A conference in his honor was held on his retirement in 1996, the proceedings of which were published as a festschrift. The 18th British Combinatorial Conference, held in Sussex in July 2001, was dedicated to his memory.
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Claspers may also be interpreted algebraically, as a diagram calculus for the braided strict monoidal category Cob of oriented connected surfaces with connected boundary. Additionally, most crucially, claspers may be roughly viewed as a topological realization of Jacobi diagrams, which are purely combinatorial objects. This explains the Lie algebra structure of the graded vector space of Jacobi diagrams in terms of the Hopf algebra structure of Cob.
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A simple example of the Ehrenfeucht–Fraïssé game is given in one of Ivars Peterson's MathTrek columns.Example of the Ehrenfeucht-Fraïssé game. Phokion Kolaitis' slidesCourse on combinatorial games in finite model theory by Phokion Kolaitis (.ps file) and Neil Immerman's book chapter on Ehrenfeucht–Fraïssé games discuss applications in computer science, the methodology for proving inexpressibility results, and several simple inexpressibility proofs using this methodology.
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The idea is that with each granule cell receiving input from only 4–5 mossy fibers, a granule cell would not respond if only a single one of its inputs was active, but would respond if more than one were active. This "combinatorial coding" scheme would potentially allow the cerebellum to make much finer distinctions between input patterns than the mossy fibers alone would permit.
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He formulated Falconer's conjecture on the dimension of distance sets and conceived the notion of a digital sundial. In combinatorial geometry he established a lower bound of 5 for the chromatic number of the plane in the Lebesgue measurable case. Falconer was born at Bearsted Memorial Maternity Hospital outside Hampton Court Palace. He was educated at Kingston Grammar School, Kingston upon Thames and Corpus Christi College, Cambridge.
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These sizes moreover do not come in any definite order, while the same size may occur more than once; one may choose to arrange them into a weakly decreasing list of numbers, whose sum is the number n. This gives the combinatorial notion of a partition of the number n, into exactly x (for surjective ƒ) or at most x (for arbitrary ƒ) parts.
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The combinatorial power of carbon is manifested in the composition of the molecular populations detected in circum- and interstellar media (see the Astrochemistry.net web site). The number and the complexity of carbon-containing molecules are significantly higher than those of inorganic compounds, presumably all over the Universe. One of the most abundant C-containing three-atoms molecule observed in space is hydrogen cyanide (HCN).
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Over the centuries there were many attempts to explain this sequence. Some basic elements are obvious: each symbol is paired with an "upside-down" neighbor, except for 1, 27, 29, and 61 which are "vertically" symmetrical and paired with "inversed" neighbors. A combinatorial mathematical basis was explained for the first time in 2006. STEDT Monograph 5: Classical Chinese Combinatorics: Derivation of the Book of Changes Hexagram Sequence.
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In mathematics, a pseudomanifold is a special type of topological space. It looks like a manifold at most of its points, but it may contain singularities. For example, the cone of solutions of z^2=x^2+y^2 forms a pseudomanifold. Figure 1: A pinched torus A pseudomanifold can be regarded as a combinatorial realisation of the general idea of a manifold with singularities.
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Friedrich Eisenbrand Friedrich Eisenbrand (born 3 July 1971 in Quierschied, Saarland) is a German mathematician and computer scientist. He is a professor at EPFL Lausanne working in discrete mathematics, linear programming, combinatorial optimization and algorithmic geometry of numbers. Eisenbrand received his Ph.D. at Saarland University in 2000.Mathematics Genealogy Project He gave a talk at the International Congress of Mathematicians in Seoul in 2014.
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One potential disadvantage is an increases job shortages as automation may replace staff members who do tasks easily replicated by a robot. Some systems require the use of programming languages such as C++ or Visual Basic to run more complicated tasks.Cargill, J. F.; Lebl, M., New methods in combinatorial chemistry — robotics and parallel synthesis. Current Opinion in Chemical Biology 1997, 1 (1), 67-71.
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One of the most notable features of categorical quantum mechanics is that the compositional structure can be faithfully captured by a purely diagrammatic calculus.B. Coecke, Quantum picturalism, Contemporary Physics 51, 59–83 (2010). These diagrammatic languages can be traced back to Penrose graphical notation, developed in the early 1970s.R. Penrose, Applications of negative dimensional tensors, In: Combinatorial Mathematics and its Applications, D.~Welsh (Ed), pages 221–244.
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Hopf spent the year after his doctorate at the University of Göttingen, where David Hilbert, Richard Courant, Carl Runge, and Emmy Noether were working. While there he met Paul Alexandrov and began a lifelong friendship. In 1926 Hopf moved back to Berlin, where he gave a course in combinatorial topology. He spent the academic year 1927/28 at Princeton University on a Rockefeller fellowship with Alexandrov.
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However, their subsequent investigations identified that both the disulfide and hydrazone covalent bonds exhibit effective component exchange processes and so present a reliable means of generating dynamic combinatorial libraries capable of thermodynamic templation. This chemistry now forms the basis of much research in the developing field of dynamic covalent chemistry, and has in recent years emerged as a powerful tool for the discovery of molecular receptors.
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John Alan Robinson (9 March 1930 – 5 August 2016) was a philosopher, mathematician, and computer scientist. He was a professor emeritus at Syracuse University. Alan Robinson's major contribution is to the foundations of automated theorem proving. His unification algorithm eliminated one source of combinatorial explosion in resolution provers; it also prepared the ground for the logic programming paradigm, in particular for the Prolog language.
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Peptide combinatorial libraries of tri-, tetra-, and pentapeptides with various amino acid compositions were screened as potential sources of inhibitors, to see if it serves as either pure or mixed competitive inhibitor for the hAANAT enzyme. Molecular modeling and structure-activity relationship studies made it possible to pinpoint the amino acid residue of the pentapeptide inhibitor S 34461 that interacts with the cosubstrate-binding site.
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A trichrome map-coloring game in progress, on a map of the United States. On their turn, a player may choose any of the three colors to shade an unshaded state, so long as it would not share a color with a bordering state. Three states have become unshadeable, being surrounded by all three colors. Several map-coloring games are studied in combinatorial game theory.
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The process is also called posterior decoding. The algorithm computes probability much more efficiently than the naive approach, which very quickly ends up in a combinatorial explosion. Together, they can provide the probability of a given emission/observation at each position in the sequence of observations. It is from this information that a version of the most likely state path is computed ("posterior decoding").
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Oriented-matroid theory allows a combinatorial approach to the max-flow min- cut theorem. A network with the value of flow equal to the capacity of an s-t cut An oriented matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane arrangements over ordered fields.R. Tyrrell Rockafellar 1969. Anders Björner et alia, Chapters 1-3.
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In addition he wrote a book on combinatorial analysis (Vienna 1826). In 1866, he retired. Among his lasting impacts in mathematics is the introduction of the notation \tbinom nk for the binomial coefficient, which is the coefficient of xk in the expansion of the binomial (x+1)n and, more generally, the number of k-element subsets of an n-element set. From p.
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In 2005 all chess game endings with six pieces or less were solved, showing the result of each position if played perfectly. It took ten more years to complete the tablebase with one more chess piece added, thus completing a 7-piece tablebase. Adding one more piece to a chess ending (thus making an 8-piece tablebase) is considered intractable due to the added combinatorial complexity.
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In mathematical set theory, a square principle is a combinatorial principle asserting the existence of a cohering sequence of short closed unbounded (club) sets so that no one (long) club set coheres with them all. As such they may be viewed as a kind of incompactness phenomenon. Section 4. They were introduced by Ronald Jensen in his analysis of the fine structure of the constructible universe L.
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Three appendices provide background on combinatorics and asymptotics, in complex analysis, and in probability theory. The combinatorial structures that are investigate throughout the book range widely over sequences, formal languages, partitions and compositions, permutations, graphs and paths in graphs, and lattice paths. With these topics, the analysis in the book connects to applications in other areas including abstract algebra, number theory, and the analysis of algorithms.
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However, this technique won't work if it the program does not know the next data chunk to retrieve from memory. In other words, it won't work in combinatorial algorithms, such as tree spanning or random list ranking. In addition, multi-buffering assumes that memory latency is constant and can be hidden by statically. However, reality shows that memory latency changes from application to another.
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Fischer et al. (2003) proposed an exact solver, though the algorithm does not have a polynomial running time in the worst case. The algorithm is purely combinatorial and implements a pivoting scheme similar to the simplex method for linear programming, used earlier in some heuristics. It starts with a large sphere that covers all points and gradually shrinks it until it cannot be shrunk further.
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"Jack's and Jill's children"). Some grammars make no distinction in meaning between the two forms. Some publishers' style guides, however, make a distinction, assigning the "segregatory" (or "distributive") meaning to the form "John's and Mary's" and the "combinatorial" (or "joint") meaning to the form "John and Mary's". A third alternative is a construction of the form "Jack's children and Jill's", which is always distributive, i.e.
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In computer science, all-pairs testing or pairwise testing is a combinatorial method of software testing that, for each pair of input parameters to a system (typically, a software algorithm), tests all possible discrete combinations of those parameters. Using carefully chosen test vectors, this can be done much faster than an exhaustive search of all combinations of all parameters, by "parallelizing" the tests of parameter pairs.
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Such codes have applications in electrical engineering, coding theory, and computer network topologies. In these applications, it is important to devise as long a code as is possible for a given dimension of hypercube. The longer the code, the more effective are its capabilities. Finding the longest snake or coil becomes notoriously difficult as the dimension number increases and the search space suffers a serious combinatorial explosion.
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Butler's 1986 dissertation was Combinatorial Properties of Partially Ordered Sets associated with Partitions and Finite Abelian Groups. She subsequently published some of this research as "A unimodality result in the enumeration of subgroups of a finite abelian group" (Proc. AMS 1987), which concerned applications of algebraic combinatorics in group theory. Her work in this line of research also included her book Subgroup Lattices and Symmetric Functions (Mem.
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Haim Hanani ( as Chaim Chojnacki- April 8, 1991)BnF: Haim Hanani (1912-1991) was a Polish-born Israeli mathematician, known for his contributions to combinatorial design theory, in particular for the theory of pairwise balanced designs and for the proof of an existence theorem for Steiner quadruple systems. He is also known for the Hanani–Tutte theorem on odd crossings in non-planar graphs.
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Thus, a biomolecule or complex of biomolecules can often adopt a very large number of functionally distinct states. The number of states scales exponentially with the number of possible modifications, a phenomenon known as "combinatorial explosion". This is of concern for computational biologists who model or simulate such biomolecules, because it raises questions about how such large numbers of states can be represented and simulated.
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Furthermore, genes are often flanked by several binding sites for distinct transcription factors, and efficient expression of each of these genes requires the cooperative action of several different transcription factors (see, for example, hepatocyte nuclear factors). Hence, the combinatorial use of a subset of the approximately 2000 human transcription factors easily accounts for the unique regulation of each gene in the human genome during development.
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Through combinatorial studies of viral and bacterial systems, he has identified targets for novel pharmacological studies. Later in the 1980s, Keene identified RNA recognition motif (RRM) proteins. RRM proteins are the largest family of RNA-binding proteins and the seventh largest protein family of the human genome. RRM is a prevalent RNA- binding fold involving proteins implicated in RNA biogenesis, processing, transport, and degradation.
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If the synthesized molecules of a combinatorial library are cleaved from the solid support a soluble mixture forms. In such solution, millions of different compounds may be found. When this synthetic method was developed, it first seemed impossible to identify the molecules, and to find molecules with useful properties. Strategies for identification of the useful components had been developed, however, to solve the problem.
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Many computer algorithms, such as backtracking and dancing links can solve most 9×9 puzzles efficiently, but combinatorial explosion occurs as n increases, creating limits to the properties of Sudokus that can be constructed, analyzed, and solved as n increases. A Sudoku puzzle can be expressed as a graph coloring problem.Lewis, R. A Guide to Graph Colouring: Algorithms and Applications. Springer International Publishers, 2015.
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Several schemes assist managing programmer activity so that fewer bugs are produced. Software engineering (which addresses software design issues as well) applies many techniques to prevent defects. For example, formal program specifications state the exact behavior of programs so that design bugs may be eliminated. Unfortunately, formal specifications are impractical for anything but the shortest programs, because of problems of combinatorial explosion and indeterminacy.
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Ryser was born to the family of Fred G. and Edna (Huels) Ryser. He received the B.A. (1945), M.A. (1947), and Ph.D. (1948) from the University of Wisconsin.Richard A. Brualdi: "In memoriam: Herbert J. Ryser", Journal of Combinatorial Theory, Series A 47(1) (January 1988), pp. 1–5 His doctoral thesis "Rational Vector Spaces" was supervised by Cornelius Joseph Everett, Jr. and Cyrus C. MacDuffee.
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A simplicial 3-complex. A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.
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In knot theory, the Milnor conjecture says that the slice genus of the (p, q) torus knot is :(p-1)(q-1)/2. It is in a similar vein to the Thom conjecture. It was first proved by gauge theoretic methods by Peter Kronheimer and Tomasz Mrowka.. Jacob Rasmussen later gave a purely combinatorial proof using Khovanov homology, by means of the s-invariant..
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The symphony is in three movements in the traditional fast-slow-fast pattern. The second and third movements are performed without a break: #Allegro—Impetuoso—tranquillo #Con movimento adagio—doppio movimento quasi allegretto #Allegro vivace Although a single twelve-tone row forms the basis of the entire symphony, a second, related row is also used in the first movement only . This main row is: C, C, G, F, G, A / D, B, D, E, B F The two hexachords of this row are combinatorial by inversion at T3 (transposition by a minor third) . The secondary row in the first movement is: A–G–F–F–C–A / B–D–B–E–E–C However, Sessions's free treatment of the combinatorial hexachords and of various trichord (particularly 014, 016, and 026) tends to displace textbook twelve-tone technique, producing a complex but coherent network of pitch-class sets .
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Grassmann then showed that once geometry is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial dimensions; the number of possible dimensions is in fact unbounded. Fearnley-Sander (1979) describes Grassmann's foundation of linear algebra as follows:Hermann Grassmann and the Creation of Linear Algebra Following an idea of Grassmann's father, A1 also defined the exterior product, also called "combinatorial product" (in German: äußeres ProduktTr. outer product or kombinatorisches ProduktTr. combinatorial product), the key operation of an algebra now called exterior algebra. (One should keep in mind that in Grassmann's day, the only axiomatic theory was Euclidean geometry, and the general notion of an abstract algebra had yet to be defined.) In 1878, William Kingdon Clifford joined this exterior algebra to William Rowan Hamilton's quaternions by replacing Grassmann's rule epep = 0 by the rule epep = 1.
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The main part of the book is organized into three parts. The first part, covering three chapters and roughly the first quarter of the book, concerns the symbolic method in combinatorics, in which classes of combinatorial objects are associated with formulas that describe their structures, and then those formulas are reinterpreted to produce the generating functions or exponential generating functions of the classes, in some cases using tools such as the Lagrange inversion theorem as part of the reinterpretation process. The chapters in this part divide the material into the enumeration of unlabeled objects, the enumeration of labeled objects, and multivariate generating functions. The five chapters of the second part of the book, roughly half of the text and "the heart of the book", concern the application of tools from complex analysis to the generating function, in order to understand the asymptotics of the numbers of objects in a combinatorial class.
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Dominique Foata (born October 12, 1934) is a mathematician who works in enumerative combinatorics. With Pierre Cartier and Marcel-Paul Schützenberger he pioneered the modern approach to classical combinatorics, that lead, in part, to the current blossoming of algebraic combinatorics. His pioneering work on permutation statistics, and his combinatorial approach to special functions, are especially notable. Foata gave an invited talk at the International Congress of Mathematicians in Warsaw (1983).
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Bust of Walther von Dyck at his grave in Munich. Walther Franz Anton von Dyck (6 December 1856 in Munich – 5 November 1934 in Munich), born Dyck and later ennobled, was a German mathematician. He is credited with being the first to define a mathematical group, in the modern sense in . He laid the foundations of combinatorial group theory, being the first to systematically study a group by generators and relations.
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Many Ig superfamily molecules bind homophilically and heterophilically, and Dscam/DSCAM proteins are no exception. Vertebrate DSCAMs and DSCAML1s have not only been shown to bind homophilically (i.e., DSCAM–DSCAM or DSCAML1–DSCAML1, and not DSCAM–DSCAML1), but also have cell-type specific, mutually exclusive, expression patterns. Due to the combinatorial use of alternative exons, the homophilic binding specificity of Drosophila Dscam is amplified to tens of thousands of potential homodimers.
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The simplest case yields invariants of Legendrian knots inside contact three-manifolds. The relative contact homology has been shown to be a strictly more powerful invariant than the "classical invariants", namely Thurston-Bennequin number and rotation number (within a class of smooth knots). Yuri Chekanov developed a purely combinatorial version of relative contact homology for Legendrian knots, i.e. a combinatorially defined invariant that reproduces the results of relative contact homology.
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Maurice Peter Herlihy (born 4 January 1954) is a computer scientist active in the field of multiprocessor synchronization. Herlihy has contributed to areas including theoretical foundations of wait-free synchronization, linearizable data structures, applications of combinatorial topology to distributed computing, as well as hardware and software transactional memory. He is the An Wang Professor of Computer Science at Brown University, where he has been a member of the faculty since 1994.
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In this 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. In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk-Ulam theorem, named after Albert W. Tucker. Let T be a triangulation of the closed n-dimensional ball B_n.
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The Geometric Satake equivalence establishes an equivalence between representations of the Langlands dual group {}^L G of a reductive group G and certain equivariant perverse sheaves on the affine Grassmannian associated to G. This equivalence provides a non-combinatorial construction of the Langlands dual group. It is proved by showing that the mentioned category of perverse sheaves is a Tannakian category and identifying its Tannaka dual group with {}^L G.
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The house edge of casino games varies greatly with the game. Keno can have house edges up to 25% and slot machines can have up to 15%, while most Australian Pontoon games have house edges between 0.3% and 0.4%. The calculation of the Roulette house edge was a trivial exercise; for other games, this is not usually the case. Combinatorial analysis and/or computer simulation is necessary to complete the task.
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He discovered Kripke–Joyal semantics,Robert Goldblatt, A Kripke-Joyal semantics for noncommutative logic in quantales; Advances in Modal Logic 6, 209--225, Coll. Publ., London, 2006; the theory of combinatorial species and with Myles Tierney a generalization of the Galois theory of Alexander GrothendieckA. Joyal, M. Tierney, An extension of the Galois theory of Grothendieck, Memoirs of the American Mathematical Society 51 (1984), no. 309, vii+71 pp.
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A cyclic order on a finite set can be determined by an injection into the unit circle, . There are many possible functions that induce the same cyclic order—in fact, infinitely many. In order to quantify this redundancy, it takes a more complex combinatorial object than a simple number. Examining the configuration space of all such maps leads to the definition of an polytope known as a cyclohedron.
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In some approaches the floorplan may be a partition of the whole chip area into axis aligned rectangles to be occupied by IC blocks. This partition is subject to various constraints and requirements of optimization: block area, aspect ratios, estimated total measure of interconnects, etc. Finding good floorplans has been a research area in combinatorial optimization. Most of the problems related to finding optimal floorplans are NP-hard, i.e.
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In 2002 Omer Reingold, Salil Vadhan, and Avi Wigderson gave a simple, explicit combinatorial construction of constant-degree expander graphs. The construction is iterative, and needs as a basic building block a single, expander of constant size. In each iteration the zigzag product is used in order to generate another graph whose size is increased but its degree and expansion remains unchanged. This process continues, yielding arbitrarily large expanders.
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Similar expressions can be found for r > 1 with increasing complexity as r increases. The numbers F^{(1)}_n are the row sums of Hosoya's triangle. As with Fibonacci numbers, there are several combinatorial interpretations of these sequences. For example F^{(1)}_n is the number of ways n - 2 can be written as an ordered sum involving only 0, 1, and 2 with 0 used exactly once.
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An example of change ringing (with six bells), one of the earliest nontrivial results in graph theory. Basic combinatorial concepts and enumerative results appeared throughout the ancient world. In the 6th century BCE, ancient Indian physician Sushruta asserts in Sushruta Samhita that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc., thus computing all 26 − 1 possibilities.
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The Alday-Gaiotto-Tachikawa relation between two- dimensional conformal field theory and supersymmetric gauge theory, more specifically, between the conformal blocks of Liouville theory and Nekrasov partition functions of supersymmetric gauge theories in four dimensions, leads to combinatorial expressions for conformal blocks as sums over Young diagrams. Each diagram can be interpreted as a state in a representation of the Virasoro algebra, times an abelian affine Lie algebra.
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Subdivision rules give a sequence of tilings of a surface, and tilings give an idea of distance, length, and area (by letting each tile have length and area 1). In the limit, the distances that come from these tilings may converge in some sense to an analytic structure on the surface. The Combinatorial Riemann Mapping Theorem gives necessary and sufficient conditions for this to occur. Its statement needs some background.
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The Bruhat order on the Schubert varieties of a flag manifold or a Grassmannian was first studied by , and the analogue for more general semisimple algebraic groups was studied by . started the combinatorial study of the Bruhat order on the Weyl group, and introduced the name "Bruhat order" because of the relation to the Bruhat decomposition introduced by François Bruhat. The left and right weak Bruhat orderings were studied by .
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Hsu made significant contributions to asymptotic analysis, approximation theory, and combinatorics. He began collaborating with American mathematician Henry W. Gould in 1965, years before US President Richard Nixon established official relations with the People's Republic of China. Their collaboration resulted in the establishment of the Gould–Hsu Matrix Inversion Formula in 1973, which is important for computing combinatorial identities. Hsu published over 170 research papers and more than 10 monographs.
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This result (now known as Fleischner's theorem) had been submitted in 1971 and was published in 1974.Herbert Fleischner: The square of every two- connected graph is Hamiltonian. In: Journal of Combinatorial Theory, Series B. 16 (1974): 29–34. Another milestone in his research was the solution of the „Cycle plus Triangles Problems“ posed by Paul Erdős; its solution came about in cooperation with Michael Stiebitz (TU Ilmenau).
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The max-flow min-cut theorem states that the maximum flow through a network is exactly the capacity of its minimum cut. This theorem can be proved using the criss-cross algorithm for oriented matroids. The criss-cross algorithm is often studied using the theory of oriented matroids (OMs), which is a combinatorial abstraction of linear-optimization theory.The theory of oriented matroids was initiated by R. Tyrrell Rockafellar.
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The cells then grow in a petri dish for several days and are inserted into the early-stage embryos. Lastly, the embryos are placed into the adult female’s uterus where it can grow into its offspring.[9] Some alleles in this project cannot be knocked out using traditional methods and require the specificity of the conditional gene knockout technique. Other combinatorial methods are needed to knockout the last remaining alleles.
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The Australasian Journal of Combinatorics is a triannual peer-reviewed open- access scientific journal covering combinatorics. It was established in 1990 and is published by the Centre for Discrete Mathematics and Computing (University of Queensland) on behalf of the Combinatorial Mathematics Society of Australasia. Originally published biannually, it has been published three times per year since 2005. The editor-in-chief is Elizabeth J. Billington (University of Queensland).
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In financial markets, together with Eric Budish and John Shim, he has been a critic of high-frequency trading. His book, Combinatorial Auctions, edited with Yoav Shoham and Richard Steinberg, has more than 1,300 citations. The book explains why and how to conduct auctions with package bidding. He has provided advice on electricity auctions and electricity market restructuring in New England, Alberta, Colombia, the UK, France and New Zealand.
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Since 2013, development of the CRISPR/Cas9 technology, based on a prokaryotic viral defense system, has allowed for the editing or mutagenesis of a genome in vivo. Site-directed mutagenesis has proved useful in situations that random mutagenesis is not. Other techniques of mutagenesis include combinatorial and insertional mutagenesis. Mutagenesis that is not random can be used to clone DNA, investigate the effects of mutagens, and engineer proteins.
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These strengths are characterized by the strong, weak, electromagnetic (fine-structure constant), and gravitational coupling constants.Ted Bastin and C.W. Kilmister, Combinatorial Physics, World Scientific 1995, Other leading contributors in the field include H. Pierre Noyes, Ted Bastin, Clive W. Kilmister, John Amson, Mike Manthey, and David McGoveran. As described by Bastin et al., the hierarchy is generated as the cumulative sum of the sequence 3, 7, 127, 2127 − 1\.
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Issued August 29, 1995 Light-directed combinatorial synthesis has been used by Stephen Fodor and coworkers at Affymetrix to make DNA arrays containing millions of different sequences for genetic analyses. Starting in 1975, Stryer authored eight editions of a textbook entitled Biochemistry.Latchman,D.S. (1995) Trends Biochem. Sci. 20:488. Stryer also chaired a National Research Council committee that produced a report entitled Bio2010: Transforming Undergraduate Education for Future Research Biologists.
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Gould has published extensive bibliographies on combinatorial topics and on Cauchy's integral theorem. In 1977, he received the J. Shelton Horsley Research Award from the Virginia Academy of Science. In March 1988, Gould received the Benedum Distinguished Scholar Award for Physical Sciences and Technology at West Virginia University.Benedum Distinguished Scholar Award Past Recipients In 1990, Gould was elected a Foundation Fellow of the Institute of Combinatorics and its Applications.
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A reviewer notes that "The author does a beautiful job showing and developing the practical applicability of the fascinating area of finite field theory". In 1999 his book Graphs, Networks and Algorithms appeared as translation of the 1994 German version. A reviewer calls it a "first class textbook" and indispensable for teachers of combinatorial optimization. The second edition appeared in 2005, the third in 2008, and the fourth in 2013.
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Rado made contributions in combinatorics and graph theory including 18 papers with Paul Erdős. In graph theory, the Rado graph, a countably infinite graph containing all countably infinite graphs as induced subgraphs, is named after Rado. He rediscovered it in 1964 after previous works on the same graph by Wilhelm Ackermann, Paul Erdős, and Alfréd Rényi. In combinatorial set theory, the Erdős–Rado theorem extends Ramsey's theorem to infinite sets.
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ILAS was founded in 1989. Its genesis occurred at the Combinatorial Matrix Analysis Conference held at the University of Victoria in British Columbia, Canada, May 20–23, 1987, hosted by Dale Olesky and Pauline van den Driessche. ILAS was initially known as the International Matrix Group, founded in 1987. The founding officers of ILAS were Hans Schneider, President; Robert C. Thompson, Vice President; Daniel Hershkowitz, Secretary; and James R. Weaver, Treasurer.
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This article is about Steiner's taxonomy of tasks. In his book Group Processes and Productivity, Ivan Dale Steiner identified a taxonomy of group tasks to be a key source of coordination problems in groups, contributing to process losses within those groups. These tasks are divided into three categories: Component (or divisibility), Focus (quantity or quality), and Interdependence (combinatorial strategies), with an overlap of tasks between categories.Forsyth, D. R. (2010, 2006).
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In mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory, namely the strengthened finite Ramsey theorem, is true, but not provable in Peano arithmetic. This was the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano arithmetic; it was already known that such statements existed by Gödel's first incompleteness theorem.
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Players and investigators may use a wide range of computer algorithms to solve Sudokus, study their properties, and make new puzzles, including Sudokus with interesting symmetries and other properties. There are several computer algorithms that will solve most 9×9 puzzles (=9) in fractions of a second, but combinatorial explosion occurs as increases, creating limits to the properties of Sudokus that can be constructed, analyzed, and solved as increases.
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God's algorithm is a notion originating in discussions of ways to solve the Rubik's Cube puzzle,Paul Anthony Jones, Jedburgh Justice and Kentish Fire: The Origins of English in Ten Phrases and Expressions, Hachette UK, 2014 . but which can also be applied to other combinatorial puzzles and mathematical games.See e.g. Rubik's Cubic Compendium by Ernö Rubik, Tamás Varga, Gerzson Kéri, György Marx, and Tamás Vekerdy (1987, Oxford University Press, ), p.
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Later collaborative work with ultra-fast laser laboratories helped yield evidence for a vibrational coherence in the sub-picosecond processes of photosynthetic charge separation. International collaborative work was funded by a Human Frontier Science Program Award. hydropathy in the Genetic Code. Expanded View Youvan and his students have also worked in the field of combinatorial mutagenesis which can be used for directed evolution of proteins, such as enzymes.
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Cost transfer algorithms have been shown to be particularly efficient to solve real-world problem when soft constraints are binary or ternary (maximal arity of constraints in the problem is equal to 2 or 3). For soft constraints of large arity, cost transfer becomes a serious issue because the risk of combinatorial explosion has to be controlled. An algorithm, called GAC^w-WSTR,C. Lecoutre, N. Paris, O. Roussel, S. Tabary.
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Mavis McIntosh Riordan, 83; Represented Noted Writers, The New York Times obituary, August 6, 1986. Riordan's long professional career was at Bell Labs, which he joined in 1926 (a year after its foundation) and where he remained, publishing over a hundred scholarly papers on combinatorial analysis, until he retired in 1968. He then joined the faculty at Rockefeller University as professor emeritus. A Festschrift was published in his honor in 1978.
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A simple combinatorial rule for defining the winding number was proposed by August Ferdinand Möbius in 1865 and again independently by James Waddell Alexander II in 1928. Any curve partitions the plane into several connected regions, one of which is unbounded. The winding numbers of the curve around two points in the same region are equal. The winding number around (any point in) the unbounded region is zero.
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Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming. Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.
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From left: , Tanja Eisner, Tamar Ziegler, Vitaly Bergelson, Markus Haase, Terence C. Tao, Balint Farkas, and Nikos Frantzikinakis, at the 2012 MFO Study Group Ergodic Theory and Combinatorial Number Theory Tatjana (Tanja) Eisner (née Lobova, born 1980) is a German and Ukrainian mathematician specializing in functional analysis, operator theory as well as ergodic theory and its connection to number theory. She is a professor of mathematics at Leipzig University.
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Robotics have applications with Combinatorial Chemistry which has great impact on the pharmaceutical industry. The use of robotics has allowed for the use of much smaller reagent quantities and mass expansion of chemical libraries. The "parallel synthesis" method can be improved upon with automation. The main disadvantage to "parallel-synthesis" is the amount of time it takes to develop a library, automation is typically applied to make this process more efficient.
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H. Pierre Noyes (December 10, 1923 – September 30, 2016) was an American theoretical physicist. He was a member of the faculty at the SLAC National Accelerator Laboratory at Stanford University since 1962. Noyes specialized in several areas of research, including the relativistic few-body problem in nuclear and particle physics; foundations of physics; combinatorial hierarchy; and bit-string physics: a discrete model for masses, coupling constants, and cosmology from first principles.
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Wendl examined a number of matching and covering problems in combinatorial probability, especially as these problems apply to molecular biology. He determined the distribution of match counts of pairs of integer multisets in terms of Bell polynomials,Wendl MC (2005) Probabilistic assessment of clone overlaps in DNA fingerprint mapping via a priori models, J. Comp. Biol. 12(3), 283-297. a problem directly relevant to physical mapping of DNA.
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In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, they are named after Eric Temple Bell, who wrote about them in the 1930s. The Bell numbers are denoted Bn, where n is an integer greater than or equal to zero.
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Esther M. (Estie) Arkin is an Israeli–American mathematician and computer scientist whose research interests include operations research, computational geometry, combinatorial optimization, and the design and analysis of algorithms. She is a professor of applied mathematics and statistics at Stony Brook University. At Stony Brook, she also directs the undergraduate program in applied mathematics and statistics, and is an affiliated faculty member with the department of computer science.
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Early twenty-first century pesticide research has focused on developing molecules that combine low use rates and that are more selective, safer, resistance-breaking and cost-effective. Obstacles include increasing pesticide resistance and an increasingly stringent regulatory environment. The sources of new molecules employ natural products, competitors, universities, chemical vendors, combinatorial chemistry libraries, intermediates from projects in other indications and compound collections from pharmaceutical and animal health companies.
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One may talk about balls in any topological space , not necessarily induced by a metric. An (open or closed) -dimensional topological ball of is any subset of which is homeomorphic to an (open or closed) Euclidean -ball. Topological -balls are important in combinatorial topology, as the building blocks of cell complexes. Any open topological -ball is homeomorphic to the Cartesian space and to the open unit -cube (hypercube) .
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The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. Indeed, in algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.
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In a weighted bipartite graph, the optimization problem is to find a maximum-weight matching; a dual problem is to find a minimum-weight matching. This problem is often called maximum weighted bipartite matching, or the assignment problem. The Hungarian algorithm solves the assignment problem and it was one of the beginnings of combinatorial optimization algorithms. It uses a modified shortest path search in the augmenting path algorithm.
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The European Journal of Combinatorics is a peer-reviewed scientific journal for combinatorics. It is an international, bimonthly journal of discrete mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and the theories of computing. The journal includes full-length research papers, short notes, and research problems on several topics.
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Apple's RIP was of its own design, and was implemented using remarkably few ICs, including PALs for most combinatorial logic, with the subsystem timing, DRAM refreshing, and rasterization functions being implemented in very few medium-scale-integration PALs. Apple's competitors (i.e., QMS, NEC, and others) generally used a variation of one of Adobe's RIPs with their large quantity of small-scale-integration (i.e., Texas Instruments' 7400 series) ICs.
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In geometry, isotoxal polyhedra and tilings are defined by the property that they have symmetries taking any edge to any other edge.Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, , p. 371 Polyhedra with this property can also be called "edge-transitive", but they should be distinguished from edge-transitive graphs, where the symmetries are combinatorial rather than geometric. Regular polyhedra are isohedral (face- transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive).
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31, 365–368 (2012). Beside these, MMPA might pose some limitations in terms of computational resources, especially when dealing with databases of compounds with a large number of breakable bonds. Further, more atoms in the variable part of the molecule also leads to combinatorial explosion problems. To deal with this, the number of breakable bonds and number of atoms in the variable part can be used to pre-filter the database.
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Sujiko is a logic-based, combinatorial number-placement puzzle created by Jai Gomer of Kobayaashi Studios. A completed Sujiko puzzle. The puzzle takes place on a 3x3 grid with four circled number clues at the centre of each quadrant which indicate the sum of the four numbers in that quadrant. The numbers 1-9 must be placed in the grid, in accordance with the circled clues, to complete the puzzle.
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Bartoszyński studied mathematics at the University of Warsaw from 1976 to 1981, and worked there from 1981 to 1987. In 1984 he defended his Ph.D. thesis Combinatorial aspects of measure and category; his advisor was Wojciech Guzicki. In 2004 he received his habilitation from the Polish Academy of Sciences. From 1986 on he worked in the United States: he taught at University of California in Berkeley and Davis.
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They show that using pseudotriangulations in place of triangulations allows their algorithms to maintain these structures with relatively few combinatorial changes as the inputs move, and they use these dynamic pseudotriangulations to perform collision detection among the moving objects. Gudmundsson et al. (2004) consider the problem of finding a pseudotriangulation of a point set or polygon with minimum total edge length, and provide approximation algorithms for this problem.
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In combinatorial number theory, the Lambek–Moser theorem is a generalization of Beatty's theorem that defines a partition of the positive integers into two subsets from any monotonic integer-valued function. Conversely, any partition of the positive integers into two subsets may be defined from a monotonic function in this way. The theorem was discovered by Leo Moser and Joachim Lambek. provides a visual proof of the result.
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Several data structures that are combinatorial maps have been developed to store boundary representations of solids. In addition to planar faces, modern systems provide the ability to store quadrics and NURBS surfaces as a part of the boundary representation. Boundary representations have evolved into a ubiquitous representation scheme of solids in most commercial geometric modelers because of their flexibility in representing solids exhibiting a high level of geometric complexity.
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A De Bruijn torus. Each 2-by-2 binary matrix can be found within it exactly once. In combinatorial mathematics, a De Bruijn torus, named after Nicolaas Govert de Bruijn, is an array of symbols from an alphabet (often just 0 and 1) that contains every m-by-n matrix exactly once. It is a torus because the edges are considered wraparound for the purpose of finding matrices.
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Among the approaches that have been proposed to tackle combinatorial complexity in multi-state modeling, some are mainly concerned with addressing the specification problem, some are focused on finding effective methods of computation. Some tools address both specification and computation. The sections below discuss rule-based approaches to the specification problem and particle-based approaches to solving the computation problem. A wide range of computational tools exist for multi-state modeling.
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An abstract strategy game is a board, card or other game where game play does not simulate a real world theme, and a player's decisions affect the outcome. Many abstract strategy games are also combinatorial, i.e. they provide perfect information, and do not rely on physical dexterity nor on random elements such as rolling dice or drawing cards or tiles. Some board games can also be played on pen and paper.
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Heegaard diagrams, which are simple combinatorial descriptions of Heegaard splittings, have been used extensively to construct invariants of three- manifolds. The most recent example of this is the Heegaard Floer homology of Peter Ozsvath and Zoltán Szabó. The theory uses the g^{th} symmetric product of a Heegaard surface as the ambient space, and tori built from the boundaries of meridian disks for the two handlebodies as the Lagrangian submanifolds.
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Although this technique is very powerful in that many sensors can be created simultaneously, it is currently only feasible for creating short DNA strands (15-25 nucleotides). Reliability and cost factors limit the number of photolithography steps that can be done. Furthermore, light-directed combinatorial synthesis techniques are not currently possible for proteins or other sensing molecules. As noted above, most microarrays consist of a Cartesian grid of sensors.
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Patrick Michael Grundy (16 November 1917, Yarmouth, Isle of Wight – 4 November 1959) was an English mathematician and statistician. He was one of the eponymous co-discoverers of the Sprague–Grundy function and its application to the analysis of a wide class of combinatorial games.Except where otherwise indicated by alternative citations, the sources for the material in this article are the obituaries by Goddard (1960) and Smith (1960).
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It also often serves as a popular benchmark for comparing new graph models in network science. In 2006, the American Mathematics Society and the Conference Board of the Mathematical Sciences co- published Fan Chung and Linyuan Lu's book Complex Graphs and Networks. The book gave a well-structured exposition for using combinatorial, probabilistic, spectral methods as well as other new and improved tools to analyze real-world large information networks.
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An illustration of the COMP algorithm. COMP identifies item a as being defective and item b as being non-defective. However, it incorrectly labels c as a defective, since it is “hidden” by defective items in every test in which it appears. Combinatorial Orthogonal Matching Pursuit, or COMP, is a simple non-adaptive group-testing algorithm that forms the basis for the more complicated algorithms that follow in this section.
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The theory of lexicographic codes is closely connected to combinatorial game theory. In particular, the codewords in a binary lexicographic code of distance d encode the winning positions in a variant of Grundy's game, played on a collection of heaps of stones, in which each move consists of replacing any one heap by at most d − 1 smaller heaps, and the goal is to take the last stone.
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A 1996 paper formulated a peg solitaire problem as a combinatorial optimization problem and discussed the properties of the feasible region called 'a solitaire cone'. In 1999 peg solitaire was completely solved on a computer using an exhaustive search through all possible variants. It was achieved making use of the symmetries, efficient storage of board constellations and hashing. In 2001 an efficient method for solving peg solitaire problems was developed.
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An object is weakly labelled if each of its atoms has a nonnegative integer label, and each of these labels is distinct. An object is (strongly or well) labelled, if furthermore, these labels comprise the consecutive integers [1 \ldots n]. Note: some combinatorial classes are best specified as labelled structures or unlabelled structures, but some readily admit both specifications. A good example of labelled structures is the class of labelled graphs.
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Anthony W. Czarnik (born 1957) is an American chemist and inventor. He is best known for pioneering studies in the field of fluorescent chemosensors“University of Malta :"Themed collection on chemosensors and molecular logic”“Justia Patents:"Patents by Inventor Anthony W. Czarnik” and co-founding Illumina, Inc., a biotechnology company in San Diego.“Illumina, Inc. history, profile and corporate video” Czarnik was also the founding editor of ACS Combinatorial Science.
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Vera Molnar, born 1924 in Hungary, is one of the pioneers of computer and algorithmic arts. Trained as a traditional artist, Molnar studied for a diploma in art history and aesthetics at the Budapest College of Fine Arts. She iterated combinatorial images from as early as 1959. In 1968 she began working with computers, where she began to create algorithmic paintings based on simple geometric shapes and geometrical themes.
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Chapters four and five concern the combinatorial enumeration of completed Sudoku puzzles, before and after factoring out the symmetries and equivalence classes of these puzzles using Burnside's lemma in group theory. Chapter six looks at combinatorial search techniques for finding small systems of givens that uniquely define a puzzle solution; soon after the book's publication, these methods were used to show that the minimum possible number of givens is 17. The next two chapters look at two different mathematical formalizations of the problem of going from a Sudoku problem to its solution, one involving graph coloring (more precisely, precoloring extension of the Sudoku graph) and another involving using the Gröbner basis method to solve systems of polynomial equations. The final chapter studies questions in extremal combinatorics motivated by Sudoku, and (although 76 Sudoku puzzles of various types are scattered throughout the earlier chapters) the epilogue presents a collection of 20 additional puzzles, in advanced variations of Sudoku.
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A clutter is a family of subsets of a finite set such that none contains any other; that is, it is a Sperner family. The difference is in the questions typically asked. Clutters are an important structure in the study of combinatorial optimization. (In more complicated language, a clutter is a hypergraph (V,E) with the added property that A ot\subseteq B whenever A,B \in E and A eq B (i.e.
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An example of a computational problem is how to efficiently determine the allocation once the bids have been submitted to the auctioneer. This is called the winner determination problem. The winner determination problem can be stated as follows: given a set of bids in a combinatorial auction, find an allocation of items to bidders—including the possibility that the auctioneer retains some items—that maximizes the auctioneer’s revenue. This problem is difficult for large instances.
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In the mathematical fields of graph theory and combinatorial optimization, the bipartite dimension or biclique cover number of a graph G = (V, E) is the minimum number of bicliques (that is complete bipartite subgraphs), needed to cover all edges in E. A collection of bicliques covering all edges in G is called a biclique edge cover, or sometimes biclique cover. The bipartite dimension of G is often denoted by the symbol d(G).
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All Set was commissioned by the 1957 Brandeis University Creative Arts Festival, which in that year was a jazz festival. It was premiered there by the Bill Evans Orchestra in a performance that was recorded and released on a Columbia Records LP in 1963. The title is a play on words referring to the all-combinatorial twelve-tone series Babbitt used in composing the work . The published score is dedicated to Gunther Schuller .
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Dynamic combinatorial chemistry has been used as a method to develop ligands for biomolecules and receptors for small molecules. Ligands that can recognize biomolecules are being identified by preparing libraries of potential ligands in the presence of a target biomacromolecule. This is relevant for application as biosensors for fast monitoring of imbalances and illnesses and therapeutic agents. Individual components of certain chemical system will self-assemble to form receptors which are complementary to target molecule.
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Games like nim also admit a rigorous analysis using combinatorial game theory. Whether a game is solved is not necessarily the same as whether it remains interesting for humans to play. Even a strongly solved game can still be interesting if its solution is too complex to be memorized; conversely, a weakly solved game may lose its attraction if the winning strategy is simple enough to remember (e.g. Maharajah and the Sepoys).
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Multiunit auctions sell more than one identical item at the same time, rather than having separate auctions for each. This type can be further classified as either a uniform price auction or a discriminatory price auction. An example for them is spectrum auctions. Combinatorial auction is any auction for the simultaneous sale of more than one item where bidders can place bids on an "all-or-nothing" basis on "packages" rather than just individual items.
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Hallard T. Croft and Paul Erdős proved tk > c n2 / k3, where n is the number of points and tk is the number of k-point lines.The Handbook of Combinatorics, edited by László Lovász, Ron Graham, et al, in the chapter titled Extremal Problems in Combinatorial Geometry by Paul Erdős and George B. Purdy. Their construction contains some m-point lines, where m > k. One can also ask the question if these are not allowed.
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Eric Katz is a mathematician working in combinatorial algebraic geometry and arithmetic geometry. He is currently an associate professor in the Department of Mathematics at Ohio State University. In joint work with Karim Adiprasito and June Huh, he resolved the Heron–Rota–Welsh conjecture on the log-concavity of the characteristic polynomial of matroids. With Joseph Rabinoff and David Zureick-Brown, he has given bounds on rational and torsion points on curves.
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Electrical engineering uses drawn symbols and connect them with lines that stand for the mathematicals act of substitution and replacement. They then verify their drawings with truth tables and simplify the expressions as shown below by use of Karnaugh maps or the theorems. In this way engineers have created a host of "combinatorial logic" (i.e. connectives without feedback) such as "decoders", "encoders", "mutifunction gates", "majority logic", "binary adders", "arithmetic logic units", etc.
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He had earlier discovered a similar polytope, the 57-cell (Coxeter 1982, 1984). By 1994 Grünbaum was considering polytopes abstractly as combinatorial sets of points or vertices, and was unconcerned whether faces were planar. As he and others refined these ideas, such sets came to be called abstract polytopes. An abstract polytope is defined as a partially ordered set (poset), whose elements are the polytope's faces (vertices, edges, faces etc.) ordered by containment.
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The chaperone code refers to post-translational modifications of molecular chaperones that control protein folding. Whilst the genetic code specifies how DNA makes proteins, and the histone code regulates histone-DNA interactions, the chaperone code controls how proteins are folded to produce a functional proteome. The chaperone code refers to the combinatorial array of post- translational modifications (enzymes add chemical modifications to amino acids that change their properties) —i.e. phosphorylation, acetylation, ubiquitination, methylation, etc.
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A vertex-cover (transversal) T is called minimal if no proper subset of T is a transversal. The transversal hypergraph of H is the hypergraph (X, F) whose hyperedge set F consists of all minimal-transversals of H. Computing the transversal hypergraph has applications in combinatorial optimization, in game theory, and in several fields of computer science such as machine learning, indexing of databases, the satisfiability problem, data mining, and computer program optimization.
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In some problems, each variable can take one of several discrete values, or the range of possible values is divided to give a finite number of possibilities. Taking the variables together, a huge number of combinations of values must be considered. This effect is also known as the combinatorial explosion. Even in the simplest case of d binary variables, the number of possible combinations already is 2^d, exponential in the dimensionality.
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The de Bruijn sequence for alphabet size and substring length . In general there are many sequences for a particular n and k but in this example it is unique, up to cycling. In combinatorial mathematics, a de Bruijn sequence of order n on a size-k alphabet A is a cyclic sequence in which every possible length-n string on A occurs exactly once as a substring (i.e., as a contiguous subsequence).
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Neyman has made numerous fundamental contributions to the theory of the value. In a "remarkable tour- de-force of combinatorial reasoning",Aumann, R.J. (1980), "Recent Developments in the Theory of the Shapley Value", Proceedings of the International Congress of Mathematicians, Helsinki, 1978, pp. 995–1003, Academia Scientiarum Fennica he proved the existence of an asymptotic value for weighted majority games.Neyman, A., 1981, "Singular games have asymptotic values," Mathematics of Operations Research, 6, pp 205–212.
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SciCast was an outgrowth of an earlier project called DAGGRE (Decomposition-Based Elicitation and Aggregation), also an IARPA project that implemented GMU economist Robin Hanson's idea of combinatorial prediction tech markets, and was a participant in the IARPA Aggregative Contingent Estimation tournament. The launch of SciCast itself was announced by Robin Hanson on January 3, 2014, and the official announcements were made on January 10, 2014. The launch received some news coverage in March 2014.
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Magnanti's teaching and research interests are in applied and theoretical aspects of large-scale optimization and operations research, specifically on the theory and application of large-scale optimization, particularly in the areas of network flows, nonlinear programming, and combinatorial optimization. He has conducted research on such topics as production planning and scheduling, transportation planning, facility location, logistics, and communication systems design. He is also known for pioneering an educational philosophy that combines engineering and management.
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He started the Journal of Algebraic Combinatorics, and was the Editor-in-Chief of the Electronic Journal of Combinatorics from 2004 to 2008. He is also on the editorial board of the Journal of Combinatorial Theory Series B and Combinatorica.Chris Godsil's CV . He obtained his Ph.D. in 1979 at the University of Melbourne under the supervision of Derek Alan Holton.. He wrote a paper with Paul Erdős, so making his Erdős number equal to 1.
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If moreover the intervals of the integers are taken to start at 0, then the k-combination at a given place i in the enumeration can be computed easily from i, and the bijection so obtained is known as the combinatorial number system. It is also known as "rank"/"ranking" and "unranking" in computational mathematics. There are many ways to enumerate k combinations. One way is to visit all the binary numbers less than 2n.
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Since 2007 he has been Li Kuo-Ting Forum Professor at National Cheng Kung University.Seminar with Academician Chung-Laung Liu: ‘My Learning Experiences’ , National Cheng Kung University, October 7, 2009, retrieved January 26, 2017. He is the author and co-author of seven books and monographs, and over 180 technical papers. His research interests include computer-aided design of VLSI circuits, real-time systems, computer-aided instruction, combinatorial optimization, and discrete mathematics.
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'After being mentored by Tom Cech, Bevilacqua became interested in the folding of RNA and its interactions with chemistry. His research looks at how RNA affects biological processes. He studies viral replication in humans and the responses to abiotic stresses in plants. Some approaches that his lab uses are rapid mixing kinetics, fluorescence spectroscopy, UV melting, site-directed mutagenesis, combinatorial selection of RNA (or SELEX), Raman spectroscopy, NMR, SAXS, and X-ray crystallography.
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On the issue of selecting appropriate individuals among the EA population that should undergo individual learning, fitness-based and distribution-based strategies were studied for adapting the probability of applying individual learning on the population of chromosomes in continuous parametric search problems with Land extending the work to combinatorial optimization problems. Bambha et al. introduced a simulated heating technique for systematically integrating parameterized individual learning into evolutionary algorithms to achieve maximum solution quality.
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Inversion is an interesting operator, especially powerful for combinatorial optimization. It consists of inverting a small sequence within a chromosome. In gene expression programming it can be easily implemented in all gene domains and, in all cases, the offspring produced is always syntactically correct. For any gene domain, a sequence (ranging from at least two elements to as big as the domain itself) is chosen at random within that domain and then inverted.
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Egerváry's interests spanned the theory of algebraic equations, geometry, differential equations, and matrix theory. In what later became a classic result in the field of combinatorial optimization, Egerváry generalized Kőnig's theorem to the case of weighted graphs. This contribution was translated and published in 1955 by Harold W. Kuhn, who also showed how to apply Kőnig's and Egerváry's method to solve the assignment problem; the resulting algorithm has since been known as the "Hungarian method".
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Consequently, an isomorphism between two given well-ordered sets will be unique. #Cauchy's theorem on geometry of convex polytopes states that a convex polytope is uniquely determined by the geometry of its faces and combinatorial adjacency rules. #Alexandrov's uniqueness theorem states that a convex polyhedron in three dimensions is uniquely determined by the metric space of geodesics on its surface. #Rigidity results in K-theory show isomorphisms between various algebraic K-theory groups.
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He held various exhibitions in Europe. His first exhibition in Kuwait took place in 1982. He produced the series Desperta Ferro, ten pieces on a theme that anticipated the Almogávares, and completed Neon - a sculpture with computer-controlled neon circuits, which gave life to countless combinatorial possibilities of equestrian figures. He undertook his first collaboration with the architect Núñez Yanowski, for the design for the Place Picasso in Marne-la-Vallée, Paris.
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Farisi made a number of important contributions to number theory. His most impressive work in number theory is on amicable numbers. In Tadhkira al-ahbab fi bayan al-tahabb ("Memorandum for friends on the proof of amicability") introduced a major new approach to a whole area of number theory, introducing ideas concerning factorization and combinatorial methods. In fact Farisi's approach is based on the unique factorization of an integer into powers of prime numbers.
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Other editorial board memberships include the Journal of Computer and System Sciences, the Journal of Combinatorial Optimization, and the Journal of Complexity. He has also served on conference committees and chaired various conferences, such as the ACM Symposium on Principles of Database Systems and the IEEE Symposium on Foundations of Computer Science. As of June 2020, his publications have been cited close to 35,000 times, and he has an h-index of 93.
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Kiss et al described a new construction for the Bloom filter that avoids false positives in addition to the typical non- existence of false negatives. The construction applies to a finite universe from which set elements are taken. It relies on existing non-adaptive combinatorial group testing scheme by Eppstein, Goodrich and Hirschberg. Unlike the typical Bloom filter, elements are hashed to a bit array through deterministic, fast and simple-to-calculate functions.
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His first article on this topic appeared in 1894. His research in geometry led to the abstract topological definition of homotopy and homology. He also first introduced the basic concepts and invariants of combinatorial topology, such as Betti numbers and the fundamental group. Poincaré proved a formula relating the number of edges, vertices and faces of n-dimensional polyhedron (the Euler–Poincaré theorem) and gave the first precise formulation of the intuitive notion of dimension.
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MAGI developed a software program called Synthavision to create CGI images and films. Synthavision was one of the first systems to implement a ray-tracing algorithmic approach to hidden surface removal in rendering images. The software was a constructive solid geometry (CSG) system, in that the geometry was solid primitives with combinatorial operators (such as Boolean operators). Synthavision's modeling method does not use polygons or wireframe meshes that most CGI companies use today.
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In nondeterministic communication complexity, Alice and Bob have access to an oracle. After receiving the oracle's word, the parties communicate to deduce f(x,y). The nondeterministic communication complexity is then the maximum over all pairs (x,y) over the sum of number of bits exchanged and the coding length of the oracle word. Viewed differently, this amounts to covering all 1-entries of the 0/1-matrix by combinatorial 1-rectangles (i.e.
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Eugenia Malinnikova (born 23 April 1974) is a mathematician, winner of the 2017 Clay Research Award which she shared with Aleksandr Logunov "in recognition of their introduction of a novel geometric combinatorial method to study doubling properties of solutions to elliptic eigenvalue problems".Aleksandr Logunov and Eugenia Malinnikova from www.claymath.org, last read April 19, 2017. She competed three times in the International Mathematical Olympiad, winning three Gold medals (including two perfect scores).
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The Electronic Journal of Combinatorics is a peer-reviewed open access scientific journal covering research in combinatorial mathematics. The journal was established in 1994 by Herbert Wilf (University of Pennsylvania) and Neil Calkin (Georgia Institute of Technology).Wilf, H.S. About the Electronic Journal of Combinatorics The Electronic Journal of Combinatorics is a founding member of the Free Journal Network. According to the Journal Citation Reports, the journal had a 2017 impact factor of 0.762.
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A molecular logic gate is a molecule that performs a logical operation based on one or more physical or chemical inputs and a single output. The field has advanced from simple logic systems based on a single chemical or physical input to molecules capable of combinatorial and sequential operations such as arithmetic operations i.e. moleculators and memory storage algorithms. For logic gates with a single input, there are four possible output patterns.
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At some point SIAM took it over. The first SIAM selection committee is listed as 2002. The prize is given every two years, alternately in two categories: (1) for a notable application of combinatorial theory; (2) for a notable contribution in another area of interest to George Pólya such as approximation theory, complex analysis, number theory, orthogonal polynomials, probability theory, or mathematical discovery and learning. The prize is broadly intended to recognize specific recent work.
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Applied mathematics has substantial overlap with the discipline of statistics. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions. Statistical theory relies on probability and decision theory, and makes extensive use of scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra and combinatorial design. Applied mathematicians and statisticians often work in a department of mathematical sciences (particularly at colleges and small universities).
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The ABC model of flower development The molecular control of floral organ identity determination appears to be fairly well understood in some species. In a simple model, three gene activities interact in a combinatorial manner to determine the developmental identities of the organ primordia within the floral meristem. These gene functions are called A, B and C-gene functions. In the first floral whorl only A-genes are expressed, leading to the formation of sepals.
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Cohen is originally from Tel Aviv, where her father was a banker. She earned a bachelor's degree in 1985 and a master's degree in 1986 from Tel Aviv University; her master's thesis was supervised by Michael Tarsi. She moved to Stanford University for her doctoral studies, and completed her Ph.D. in 1991 with Andrew V. Goldberg as her doctoral advisor and Nimrod Megiddo as an unofficial mentor. Her dissertation was Combinatorial Algorithms for Optimization Problems.
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The first solution was published by Arthur Cayley. This was shortly followed by Kirkman's own solution which was given as a special case of his considerations on combinatorial arrangements published three years prior. J. J. Sylvester also investigated the problem and ended up declaring that Kirkman stole the idea from him. The puzzle appeared in several recreational mathematics books at the turn of the century by Lucas, Rouse Ball, Ahrens, and Dudeney.
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In mathematics, a differential poset is a partially ordered set (or poset for short) satisfying certain local properties. (The formal definition is given below.) This family of posets was introduced by as a generalization of Young's lattice (the poset of integer partitions ordered by inclusion), many of whose combinatorial properties are shared by all differential posets. In addition to Young's lattice, the other most significant example of a differential poset is the Young–Fibonacci lattice.
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In his spare time, Brokenshire became a world-class puzzling expert. Specialising in combinatorial and mechanical puzzles, he was in regular contact with puzzle researchers, designers, makers, enthusiasts and other specialists around the world. He introduced some novel solutions to existing problems, and was exceptionally quick to solve new problems. He was retained by a number of major puzzles companies as a consultant to offer an assessment on the viability of proposed puzzles.
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Scott A. Vanstone was a mathematician and cryptographer in the University of Waterloo Faculty of Mathematics. He was a member of the school's Centre for Applied Cryptographic Research, and was also a founder of the cybersecurity company Certicom. He received his PhD in 1974 at the University of Waterloo, and for about a decade worked principally in combinatorial design theory, finite geometry, and finite fields. In the 1980s he started working in cryptography.
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Shor's algorithm requires a universal quantum computer. D-Wave claims only to do quantum annealing. "A cross-disciplinary introduction to quantum annealing-based algorithms" presents an introduction to combinatorial optimization (NP-hard) problems, the general structure of quantum annealing-based algorithms and two examples of this kind of algorithms for solving instances of the max-SAT and Minimum Multicut problems, together with an overview of the quantum annealing systems manufactured by D-Wave Systems.
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In contrast to high-throughput screening, virtual screening involves computationally screening in silico libraries of compounds, by means of various methods such as docking, to identify members likely to possess desired properties such as biological activity against a given target. In some cases, combinatorial chemistry is used in the development of the library to increase the efficiency in mining the chemical space. More commonly, a diverse library of small molecules or natural products is screened.
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A series may be divided into subsets, and the members of the aggregate not part of a subset are said to be its complement. A subset is self-complementing if it contains half of the set and its complement is also a permutation of the original subset. This is most commonly seen with hexachords, six-note segments of a tone row. A hexachord that is self-complementing for a particular permutation is called prime combinatorial.
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The strategy can be proven to work because the time it takes the devil to convert a safe cube in the angel's path to an unsafe cube is longer than the time it takes the angel to get to that cube. This proof was published by Imre Leader and Béla Bollobás in 2006.B. Bollobás and I. Leader, The angel and the devil in three dimensions. Journal of Combinatorial Theory, Series A. vol.
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Michael Henry Albert (born September 20, 1962) is a mathematician and computer scientist, originally from Canada, and currently a professor and the head of the computer science department at the University of Otago in Dunedin, New Zealand. His varied research interests include combinatorics and combinatorial game theory. He received his B.Math in 1981 from the University of Waterloo. In that year Albert received the Rhodes Scholarship, and he completed his D. Phil.
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A VCG mechanism can also be used in a double auction. It is the most general form of incentive-compatible double-auction since it can handle a combinatorial auction with arbitrary value functions on bundles. Unfortunately, it is not budget-balanced: the total value paid by the buyers is smaller than the total value received by the sellers. Hence, in order to make it work, the auctioneer has to subsidize the trade.
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Let f : Γ → Γ be a combinatorial map. A turn is an unordered pair e, h of oriented edges of Γ (not necessarily distinct) having a common initial vertex. A turn e, h is degenerate if e = h and nondegenerate otherwise. A turn e, h is illegal if for some n ≥ 1 the paths fn(e) and fn(h) have a nontrivial common initial segment (that is, they start with the same edge).
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In discrete mathematics, especially combinatorics and finite probability theory, -tuples arise in the context of various counting problems and are treated more informally as ordered lists of length . -tuples whose entries come from a set of elements are also called arrangements with repetition, permutations of a multiset and, in some non-English literature, variations with repetition. The number of -tuples of an -set is . This follows from the combinatorial rule of product.
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He was also a consultant for the Rand Corporation (1966–71) and for the Brookhaven National Laboratory (1969-1973). Rota was elected to the National Academy of Sciences in 1982, was vice president of the American Mathematical Society (AMS) from 1995–97, and was a member of numerous other mathematical and philosophical organizations. He taught a difficult but very popular course in probability. He also taught Applications of Calculus, differential equations, and Combinatorial Theory.
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It mimics the food foraging behaviour of honey bee colonies. In its basic version the algorithm performs a kind of neighbourhood search combined with global search, and can be used for both combinatorial optimization and continuous optimization. The only condition for the application of the bees algorithm is that some measure of distance between the solutions is defined. The effectiveness and specific abilities of the bees algorithm have been proven in a number of studies.
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A fundamental problem in combinatorial optimization is finding a maximum matching. This problem has various algorithms for different classes of graphs. In an unweighted bipartite graph, the optimization problem is to find a maximum cardinality matching. The problem is solved by the Hopcroft-Karp algorithm in time time, and there are more efficient randomized algorithms, approximation algorithms, and algorithms for special classes of graphs such as bipartite planar graphs, as described in the main article.
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The process helps ensure that the students who are most passionate about math come to camp. Admission is selective: in 2016, the acceptance rate was 15%. Classes at Mathcamp come in four designations of pace and difficulty. The milder classes often include basic proof techniques, number theory, graph theory, and combinatorial game theory, while the spicier classes cover advanced topics in abstract algebra, topology, theoretical computer science, category theory, and mathematical analysis.
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In axiomatic set theory, a mathematical discipline, a morass is an infinite combinatorial structure, used to create "large" structures from a "small" number of "small" approximations. They were invented by Ronald Jensen for his proof that cardinal transfer theorems hold under the axiom of constructibility. A far less complex but equivalent variant known as a simplified morass was introduced by Velleman, and the term morass is now often used to mean these simpler structures.
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Many metaheuristics implement some form of stochastic optimization, so that the solution found is dependent on the set of random variables generated. In combinatorial optimization, by searching over a large set of feasible solutions, metaheuristics can often find good solutions with less computational effort than optimization algorithms, iterative methods, or simple heuristics. As such, they are useful approaches for optimization problems. Several books and survey papers have been published on the subject.
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Hops over an empty square, a toad, or more than one square are not allowed. Analogous rules apply for Right: on a turn, the Right player may move a frog left into a neighboring empty space, or hop a frog over a single toad into an empty square immediately to the toad's left. Under the normal play rule conventional for combinatorial game theory, the first player to be unable to move on his turn loses.
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In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900... See in particular p. 133. They were then applied to the study of the symmetric group by Georg Frobenius in 1903.
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In combinatorial mathematics, a Dowling geometry, named after Thomas A. Dowling, is a matroid associated with a group. There is a Dowling geometry of each rank for each group. If the rank is at least 3, the Dowling geometry uniquely determines the group. Dowling geometries have a role in matroid theory as universal objects (Kahn and Kung, 1982); in that respect they are analogous to projective geometries, but based on groups instead of fields.
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Foundations for a Discrete Physics. SLAC-PUB-4526. Stanford, CA: SLAC Theory Group, Stanford University. a pregeometry and purely discrete and finite justification for differential geometry (called the ordering operator calculus) is developed from first principles and applied it to physics. The work includes a purely combinatorial derivation of the parallel transport operator, shows that the construction of certain discrete analogs to velocity intrinsically obey Lorentz invariance, while giving commutation relations, and the uncertainty principle.
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Branch and cut is a method of combinatorial optimization for solving integer linear programs (ILPs), that is, linear programming (LP) problems where some or all the unknowns are restricted to integer values. Branch and cut involves running a branch and bound algorithm and using cutting planes to tighten the linear programming relaxations. Note that if cuts are only used to tighten the initial LP relaxation, the algorithm is called cut and branch.
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The problem of combinatorial explosion is also relevant to synthetic biology, with a recent model of a relatively simple synthetic eukaryotic gene circuit featuring 187 species and 1165 reactions. Of course, not all of the possible states of a multi-state molecule or complex will necessarily be populated. Indeed, in systems where the number of possible states is far greater than that of molecules in the compartment (e.g. the cell), they cannot be.
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Wythoff is known in combinatorial game theory and number theory for his study of Wythoff's game, whose solution involves the Fibonacci numbers. The Wythoff array, a two-dimensional array of numbers related to this game and to the Fibonacci sequence, is also named after him... In geometry, Wythoff is known for the Wythoff construction of uniform tilings and uniform polyhedra and for the Wythoff symbol used as a notation for these geometric objects.
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Due to , : :For any abstract elliptic operator on a closed, oriented, topological manifold, the analytical index equals the topological index. The proof of this result goes through specific considerations, including the extension of Hodge theory on combinatorial and Lipschitz manifolds , , the extension of Atiyah–Singer's signature operator to Lipschitz manifolds , Kasparov's K-homology and topological cobordism . This result shows that the index theorem is not merely a differentiable statement, but rather a topological statement.
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There are however several books on combinatorial analysis and topology which contain a chapter on graph theory. There has recently been a resurgence of interest in both the theory and application of graphs, whence the author obtains the title of his book. The book contains a considerable number of new results on graph theory which have been discovered since the book of Denes König, and is therefore a most welcome addition to the mathematical literature.
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502/504 Environmental science. Conservation of natural resources. Threats to the environment and protection against them 502 The environment and its protection 504 Threats to the environment 51 Mathematics 510 Fundamental and general considerations of mathematics 511 Number theory 512 Algebra 514 Geometry 517 Analysis 519.1 Combinatorial analysis. Graph theory 519.2 Probability. Mathematical statistics 519.6 Computational mathematics. Numerical analysis 519.7 Mathematical cybernetics 519.8 Operational research (OR): mathematical theories and methods 52 Astronomy. Astrophysics.
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In practice, it is often possible to achieve substantial improvement over 2-opt without the combinatorial cost of the general 3-opt by restricting the 3-changes to this special subset where two of the removed edges are adjacent. This so-called two-and-a-half-opt typically falls roughly midway between 2-opt and 3-opt, both in terms of the quality of tours achieved and the time required to achieve those tours.
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Penrose graphical notation (tensor diagram notation) of a matrix product state of five particles. In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971.Roger Penrose, "Applications of negative dimensional tensors," in Combinatorial Mathematics and its Applications, Academic Press (1971). See Vladimir Turaev, Quantum invariants of knots and 3-manifolds (1994), De Gruyter, p.
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The quadratic assignment problem (QAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics, from the category of the facilities location problems first introduced by Koopmans and BeckmannKoopmans TC, Beckmann M (1957). Assignment problems and the location of economic activities. Econometrica 25(1):53-76. The problem models the following real-life problem: :There are a set of n facilities and a set of n locations.
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7 (1957), 1073-1082.E. Barcucci, S. Brunetti, A. Del Lungo, M. Nivat, Reconstruction of lattice sets from their horizontal, vertical and diagonal X-rays, Discrete Mathematics 241(1-3): 65-78 (2001). In fact, a number of discrete tomography problems were first discussed as combinatorial problems. In 1957, H. J. Ryser found a necessary and sufficient condition for a pair of vectors being the two orthogonal projections of a discrete set.
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Andres Luure Andres Luure (born 22 May 1959 in Tallinn) is an Estonian philosopher and translator, and a researcher at Tallinn University. Luure graduated from the Moscow State University in 1983, majoring in mathematics. In 1998, he successfully defended his MA thesis titled "A combinatorial model of referring". In 2006, he successfully defended his Ph. D. thesis titled "Duality and sextets: a new structure of categories" in semiotics at the University of Tartu.
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BBO has been extended to noisy functions (that is, functions whose fitness evaluation is corrupted by noise); constrained functions; combinatorial functions; and multi-objective functions. Moreover, a micro biogeography-inspired multi-objective optimization algorithm (μBiMO) was implemented: it is suitable for solving multi-objective optimisations in the field of industrial design because it is based on a small number of islands (hence the name μBiMO), i.e. few objective function calls are required.
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The nerve of a category is often used to construct topological versions of moduli spaces. If X is an object of C, its moduli space should somehow encode all objects isomorphic to X and keep track of the various isomorphisms between all of these objects in that category. This can become rather complicated, especially if the objects have many non-identity automorphisms. The nerve provides a combinatorial way of organizing this data.
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Conway was widely known for his contributions to combinatorial game theory (CGT), a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays. He also wrote the book On Numbers and Games (ONAG) which lays out the mathematical foundations of CGT. He was also one of the inventors of sprouts, as well as philosopher's football.
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Herzog has published several notable books which are considered as the main sources in the fields of Commutative Algebra and Combinatorial Commutative Algebra, Cohen-Macaulay rings (1993), Monomial Ideals (2011), Binomial Ideals (2018). He has published over 220 research articles in mathematics and served as thesis advisor to more than 18 doctoral students, many of whom have had distinguished careers in Commutative Algebra. Since 2000 he is the corresponding member of the Academia Peloritana dei Pericolanti di Messina.
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In many sources the Švarc–Milnor lemma is stated under a slightly more restrictive assumption that the space X be a geodesic metric space (rather that a length space), and most applications concern this context. Sometimes a properly discontinuous cocompact isometric action of a group G on a proper geodesic metric space X is called a geometric action.I. Kapovich, and N. Benakli, Boundaries of hyperbolic groups. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp.
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In combinatorial mathematics, the necklace polynomial, or Moreau's necklace- counting function, introduced by , counts the number of distinct necklaces of n colored beads chosen out of α available colors. The necklaces are assumed to be aperiodic (not consisting of repeated subsequences), and the counting is done "without flipping over" (without reversing the order of the beads). This counting function describes, among other things, the number of free Lie algebras and the number of irreducible polynomials over a finite field.
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Peptide aptamers are artificial proteins selected or engineered to bind specific target molecules. These proteins consist of one or more peptide loops of variable sequence displayed by a protein scaffold. They are typically isolated from combinatorial libraries and often subsequently improved by directed mutation or rounds of variable region mutagenesis and selection. In vivo, peptide aptamers can bind cellular protein targets and exert biological effects, including interference with the normal protein interactions of their targeted molecules with other proteins.
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X-Aptamers are able to explore new features by utilizing a new selection process. Unlike SELEX, X-Aptamer selection does not rely on multiple repeated rounds of PCR amplification but rather involves a two-step bead-based discovery process. In the primary selection process, combinatorial libraries are created where each bead will carry approximately 10^12 copies of a single sequence. The beads operate as carriers, where the bound sequences will ultimately be detached into solution.
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It was given its first definite formal expression under the name "Constellations" by A. Jacques Jacques A., Constellations et Graphes Topologiques, Colloque Math. Soc. János Bolyai, p. 657-672, 1970 but the concept was already extensively used under the name "rotation" by Gerhard RingelRingel G., Map Color Theorem, Springer-Verlag, Berlin 1974 and J.W.T. Youngs in their famous solution of the Heawood map-coloring problem. The term "constellation" was not retained and instead "combinatorial map" was favored.
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Induction and maintenance of general anesthesia, and the control of the various physiological side effects is typically achieved through a combinatorial drug approach. Individual general anesthetics vary with respect to their specific physiological and cognitive effects. While general anesthesia induction may be facilitated by one general anesthetic, others may be used in parallel or subsequently to achieve and maintain the desired anesthetic state. The drug approach utilized is dependent upon the procedure and the needs of the healthcare providers.
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There is a natural partial order on partitions given by inclusion of Young diagrams. This partially ordered set is known as Young's lattice. The lattice was originally defined in the context of representation theory, where it is used to describe the irreducible representations of symmetric groups Sn for all n, together with their branching properties, in characteristic zero. It also has received significant study for its purely combinatorial properties; notably, it is the motivating example of a differential poset.
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R. Grossi, A. Gupta, and J. S. Vitter, High-order entropy- compressed text indexes, Proceedings of the 14th Annual SIAM/ACM Symposium on Discrete Algorithms (SODA), January 2003, 841-850. P. Ferragina, R. Giancarlo, G. Manzini, The myriad virtues of Wavelet Trees, Information and Computation, Volume 207, Issue 8, August 2009, Pages 849-866 G. Navarro, Wavelet Trees for All, Proceedings of 23rd Annual Symposium on Combinatorial Pattern Matching (CPM), 2012 H.-L. Chan, W.-K. Hon, T.-W.
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A longstanding question in combinatorial game theory asks whether there is a game of beggar-my-neighbour that goes on forever. This can happen only if the game is eventually periodic—that is, if it eventually reaches some state it has been in before. Some smaller decks of cards have infinite games, while others do not. John Conway once listed this among his anti-Hilbert problems, open questions whose pursuit should emphatically not drive the future of mathematical research.
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With Conway, Guy found the complete solution to the Soma cube of Piet Hein. Also with Conway, an enumeration led to the discovery of the grand antiprism, an unusual uniform polychoron in four dimensions. The two had met at Gonville and Caius College, Cambridge, where Guy was an undergraduate student from 1960, and Conway was a graduate student. It was through Michael that Conway met Richard Guy, who would become a co-author of works in combinatorial game theory.
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Martin Aigner (born February 28, 1942 in Linz) is an Austrian mathematician, professor at Freie Universität Berlin since 1974, with interests in combinatorial mathematics and graph theory.FU-berlin.de He received Ph.D from the University of Vienna. His book Proofs from THE BOOK (co-written with Günter M. Ziegler) has been translated into 12 languages.Google Books He is a recipient of a 1996 Lester R. Ford Award from the MAA for his expository article Turán's Graph Theorem.
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Other techniques include the 'multiple randomly populated supercell' approach, which better describes the random population of a true solid solution (although is far more computationally demanding). This method has also been used to model glassy/amorphous (including bulk metallic glasses) systems without a crystal lattice. Further, modeling techniques are being used to suggest new HEAs for targeted applications. The use of modeling techniques in this 'combinatorial explosion' is necessary for targeted and rapid HEA discovery and application.
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In 1918, Mandelstam theoretically predicted the fine structure splitting in Rayleigh scattering due to light scattering on thermal acoustic waves. Beginning from 1926, Mandelstam and Landsberg initiated experimental studies on vibrational scattering of light in crystals at the Moscow State University. As a result of this research, Landsberg and Mandelstam discovered the effect of the combinatorial scattering of light on 21 February 1928. They presented this fundamental discovery for the first time at a colloquium on 27 April 1928.
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In combinatorial mathematics, block walking is a method useful in thinking about sums of combinations graphically as "walks" on Pascal's triangle. As the name suggests, block walking problems involve counting the number of ways an individual can walk from one corner A of a city block to another corner B of another city block given restrictions on the number of blocks the person may walk, the directions the person may travel, the distance from A to B, et cetera.
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The lengths of the fragments are measured by electrophoresis, which can be used to deduce their distance along the map according to the restriction site, the markers of a physical map. The progress involves combinatorial algorithms. During the progress, a chromosome is obtained from a hybrid cell and cut at rare restriction site to produce large fragments. The fragments will be separated by size and undergo hybridization, forming the macrorestriction map and different contiguous blocks (i.e. contigs).
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A square pyramid and the associated abstract polytope. In mathematics, an abstract polytope is an algebraic partially ordered set or poset which captures the combinatorial properties of a traditional polytope without specifying purely geometric properties such as angles or edge lengths. A polytope is a generalisation of polygons and polyhedra into any number of dimensions. An ordinary geometric polytope is said to be a realization in some real N-dimensional space, typically Euclidean, of the corresponding abstract polytope.
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This approach (often referred to as the probabilistic method) proved highly effective in applications to extremal combinatorics and graph theory. A closely related area is the study of finite Markov chains, especially on combinatorial objects. Here again probabilistic tools are used to estimate the mixing time. Often associated with Paul Erdős, who did the pioneering work on the subject, probabilistic combinatorics was traditionally viewed as a set of tools to study problems in other parts of combinatorics.
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In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. In other words, the only difference between player 1 and player 2 is that player 1 goes first. The game is played until a terminal position is reached. A terminal position is one from which no moves are possible.
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The Gale–Ryser theorem is a result in graph theory and combinatorial matrix theory, two branches of combinatorics. It provides one of two known approaches to solving the bipartite realization problem, i.e. it gives a necessary and sufficient condition for two finite sequences of natural numbers to be the degree sequence of a labeled simple bipartite graph; a sequence obeying these conditions is called "bigraphic". It is an analog of the Erdős–Gallai theorem for simple graphs.
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She graduated in 2003, and went to Boston University for graduate school, where she completed her Ph.D. in 2008. Her dissertation, Growth Estimates for Dyson-Schwinger Equations, was supervised by Dirk Kreimer. In 2016 she was awarded a Humboldt Fellowship to visit Kreimer at the Humboldt University of Berlin. Yeats is the author of the books Rearranging Dyson–Schwinger Equations (Memoirs of the American Mathematical Society, 2011) and A Combinatorial Perspective on Quantum Field Theory (Springer, 2017).
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Willard H. Clatworthy (October 16, 1915 – February 15, 2010) was a professor emeritus from University at Buffalo and a World War II veteran from Williamsville, New York. He is known for his work in BIBD designs and combinatorial mathematics. Clatworthy received his Ph.D. in the year 1952 from the University of North Carolina at Chapel Hill under the direction of R. C. Bose; Clatworthy authored papers with S. S. Shrikhande, and J. M. Cameron on BIBD designs.
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The first purely combinatorial algorithm for linear programming was devised by Michael J. Todd. Todd's algorithm was developed not only for linear-programming in the setting of oriented matroids, but also for quadratic-programming problems and linear-complementarity problems. Todd's algorithm is complicated even to state, unfortunately, and its finite-convergence proofs are somewhat complicated. The criss-cross algorithm and its proof of finite termination can be simply stated and readily extend the setting of oriented matroids.
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In combinatorics and computer science, covering problems are computational problems that ask whether a certain combinatorial structure 'covers' another, or how large the structure has to be to do that. Covering problems are minimization problems and usually linear programs, whose dual problems are called packing problems. The most prominent examples of covering problems are the set cover problem, which is equivalent to the hitting set problem, and its special cases, the vertex cover problem and the edge cover problem.
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Conversely, metaheuristics provide sub-optimal (sometimes optimal) solutions in a reasonable time. Thus, metaheuristics usually allow to meet the resolution delays imposed in the industrial field as well as they allow to study general problem classes instead that particular problem instances. In general, many of the best performing techniques in precision and effort to solve complex and real-world problems are metaheuristics. Their fields of application range from combinatorial optimization, bioinformatics, and telecommunications to economics, software engineering, etc.
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Other examples of four-helix bundles include cytochrome, ferritin, human growth hormone, cytokine, and Lac repressor C-terminal. The four-helix bundle fold has proven an attractive target for de novo protein design, with numerous de novo four-helix bundle proteins having been successfully designed by rational and by combinatorial methods. Although sequence is not conserved among four-helix bundles, sequence patterns tend to mirror those of coiled-coil structures in which every fourth and seventh residue is hydrophobic.
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In the area of mathematics called combinatorial group theory, the Schreier coset graph is a graph associated with a group G, a generating set {xi : i in I} of G, and a subgroup H ≤ G. The Schreier graph encodes the abstract structure of a group modulo an equivalence relation formed by the coset. The graph is named after Otto Schreier, who used the term “Nebengruppenbild”. An equivalent definition was made in an early paper of Todd and Coxeter.
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Apart from likelihood estimates, graph-cut using maximum flowS. Vicente, V. Kolmogorov and C. Rother (2008): "Graph cut based image segmentation with connectivity priors", CVPR and other highly constrained graph based methodsCorso, Z. Tu, and A. Yuille (2008): "MRF Labelling with Graph-Shifts Algorithm", Proceedings of International workshop on combinatorial Image AnalysisB. J. Frey and D. MacKayan (1997): "A Revolution: Belief propagation in Graphs with Cycles", Proceedings of Neural Information Processing Systems (NIPS) exist for solving MRFs.
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Arratia developed the ideas of interlace polynomials with Béla Bollobás and Gregory Sorkin,. found an equivalent formulation of the Stanley–Wilf conjecture as the convergence of a limit, and was the first to investigate the lengths of superpatterns of permutations. He has also written highly cited papers on the Chen–Stein method on distances between probability distributions,.. on random walks with exclusion,. and on sequence alignment... He is a coauthor of the book Logarithmic Combinatorial Structures: A Probabilistic Approach....
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Hypothetical zeolites are combinatorial models of potential structures of the minerals known as zeolites. They are four-regular periodic graphs, with vertices representing the tetrahedral atom (usually Si or Al) and the edges representing the bridging oxygen atoms. Alternatively, by ignoring the tetrahedral atoms, zeolites can be represented by a six-valent periodic graph of oxygen atoms, with each O-O edge defining an edge of a tetrahedron. At present there are millions of hypothetical zeolite frameworks known.
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Axons express patterns of cell-surface adhesion molecules that allow them to match with specific layer targets. An important family of adhesion molecules is constituted by the cadherins, whose different combination on targeting cells allow the traction and guidance of the forming axons. A typical example of layers with combinatorial expression of these molecules is the tectal laminae in the chick tectum, where the N-cadherin molecule is present only in those layers that receive axons form the retina.
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Multiattributive and combinatorial auction mechanisms are emerging to allow further types of negotiation. Support for complex multi-attribute negotiations is a critical success factor for the next generation of electronic markets and, more generally, for all types of electronic exchanges. This is what the second type of electronic negotiation, namely Negotiation Support, addresses. While auctions are essentially mechanisms, bargaining is often the only choice in complex cases or those cases where no choice of partners is given.
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Notes of his from 1886 to 1912 were never put together into a finished book as he had intended. His precepts included the maxim, "Nouns, nouns and more nouns." Some particularly intriguing fragments describe Prus's combinatorial calculations of the millions of potential "individual types" of human characters, given a stated number of "individual traits." A curious comparative-literature aspect has been noted to Prus's career, which paralleled that of his American contemporary, Ambrose Bierce (1842–1914).
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The Combinatorial Multiarmed Bandit (CMAB) problem arises when instead of a single discrete variable to choose from, an agent needs to choose values for a set of variables. Assuming each variable is discrete, the number of possible choices per iteration is exponential in the number of variables. Several CMAB settings have been studied in the literature, from settings where the variables are binary to more general setting where each variable can take an arbitrary set of values.
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Further, Krichevskii became a doctor of physical and mathematical sciences (1988) and professor (1991), specializing in the field of mathematical cybernetics and information theory. From 1962 to 1996 he worked at the Sobolev Institute of Mathematics. In the late 90s he worked in University of California, Riverside, US. His main publications are in the fields of universal source coding, optimal hashing, combinatorial retrieval and error-correcting codes. Krichevsky–Trofimov estimator is widely used in source coding and bioinformatics.
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There are many classes of infinite-dimensional Lie algebras whose representations have been studied. Among these, an important class are the Kac–Moody algebras.. They are named after Victor Kac and Robert Moody, who independently discovered them. These algebras form a generalization of finite-dimensional semisimple Lie algebras, and share many of their combinatorial properties. This means that they have a class of representations that can be understood in the same way as representations of semisimple Lie algebras.
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Coxeter developed the theory further. The conceptual issues raised by complex polytopes, non-convexity, duality and other phenomena led Grünbaum and others to the more general study of abstract combinatorial properties relating vertices, edges, faces and so on. A related idea was that of incidence complexes, which studied the incidence or connection of the various elements with one another. These developments led eventually to the theory of abstract polytopes as partially ordered sets, or posets, of such elements.
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The Hauptvermutung (German for main conjecture) of geometric topology is the question of whether any two triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i.e. the subdivided triangulations are built up in the same combinatorial pattern. It was originally formulated as a conjecture in 1908 by Ernst Steinitz and Heinrich Franz Friedrich Tietze, but it is now known to be wrong. The non-manifold version was disproved by John Milnor in 1961 using Reidemeister torsion.
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Furthermore it follows that any TU matrix has only 0, +1 or −1 entries. The opposite is not true, i.e., a matrix with only 0, +1 or −1 entries is not necessarily unimodular. A matrix A is TU if and only if AT is TU. Totally unimodular matrices are extremely important in polyhedral combinatorics and combinatorial optimization since they give a quick way to verify that a linear program is integral (has an integral optimum, when any optimum exists).
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Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometric graphs, meaning graphs drawn in the Euclidean plane with possibly intersecting straight-line edges, and topological graphs, where the edges are allowed to be arbitrary continuous curves connecting the vertices, thus it is "the theory of geometric and topological graphs" (Pach 2013).
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The Cayley graph of the free group on two generators a and b In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley) and uses a specified, usually finite, set of generators for the group. It is a central tool in combinatorial and geometric group theory.
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AMS, doi:10.1090/S0273-0979-1994-00481-1; D. E. Cohen, Bull LMS, doi:10.1112/blms/25.6.614; Richard M. Thomas, which introduced, formalized and developed the theory of automatic groups. The theory of automatic groups brought new computational ideas from computer science to geometric group theory and played an important role in the development of the subject in 1990s. A 1994 paper of Cannon gave a proof of the "combinatorial Riemann mapping theorem"James W. Cannon.
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Todd received an MA in Natural Sciences from the University of Cambridge in 1995. He obtained his PhD in Organic Chemistry at the same institution in 1999, working with Chris Abell on encoding and linker strategies for combinatorial chemistry. Todd was a Wellcome Trust Postdoctoral Research Fellow at the University of California, Berkeley from 1999 to 2000, working with Paul A. Bartlett on synthesis of amino acid-derived heterocycles by Lewis acid catalysis and radical cyclisations from peptide acetals.
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Scott encoding appears first in a set of unpublished lecture notes by Dana ScottScott, Dana, A system of functional abstraction (1968). Lectures delivered at University of California, Berkeley, (1962) whose first citation occurs in the book Combinatorial Logic, Volume II . Michel Parigot gave a logical interpretation of and strongly normalizing recursor for Scott-encoded numerals, referring to them as the "Stack type" representation of numbers. Torben Mogensen later extended Scott encoding for the encoding of Lambda terms as data.
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A plane graph (in blue) and its medial graph (in red). In the mathematical discipline of graph theory, the medial graph of plane graph G is another graph M(G) that represents the adjacencies between edges in the faces of G. Medial graphs were introduced in 1922 by Ernst Steinitz to study combinatorial properties of convex polyhedra, although the inverse construction was already used by Peter Tait in 1877 in his foundational study of knots and links.
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One sheet of the Cayley graph of the Baumslag–Solitar group . Red edges correspond to and blue edges correspond to . The sheets of the Cayley graph of the Baumslag-Solitar group fit together into an infinite binary tree. In the mathematical field of group theory, the Baumslag–Solitar groups are examples of two-generator one-relator groups that play an important role in combinatorial group theory and geometric group theory as (counter)examples and test-cases.
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Scott Huxtable, David G. Cahill, Vincent Fauconnier, Jeffrey O. White, and Ji-Cheng Zhao, "Thermal conductivity imaging at micrometre-scale resolution for combinatorial studies of materials", Nature Materials 3 298-301 (2004), . Thus, lasers such as titanium sapphire (Ti:Al2O3) laser, with pulse width of ~200 fs, are used to monitor the characteristics of the interface. Other type of lasers include Yb:fiber, Yb:tungstate, Er:fiber, Nd:glass. Second-harmonic generation may be utilized to achieve frequency of double or higher.
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Submodular and supermodular set functions are also studied in combinatorial optimization. Many of the results in have analogues in , where submodular functions were first presented as generalizations of matroids. In this context, the core of a convex cost game is called the base polyhedron, because its elements generalize base properties of matroids. However, the optimization community generally considers submodular functions to be the discrete analogues of convex functions , because the minimization of both types of functions is computationally tractable.
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In combinatorial optimization, a field within mathematics, the linear bottleneck assignment problem (LBAP) is similar to the linear assignment problem.Assignment Problems, by Rainer Burkard, Mauro Dell'Amico, Silvano Martello, 2009, Chapter 6.2 "Linear Bottleneck Assignment Problem" (p. 172) In plain words the problem is stated as follows: :There are a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.
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There is an explicit function f that has been proved to be one-way, if and only if one-way functions exist. In other words, if any function is one-way, then so is f. Since this function was the first combinatorial complete one-way function to be demonstrated, it is known as the "universal one-way function". The problem of finding a one way function is thus reduced to proving that one such function exists.
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In number theory, the larger sieve is a sieve invented by Patrick X. Gallagher. The name denotes a heightening of the large sieve. Combinatorial sieves like the Selberg sieve are strongest, when only a few residue classes are removed, while the term large sieve means that this sieve can take advantage of the removal of a large number of up to half of all residue classes. The larger sieve can exploit the deletion of an arbitrary number of classes.
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Luby and Staddon have used a combinatorial approach to study the trade-offs for some general classes of broadcast encryption algorithms. A particularly efficient tree-based construction is the "subset difference" scheme, which is derived from a class of so-called subset cover schemes. The subset difference scheme is notably implemented in the AACS for HD DVD and Blu-ray Disc encryption. A rather simple broadcast encryption scheme is used for the CSS for DVD encryption.
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He is a member of the Norwegian Academy of Science and Letters. Tverberg, in 1965, proved a result on intersection patterns of partitions of point configurations that has come to be known as Tverberg's partition theorem. It inaugurated a new branch of combinatorial geometry, with many variations and applications. An account by Günter M. Ziegler of Tverberg's work in this direction appeared in the issue of the Notices of the American Mathematical Society for April, 2011.
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In mathematics, Belyi's theorem on algebraic curves states that any non- singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only. This is a result of G. V. Belyi from 1979. At the time it was considered surprising, and it spurred Grothendieck to develop his theory of dessins d'enfant, which describes nonsingular algebraic curves over the algebraic numbers using combinatorial data.
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For LEDA's planarity testing function, If the graph is planar, a combinatorial embedding is produced as a witness. If not, a Kuratowski subgraph is returned. These values can then be passed directly to checker functions to confirm their validity. A developer only needs to understand the inner-workings of these checker functions to be confident that the result is correct, which greatly reduces the learning curve compared to gaining a full understanding of LEDA's planarity testing algorithm.
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In electronics, an adder is a combinatorial or sequential logic element which computes the n-bit sum of two numbers. The family of Ling adders is a particularly fast adder and is designed using H. Ling's equations and generally implemented in BiCMOS. Samuel Naffziger of Hewlett Packard presented an innovative 64 bit adder in 0.5 μm CMOS based on Ling's equations at ISSCC 1996. The Naffziger adder's delay was less than 1 nanosecond, or 7 FO4.
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Young tableaux have numerous applications in combinatorics, representation theory, and algebraic geometry. Various ways of counting Young tableaux have been explored and lead to the definition of and identities for Schur functions. Many combinatorial algorithms on tableaux are known, including Schützenberger's jeu de taquin and the Robinson–Schensted–Knuth correspondence. Lascoux and Schützenberger studied an associative product on the set of all semistandard Young tableaux, giving it the structure called the plactic monoid (French: le monoïde plaxique).
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The ellipsoid method is used on low-dimensional problems, such as planar location problems, where it is numerically stable. On even "small"-sized problems, it suffers from numerical instability and poor performance in practice. However, the ellipsoid method is an important theoretical technique in combinatorial optimization. In computational complexity theory, the ellipsoid algorithm is attractive because its complexity depends on the number of columns and the digital size of the coefficients, but not on the number of rows.
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Graham Brightwell is a British mathematician working in the field of discrete mathematics. He was a research student at the University of Cambridge and obtained his PhD in 1988 writing on "Linear Extensions of Partially Ordered Sets" under the supervision of Béla Bollobás. He has published nearly 100 papers in pure mathematics, including over a dozen with Béla Bollobás. His research interests include random combinatorial structures; partially ordered sets; algorithms; random graphs; discrete mathematics and graph theory.
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In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication. It is faster than the standard matrix multiplication algorithm and is useful in practice for large matrices, but would be slower than the fastest known algorithms for extremely large matrices. Strassen's algorithm works for any ring, such as plus/multiply, but not all semirings, such as min-plus or boolean algebra, where the naive algorithm still works, and so called combinatorial matrix multiplication.
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Alan Jerome Hoffman (born May 30, 1924) is an American mathematician and IBM Fellow emeritus, T. J. Watson Research Center, IBM, in Yorktown Heights, New York. He is the founding editor of the journal Linear Algebra and its Applications, and holds several patents. He has contributed to combinatorial optimization and the eigenvalue theory of graphs. Hoffman and Robert Singleton constructed the Hoffman–Singleton graph, which is the unique Moore graph of degree 7 and diameter 2.
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In Piaget’s model of intellectual development, the fourth and final stage is the formal operational stage. In the classic book “The Growth of Logical Thinking from Childhood to Adolescence” by Jean Piaget and Barbel Inhelder formal operational reasoning takes many forms, including propositional reasoning, deductive logic, separation and control of variables, combinatorial reasoning, and proportional reasoning. Robert Karplus, a science educator in the 1960s and 1970s, investigated all these forms of reasoning in adolescents & adults. Mr. Tall-Mr.
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The term "algebraic combinatorics" was introduced in the late 1970s.Algebraic Combinatorics by Eiichi Bannai Through the early or mid-1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association schemes, strongly regular graphs, posets with a group action) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric functions, Young tableaux). This period is reflected in the area 05E, Algebraic combinatorics, of the AMS Mathematics Subject Classification, introduced in 1991.
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Conversely, by letting q vary and seeing q-analogs as deformations, one can consider the combinatorial case of as a limit of q-analogs as (often one cannot simply let in the formulae, hence the need to take a limit). This can be formalized in the field with one element, which recovers combinatorics as linear algebra over the field with one element: for example, Weyl groups are simple algebraic groups over the field with one element.
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An Apollonian network The Goldner–Harary graph, a non-Hamiltonian Apollonian network In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.
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In topology, a surface is generally defined as a manifold of dimension two. This means that a topological surface is a topological space such that every point has a neighborhood that is homeomorphic to an open subset of a Euclidean plane. Every topological surface is homeomorphic to a polyhedral surface such that all facets are triangles. The combinatorial study of such arrangements of triangles (or, more generally, of higher-dimensional simplexes) is the starting object of algebraic topology.
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"A Combinatorial Approach to Materials Discovery" Science 268 (1995) 1738 in the context of luminescent materials obtained by co-deposition of elements on a silicon substrate. His work was preceded by J. J. Hanak in 1970J.J. Hanak, J. Mater. Sci, 1970, 5, 964-971 but the computer and robotics tools were not available for the method to spread at the time. Work has been continued by several academic groupsCombinatorial methods for development of sensing materials, Springer, 2009.
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Process network synthesis (PNS) is a method to represent a process structure in a 'directed bipartite graph'. Process network synthesis uses the P-graph method to create a process structure. The scientific aim of this method is to find optimum structures. Process network synthesis uses a bipartite graph method P-graphP-graph method and employs combinatorial rules to find all feasible network solutions (maximum structure) and links raw materials to desired products related to the given problem.
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In topology, a graph manifold (in German: Graphenmannigfaltigkeit) is a 3-manifold which is obtained by gluing some circle bundles. They were invented and classified by the German topologist Friedhelm Waldhausen in 1967. This definition allows a very convenient combinatorial description as a graph whose vertices are the fundamental parts and (decorated) edges stand for the description of the gluing, hence the name. Two very important classes of examples are given by the Seifert bundles and the Solv manifolds.
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Heuristically, it would be expected that evolution between spin network states might be described by discrete combinatorial operations on the spin networks, which would then trace a two-dimensional skeleton of spacetime. This approach is related to state-sum models of statistical mechanics and topological quantum field theory such as the Turaeev–Viro model of 3D quantum gravity, and also to the Regge calculus approach to calculate the Feynman path integral of general relativity by discretizing spacetime.
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Bruce Hajek is the Center for Advanced Study Professor of Electrical and Computer Engineering, Professor in the Coordinated Science Laboratory, and Hoeft Chair in Engineering at the University of Illinois at Urbana-Champaign. His research spans communication networks, auction theory, stochastic analysis, combinatorial optimization, machine learning, information theory, as well as bioinformatics. He was elected into the National Academy of Engineering in 1999. and is the 2003 winner of the IEEE Koji Kobayashi Computers and Communications Award.
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He was appointed Lecturer in the Faculty of Mathematics, and therefore received recognition as a Cambridge MA by 'Special Grace' on 24 November 1928. He worked mainly on combinatorial methods and questions in real analysis, such as the Kakeya needle problem and the Hausdorff-Besicovitch dimension. These two particular areas have proved increasingly important as the years have gone by. The Kovner–Besicovitch measure of the central symmetry of planar convex sets is also named after him.
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A portion of the Rhind papyrus. The earliest recorded use of combinatorial techniques comes from problem 79 of the Rhind papyrus, which dates to the 16th century BCE. The problem concerns a certain geometric series, and has similarities to Fibonacci's problem of counting the number of compositions of 1s and 2s that sum to a given total. In Greece, Plutarch wrote that Xenocrates of Chalcedon (396–314 BC) discovered the number of different syllables possible in the Greek language.
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J. N. Srivastava received a Ph.D. in 1962 Srivastava's Ph.D. from the University of North Carolina at Chapel Hill. Prof. R. C. Bose was Srivastava's advisor. He joined Colorado State University in 1966.Some Prehistory of the Department of Statistics and Statistical Laboratory at Colorado State University He was known for his contributions in design of experiments as well as in multivariate analysis, survey sampling, reliability, coding theory, combinatorial theory, and other areas of statistics and mathematics.
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In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spacetimes and simplicial sets. The basic goal is to find algebraic invariants that classify directed spaces up to directed analogues of homotopy equivalence. For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental n-categories of directed spaces.
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A k-combination of a set S is a subset of S with k (distinct) elements. The main purpose of the combinatorial number system is to provide a representation, each by a single number, of all \tbinom nk possible k-combinations of a set S of n elements. Choosing, for any n, } as such a set, it can be arranged that the representation of a given k-combination C is independent of the value of n (although n must of course be sufficiently large); in other words considering C as a subset of a larger set by increasing n will not change the number that represents C. Thus for the combinatorial number system one just considers C as a k-combination of the set N of all natural numbers, without explicitly mentioning n. In order to ensure that the numbers representing the k-combinations of } are less than those representing k-combinations not contained in }, the k-combinations must be ordered in such a way that their largest elements are compared first.
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The goal was to understand when an action of a group by homeomorphisms on a 2-sphere is (up to a topological conjugation) an action on the standard Riemann sphere by Möbius transformations. The "combinatorial Riemann mapping theorem" of Cannon gave a set of sufficient conditions when a sequence of finer and finer combinatorial subdivisions of a topological surface determine, in the appropriate sense and after passing to the limit, an actual conformal structure on that surface. This paper of Cannon led to an important conjecture, first explicitly formulated by Cannon and Swenson in 1998 (but also suggested in implicit form in Section 8 of Cannon's 1994 paper) and now known as Cannon's conjecture, regarding characterizing word-hyperbolic groups with the 2-sphere as the boundary. The conjecture (Conjecture 5.1 in ) states that if the ideal boundary of a word-hyperbolic group G is homeomorphic to the 2-sphere, then G admits a properly discontinuous cocompact isometric action on the hyperbolic 3-space (so that G is essentially a 3-dimensional Kleinian group).
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Parkes founded the EconCS research group within Harvard School of Engineering and Applied Sciences. Known for his work on incentive engineering for computational systems, early research contributed to the design of combinatorial auctions, procedures for selling complex packages of goods. ge has worked on decentralized mechanism design as well as mechanism design in dynamic environments, where resources, participants, and information local to participants vary over time to embrace the real-world uncertainty. He served as technical advisor to CombineNet, Inc.
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If it is A's turn to move, A gives a value to each of their legal moves. A possible allocation method consists in assigning a certain win for A as +1 and for B as −1. This leads to combinatorial game theory as developed by John Horton Conway. An alternative is using a rule that if the result of a move is an immediate win for A it is assigned positive infinity and if it is an immediate win for B, negative infinity.
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Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician and ordained minister of the Church of England. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, and was listed by Alexander Macfarlane as one of ten leading 19th-century British mathematicians... In the 1840s, he obtained an existence theorem for Steiner triple systems that founded the field of combinatorial design theory, while the related Kirkman's schoolgirl problem is named after him...
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Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology computation,Perseus: the Persistent Homology software. denoising,U. Bauer, C. Lange, and M. Wardetzky: Optimal Topological Simplification of Discrete Functions on Surfaces mesh compression,T Lewiner, H Lopez and G Tavares: Applications of Forman's discrete Morse theory to topological visualization and mesh compression and topological data analysis.
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Ralph Peter Grimaldi (born January 1943) is an American mathematician specializing in discrete mathematics who is a full professor at Rose-Hulman Institute of Technology. He is known for his textbook Discrete and Combinatorial Mathematics: An Applied Introduction , first published in 1985 and now in its fifth edition, and his numerous research papers. He was born and raised in New York City and graduated from what is now the State University of New York at Albany in 1964 (B.S.) and 1965 (M.
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After one year of legal studies, he was awarded his bachelor's degree in Law on 28 September 1665.Hubertus Busche, Leibniz' Weg ins perspektivische Universum: Eine Harmonie im Zeitalter der Berechnung, Meiner Verlag, 1997, p. 120. His dissertation was titled De conditionibus (On Conditions). In early 1666, at age 19, Leibniz wrote his first book, De Arte Combinatoria (On the Combinatorial Art), the first part of which was also his habilitation thesis in Philosophy, which he defended in March 1666.
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Three compositional ideas, initially independent but often intertwined, are developed throughout Alberto Posadas’ work. The first one is based on the use of mathematical and physical procedures, starting with systems of combinatorial mathematics and evolving towards the application of ”fractal theory”. This fusion between mathematics and music comes from a desire to put into music the systems that regulate nature. This approach has been applied to works like Apeiron for orchestra, A silentii sonitu for string quartet and Invarianza for ensemble.
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RCASE is non-statistical and thus does not require any hypotheses. If the key parameters causing the issue or fault in a process are not present in a dataset, it will still narrow the search space and advise where the root cause may lie. This is a different approach to statistical theories which try to find a best fit. RCASE is based on optimised combinatorial theory and runs on either a grid cluster or a high performance in-memory database.
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Graphs are drawn as rods connected by rotating hinges. The cycle graph C4 drawn as a square can be tilted over by the blue force into a parallelogram, so it is a flexible graph. K3, drawn as a triangle, cannot be altered by any force that is applied to it, so it is a rigid graph. In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges.
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Combinatorial Geometry in the Plane is a book in discrete geometry. It was translated from a German-language book, Kombinatorische Geometrie in der Ebene, which its authors Hugo Hadwiger and Hans Debrunner published through the University of Geneva in 1960, expanding a 1955 survey paper that Hadwiger had published in L'Enseignement mathématique. Victor Klee translated it into English, and added a chapter of new material. It was published in 1964 by Holt, Rinehart and Winston, and republished in 1966 by Dover Publications.
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Experiments targeting selective phenotypic markers are screened and identified by plating the cells on differential medias. Each cycle ultimately takes 2.5 hours to process, with additional time required to grow isogenic cultures and characterize mutations. By iteratively introducing libraries of mutagenic ssDNAs targeting multiple sites, MAGE can generate combinatorial genetic diversity in a cell population. There can be up to 50 genome edits, from single nucleotide base pairs to whole genome or gene networks simultaneously with results in a matter of days.
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Nisan is the author of Using Hard Problems to Create Pseudorandom Generators (MIT Press, ACM Distinguished Dissertation Series, 1992), co-author with Eyal Kushilevitz of the book Communication Complexity (Cambridge University Press, 1997), and co-author with Shimon Schocken of The Elements of Computing Systems: Building a Modern Computer from First Principles (The MIT Press, 2005). In 2007 he co-edited the book Algorithmic Game Theory (Cambridge University Press, 2007). He has written highly cited papers on mechanism design,. combinatorial auctions,.
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The objective of the VRP is to minimize the total route cost. In 1964, Clarke and Wright improved on Dantzig and Ramser's approach using an effective greedy approach called the savings algorithm. Determining the optimal solution to VRP is NP-hard, so the size of problems that can be solved, optimally, using mathematical programming or combinatorial optimization may be limited. Therefore, commercial solvers tend to use heuristics due to the size and frequency of real world VRPs they need to solve.
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With Hua Luogeng (华罗庚, alternatively Hua Loo-Keng), he developed high-dimensional combinatorial designs for numerical integration on the unit cube. Their work came to the attention of the statistician Kai-Tai Fang, who realized that their results could be used in the design of experiments. In particular, their results could be used to investigate interaction, for example, in factorial experiments and response surface methodology. Collaborating with Fang led to uniform designs, which have been used also in computer simulations.
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In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali.. The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset E of Rd by a disjoint family extracted from a Vitali covering of E.
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The 11-category, game theoretical taxonomy of chess includes: two player, no-chance, combinatorial, Markov state (present state is all a player needs to move; although past state led up to that point, knowledge of the sequence of past moves is not required to make the next move, except to take into account of en passant and castling, which do depend on the past moves), zero sum, symmetric, perfect information, non-cooperative, discrete, extensive form (tree decisions, not payoff matrices), and sequential.
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In mathematics, a distance-regular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance j from v and at distance k from w depends only upon j, k, and i = d(v, w). Every distance- transitive graph is distance-regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large automorphism group.
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In 2010, Lee and his co-authors won the ICS Prize of the INFORMS Computing Society for their work showing that many combinatorial feasibility problems could be recast as systems of polynomial equations in complex variables and then shown to be infeasible by applying Hilbert's Nullstellensatz and a wide variety of computational techniques.INFORMS Computing Society Prize , INFORMS Computing Society, retrieved 2018-01-25. In 2013, Lee was elected as a Fellow of INFORMS.INFORMS Fellows Class of 2013, INFORMS, retrieved 2018-01-25.
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Vladimir Iosifovich Levenshtein (; March 20, 1935 – September 6, 2017) was a Russian scientist who did research in information theory, error-correcting codes, and combinatorial design. Among other contributions, he is known for the Levenshtein distance and a Levenshtein algorithm, which he developed in 1965. He graduated from the Department of Mathematics and Mechanics of Moscow State University in 1958 and worked at the Keldysh Institute of Applied Mathematics in Moscow ever since. He was a fellow of the IEEE Information Theory Society.
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The \Delta-lemma states that every uncountable collection of finite sets contains an uncountable \Delta- system. The \Delta-lemma is a combinatorial set-theoretic tool used in proofs to impose an upper bound on the size of a collection of pairwise incompatible elements in a forcing poset. It may for example be used as one of the ingredients in a proof showing that it is consistent with Zermelo–Fraenkel set theory that the continuum hypothesis does not hold. It was introduced by .
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Cabezón obtained his PhD with the thesis "Combinatorial Koszul homology: computations and applications" for which he obtained the grade of outstanding cum laude unanimously of the court. His thesis is framed within the area of computational algebra. In it, the homology of Koszul for monomial ideals is studied. In the thesis, Cabezón described the structure of this type of ideals based on his Koszul homology, described algorithms for the calculation of this homology, and implemented algorithms that show to be effective.
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It is also noteworthy that many combinatorial statements are exactly equiconsistent with some large cardinal rather than, say, being intermediate between them. The order of consistency strength is not necessarily the same as the order of the size of the smallest witness to a large cardinal axiom. For example, the existence of a huge cardinal is much stronger, in terms of consistency strength, than the existence of a supercompact cardinal, but assuming both exist, the first huge is smaller than the first supercompact.
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Away from the area of large cardinals, Kunen is known for intricate forcing and combinatorial constructions. He proved that it is consistent that Martin's axiom first fails at a singular cardinal and constructed under the continuum hypothesis a compact L-space supporting a nonseparable measure. He also showed that P(\omega)/Fin has no increasing chain of length \omega_2 in the standard Cohen model where the continuum is \aleph_2. The concept of a Jech–Kunen tree is named after him and Thomas Jech.
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When generating well defined styles, music can be seen as a combinatorial optimization problem, whereby the aim is to find the right combination of notes such that the objective function is minimized. This objective function typically contains rules of a particular style, but could be learned using machine learning methods such as Markov models. Researchers have generated music using a myriad of different optimization methods, including integer programming, variable neighbourhood search, and evolutionary methods as mentioned in the next subsection.
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In this case, they usually capture a typical structure of combinatorial problems. For instance, the `regular` constraint expresses that a sequence of variables is accepted by a deterministic finite automaton. Global constraints are used to simplify the modeling of constraint satisfaction problems, to extend the expressivity of constraint languages, and also to improve the constraint resolution: indeed, by considering the variables altogether, infeasible situations can be seen earlier in the solving process. Many of the global constraints are referenced into an online catalog.
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Duality of polytopes and order-theoretic duality are both involutions: the dual polytope of the dual polytope of any polytope is the original polytope, and reversing all order-relations twice returns to the original order. Choosing a different center of polarity leads to geometrically different dual polytopes, but all have the same combinatorial structure. A planar graph in blue, and its dual graph in red. From any three- dimensional polyhedron, one can form a planar graph, the graph of its vertices and edges.
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Mehlhorn is the author of several books and over 250 scientific publications,. which include fundamental contributions to data structures, computational geometry, computer algebra, parallel computing, VLSI design, computational complexity, combinatorial optimization, and graph algorithms. Mehlhorn has been an important figure in the development of algorithm engineering and is one of the developers of LEDA, the Library of Efficient Data types and Algorithms. Mehlhorn has played an important role in the establishment of several research centres for computer science in Germany.
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This technique uses barcoding to measure chromatin accessibility in thousands of individual cells; it can generate epigenomic profiles from 10,000-100,000 cells per experiment. But combinatorial cellular indexing requires additional, custom-engineered equipment or a large quantity of custom, modified Tn5. Computational analysis of scATAC-seq is based on construction of a count matrix with number of reads per open chromatin regions. Open chromatin regions can be defined, for example, by standard peak calling of pseudo bulk ATAC-seq data.
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A particularly simple combinatorial construction of the Tutte–Coxeter graph is due to Coxeter (1958b), based on work by Sylvester (1844). In modern terminology, take a complete graph on 6 vertices K6. It has 15 edges and also 15 perfect matchings. Each vertex of the Tutte–Coxeter graph corresponds to an edge or perfect matching of the K6, and each edge of the Tutte–Coxeter graph connects a perfect matching of the K6 to each of its three component edges.
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In 1985, he started at Purdue University, where he was professor of chemistry (1985–1991). He is Richard and Alice Cramer Professor of Chemistry and member of the Skaggs Institute for Chemical Biology at The Scripps Research Institute. Boger is active in the field of organic chemistry with research interests including natural product synthesis, synthetic methodology, medicinal chemistry, and combinatorial chemistry. He is also the author of a popular book on synthetic organic chemistry: Modern Organic Synthesis Lecture Notes (TSRI Press, 1999).
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Garvan is well-known for his work in the fields of q-series and integer partitions. Most famously, in 1988, Garvan and Andrews discovered a definition of the crank of a partition. The crank of a partition is an elusive combinatorial statistic similar to the rank of a partition which provides a key to the study of Ramanujan congruences in partition theory. It was first described by Freeman Dyson in a paper on ranks for the journal Eureka in 1944.
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Károly Bezdek (born May 28, 1955 in Budapest, Hungary) is a Hungarian-Canadian mathematician. He is a professor as well as a Canada Research Chair of mathematics and the director of the Centre for Computational and Discrete Geometry at the University of Calgary in Calgary, Alberta, Canada. Also he is a professor (on leave) of mathematics at the University of Pannonia in Veszprém, Hungary. His main research interests are in geometry in particular, in combinatorial, computational, convex, and discrete geometry.
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Gérard Denis Cohen is a Computer Science Professor with Telecom ParisTech (ENST) in Paris, France. Cohen was awarded with Ph.D. from the Pierre and Marie Curie University where he studied under the mentorships of Robert Fortet and Michel Deza. His dissertation thesis was on "Distance Minimale et Enumeration de Poids des Codes Lineaires Mathematics Subject Classification: 05—Combinatorics". Cohen was appointed a Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2013 for his contributions to combinatorial aspects of coding theory.
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