Not since Anthony Weiner ran for New York City mayor with a 64-point plan for saving the city has a New York politician explained himself with such enumerative zeal.
|
|
In 1620, early modern philosopher Francis Bacon repudiated the value of mere experience and enumerative induction alone. His method of inductivism required that minute and many-varied observations that uncovered the natural world's structure and causal relations needed to be coupled with enumerative induction in order to have knowledge beyond the present scope of experience. Inductivism therefore required enumerative induction as a component.
|
|
When all extremes have been analyzed, they reach the non-enumerative (true) ultimate.
|
|
The two principal methods used to reach inductive conclusions are enumerative induction and eliminative induction.
|
|
Circles of Apollonius: Eight colored circles are tangent to the three black circles. Many of the important mathematical applications of mirror symmetry belong to the branch of mathematics called enumerative geometry. In enumerative geometry, one is interested in counting the number of solutions to geometric questions, typically using the techniques of algebraic geometry. One of the earliest problems of enumerative geometry was posed around the year 200 BCE by the ancient Greek mathematician Apollonius, who asked how many circles in the plane are tangent to three given circles.
|
|
List of Emeriti - UNSW Sydney, unsw.edu.au. Retrieved 27 May 2018. His specialities included French poetry of nineteenth century and enumerative bibliography.
|
|
Philosophical investigations, Part 1 §27–34 An enumerative definition of a concept or a term is an extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question. Enumerative definitions are only possible for finite sets (and in fact only practical for relatively small sets).
|
|
For highly constrained situations such as throwing dice or experiments with decks of cards, calculating the probability of events is basically enumerative combinatorics.
|
|
Enumerative geometry saw spectacular development towards the end of the nineteenth century, at the hands of Hermann Schubert. He introduced for the purpose the Schubert calculus, which has proved of fundamental geometrical and topological value in broader areas. The specific needs of enumerative geometry were not addressed until some further attention was paid to them in the 1960s and 1970s (as pointed out for example by Steven Kleiman). Intersection numbers had been rigorously defined (by André Weil as part of his foundational programme 1942-6, and again subsequently), but this did not exhaust the proper domain of enumerative questions.
|
|
There are two primary types of classification used for information organization: enumerative and faceted. An enumerative classification contains a full set of entries for all concepts. A faceted classification system uses a set of semantically cohesive categories that are combined as needed to create an expression of a concept. In this way, the faceted classification is not limited to already defined concepts.
|
|
Vakil is an algebraic geometer and his research work spans over enumerative geometry, topology, Gromov-Witten theory, and classical algebraic geometry. He has solved several old problems in Schubert calculus. Among other results, he proved that all Schubert problems are enumerative over the real numbers, a result that resolves an issue mathematicians have worked on for at least two decades.
|
|
In the mathematical field of enumerative combinatorics, identities are sometimes established by arguments that rely on singling out one "distinguished element" of a set.
|
|
In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.
|
|
Although past work in the area has studied it through methods of enumerative combinatorics, this book instead is centered around explicit calculations related to the tropicalization of classical varieties. Although it is much more comprehensive than the two previous introductory books in this area by Itenberg et al., some topics in tropical geometry are (deliberately) omitted, including enumerative geometry and mirror symmetry. The book has six chapters.
|
|
Okasha then notes the unresolved dispute among philosophers over whether enumerative induction is a consequence of IBE, a view that Okasha, omitting Popper from mention, introduces by noting, "The philosopher Gilbert Harman has argued that IBE is more fundamental" p 32. Yet other philosophers have asserted the converse—that IBE derives from enumerative induction, more fundamental—and, although inference could in principle work both ways, the dispute is unresolved [p 32]. In a 1965 paper, now classic, Gilbert Harman explains enumerative induction as simply a masked effect of what C S Pierce had termed abduction, that is, inference to the best explanation, or IBE.
|
|
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.
|
|
Ragni Piene Ragni Piene (born 18 January 1947, Oslo)CV is a Norwegian mathematician, specializing in algebraic geometry, with particular interest in enumerative results and intersection theory.
|
|
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.
|
|
She is known for applying string theory to various problems in mathematics, including knot theory (refined Chern–Simons theory), enumerative geometry, mirror symmetry, and the geometric Langlands correspondence.
|
|
In mathematics, Topological Recursion is a recursive definition of invariants of spectral curves. It has applications in enumerative geometry, random matrix theory, mathematical physics, string theory, knot theory.
|
|
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.
|
|
In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of several more modern theories, for example characteristic classes, and in particular its algorithmic aspects are still of current interest. The phrase "Schubert calculus" is sometimes used to mean the enumerative geometry of linear subspaces, roughly equivalent to describing the cohomology ring of Grassmannians, and sometimes used to mean the more general enumerative geometry of nonlinear varieties. Even more generally, “Schubert calculus” is often understood to encompass the study of analogous questions in generalized cohomology theories.
|
|
Splitting the question, as now it would be understood, into Schubert calculus and enumerative geometry, the former is well-founded on the basis of the topology of Grassmannians, and intersection theory. The latter has status that is less clear, if clarified with respect to the position in 1900. While enumerative geometry made no connection with physics during the first century of its development, it has since emerged as a central element of string theory.
|
|
Enumerative induction is an inductive method in which a conclusion is constructed based upon the number of instances that support it. The more supporting instances, the stronger the conclusion. The most basic form of enumerative induction reasons from particular instances to all instances, and is thus an unrestricted generalization. If one observes 100 swans, and all 100 were white, one might infer a universal categorical proposition of the form All swans are white.
|
|
Hieronymus Georg Zeuthen (15 February 1839 – 6 January 1920) was a Danish mathematician. He is known for work on the enumerative geometry of conic sections, algebraic surfaces, and history of mathematics.
|
|
As this reasoning form's premises, even if true, do not entail the conclusion's truth, this is a form of inductive inference. The conclusion might be true, and might be thought probably true, yet it can be false. Questions regarding the justification and form of enumerative inductions have been central in philosophy of science, as enumerative induction has a pivotal role in the traditional model of the scientific method. :All life forms so far discovered are composed of cells.
|
|
Stanley, Richard P. (1999) Enumerative Combinatorics, Volume 2, p. 289. Cambridge University Press. . The Durfee symbol consists of the two partitions represented by the points to the right or below the Durfee square.
|
|
In its milder variant, Rudolf Carnap tried, but always failed, to formalize an inductive logic whereby a universal law's truth via observational evidence could be quantified by "degree of confirmation". Asserting a type of hypotheticodeductivism termed falsificationism, Karl Popper from the 1930s onward especially attacked inductivism and its positivist variants as untenable. In 1963, Popper called enumerative induction "a myth", a deductive inference from a tacit theory explanatory. Gilbert Harman soon explained enumerative induction as a masked outcome of IBE.
|
|
Jovan Karamata, Théorèmes sur la sommabilité exponentielle et d'autres sommabilités s'y rattachant, Mathematica (Cluj) 9 (1935), pp, 164–178. The notation S(n, k) was used by Richard Stanley in his book Enumerative Combinatorics.
|
|
Hilbert's fifteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. The problem is to put Schubert's enumerative calculus on a rigorous foundation.
|
|
Aaron C. Pixton (born January 13, 1986) is an American mathematician at the University of Michigan. He works in enumerative geometry, and is also known for his chess playing, where he is a FIDE Master.
|
|
In 1990-1991, had a major impact not only on enumerative algebraic geometry but on the whole mathematics and motivated . The mirror pair of two quintic threefolds in this paper have the following Hodge diamonds.
|
|
Interpreting experience to reveal uniformity of nature, indeed, and thereby justify enumerative induction, Mill accepted positivism—the first modern philosophy of science. Also a political philosophy, it posed scientific knowledge as ultimately the only knowledge.
|
|
Comte found enumerative induction reliable as a consequence of its grounding in available experience. He asserted the use of science, rather than metaphysical truth, as the correct method for the improvement of human society. According to Comte, scientific method frames predictions, confirms them, and states laws—positive statements—irrefutable by theology or by metaphysics. Regarding experience as justifying enumerative induction by demonstrating the uniformity of nature,Wesley C Salmon, "The uniformity of Nature", Philosophy and Phenomenological Research, 1953 Sep;14(1):39–48, [39].
|
|
In 1986, geometer Sheldon Katz proved that the number of curves, such as circles, that are defined by polynomials of degree two and lie entirely in the quintic is 609,250. By the year 1991, most of the classical problems of enumerative geometry had been solved and interest in enumerative geometry had begun to diminish. According to mathematician Mark Gross, "As the old problems had been solved, people went back to check Schubert's numbers with modern techniques, but that was getting pretty stale."Yau and Nadis 2010, p.
|
|
The Gopakumar-Vafa invariants do not presently have a rigorous mathematical definition, and this is one of the major problems in the subject. The Gromov-Witten invariants of smooth projective varieties can be defined entirely within algebraic geometry. The classical enumerative geometry of plane curves and of rational curves in homogeneous spaces are both captured by GW invariants. However, the major advantage that GW invariants have over the classical enumerative counts is that they are invariant under deformations of the complex structure of the target.
|
|
The generalization of GAGA for stacky curves is used in the derivation of algebraic structure theory of rings of modular forms. The study of stacky curves is used extensively in equivariant Gromov–Witten theory and enumerative geometry.
|
|
Kleiman is known for his work in algebraic geometry and commutative algebra. He has made seminal contributions in motivic cohomology, moduli theory, intersection theory and enumerative geometry. A 2002 study of 891 academic collaborations in enumerative geometry and intersection theory covered by Mathematical Reviews found that he was not only the most prolific author in those areas, but also the one with the most collaborative ties, and the most central author of the field in terms of closeness centrality; the study's authors proposed to name the collaboration graph of the field in his honor..
|
|
523 The Calabi–Yau manifolds used in string theory are of interest in pure mathematics, and mirror symmetry allows mathematicians to solve problems in enumerative algebraic geometry, a branch of mathematics concerned with counting the numbers of solutions to geometric questions. A classical problem of enumerative geometry is to enumerate the rational curves on a Calabi–Yau manifold such as the one illustrated above. By applying mirror symmetry, mathematicians have translated this problem into an equivalent problem for the mirror Calabi–Yau, which turns out to be easier to solve.
|
|
In 1991, Edward Witten conjectured a use of Gromov's theory to define enumerative invariants.Witten, Edward. Two- dimensional gravity and intersection theory on moduli space. Surveys in differential geometry (Cambridge, MA, 1990), 243–310, Lehigh Univ., Bethlehem, PA, 1991.
|
|
In 2010 he was an invited speaker at the International Congress of Mathematicians in Hyderabad and gave a talk The tangent space to an enumerative problem. In December 2014 he was elected a Fellow of the American Mathematical Society.
|
|
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.
|
|
Thue–Morse infinite word. Combinatorics on words deals with formal languages. It arose independently within several branches of mathematics, including number theory, group theory and probability. It has applications to enumerative combinatorics, fractal analysis, theoretical computer science, automata theory, and linguistics.
|
|
Her doctorate involved matroid theory and enumerative combinatorics, and was supervised by Thomas Allan Dowling. She was the 25th black woman to earn a Ph.D. in mathematics in the U.S.Carolyn Mahoney, Ohio Women's Hall of Fame, retrieved 2015-02-21.
|
|
171 Although the methods used in this work were based on physical intuition, mathematicians have gone on to prove rigorously some of the predictions of mirror symmetry. In particular, the enumerative predictions of mirror symmetry have now been rigorously proven.
|
|
In some contexts, such as enumerative combinatorics, the term enumeration is used more in the sense of counting – with emphasis on determination of the number of elements that a set contains, rather than the production of an explicit listing of those elements.
|
|
Lucia Caporaso in Oberwolfach, 2006 Lucia Caporaso is an Italian mathematician, holding a professorship in mathematics at Roma Tre University. She was born in Rome, Italy, on May 25,1965. Her research includes work in algebraic geometry, arithmetic geometry, tropical geometry and enumerative geometry.
|
|
Bruce Sagan Bruce E. Sagan (born March 29, 1954, Chicago, Illinois) is a Professor of Mathematics at Michigan State University. He specializes in enumerative, algebraic, and topological combinatorics. He is also known as a musician, playing music from Scandinavia and the Balkans.
|
|
Hana, Jiri. Czech Clitics in Higher Order Grammar. Diss. The Ohio State University, 2007. It can be viewed simultaneously as generative-enumerative (like categorial grammar and principles and parameters) or model theoretic (like head-driven phrase structure grammar or lexical functional grammar).
|
|
In mathematics, the MacMahon Master theorem (MMT) is a result in enumerative combinatorics and linear algebra. It was discovered by Percy MacMahon and proved in his monograph Combinatory analysis (1916). It is often used to derive binomial identities, most notably Dixon's identity.
|
|
Her dissertation, On the Enumerative Geometry of Stationary Multiple- points, was supervised by Steven Kleiman. She became editor of the American Mathematical Monthly beginning in 2017. Colley is the author of the textbook Vector Calculus (Prentice Hall, 1997; 4th ed., Pearson, 2011).
|
|
Normaliz also computes enumerative data, such as multiplicities (volumes) and Hilbert series. The kernel of Normaliz is a templated C++ class library. For multivariate polynomial arithmetic it uses CoCoALib. Normaliz has interfaces to several general computer algebra systems: CoCoA, GAP, Macaulay2 and Singular.
|
|
Georges-Henri Halphen (; 30 October 1844, Rouen – 23 May 1889, Versailles) was a French mathematician. He was known for his work in geometry, particularly in enumerative geometry and the singularity theory of algebraic curves, in algebraic geometry. He also worked on invariant theory and projective differential geometry.
|
|
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.
|
|
The empiricist David Hume's 1740 stance found enumerative induction to have no rational, let alone logical, basis but instead induction was a custom of the mind and an everyday requirement to live. While observations, such as the motion of the sun, could be coupled with the principle of the uniformity of nature to produce conclusions that seemed to be certain, the problem of induction arose from the fact that the uniformity of nature was not a logically valid principle. Hume was skeptical of the application of enumerative induction and reason to reach certainty about unobservables and especially the inference of causality from the fact that modifying an aspect of a relationship prevents or produces a particular outcome.
|
|
This study had many implications for theoretical chemistry. The techniques he used mainly concern the enumeration of graphs with particular properties. Enumerative graph theory then arose from the results of Cayley and the fundamental results published by Pólya between 1935 and 1937. These were generalized by De Bruijn in 1959.
|
|
Mireille Bousquet-Mélou at Oberwolfach in 2014. Mireille Bousquet-Mélou (born May 12, 1967) is a French mathematician who specializes in enumerative combinatorics and who works as a senior researcher for the Centre national de la recherche scientifique (CNRS) at the computer science department (LaBRI) of the University of Bordeaux..
|
|
University of Florida, Academy of Distinguished Teaching Scholars Bóna's main fields of research include the combinatorics of permutations, as well as enumerative and analytic combinatorics. Since 2010, he has been one of the editors-in-chief of the Electronic Journal of Combinatorics.Editorial Team, Electronic Journal of Combinatorics, accessed 2013-06-03.
|
|
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics, as well as applied mathematics.
|
|
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.
|
|
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.
|
|
Accurate specification of reliable, complex systems requires a language that is executable (for enumerative verification) and has formal semantics (to avoid any as language ambiguities that could lead to interpretation divergences between designers and implementors). Formal semantics are also required when it is necessary to establish the correctness of an infinite system; this cannot be done using enumerative techniques because they deal only with finite abstractions, so must be done using theorem proving techniques, which only apply to languages with a formal semantics. CADP acts on a LOTOS description of the system. LOTOS is an international standard for protocol description (ISO/IEC standard 8807:1989), which combines the concepts of process algebras (in particular CCS and CSP and algebraic abstract data types.
|
|
He was then a postdoctoral fellow at the IBM Watson Research Center and MIT. He then joined Brandeis University faculty in 1984. He was promoted to Professor of Mathematics and Computer Science in 1990, became a chair in 1996–98, and Professor Emeritus in 2015. Gessel is a prolific contributor to enumerative and algebraic combinatorics.
|
|
By taking conjugates, the number of partitions of into exactly k parts is equal to the number of partitions of in which the largest part has size . The function satisfies the recurrence : with initial values and if and and are not both zero.Richard Stanley, Enumerative Combinatorics, volume 1, second edition. Cambridge University Press, 2012. Chapter 1, section 1.7.
|
|
Iconclass (art), British Catalogue of Music Classification, and Dickinson classification (music), or the NLM Classification (medicine). ; National schemes: Specially created for certain countries, e.g. the Swedish library classification system, SAB (Sveriges Allmänna Biblioteksförening). In terms of functionality, classification systems are often described as: ; Enumerative: Subject headings are listed alphabetically, with numbers assigned to each heading in alphabetical order.
|
|
A faceted classification is a classification scheme used in organizing knowledge into a systematic order. A faceted classification uses semantic categories, either general or subject-specific, that are combined to create the full classification entry. Many library classification systems use a combination of a fixed, enumerative taxonomy of concepts with subordinate facets that further refine the topic.
|
|
Mair 2012:35-36) Huijiao authoritatively explains the original meaning of geyi as correlating Indian Buddhist shishu 事數 "enumerative categories (or categorized enumeration) of things/items, i.e., (technical) terms" with comparable material from Chinese sources. Shishu has two synonyms of fashu 法數 "categories of Buddhist concepts" and mingshu 名數 "numbered groups of Buddhist terms".
|
|
Susan Jane Colley (née Morris, born 1959) is an American mathematician. She is Andrew and Pauline Delaney Professor of Mathematics at Oberlin College, and editor-in-chief of the American Mathematical Monthly. Her mathematical research specialty is enumerative geometry. Colley went to the Massachusetts Institute of Technology (MIT) as an undergraduate, and earned her Ph.D. at MIT in 1983.
|
|
In mathematics, a stacky curve is an object in algebraic geometry that is roughly an algebraic curve with potentially "fractional points" called stacky points. A stacky curve is a type of stack used in studying Gromov–Witten theory, enumerative geometry, and rings of modular forms. Stacky curves are deeply related to 1-dimensional orbifolds and therefore sometimes called orbifold curves or orbicurves.
|
|
Greek historian Plutarch discusses an argument between Chrysippus (3rd century BCE) and Hipparchus (2nd century BCE) of a rather delicate enumerative problem, which was later shown to be related to Schröder–Hipparchus numbers.Stanley, Richard P.; "Hipparchus, Plutarch, Schröder, and Hough", American Mathematical Monthly 104 (1997), no. 4, 344–350. In the Ostomachion, Archimedes (3rd century BCE) considers a tiling puzzle.
|
|
Matroid theory abstracts part of geometry. It studies the properties of sets (usually, finite sets) of vectors in a vector space that do not depend on the particular coefficients in a linear dependence relation. Not only the structure but also enumerative properties belong to matroid theory. Matroid theory was introduced by Hassler Whitney and studied as a part of order theory.
|
|
More precisely, they intersect if they are connected via one or more pseudoholomorphic curves. Gromov–Witten invariants, which count these curves, appear as coefficients in expansions of the quantum cup product. Because it expresses a structure or pattern for Gromov–Witten invariants, quantum cohomology has important implications for enumerative geometry. It also connects to many ideas in mathematical physics and mirror symmetry.
|
|
In enumerative geometry, Steiner's conic problem is the problem of finding the number of smooth conics tangent to five given conics in the plane in general position. If the problem is considered in the complex projective plane CP2, the correct solution is 3264 (). The problem is named after Jakob Steiner who first posed it and who gave an incorrect solution in 1848.
|
|
In Popper's schema, enumerative induction is "a kind of optical illusion" cast by the steps of conjecture and refutation during a problem shift. An imaginative leap, the tentative solution is improvised, lacking inductive rules to guide it. The resulting, unrestricted generalization is deductive, an entailed consequence of all explanatory considerations. Controversy continued, however, with Popper's putative solution not generally accepted.
|
|
The detailed study of the Grassmannians uses a decomposition into subsets called Schubert cells, which were first applied in enumerative geometry. The Schubert cells for are defined in terms of an auxiliary flag: take subspaces , with . Then we consider the corresponding subset of , consisting of the having intersection with of dimension at least , for . The manipulation of Schubert cells is Schubert calculus.
|
|
Wendy Joanne Myrvold is a Canadian mathematician and computer scientist known for her work on graph algorithms, planarity testing, and algorithms in enumerative combinatorics. She is a professor emeritus of computer science at the University of Victoria. Myrvold completed her Ph.D. in 1988 at the University of Waterloo. Her dissertation, The Ally and Adversary Reconstruction Problems, was supervised by Charles Colbourn.
|
|
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.
|
|
J S Mill thought, unlike Comte, that scientific laws were susceptible to recall or revision. And Mill withheld from Comte's Religion of Humanity. Still, regarding experience to justify enumerative induction by having shown, indeed, the uniformity of nature,Wesley C Salmon, "The uniformity of Nature", Philosophy and Phenomenological Research, 1953 Sep;14(1):39–48, p 39. Mill commended Comte's positivism.
|
|
While enumerative methods often resort to regular expression representation of binding sites, PSFM and their formal treatment under Information Theory methods are the representation of choice for both deterministic and stochastic methods. Hybrid methods, e.g. ChIPMunk that combines greedy optimization with subsampling, also use PSFM. Recent advances in sequencing have led to the introduction of comparative genomics approaches to DNA binding motif discovery, as exemplified by PhyloGibbs.
|
|
Title page, Fowre Hymnes, by Edmund Spenser, published by William Ponsonby, London, 1596 William Ponsonby (1546? – 1604) was a prominent London publisher of the Elizabethan era. Active in the 1577-1603 period, Ponsonby published the works of Edmund Spenser, Sir Philip Sidney, and other members of the Sidney circle;Michael Brennan, "William Ponsonby: Elizabethan Stationer," Analytical and Enumerative Bibliography Vol. 7 No. 3 (1983), p. 91.
|
|
In general, the solution to the problem of Apollonius is that there are eight such circles.Yau and Nadis 2010, p. 166 The Clebsch cubic Enumerative problems in mathematics often concern a class of geometric objects called algebraic varieties which are defined by the vanishing of polynomials. For example, the Clebsch cubic (see the illustration) is defined using a certain polynomial of degree three in four variables.
|
|
In addition to its applications in enumerative geometry, mirror symmetry is a fundamental tool for doing calculations in string theory. In the A-model of topological string theory, physically interesting quantities are expressed in terms of infinitely many numbers called Gromov–Witten invariants, which are extremely difficult to compute. In the B-model, the calculations can be reduced to classical integrals and are much easier.Zaslow 2008, pp.
|
|
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).
|
|
Figure 11: An Apollonius problem with no solutions. A solution circle (pink) must cross the dashed given circle (black) to touch both of the other given circles (also black). The problem of counting the number of solutions to different types of Apollonius' problem belongs to the field of enumerative geometry. The general number of solutions for each of the ten types of Apollonius' problem is given in Table 1 above.
|
|
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.
|
|
The computing science of syntax and algorithms are used to form search results from a database. Content management systems and frequent searches can assist software engineers in optimizing more refined queries with methods of parameters and subroutines. Suggestions can be results for the current query or related queries by words, time and dates, categories and tags. The suggestion list may be reordered by other options, as enumerative, hierarchical or faceted.
|
|
In other words, every instance of a problem in the complexity class #P can be reduced to an instance of the #SAT problem. This is an important result because many difficult counting problems arise in Enumerative Combinatorics, Statistical physics, Network Reliability, and Artificial intelligence without any known formula. If a problem is shown to be hard, then it provides a complexity theoretic explanation for the lack of nice looking formulas.
|
|
Springer-Verlag, New York, 1986 Yau is considered as one of the major contributors to the development of modern differential geometry and geometric analysis. The impact of Yau's work can be seen in the mathematical and physical fields of differential geometry, partial differential equations, convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work has also touched upon applied mathematics, engineering, and numerical analysis.
|
|
Thus, the view that Popper was obviously wrong—since scientists use induction in effort to "prove" their theories true—is structured by conflicting semantics.Greenland, "Induction versus Popper", Int J Epidemiol, 1998;27(4):543–8. By now, enumerative induction has been shown to exist, but is found rarely, as in programs of machine learning in Artificial Intelligence. Likewise, machines can be programmed to operate on probabilistic inference of near certainty.
|
|
The entirety of the original problem statement is as follows: > The problem consists in this: To establish rigorously and with an exact > determination of the limits of their validity those geometrical numbers > which Schubert especially has determined on the basis of the so-called > principle of special position, or conservation of number, by means of the > enumerative calculus developed by him. Although the algebra of today > guarantees, in principle, the possibility of carrying out the processes of > elimination, yet for the proof of the theorems of enumerative geometry > decidedly more is requisite, namely, the actual carrying out of the process > of elimination in the case of equations of special form in such a way that > the degree of the final equations and the multiplicity of their solutions > may be foreseen.Hilbert, David, "Mathematische Probleme" Göttinger > Nachrichten, (1900), pp. 253-297, and in Archiv der Mathematik und Physik, > (3) 1 (1901), 44-63 and 213-237.
|
|
Instead of passing through points, a different condition on a curve is being tangent to a given line. Being tangent to five given lines also determines a conic, by projective duality, but from the algebraic point of view tangency to a line is a quadratic constraint, so naive dimension counting yields 25 = 32 conics tangent to five given lines, of which 31 must be ascribed to degenerate conics, as described in fudge factors in enumerative geometry; formalizing this intuition requires significant further development to justify. Another classic problem in enumerative geometry, of similar vintage to conics, is the Problem of Apollonius: a circle that is tangent to three circles in general determines eight circles, as each of these is a quadratic condition and 23 = 8\. As a question in real geometry, a full analysis involves many special cases, and the actual number of circles may be any number between 0 and 8, except for 7\.
|
|
In other cases, she adapted popular contemporary American songs. Bolduc often used the technique of the enumerative song, which lists something such as foods or tasks. This technique was traditional in French-Canadian folk songs, derived from similar French traditions. Bolduc also employed the traditional French folk song style of the dialogue song, usually a duet with a man, where the song is a conversation or debate between the man and the woman.
|
|
Analytic induction is a research strategy in sociology aimed at systematically developing causal explanations for types of phenomena. It was first outlined by Florian Znaniecki in 1934. He contrasted it with the kind of enumerative induction characteristic of statistical analysis. Where the latter was satisfied with probabilistic correlations, Znaniecki insisted that science is concerned with discovering causal universals, and that in social science analytic induction is the means of discovering these. Chapters 7 and 8.
|
|
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.
|
|
In 1934 he formulated the principle of analytic induction, designed to identify universal propositions and causal laws. He contrasted it with enumerative research, which provided mere correlations and could not account for exceptions in statistical relationships. He was also critical of the statistical method, which he did not see as very useful. In addition to the science of sociology, Znaniecki was also deeply interested in the larger field of the sociology of science.
|
|
Rahul Pandharipande (born 1969) is a mathematician who is currently a professor of mathematics at the Swiss Federal Institute of Technology Zürich (ETH) working in algebraic geometry. His particular interests concern moduli spaces, enumerative invariants associated to moduli spaces, such as Gromov–Witten invariants and Donaldson–Thomas invariants, and the cohomology of the moduli space of curves. His father Vijay Raghunath Pandharipande was a renowned theoretical physicist who worked in the area of nuclear physics.
|
|
By Hume's fork, the two categories never cross. Any treatises containing neither can contain only "sophistry and illusion". (Flew, Dictionary, "Hume's fork", p 156). highlighted the problem of induction,Not privy to the world's either necessities or impossibilities, but by force of habit or mental nature, humans experience sequence of sensory events, find seeming constant conjunction, make the unrestricted generalization of an enumerative induction, and justify it by presuming uniformity of nature.
|
|
163 Mathematicians became interested in mirror symmetry around 1990 when physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that mirror symmetry could be used to solve problems in enumerative geometryCandelas et al. 1991 that had resisted solution for decades or more.Yau and Nadis 2010, p. 165 These results were presented to mathematicians at a conference at the Mathematical Sciences Research Institute (MSRI) in Berkeley, California in May 1991.
|
|
Inductivism infers from observations of similar effects to similar causes, and generalizes unrestrictedly—that is, by enumerative induction—to a universal law. Extending inductivism, Comtean positivism explicitly aims to oppose metaphysics, shuns imaginative theorizing, emphasizes observation, then making predictions, confirming them, and stating laws. Logical positivism, rather, would accept hypotheticodeductivsm in theory development, but nonetheless sought an inductive logic to objectively quantity a theory's confirmation by empirical evidence and, additionally, objectively compare rival theories.
|
|
In contrast to the generative-enumerative (proof- theoretic) approach to syntax assumed by transformational grammar, arc pair grammar takes a model-theoretic approach. In arc pair grammar, linguistic laws and language-specific rules of grammar are formalized as axiomatic logical statements. Sentences of a language, understood as structures of a certain type, follow the set of linguistic laws and language-specific statements. This reduces grammaticality to the logically satisfiable notion of model-theoretic satisfaction.
|
|
William Yong-Chuan "Bill" Chen is a Chinese mathematician, notable for his work in enumerative and algebraic combinatorics, and for his role in introducing algebraic combinatorics to China. He received his doctorate from the Massachusetts Institute of Technology in 1991, under the direction of Gian-Carlo Rota. He is the director of the Center for Combinatorics at Nankai University.Bill Chen Between 1991 and 1996, Chen was a research fellow at Los Alamos National Laboratory, working with James D. Louck.
|
|
1985 In the late 1980s, it was noticed that such a Calabi-Yau manifold does not uniquely determine the physics of the theory. Instead, one finds that there are two Calabi-Yau manifolds that give rise to the same physics.Dixon 1988; Lerche, Vafa, and Warner 1989 These manifolds are said to be "mirror" to one another. This mirror duality is an important computational tool in string theory, and it has allowed mathematicians to solve difficult problems in enumerative geometry.
|
|
Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth. Works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J.J. Sylvester (late 19th century) and Percy MacMahon (early 20th century) helped lay the foundation for enumerative and algebraic combinatorics.
|
|
These methods rely on the hypothesis that a set of sequences share a binding motif for functional reasons. Binding motif discovery methods can be divided roughly into enumerative, deterministic and stochastic. MEME and Consensus are classical examples of deterministic optimization, while the Gibbs sampler is the conventional implementation of a purely stochastic method for DNA binding motif discovery. Another instance of this class of methods is SeSiMCMC that is focused of weak TFBS sites with symmetry.
|
|
Although he had only a slight knowledge of the language, Cordier made major contributions to Sinology. "Cordier," as the Bibliotheca Sinica "is sometimes affectionately referred to," is "the standard enumerative bibliography" of 70,000 works on China up to 1921. Even though the author did not know Chinese, he was thorough and highly familiar with European publications. Endymion Wilkinson also praises Cordier for including the full titles, often the tables of contents, and reviews of most books.
|
|
Otto Albin Frostman (3 January 1907 - 29 December 1977) was a Swedish mathematician, known for his work in potential theory and complex analysis. Frostman earned his Ph.D. in 1935 at Lund University under the Hungarian-born mathematician Marcel Riesz, the younger brother of F. Riesz. In potential theory, Frostman's lemma is named after him. He supervised the 1971 Stockholm University Ph.D. thesis of Bernt Lindström, which initiated the "Stockholm School" of topological combinatorics (combining simplicial homology and enumerative combinatorics).
|
|
Textual scholars produce their own editions of what they discovered. Disciplines of textual scholarship include, among others, textual criticism, stemmatology, paleography, genetic criticism, bibliography and history of the book. Textual scholar David Greetham has described textual scholarship as a term encompassing "the procedures of enumerative bibliographers, descriptive, analytical, and historical bibliographers, paleographers and codicologists, textual editors, and annotators-cumulatively and collectively". Some disciplines of textual scholarship focus on certain material sources or text genres, such as epigraphy, codicology and diplomatics.
|
|
Belkale received his Ph.D. in 1999 from the University of Chicago with thesis advisor Madhav Nori. In 2003, together with Patrick Brosnan, Belkale disproved Maxim Kontsevich's Spanning-Tree Conjecture (first published in 1997). Belkale works on enumerative algebraic geometry, quantum cohomology and moduli spaces of vector bundles on curves (conformal blocks and strange duality), and the Schubert calculus and its connections to intersection theory and representation theory. He is a professor at the University of North Carolina at Chapel Hill.
|
|
Some commonly used general-purpose facets are time, place, and form. There are few purely faceted classifications; the best known of these is the Colon Classification of S. R. Ranganathan, a general knowledge classification for libraries. Some other faceted classifications are specific to special topics, such as the Art and Architecture Thesaurus and the faceted classification of occupational safety and health topics created by D. J. Foskett for the International Labour Organization. Many library classifications combine the enumerative and faceted classification techniques.
|
|
An important simple property of Kostka numbers is that Kλμ does not depend on the order of entries of μ. For example, K(3, 2) (1, 1, 2, 1) = K(3, 2) (1, 1, 1, 2). This is not immediately obvious from the definition but can be shown by establishing a bijection between the sets of semistandard Young tableaux of shape λ and weights μ and μ', where μ and μ' differ only by swapping two entries.Stanley, Enumerative combinatorics, volume 2, p. 311.
|
|
In 1962, Good found a short proof of Dixon's identity from MMT. In 1969, Cartier and Foata found a new proof of MMT by combining algebraic and bijective ideas (built on Foata's thesis) and further applications to combinatorics on words, introducing the concept of traces. Since then, MMT has become a standard tool in enumerative combinatorics. Although various q-Dixon identities have been known for decades, except for a Krattenthaler–Schlosser extension (1999), the proper q-analog of MMT remained elusive.
|
|
The classification structure is hierarchical and the notation follows the same hierarchy. Libraries not needing the full level of detail of the classification can trim right-most decimal digits from the class number to obtain more general classifications.Chan (2007), pp. 326–331 For example: :500 Natural sciences and mathematics :: 510 Mathematics ::: 516 Geometry :::: 516.3 Analytic geometries ::::: 516.37 Metric differential geometries :::::: 516.375 Finsler geometry The classification was originally enumerative, meaning that it listed all of the classes explicitly in the schedules.
|
|
Hume thus placed his own theory of knowledge on par with Newton's theory of motion (Buckle pp 70–71, Redman pp 182–83, Schliesser § abstract). Hume found enumerative induction an unavoidable custom required for one to live (Gattei pp 28–29). Hume found constant conjunction to reveal a modest causality type: counterfactual causality. Silent as to causal role—whether necessity, sufficiency, component strength, or mechanism—counterfactual causality is simply that alteration of a factor prevents or produces the event of interest.
|
|
In general, we can not use algebraic methods to optimize the quality control procedures. Usage of enumerative methods would be very tedious, especially with multi-rule procedures, as the number of the points of the parameter space to be searched grows exponentially with the number of the parameters to be optimized. Optimization methods based on the genetic algorithms offer an appealing alternative. Furthermore, the complexity of the design process of novel quality control procedures is obviously greater than the complexity of the optimization of predefined ones.
|
|
After returning to Copenhagen, Zeuthen submitted his doctoral dissertation on a new method to determine the characteristics of conic systems in 1865. Enumerative geometry remained his focus up until 1875. In 1871 he was appointed as an extraordinary professor at the University of Copenhagen, as well as becoming an editor of Matematisk Tidsskrift, a position he held for 18 years. For 39 years he served as secretary of the Royal Danish Academy of Sciences and Letters, during which he also lectured at the Polytechnic Institute.
|
|
For the Berlekamp–van Lint–Seidel and Games graphs, see In enumerative combinatorics, there are signed subsets of a set of elements. In polyhedral combinatorics, the hypercube and all other Hanner polytopes have a number of faces (not counting the empty set as a face) that is a power of three. For example, a 2-cube, or square, has 4 vertices, 4 edges and 1 face, and . Kalai's conjecture states that this is the minimum possible number of faces for a centrally symmetric polytope.
|
|
Review articles, also called "reviews of progress," are checks on the research published in journals. Some journals are devoted entirely to review articles, some contain a few in each issue, and others do not publish review articles. Such reviews often cover the research from the preceding year, some for longer or shorter terms; some are devoted to specific topics, some to general surveys. Some reviews are enumerative, listing all significant articles in a given subject; others are selective, including only what they think worthwhile.
|
|
In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the essence of the Gromov–Witten invariants, which find application in enumerative geometry and type IIA string theory. The idea of stable map was proposed by Maxim Kontsevich around 1992 and published in . Because the construction is lengthy and difficult, it is carried out here rather than in the Gromov–Witten invariants article itself.
|
|
Procesi studies noncommutative algebra, algebraic groups, invariant theory, enumerative geometry, infinite dimensional algebras and quantum groups, polytopes, braid groups, cyclic homology, geometry of orbits of compact groups, arrangements of subspaces and tori. Procesi proved that the polynomial invariants of n \times n matrices over a field K all come from the Hamilton-Cayley theorem, which says that a square matrix satisfies its own characteristic polynomial. In 1981 he was awarded the Medal of the Accademia dei Lincei, of which he is a member since 1987.
|
|
In the 1950s, foundationalism fell into decline – largely due to the influence of Willard Van Orman Quine, whose ontological relativity found any belief networked to one's beliefs on all of reality, while auxiliary beliefs somewhere in the vast network are readily modified to protect desired beliefs. Classically, foundationalism had posited infallibility of basic beliefs and deductive reasoning between beliefs—a strong foundationalism. Around 1975, weak foundationalism emerged. Thus recent foundationalists have variously allowed fallible basic beliefs, and inductive reasoning between them, either by enumerative induction or by inference to the best explanation.
|
|
It is immediately apparent from the Leibniz quote above that there are implications for sampling. Deming observed that in any forecasting activity, the population is that of future events while the sampling frame is, inevitably, some subset of historical events. Deming held that the disjoint nature of population and sampling frame was inherently problematic once the existence of special-cause variation was admitted, rejecting the general use of probability and conventional statistics in such situations. He articulated the difficulty as the distinction between analytic and enumerative statistical studies.
|
|
In general, T-duality relates two theories with different spacetime geometries. In this way, T-duality suggests a possible scenario in which the classical notions of geometry break down in a theory of Planck scale physics.Seiberg 2006 The geometric relationships suggested by T-duality are also important in pure mathematics. Indeed, according to the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, T-duality is closely related to another duality called mirror symmetry, which has important applications in a branch of mathematics called enumerative algebraic geometry.
|
|
Plutarch's dialogue Table Talk contains the line: :Chrysippus says that the number of compound propositions that can be made from only ten simple propositions exceeds a million. (Hipparchus, to be sure, refuted this by showing that on the affirmative side there are 103,049 compound statements, and on the negative side 310,952.) This statement went unexplained until 1994, when David Hough, a graduate student at George Washington University, observed that there are 103049 ways of inserting parentheses into a sequence of ten items.Stanley, Richard P. (1997, 1999), Enumerative Combinatorics, Cambridge University Press. Exercise 1.45, vol.
|
|
English-Canadian Literary Anthologies: An Enumerative Bibliography (1997) English-Canadian Literary Anthologies is first detailed bibliography of Canadian anthologies from 1837 to the present. It lists approximately 2000 anthologies and is the departure point for any comprehensive commentary on anthology formation in Canada. Making It Real: The Canonization of English- Canadian Literature (1995) Eight wide-ranging essays are brought together in this study of the origins and development of Canadian literary canons. Lecker explores many of the myths surrounding the teaching, studying, publishing, and promotion of Canadian literature.
|
|
In the context of continuous optimization, individual learning exists in the form of local heuristics or conventional exact enumerative methods. Examples of individual learning strategies include the hill climbing, Simplex method, Newton/Quasi-Newton method, interior point methods, conjugate gradient method, line search, and other local heuristics. Note that most of the common individual learning methods are deterministic. In combinatorial optimization, on the other hand, individual learning methods commonly exist in the form of heuristics (which can be deterministic or stochastic) that are tailored to a specific problem of interest.
|
|
Similarly the number of genes per enumerative bin was found to obey a Tweedie compound Poisson–gamma distribution. This probability distribution was deemed compatible with two different biological models: the microarrangement model where the number of genes per unit genomic length was determined by the sum of a random number of smaller genomic segments derived by random breakage and reconstruction of protochormosomes. These smaller segments would be assumed to carry on average a gamma distributed number of genes. In the alternative gene cluster model, genes would be distributed randomly within the protochromosomes.
|
|
Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of several more modern theories, for example characteristic classes, and in particular its algorithmic aspects are still of current interest. The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For details see Schubert variety.
|
|
Eliminative induction, also called variative induction, is an inductive method in which a conclusion is constructed based on the variety of instances that support it. Unlike enumerative induction, eliminative induction reasons based on the various kinds of instances that support a conclusion, rather than the number of instances that support it. As the variety of instances increases, the more possible conclusions based on those instances can be identified as incompatible and eliminated. This, in turn, increases the strength of any conclusion that remains consistent with the various instances.
|
|
In this manner, there is the possibility of moving from general statements to individual instances (for example, statistical syllogisms). Note that the definition of inductive reasoning described here differs from mathematical induction, which, in fact, is a form of deductive reasoning. Mathematical induction is used to provide strict proofs of the properties of recursively defined sets. The deductive nature of mathematical induction derives from its basis in a non-finite number of cases, in contrast with the finite number of cases involved in an enumerative induction procedure like proof by exhaustion.
|
|
Kefeng Liu (Chinese: 刘克峰; born 12 December 1965), is a Chinese-American mathematician who is known for his contributions to geometric analysis, particularly the geometry, topology and analysis of moduli spaces of Riemann surfaces and Calabi-Yau manifolds. He is a professor of mathematics at University of California, Los Angeles, as well as the Executive Director of the Center of Mathematical Sciences at Zhejiang University. He is best-known for his collaboration with Bong Lian and Shing-Tung Yau in which they establish some enumerative geometry conjectures motivated by mirror symmetry.
|
|
In the design of experiments, consecutive sampling, also known as total enumerative sampling, is a sampling technique in which every subject meeting the criteria of inclusion is selected until the required sample size is achieved. Along with convenience sampling and snowball sampling, consecutive sampling is one of the most commonly used kinds of nonprobability sampling. Consecutive sampling is typically better than convenience sampling in controlling sampling bias. Care needs to be taken with consecutive sampling, however, in the case that the quantity of interest has temporal or seasonal trends.
|
|
373-426, Chapter XXI, Of The Evidence Of The Law Of Universal Causation, pp. 95-111 Also popular proof and answer to skepticism (for instance that of David Hume) is that PUC has been true in so many cases, that (using basic scientific method enumerative inductive reasoning)Andersen, Hanne and Brian Hepburn, "Scientific Method", The Stanford Encyclopedia of Philosophy (Summer 2016 Edition), Edward N. Zalta (ed.) it is reasonable to say that it is true in every case, moreover counter-example i.e. event that does not have a cause is hard to conceive.Castell, A. (1972).
|
|
Thus shielding Newtonian physics by discarding scientific realism, Kant's view limited science to tracing appearances, mere phenomena, never unveiling external reality, the noumena. Kant's transcendental idealism launched German idealism, a group of speculative metaphysics. While philosophers widely continued awkward confidence in empirical sciences as inductive, John Stuart Mill, in England, proposed five methods to discern causality, purportedly how genuine inductivism exceeds mere enumerative induction. Meanwhile, in the 1830s, opposing metaphysics, Auguste Comte, in France, explicated positivism, which, unlike Bacon's model, emphasizes predictions, confirming them, and laying scientific laws, irrefutable by theology or metaphysics.
|
|
The basic musical unit in Blackfoot music is the song, and musicians, people who sing and drum, are called singers or drummers with both words being equivalent and referring to both activities (p. 49). Women, though increasingly equal participants, are not called singers or drummers and it is considered somewhat inappropriate for women to sing loudly or alone. Páskani – "dance" or "ceremony" – often implicitly includes music and is often applied to ceremonies with little dancing and much singing. (Nettl, 1989) Blackfoot musical thought is also more enumerative than European influenced musical thought which tends to be more hierarchical.
|
|
Circles of Apollonius The problem of Apollonius is one of the earliest examples of enumerative geometry. This problem asks for the number and construction of circles that are tangent to three given circles, points or lines. In general, the problem for three given circles has eight solutions, which can be seen as 23, each tangency condition imposing a quadratic condition on the space of circles. However, for special arrangements of the given circles, the number of solutions may also be any integer from 0 (no solutions) to six; there is no arrangement for which there are seven solutions to Apollonius' problem.
|
|
Archives of Natural History provides an avenue for the publication of papers on the history and bibliography of natural history in its broadest sense, and in all periods and all cultures. This includes botany, geology, palaeontology and zoology, the lives of naturalists, their publications, correspondence and collections, and the institutions and societies to which they belong. Bibliographical papers concerned with the study of rare books, manuscripts and illustrative material, and analytical and enumerative bibliographies are also published. From time to time, the Society also publishes other works of interest, the most recent being Darwin in the Archives.
|
|
Igor Pak () (born 1971, Moscow, Soviet Union) is a professor of mathematics at the University of California, Los Angeles, working in combinatorics and discrete probability. He formerly taught at the Massachusetts Institute of Technology, and the University of Minnesota and is best known for his bijective proof of the hook-length formula for the number of Young tableaux, and his work on random walks. He was a keynote speaker alongside George Andrews and Doron Zeilberger at the 2006 Harvey Mudd College Mathematics Conference on Enumerative Combinatorics. Pak is an Associate Editor for the journal Discrete MathematicsEditorial Board, Discrete Mathematics, Elsevier.
|
|
The Irish proverbial material is almost devoid of any national blasons populaires, with the possible exception of the multi-group international comparison. These comparisons are often manifested in epigrammatic form in European languages, with the most salient and representative stereotypical trait being attributed to the nations involved (what Billig (1995) refers to as ‘banal nationalism’). Enumerative structures, usually tri- or quadripartite formulas, are the favoured apparatus. The syntactic and semantic juxtaposition of negative traits for comparative purposes is then counter-balanced by the positive representation of one nation, usually in final position, most commonly the in-group that invokes the comparison.
|
|
Safeguarding metaphysics, too, it found the mind's constants holding also universal moral truths,Whereas a hypothetical imperative is practical, simply what one ought to do if one seeks a particular outcome, the categorical imperative is morally universal, what everyone always ought to do. and launched German idealism, increasingly speculative. Auguste Comte found the problem of induction rather irrelevant since enumerative induction is grounded on the empiricism available, while science's point is not metaphysical truth. Comte found human knowledge had evolved from theological to metaphysical to scientific—the ultimate stage—rejecting both theology and metaphysics as asking questions unanswerable and posing answers unverifiable.
|
|
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory. Mirror symmetry was originally discovered by physicists. Mathematicians became interested in this relationship around 1990 when Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that it could be used as a tool in enumerative geometry, a branch of mathematics concerned with counting the number of solutions to geometric questions.
|
|
This form of inference appeals to explanatory considerations to justify belief. One infers, for example, that two students copied answers from a third because this is the best explanation of the available data—they each make the same mistakes and the two sat in view of the third. Alternatively, in a more theoretical context, one infers that there are very small unobservable particles because this is the best explanation of Brownian motion. Let us call 'liberal inductivism' any view that accepts the legitimacy of a form of inference to the best explanation that is distinct from enumerative induction.
|
|
Hume noted the formal illogicality of enumerative induction—unrestricted generalization from particular instances to all instances, and stating a universal law—since humans observe sequences of sensory events, not cause and effect. Perceiving neither logical nor natural necessity or impossibility among events, humans tacitly postulate uniformity of nature, unproved. Later philosophers would select, highlight, and nickname Humean principles—Hume's fork, the problem of induction, and Hume's law—although Hume respected and accepted the empirical sciences as inevitably inductive, after all. Immanuel Kant, in Germany, alarmed by Hume's seemingly radical empiricism, identified its apparent opposite, rationalism, in Descartes, and sought a middleground.
|
|
The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For details see Schubert variety. The intersection theory of these cells, which can be seen as the product structure in the cohomology ring of the Grassmannian of associated cohomology classes, in principle allows the prediction of the cases where intersections of cells results in a finite set of points, which are potentially concrete answers to enumerative questions. A supporting theoretical result is that the Schubert cells (or rather, their classes) span the whole cohomology ring.
|
|
The Tweedie convergence theorem describes the convergence of certain statistical processes towards a family of statistical models known as the Tweedie distributions. These distributions are characterized by a variance to mean power law, that have been variously identified in the ecological literature as Taylor's law and in the physics literature as fluctuation scaling. When this variance to mean power law is demonstrated by the method of expanding enumerative bins this implies the presence of pink noise, and vice versa. Both of these effects can be shown to be the consequence of mathematical convergence such as how certain kinds of data will converge towards the normal distribution under the central limit theorem.
|
|
Rather than validate enumerative induction—the futile task of showing it a deductive inference—Herbert Feigl as well as Hans Reichenbach, apparently independently, sought to vindicate it by showing it simply useful, either a "good" or the "best" method for the goal at hand, making predictions.Grover Maxwell, "Induction and empiricism: A Bayesian-frequentist alternative", in pp 106–65, Maxwell & Anderson, eds (U Minnesota P, 1975), pp 111–17. Feigl posed it as a rule, thus neither a priori nor a posteriori but a fortiori. Reichenbach's treatment, similar to Pascal's wager, posed it as entailing greater predictive success versus the alternative of not using it.
|
|
The Archives of Natural History (formerly the Journal of the Society for the Bibliography of Natural History) is a peer-reviewed academic journal and the official journal of the Society for the History of Natural History. It publishes papers on the history and bibliography of natural history in its broadest sense, and in all periods and all cultures. This includes botany, geology, palaeontology and zoology, the lives of naturalists, their publications, correspondence and collections, and the institutions and societies to which they belong. Bibliographical papers concerned with the study of rare books, manuscripts and illustrative material, and analytical and enumerative bibliographies are also published.
|
|
Towards the end of the century, the Italian school of algebraic geometry (Enriques, Segre, Severi) broke out of the traditional subject matter into an area demanding deeper techniques. During the later part of the 19th century, the detailed study of projective geometry became less fashionable, although the literature is voluminous. Some important work was done in enumerative geometry in particular, by Schubert, that is now considered as anticipating the theory of Chern classes, taken as representing the algebraic topology of Grassmannians. Paul Dirac studied projective geometry and used it as a basis for developing his concepts of quantum mechanics, although his published results were always in algebraic form.
|
|
St. Virgilius in Salzburg The Prebiarum provides an enumerative response to many of the questions it poses, often in the form of a triadic utterance, including triads on greed (cupiditas)Richard Newhauser, The Early History of Greed: The Sin Of Avarice in Early Medieval Thought and Literature (Cambridge University Press, 2000), p. 112 online. and martyrdom. One pair of triads is of a type circulated in other florilegia of moral extracts:For example, the Liber exhortationis of Paulinus of Aquileia; for further examples of the "Three Utterances," see Mary F. Wack and Charles D. Wright, "A New Latin Source for the Old English 'Three Utterances' Exemplum" in Anglo-Saxon England (Cambridge University Press, 1991), vol.
|
|
Bousquet-Mélou won the bronze medal of the CNRS in 1993, and the silver medal in 2014. Linköping University gave her an honorary doctorate in 2005, and the French Academy of Sciences gave her their Charles- Louis de Saulces de Freycinet Prize in 2009. In 2006, she was an invited speaker at the International Congress of Mathematicians in the section on combinatorics.. Her presentation at the congress concerned connections between enumerative combinatorics, formal language theory, and the algebraic structure of generating functions, according to which enumeration problems whose generating functions are rational functions are often isomorphic to regular languages, and problems whose generating functions are algebraic are often isomorphic to unambiguous context-free languages.
|
|
In this strict sense of the word, scientific hypotheses can rarely, if ever be proved true by the data". Likewise, Popper maintains that properly, nor do scientists try to mislead people to believe that whichever theory, law, or principle is proved either naturally real (ontic truth) or universally true (epistemic truth). There are, more actually, strong arguments on both sides. Enumerative induction obviously occurs as a summary conclusion, but its literal operation is unclear, as it may, as Popper explains, reflect deductive inference from an underlying, unstated explanation of the observations.Okasha, Philosophy of Science (Oxford U P, 2002), p 22, summarizes that geneticists "examined a large number of DS sufferers and found that each had an additional chromosome.
|
|
Geometric, Algebraic and Topological Methods for Quantum Field Theory Villa de Leyva Summer School – 2017 Xenia de la Ossa is known for her contributions to mathematical physics with much of her work focusing on string theory and its interplay with algebraic geometry. In 1991, she coauthored "A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory", which contained remarkable predictions about the number of rational curves on a quintic threefold. This was the first work to use mirror symmetry in order to make enumerative predictions in algebraic geometry, which moreover went far beyond what could be proved at the time using the available techniques within the area. This paper was cited in the more important books about String Theory.
|
|
A "Calabi-Yau manifold" refers to a compact Kähler manifold which is Ricci-flat; according to Yau's verification of the Calabi conjecture, such manifolds are known to exist. Mirror symmetry, which is a proposal of physicists beginning in the late 80s, postulates that Calabi-Yau manifolds of complex dimension 3 can be grouped into pairs which share characteristics, such as Euler and Hodge numbers. Based on this conjectural picture, the physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes proposed a formula of enumerative geometry which, given any positive integer , encodes the number of rational curves of degree in a general quintic hypersurface of four- dimensional complex projective space.Candelas, Philip; de la Ossa, Xenia C.; Green, Paul S.; Parkes, Linda.
|
|
As an undergraduate, Whitworth became the founding editor in chief of the Messenger of Mathematics, and he continued as its editor until 1880. He published works about the logarithmic spiral and about trilinear coordinates, but his most famous mathematical publication is the book Choice and Chance: An Elementary Treatise on Permutations, Combinations, and Probability (first published in 1867 and extended over several later editions). The first edition of the book treated the subject primarily from the point of view of arithmetic calculations, but had an appendix on algebra, and was based on lectures he had given at Queen's College. Later editions added material on enumerative combinatorics (the numbers of ways of arranging items into groups with various constraints), derangements, frequentist probability, life expectancy, and the fairness of bets, among other topics.
|
|
For any partition λ, the Kostka number Kλλ is equal to 1: the unique way to fill the Young diagram of shape λ = (λ1, λ2, ..., λm) with λ1 copies of 1, λ2 copies of 2, and so on, so that the resulting tableau is weakly increasing along rows and strictly increasing along columns is if all the 1s are placed in the first row, all the 2s are placed in the second row, and so on. (This tableau is sometimes called the Yamanouchi tableau of shape λ.) The Kostka number Kλμ is positive (i.e., there exist semistandard Young tableaux of shape λ and weight μ) if and only if λ and μ are both partitions of the same integer n and λ is larger than μ in dominance order.Stanley, Enumerative combinatorics, volume 2, p. 315.
|
|
An extensional definition of a concept or term formulates its meaning by specifying its extension, that is, every object that falls under the definition of the concept or term in question. For example, an extensional definition of the term "nation of the world" might be given by listing all of the nations of the world, or by giving some other means of recognizing the members of the corresponding class. An explicit listing of the extension, which is only possible for finite sets and only practical for relatively small sets, is a type of enumerative definition. Extensional definitions are used when listing examples would give more applicable information than other types of definition, and where listing the members of a set tells the questioner enough about the nature of that set.
|
|
IN: A. Slavic, A. Akdag Salah and S. Davies (Eds.): Proceedings of the International UDC Seminar 2013: Classification & Visualization: Interfaces to Knowledge, The Hague (Netherlands), 24–25 October 2013. Wurzburg: Ergon Verlag, 2013, pp. 1-41 A means of arranging the entries would be needed, and Otlet, having heard of the Dewey Decimal Classification, wrote to Melvil Dewey and obtained permission to translate it into French. The idea outgrew the plan of mere translation, and a number of radical innovations were made, adapting the purely enumerative classification (in which all the subjects envisaged are already listed and coded) into one which allows for synthesis (that is, the construction of compound numbers to denote interrelated subjects that could never be exhaustively foreseen); various possible relations between subjects were identified, and symbols assigned to represent them.
|
|
Human knowledge had evolved from religion to metaphysics to science, explained Comte, which had flowed from mathematics to astronomy to physics to chemistry to biology to sociology—in that order—describing increasingly intricate domains, all of society's knowledge having become scientific, while questions of theology and of metaphysics remained unanswerable.Will Durant, The Story of Philosophy (New York: Pocket Books, 2006), p 458 Comte found enumerative induction reliable upon the experience available, and asserted that science's proper use is improving human society, not attaining metaphysical truth. According to Comte, scientific method constrains itself to observations, but frames predictions, confirms these, rather, and states laws—positive statements—irrefutable by theology and by metaphysics, and then lays the laws as foundation for subsequent knowledge.Antony Flew, A Dictionary of Philosophy, 2nd edn (New York: St Martin's Press, 1984), "positivism", p 283.
|
|
The 13 possible strict weak orderings on a set of three elements {a, b, c} In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the number of weak orderings on a set of n elements (orderings of the elements into a sequence allowing ties, such as might arise as the outcome of a horse race).. Because of this application, de Koninck calls these numbers "horse numbers", but this name does not appear to be in widespread use. Starting from n = 0, these numbers are :1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, ... . The ordered Bell numbers may be computed via a summation formula involving binomial coefficients, or by using a recurrence relation. Along with the weak orderings, they count several other types of combinatorial objects that have a bijective correspondence to the weak orderings, such as the ordered multiplicative partitions of a squarefree number or the faces of all dimensions of a permutohedron. (e.g.
|
|
The three semistandard Young tableaux of shape λ = (3, 2) and weight μ = (1, 1, 2, 1). They are counted by the Kostka number Kλμ = 3. In mathematics, the Kostka number Kλμ (depending on two integer partitions λ and μ) is a non- negative integer that is equal to the number of semistandard Young tableaux of shape λ and weight μ. They were introduced by the mathematician Carl Kostka in his study of symmetric functions ().Stanley, Enumerative combinatorics, volume 2, p. 398. For example, if λ = (3, 2) and μ = (1, 1, 2, 1), the Kostka number Kλμ counts the number of ways to fill a left-aligned collection of boxes with 3 in the first row and 2 in the second row with 1 copy of the number 1, 1 copy of the number 2, 2 copies of the number 3 and 1 copy of the number 4 such that the entries increase along columns and do not decrease along rows.
|
|
The 1931 census enumerated nearly 20 per cent of the world's population, spread over ; G. Findlay Shirras said in 1935 that this was the largest such exercise in the world but "also the quickest and the cheapest". Scholars such as Bernard S. Cohn, have argued that the censuses of the Raj period significantly influenced the social and spatial demarcations within India that exist today. The use of enumerative mechanisms such as the census, which were intended to bolster the colonial presence, may indeed have sown the seeds that grew to be independent India, although not everybody accepts this. Peter Gottschalk has said of this cultural influence that: The first British attempts to analyse demographic data in a social context preceded the all-India censuses and were designed with the intent of ending the practice of female infanticide and sati, both of which were distasteful to the colonial authorities and both of which they thought to be most common among the Rajputs.
|
|
Hemacandra asked how many meters existed of a certain length if a long note was considered to be twice as long as a short note, which is equivalent to finding the Fibonacci numbers. hexagram The ancient Chinese book of divination I Ching describes a hexagram as a permutation with repetitions of six lines where each line can be one of two states: solid or dashed. In describing hexagrams in this fashion they determine that there are 2^6=64 possible hexagrams. A Chinese monk also may have counted the number of configurations to a game similar to Go around 700 AD. Although China had relatively few advancements in enumerative combinatorics, around 100 AD they solved the Lo Shu Square which is the combinatorial design problem of the normal magic square of order three. Magic squares remained an interest of China, and they began to generalize their original 3\times3 square between 900 and 1300 AD. China corresponded with the Middle East about this problem in the 13th century.
|
|
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.
|
|
Jakob Steiner had proposed Steiner's conic problem of enumerating the number of conic sections tangent to each of five given conics, and had answered it incorrectly. Chasles developed a theory of characteristics that enabled the correct enumeration of the conics (there are 3264) (see enumerative geometry). He established several important theorems (all called Chasles's theorem). In kinematics, Chasles's description of a Euclidean motion in space as screw displacement was seminal to the development of the theories of dynamics of rigid bodies. Chasles was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1864. In 1865 he was awarded the Copley Medal. As described in A Treasury of Deception, by Michael Farquhar (Peguin Books, 2005), between 1861 and 1869 Chasles purchased some of the 27,000 forged letters from Frenchman Denis Vrain-Lucas. Included in this trove were letters from Alexander the Great to Aristotle, from Cleopatra to Julius Caesar, and from Mary Magdalene to a revived Lazarus, all in a fake medieval French. In 2004, the journal Critical Inquiry published a recently "discovered" 1871 letter written by Vrain-Lucas (from prison) to Chasles, conveying Vrain-Lucas's perspective on these events,Ken Alder, "History's Greatest Forger: Science, Fiction, and Fraud Along the Seine," Critical Inquiry 30 (Summer 2004): 704–716.
|
|