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https://en.wikipedia.org/wiki/Bcfg2
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Bcfg2 (pronounced "bee-config") is a configuration management tool developed in the Mathematics and Computer Science Division of Argonne National Laboratory. Bcfg2 aids in the infrastructure management lifecycle – configuration analysis, service deployment, and configuration auditing. It includes tools for visualizing configuration information, as well as reporting tools that help administrators understand configuration patterns in their environments.
Bcfg2 differs from similar configuration management tools due to its auditing capability. One of the stated design goals for Bcfg2 is to determine if interactive (direct) changes have been made to a machine and report on these extra changes. The client can optionally remove any additional configuration.
Overview
Bcfg2 is written in Python and enables system administrator to manage the configuration of a large number of computers using a central configuration model. Bcfg2 operates using a simple model of system configuration, modeling intuitive items like packages, services and configuration files (as well as the dependencies between them). This model of system configuration is used for verification and validation, allowing robust auditing of deployed systems. The Bcfg2 configuration specification is written using a declarative XML model. The entire specification can be validated using widely available XML schema validators along with the custom schemas included in Bcfg2.
Built to be cross-platform, Bcfg2 works on most Unix-like operating systems.
Architecture
Bcfg2 is based on a client-server architecture. The client is responsible for interpreting (but not processing) the configuration served by the server. This configuration is literal, so no client-side processing of the configuration is required. After completion of the configuration process, the client uploads a set of statistics to the server.
The Bcfg2 Client
The Bcfg2 client performs all client configuration or reconfiguration operations. It renders a declarative configuration specification, provided by the Bcfg2 server, into a set of configuration operations which will attempt to change the client's state into that described by the configuration specification.
The operation of the Bcfg2 client is intended to be as simple as possible. Conceptually, the sole purpose of the client is to reconcile the differences between the current client state and the state described in the specification received from the Bcfg2 server.
The Bcfg2 Server
The Bcfg2 server is responsible for taking a network description and turning it into a series of configuration specifications for particular clients. It also manages probed data and tracks statistics for clients.
Server operation
The Bcfg2 server takes information from two sources when generating client configuration specifications. The first is a pool of metadata that describes clients as members of an aspect-based classing system. That is, clients are defined in terms of aspects of their abstract
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https://en.wikipedia.org/wiki/Subgroups%20of%20cyclic%20groups
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In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order n, every subgroup's order is a divisor of n, and there is exactly one subgroup for each divisor. This result has been called the fundamental theorem of cyclic groups.
Finite cyclic groups
For every finite group G of order n, the following statements are equivalent:
G is cyclic.
For every divisor d of n, G has at most one subgroup of order d.
If either (and thus both) are true, it follows that there exists exactly one subgroup of order d, for any divisor of n.
This statement is known by various names such as characterization by subgroups. (See also cyclic group for some characterization.)
There exist finite groups other than cyclic groups with the property that all proper subgroups are cyclic; the Klein group is an example. However, the Klein group has more than one subgroup of order 2, so it does not meet the conditions of the characterization.
The infinite cyclic group
The infinite cyclic group is isomorphic to the additive subgroup Z of the integers. There is one subgroup dZ for each integer d (consisting of the multiples of d), and with the exception of the trivial group (generated by d = 0) every such subgroup is itself an infinite cyclic group. Because the infinite cyclic group is a free group on one generator (and the trivial group is a free group on no generators), this result can be seen as a special case of the Nielsen–Schreier theorem that every subgroup of a free group is itself free.
The fundamental theorem for finite cyclic groups can be established from the same theorem for the infinite cyclic groups, by viewing each finite cyclic group as a quotient group of the infinite cyclic group.
Lattice of subgroups
In both the finite and the infinite case, the lattice of subgroups of a cyclic group is isomorphic to the dual of a divisibility lattice. In the finite case, the lattice of subgroups of a cyclic group of order n is isomorphic to the dual of the lattice of divisors of n, with a subgroup of order n/d for each divisor d. The subgroup of order n/d is a subgroup of the subgroup of order n/e if and only if e is a divisor of d. The lattice of subgroups of the infinite cyclic group can be described in the same way, as the dual of the divisibility lattice of all positive integers. If the infinite cyclic group is represented as the additive group on the integers, then the subgroup generated by d is a subgroup of the subgroup generated by e if and only if e is a divisor of d.
Divisibility lattices are distributive lattices, and therefore so are the lattices of subgroups of cyclic groups. This provides another alternative characterization of the finite cyclic groups: they are exactly the finite groups whose lattices of subgroups are distributive. More generally, a finitely generated group is cyclic if and only if its lattice of subgroups is distributive and an arbitrary group is locally cyclic if and only its lattice of subgr
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https://en.wikipedia.org/wiki/Levi%27s%20lemma
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In theoretical computer science and mathematics, especially in the area of combinatorics on words, the Levi lemma states that, for all strings u, v, x and y, if uv = xy, then there exists a string w such that either
uw = x and v = wy (if |u| ≤ |x|)
or
u = xw and wv = y (if |u| ≥ |x|)
That is, there is a string w that is "in the middle", and can be grouped to one side or the other. Levi's lemma is named after Friedrich Wilhelm Levi, who published it in 1944.
Applications
Levi's lemma can be applied repeatedly in order to solve word equations; in this context it is sometimes called the Nielsen transformation by analogy with the Nielsen transformation for groups. For example, starting with an equation xα = yβ where x and y are the unknowns, we can transform it (assuming |x| ≥ |y|, so there exists t such that x=yt) to ytα = yβ, thus to tα = β. This approach results in a graph of substitutions generated by repeatedly applying Levi's lemma. If each unknown appears at most twice, then a word equation is called quadratic; in a quadratic word equation the graph obtained by repeatedly applying Levi's lemma is finite, so it is decidable if a quadratic word equation has a solution. A more general method for solving word equations is Makanin's algorithm.
Generalizations
The above is known as the Levi lemma for strings; the lemma can occur in a more general form in graph theory and in monoid theory; for example, there is a more general Levi lemma for traces originally due to Christine Duboc.
Several proofs of Levi's Lemma for traces can be found in The Book of Traces.
A monoid in which Levi's lemma holds is said to have the equidivisibility property. The free monoid of strings and string concatenation has this property (by Levi's lemma for strings), but by itself equidivisibility is not enough to guarantee that a monoid is free. However an equidivisible monoid M is free if additionally there exists a homomorphism f from M to the monoid of natural numbers (free monoid on one generator) with the property that the preimage of 0 contains only the identity element of M, i.e. . (Note that f simply being a homomorphism does not guarantee this latter property, as there could be multiple elements of M mapped to 0.) A monoid for which such a homomorphism exists is also called graded (and the f is called a gradation).
See also
String operations
String functions (programming)
Notes
Combinatorics on words
Lemmas
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https://en.wikipedia.org/wiki/Trace%20theory
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In mathematics and computer science, trace theory aims to provide a concrete mathematical underpinning for the study of concurrent computation and process calculi. The underpinning is provided by an algebraic definition of the free partially commutative monoid or trace monoid, or equivalently, the history monoid, which provides a concrete algebraic foundation, analogous to the way that the free monoid provides the underpinning for formal languages.
The power of trace theory stems from the fact that the algebra of dependency graphs (such as Petri nets) is isomorphic to that of trace monoids, and thus, one can apply both algebraic formal language tools, as well as tools from graph theory.
While the trace monoid had been studied by Pierre Cartier and Dominique Foata for its combinatorics in the 1960s, trace theory was first formulated by Antoni Mazurkiewicz in the 1970s, in an attempt to evade some of the problems in the theory of concurrent computation, including the problems of interleaving and non-deterministic choice with regards to refinement in process calculi.
References
Volker Diekert, Grzegorz Rozenberg, eds. The Book of Traces, (1995) World Scientific, Singapore
Volker Diekert, Yves Metivier, "Partial Commutation and Traces", In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, Vol. 3, Beyond Words. Springer-Verlag, Berlin, 1997.
Volker Diekert, Combinatorics on traces, LNCS 454, Springer, 1990,
Concurrent computing
Formal languages
Trace theory
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https://en.wikipedia.org/wiki/Dorothea%20Rockburne
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Dorothea Rockburne DFA (born c. 1932) is an abstract painter, drawing inspiration primarily from her deep interest in mathematics and astronomy. Her work is geometric and abstract, seemingly simple but very precise to reflect the mathematical concepts she strives to concretize. "I wanted very much to see the equations I was studying, so I started making them in my studio," she has said. "I was visually solving equations." Rockburne's attraction to Mannerism has also influenced her work.
Career
In 1950 Rockburne moved to the United States to attend Black Mountain College, where she studied with mathematician Max Dehn, a lifelong influence on her work. In addition to Dehn, she studied with Franz Kline, Philip Guston, John Cage, and Merce Cunningham. She also met fellow student Robert Rauschenberg. In 1955, Rockburne moved to New York City where she met many of the leading artists and poets of the time. She was influenced by the minimalist dances of Yvonne Rainer and the Judson Dance Theater.
Throughout her career, she created paintings that expressed mathematical concepts. In 1958, a solo show of her work was critically and financially successful but deemed "not good enough" by Rockburne herself. She did not publicly show her work again for more than a decade, turning her attention to dance and performance art by 1960. Rockburne participated in performances at the Judson Dance Theater and took classes at the American Ballet Theatre. During that time she supported her daughter, Christine, by working as a waitress and a studio manager for her friend Robert Rauschenberg. Bykert Gallery, in New York, which also represented Chuck Close and Brice Marden, began showing her work in 1970. Rockburne's series of installations, Set Theories, included works such as Intersection, which attempted to merge two of her other pieces of art (Group and Disjunction) to illustrate the mathematical concept of intersection. The series later led to her experimentation with new concepts and materials, such as Gold Section and carbon paper. In 2011, a retrospective exhibition of her work was shown at the Parrish Art Museum in Water Mill, N.Y., and in 2013, the Museum of Modern Art hosted a solo show of her drawings.
Rockburne is a member of the American Academy of Arts and Letters, National Academy of Design, and The Century Association. In 2016, Rockburne earned a doctorate degree at Bowdoin College.
Awards and honors
2016 Bowdoin College, Brunswick, ME, Doctorate Degree
2009 National Academy Museum & School of Fine Arts, Lifetime Achievement Award
2003, 2007 Pollock-Krasner Foundation, Lee Krasner Award
2003 Art Omi International, Francis J. Greenberger Award
2002 Honorary Doctor of Fine Arts, College of Creative Studies, Detroit, Michigan
2002 National Academy of Design, Pike Award for Watercolor
2002 National Academy of Design, Adolph & Clara Abrig Prize for Watercolor
2002 Pollock-Krasner Foundation Grant
2001 American Academy of Arts and Letters, Depart
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https://en.wikipedia.org/wiki/Norman%20Breslow
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Norman Edward Breslow (February 21, 1941 – December 9, 2015) was an American statistician and medical researcher. At the time of his death, he was Professor (Emeritus) of Biostatistics in the School of Public Health, of the University of Washington. He is co-author or author of hundreds of published works during 1967 to 2015.
Among his many accomplishments is his work with co-author Nicholas Day that developed and popularized the use of case-control matched sample research designs, in the two-volume work Statistical Methods in Cancer Research. This was with view that matched sample studies have a role within larger program of many types of studies, in making progress on a vast and important problem like cancer. Matched sample studies can quickly and cheaply test some hypothesized relationships, but their apparent findings are not definitive, and there's much they cannot accomplish. Their results, however, can inform the design of slow and expensive longitudinal large-cohort studies that are definitive, for example. Dose-response studies and other studies, too, are elements of a rational scientific program to address cancer. In 2015, he died of prostate cancer.
Breslow was an Honorary Fellow of the Royal Statistical Society of the U.K. His other professional awards an honors include: ``the Speigelman Gold Medal Award from the American Public Health Association (1978); the Snedecor Award (1995) and R.A. Fisher Award (1995) from the Committee of Presidents of Statistical Societies; the Nathan Mantel Award (2002) from the ASA Section on Statistics in Epidemiology; the Marvin Zelen Leadership Award in Statistical Science from Harvard University (2008); and the Medal of Honor, International Agency for Research on Cancer (2005)". He was also a fellow of the American Association for the Advancement of Science at the same time as his father, Lester Breslow.
References
Norman Breslow dies at 74; biostatistician's work led to advances in medical research, Los Angeles Times, December 28, 2015. Accessed June 28, 2018
Dr. Norman Breslow, 74, dies; UW biostatistician led advances in medical research, Seattle Times, December 31, 2015. Accessed June 28, 2018
External links
Conference on Statistical Methods in Epidemiology and Observational Studies
1941 births
2015 deaths
American statisticians
American medical researchers
Fellows of the American Statistical Association
People from Minneapolis
Fellows of the American Association for the Advancement of Science
Biostatisticians
Fellows of the Royal Statistical Society
Members of the National Academy of Medicine
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https://en.wikipedia.org/wiki/Jia%20Rongqing
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Jia Rongqing () is a Canadian mathematician of Chinese origin who is a mathematics professor at the University of Alberta researching approximation theory and wavelet analysis.
Life
He was an undergraduate student at the Zhejiang University in Hangzhou, China, where he obtained his Bachelor of Science in 1968. In 1980, he went to the University of Wisconsin–Madison and undertook M.Sc and Ph.D work under the supervision of Carl-Wilhelm de Boor, receiving his Ph.D. in 1983.
He is a professor of mathematics at the University of Alberta in Edmonton, Alberta.
Selected publications
R.Q. Jia, Smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc.351 (1999) 4089-4112.
R.Q. Jia, Shift-invariant spaces and linear operator equations, Israel Math. J. 103 (1998), 259-288.
R.Q. Jia, Approximation properties of multivariate wavelets, Mathematics of Computation 67 (1998), 647-665
R.Q. Jia, Perturbation of polynomial ideals, Advances in Applied Mathematics 17 (1996), 308-336.
R.Q. Jia, The Toeplitz theorem and its applications to approximation theory and linear PDE's, Trans. Amer. Math. Soc. 347 (1995), 2585-2594.
References
External links
Rong-Qing Jia's office website at the University of Alberta
20th-century Canadian mathematicians
Academic staff of the University of Alberta
Zhejiang University alumni
University of Wisconsin–Madison alumni
Living people
Year of birth missing (living people)
21st-century Canadian mathematicians
Chinese emigrants to Canada
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https://en.wikipedia.org/wiki/Mixed%20%28United%20Kingdom%20ethnicity%20category%29
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Mixed is an ethnic group category that was first introduced by the United Kingdom's Office for National Statistics for the 2001 Census. Colloquially it refers to British citizens or residents whose parents are of two or more different races or ethnic backgrounds. The Mixed or Multiple ethnic group in England and Wales numbered 1.7 million in the 2021 census, 2.9% of the population.
Statistics
A number of academics have pointed out that the ethnicity classification employed in the census and other official statistics in the UK since 1991 involve confusion between the concepts of ethnicity and race. Aspinall notes that sustained academic attention has been focused on "how the censuses measure ethnicity, especially the use of dimensions that many claim have little to do with ethnicity, such as skin colour, race, and nationality".
2001 was the first census which asked about mixed race identity. In that census, 677,177 classified themselves as of mixed ethnicity, making up 1.2 percent of the UK population. The 2011 Census gave the figure as 2.2% for England and Wales.
Office for National Statistics estimates suggest that 956,700 mixed-ethnicity people were resident in England (as opposed to the whole of the UK) as of mid-2009, compared to 654,000 at mid-2001. As of May 2011, this figure surpassed 1 million. It was estimated in 2007 that, by 2020, 1.24 million people in the UK would be of mixed race.
Research conducted by the BBC, however, suggests that the mixed race population could already be twice the official estimate figure - up to 2 million. According to The Economist in October 2020, the 2011 census figure "is probably an undercount, since not all children of mixed marriages will have ticked one of the mixed categories, and the number is likely to have grown since the census".
3.5 percent of all births in England and Wales in 2005 were mixed-ethnicity babies, with 0.9 percent being 'Mixed White and Black Caribbean', 0.5 percent 'Mixed White and Black African', 0.8 percent 'Mixed White and Asian', and 1.3 percent any other mixed background.
Subgroups
In England and Wales, the 2001 census included four sub-categories of mixed ethnic combinations: "Mixed White and Black Caribbean", "Mixed White and Black African", "Mixed White and Asian" and "Any other Mixed background", with the latter allowing people to write in their ethnicity. Analysis of census results shows that, in England and Wales only, 237,000 people stated their ethnicity as Mixed White and Black Caribbean, 189,000 as Mixed White and Asian, 156,000 as Other Mixed, and 79,000 Mixed White and Black African.
The estimates for mid-2009 for England only suggest that there are 301,300 people in the Mixed White and Black Caribbean category, 127,500 Mixed White and Black African, 292,400 Mixed White and Asian, and 235,500 Other Mixed. The White and Black African group grew fastest in percentage terms from 2001 to 2009, followed by White and Asian, Other Mixed and then White and Black Car
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https://en.wikipedia.org/wiki/Face%20diagonal
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In geometry, a face diagonal of a polyhedron is a diagonal on one of the faces, in contrast to a space diagonal passing through the interior of the polyhedron.
A cuboid has twelve face diagonals (two on each of the six faces), and it has four space diagonals. The cuboid's face diagonals can have up to three different lengths, since the faces come in congruent pairs and the two diagonals on any face are equal. The cuboid's space diagonals all have the same length. If the edge lengths of a cuboid are a, b, and c, then the distinct rectangular faces have edges (a, b), (a, c), and (b, c); so the respective face diagonals have lengths and
Thus each face diagonal of a cube with side length a is .
A regular dodecahedron has 60 face diagonals (and 100 space diagonals).
References
Elementary geometry
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https://en.wikipedia.org/wiki/History%20monoid
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In mathematics and computer science, a history monoid is a way of representing the histories of concurrently running computer processes as a collection of strings, each string representing the individual history of a process. The history monoid provides a set of synchronization primitives (such as locks, mutexes or thread joins) for providing rendezvous points between a set of independently executing processes or threads.
History monoids occur in the theory of concurrent computation, and provide a low-level mathematical foundation for process calculi, such as CSP the language of communicating sequential processes, or CCS, the calculus of communicating systems. History monoids were first presented by M.W. Shields.
History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of dependency graphs. As such, they are free objects and are universal. The history monoid is a type of semi-abelian categorical product in the category of monoids.
Product monoids and projection
Let
denote an n-tuple of (not necessarily pairwise disjoint) alphabets . Let denote all possible combinations of one finite-length string from each alphabet:
(In more formal language, is the Cartesian product of the free monoids of the . The superscript star is the Kleene star.) Composition in the product monoid is component-wise, so that, for
and
then
for all in . Define the union alphabet to be
(The union here is the set union, not the disjoint union.) Given any string , we can pick out just the letters in some using the corresponding string projection . A distribution is the mapping that operates on with all of the , separating it into components in each free monoid:
Histories
For every , the tuple is called the elementary history of a. It serves as an indicator function for the inclusion of a letter a in an alphabet . That is,
where
Here, denotes the empty string. The history monoid is the submonoid of the product monoid generated by the elementary histories: (where the superscript star is the Kleene star applied with a component-wise definition of composition as given above). The elements of are called global histories, and the projections of a global history are called individual histories.
Connection to computer science
The use of the word history in this context, and the connection to concurrent computing, can be understood as follows. An individual history is a record of the sequence of states of a process (or thread or machine); the alphabet is the set of states of the process.
A letter that occurs in two or more alphabets serves as a synchronization primitive between the various individual histories. That is, if such a letter occurs in one individual history, it must also occur in another history, and serves to "tie" or "rendezvous" them together.
Consider, for example, and . The union alphabet is of course . The elementary histories are , , , and . In this example, an individual history of the first
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https://en.wikipedia.org/wiki/Springer%20correspondence
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In mathematics, the Springer representations are certain representations of the Weyl group W associated to unipotent conjugacy classes of a semisimple algebraic group G. There is another parameter involved, a representation of a certain finite group A(u) canonically determined by the unipotent conjugacy class. To each pair (u, φ) consisting of a unipotent element u of G and an irreducible representation φ of A(u), one can associate either an irreducible representation of the Weyl group, or 0. The association
depends only on the conjugacy class of u and generates a correspondence between the irreducible representations of the Weyl group and the pairs (u, φ) modulo conjugation, called the Springer correspondence. It is known that every irreducible representation of W occurs exactly once in the correspondence, although φ may be a non-trivial representation. The Springer correspondence has been described explicitly in all cases by Lusztig, Spaltenstein and Shoji. The correspondence, along with its generalizations due to Lusztig, plays a key role in Lusztig's classification of the irreducible representations of finite groups of Lie type.
Construction
Several approaches to Springer correspondence have been developed. T. A. Springer's original construction proceeded by defining an action of W on the top-dimensional l-adic cohomology groups of the algebraic variety Bu of the Borel subgroups of G containing a given unipotent element u of a semisimple algebraic group G over a finite field. This construction was generalized by Lusztig, who also eliminated some technical assumptions. Springer later gave a different construction, using the ordinary cohomology with rational coefficients and complex algebraic groups.
Kazhdan and Lusztig found a topological construction of Springer representations using the Steinberg variety and, allegedly, discovered Kazhdan–Lusztig polynomials in the process. Generalized Springer correspondence has been studied by Lusztig and Spaltenstein and by Lusztig in his work on character sheaves. Borho and MacPherson gave yet another construction of the Springer correspondence.
Example
For the special linear group SLn, the unipotent conjugacy classes are parametrized by partitions of n: if u is a unipotent element, the corresponding partition is given by the sizes of the Jordan blocks of u. All groups A(u) are trivial.
The Weyl group W is the symmetric group Sn on n letters. Its irreducible representations over a field of characteristic zero are also parametrized by the partitions of n.
The Springer correspondence in this case is a bijection, and in the standard parametrizations, it is given by transposition of the partitions (so that the trivial representation of the Weyl group corresponds to the regular unipotent class, and the sign representation corresponds to the identity element of G).
Applications
Springer correspondence turned out to be closely related to the classification of primitive ideals in the universal e
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https://en.wikipedia.org/wiki/T.%20A.%20Springer
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Tonny Albert Springer (13 February 1926 – 7 December 2011) was a mathematician at Utrecht University who worked on linear algebraic groups, Hecke algebras, complex reflection groups, and who introduced Springer representations and the Springer resolution.
Springer began his undergraduate studies in 1945 at Leiden University and remained there for his graduate work in mathematics, earning his PhD in 1951 under Hendrik Kloosterman with thesis Over symplectische Transformaties. As a postdoc Springer spent the academic year 1951/1952 at the University of Nancy and then returned to Leiden University, where he was employed until 1955. In 1955 he accepted a lectureship at Utrecht University, where he became professor ordinarius in 1959 and continued in that position until 1991 when he retired as professor emeritus. Springer's visiting professorships included many institutions: the University of Göttingen (1963), the Institute for Advanced Study (1961/1962, 1969, 1983), IHES (1964, 1973, 1975, 1983), Tata Institute of Fundamental Research (1968, 1980), UCLA (1965/1966), the Australian National University, the University of Sydney, the University of Rome Tor Vergata, the University of Basel, the Erwin Schrödinger Institute in Vienna, and the University of Paris VI.
In 1964 Springer was elected to the Royal Netherlands Academy of Arts and Sciences. In 2006 in Madrid he was an invited speaker at the International Congress of Mathematicians with lecture on Some results on compactifications of semisimple groups. (At the 1962 ICM in Stockholm he made a short contribution Twisted composition algebras, but was not an invited speaker.)
Publications
Reprint of the 1973 edition.
;
References
Profile
Springer's home page.
1926 births
2011 deaths
Dutch mathematicians
Institute for Advanced Study visiting scholars
Leiden University alumni
Members of the Royal Netherlands Academy of Arts and Sciences
Academic staff of Utrecht University
Scientists from The Hague
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https://en.wikipedia.org/wiki/Historia%20Mathematica
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Historia Mathematica: International Journal of History of Mathematics is an academic journal on the history of mathematics published by Elsevier. It was established by Kenneth O. May in 1971 as the free newsletter Notae de Historia Mathematica, but by its sixth issue in 1974 had turned into a full journal.
The International Commission on the History of Mathematics began awarding the Montucla Prize, for the best article by an early career scholar in Historia Mathematica, in 2009. The award is given every four years.
Editors
The editors of the journal have been:
Kenneth O. May, 1974–1977
Joseph W. Dauben, 1977–1985
Eberhard Knobloch, 1985–1994
David E. Rowe, 1994–1996
Karen Hunger Parshall, 1996–2000
Craig Fraser and Umberto Bottazzini, 2000–2004
Craig Fraser, 2004–2007
Benno van Dalen, 2007–2009
June Barrow-Green and Niccolò Guicciardini, 2010–2013
Niccolò Guicciardini and Tom Archibald, 2013-2015
Tom Archibald and Reinhard Siegmund-Schultze, 2016–present
Abstracting and indexing
The journal is abstracted and indexed in Mathematical Reviews, SCISEARCH, and Scopus.
References
A Brief History of the International Commission on the History of Mathematics (ICHM)
External links
Historia Mathematica at the International Commission on the History of Mathematics
Elsevier academic journals
Quarterly journals
Academic journals established in 1974
English-language journals
History of mathematics journals
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https://en.wikipedia.org/wiki/New%20York%20Number%20Theory%20Seminar
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The New York Number Theory Seminar is a research seminar devoted to the theory of numbers and related parts of mathematics and physics.
The seminar began in 1982 under the founding organizers Harvey Cohn, David and Gregory Chudnovsky, and Melvyn B. Nathanson. It is held at the Graduate Center, CUNY.
Overview
The New York Number Theory Seminar began in January 1982 and was originally organized by number theorists Harvey Cohn, David and Gregory Chudnovsky ,and Melvyn B. Nathanson. Since the retirement of Cohn, Nathanson is the sole organizer. The seminar also organizes an annual Workshop on Combinatorial and Additive Number Theory (CANT) at the Graduate Center, CUNY.
Publications
Four volumes of the collected lecture notes of the seminar were published in the Lecture Notes in Mathematics series by Springer-Verlag. These volumes covered the seminar from 1982 to 1988. Three additional stand-alone books were published by Springer-Verlag under the title Number Theory, covering the seminar between 1989 and 2003.
External links
References
Mathematics education in the United States
City University of New York
Number theory
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https://en.wikipedia.org/wiki/West%20Mallee
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West Mallee is a statistical subdivision defined under the Australian Standard Geographical Classification, and therefore used by the Australian Bureau of Statistics. It is one of three subdivisions of the Mallee statistical division of the Australian state of Victoria. It consists of three statistical local areas: Buloke (S) - North, Buloke (S) - South and Mildura (RC) - Pt B.
References
External links
Demographics of Australia
Geography of Victoria (state)
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https://en.wikipedia.org/wiki/Statistical%20study%20of%20energy%20data
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Energy statistics refers to collecting, compiling, analyzing and disseminating data on commodities such as coal, crude oil, natural gas, electricity, or renewable energy sources (biomass, geothermal, wind or solar energy), when they are used for the energy they contain. Energy is the capability of some substances, resulting from their physico-chemical properties, to do work or produce heat. Some energy commodities, called fuels, release their energy content as heat when they burn. This heat could be used to run an internal or external combustion engine.
The need to have statistics on energy commodities became obvious during the 1973 oil crisis that brought tenfold increase in petroleum prices. Before the crisis, to have accurate data on global energy supply and demand was not deemed critical. Another concern of energy statistics today is a huge gap in energy use between developed and developing countries. As the gap narrows (see picture), the pressure on energy supply increases tremendously.
The data on energy and electricity come from three principal sources:
Energy industry
Other industries ("self-producers")
Consumers
The flows of and trade in energy commodities are measured both in physical units (e.g., metric tons), and, when energy balances are calculated, in energy units (e.g., terajoules or tons of oil equivalent). What makes energy statistics specific and different from other fields of economic statistics is the fact that energy commodities undergo greater number of transformations (flows) than other commodities. In these transformations energy is conserved, as defined by and within the limitations of the first and second laws of thermodynamics.
See also
Energy system
World energy resources and consumption
External links
Statistical Energy Database Review: Enerdata Yearbook 2012
International Energy Agency: Statistics
United Nations: Energy Statistics
The Oslo Group on Energy Statistics
DOE Energy Information Administration
Year of Energy 2009
European Energy Statistics & Key Indicators
Publications
Energy Statistics Yearbook 2004, United Nations, 2006
Energy Balances and Electricity Profiles 2004, United Nations, 2006
Statistical data sets
Energy measurement
Applied statistics
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https://en.wikipedia.org/wiki/Karnaugh%20map
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The Karnaugh map (KM or K-map) is a method of simplifying Boolean algebra expressions. Maurice Karnaugh introduced it in 1953 as a refinement of Edward W. Veitch's 1952 Veitch chart, which was a rediscovery of Allan Marquand's 1881 logical diagram aka Marquand diagram but with a focus now set on its utility for switching circuits. Veitch charts are also known as Marquand–Veitch diagrams or, rarely, as Svoboda charts, and Karnaugh maps as Karnaugh–Veitch maps (KV maps).
The Karnaugh map reduces the need for extensive calculations by taking advantage of humans' pattern-recognition capability. It also permits the rapid identification and elimination of potential race conditions.
The required Boolean results are transferred from a truth table onto a two-dimensional grid where, in Karnaugh maps, the cells are ordered in Gray code, and each cell position represents one combination of input conditions. Cells are also known as minterms, while each cell value represents the corresponding output value of the boolean function. Optimal groups of 1s or 0s are identified, which represent the terms of a canonical form of the logic in the original truth table. These terms can be used to write a minimal Boolean expression representing the required logic.
Karnaugh maps are used to simplify real-world logic requirements so that they can be implemented using a minimum number of logic gates. A sum-of-products expression (SOP) can always be implemented using AND gates feeding into an OR gate, and a product-of-sums expression (POS) leads to OR gates feeding an AND gate. The POS expression gives a complement of the function (if F is the function so its complement will be F'). Karnaugh maps can also be used to simplify logic expressions in software design. Boolean conditions, as used for example in conditional statements, can get very complicated, which makes the code difficult to read and to maintain. Once minimised, canonical sum-of-products and product-of-sums expressions can be implemented directly using AND and OR logic operators.
Example
Karnaugh maps are used to facilitate the simplification of Boolean algebra functions. For example, consider the Boolean function described by the following truth table.
Following are two different notations describing the same function in unsimplified Boolean algebra, using the Boolean variables , , , and their inverses.
where are the minterms to map (i.e., rows that have output 1 in the truth table).
where are the maxterms to map (i.e., rows that have output 0 in the truth table).
Construction
In the example above, the four input variables can be combined in 16 different ways, so the truth table has 16 rows, and the Karnaugh map has 16 positions. The Karnaugh map is therefore arranged in a 4 × 4 grid.
The row and column indices (shown across the top and down the left side of the Karnaugh map) are ordered in Gray code rather than binary numerical order. Gray code ensures that only one variable changes between each pa
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https://en.wikipedia.org/wiki/Gosset%E2%80%93Elte%20figures
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In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams.
The Coxeter symbol for these figures has the form ki,j, where each letter represents a length of order-3 branches on a Coxeter–Dynkin diagram with a single ring on the end node of a k length sequence of branches. The vertex figure of ki,j is (k − 1)i,j, and each of its facets are represented by subtracting one from one of the nonzero subscripts, i.e. ki − 1,j and ki,j − 1.
Rectified simplices are included in the list as limiting cases with k=0. Similarly 0i,j,k represents a bifurcated graph with a central node ringed.
History
Coxeter named these figures as ki,j (or kij) in shorthand and gave credit of their discovery to Gosset and Elte:
Thorold Gosset first published a list of regular and semi-regular figures in space of n dimensions in 1900, enumerating polytopes with one or more types of regular polytope faces. This included the rectified 5-cell 021 in 4-space, demipenteract 121 in 5-space, 221 in 6-space, 321 in 7-space, 421 in 8-space, and 521 infinite tessellation in 8-space.
E. L. Elte independently enumerated a different semiregular list in his 1912 book, The Semiregular Polytopes of the Hyperspaces. He called them semiregular polytopes of the first kind, limiting his search to one or two types of regular or semiregular k-faces.
Elte's enumeration included all the kij polytopes except for the 142 which has 3 types of 6-faces.
The set of figures extend into honeycombs of (2,2,2), (3,3,1), and (5,4,1) families in 6,7,8 dimensional Euclidean spaces respectively. Gosset's list included the 521 honeycomb as the only semiregular one in his definition.
Definition
The polytopes and honeycombs in this family can be seen within ADE classification.
A finite polytope kij exists if
or equal for Euclidean honeycombs, and less for hyperbolic honeycombs.
The Coxeter group [3i,j,k] can generate up to 3 unique uniform Gosset–Elte figures with Coxeter–Dynkin diagrams with one end node ringed. By Coxeter's notation, each figure is represented by kij to mean the end-node on the k-length sequence is ringed.
The simplex family can be seen as a limiting case with k=0, and all rectified (single-ring) Coxeter–Dynkin diagrams.
A-family [3n] (rectified simplices)
The family of n-simplices contain Gosset–Elte figures of the form 0ij as all rectified forms of the n-simplex (i + j = n − 1).
They are listed below, along with their Coxeter–Dynkin diagram, with each dimensional family drawn as a graphic orthogonal projection in the plane of the Petrie polygon of the regular simplex.
D-family [3n−3,1,1] demihypercube
Each Dn group has two Gosset–Elte figures, the n-demihypercube as 1k1, and an alternated form of the n-orthoplex, k11, constructed wit
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https://en.wikipedia.org/wiki/Amasya%20Merzifon%20Airport
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Merzifon Airport or Amasya Merzifon Airport is a military airport located in the city of Merzifon in the Amasya Province of Turkey.
Airlines and destinations
Statistics
Military usage
Merzifon is the 5th Air Wing (Ana Jet Üs or AJÜ) of the 2nd Air Force Command (Hava Kuvvet Komutanligi) of the Turkish Air Force (Türk Hava Kuvvetleri). Other wings of this command are located in Malatya/Erhaç (LTAT), Diyarbakır (LTCC) and İncirlik (LTAG).
References
External links
Airports in Turkey
Turkish Air Force bases
Buildings and structures in Amasya Province
Transport in Amasya Province
Merzifon District
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https://en.wikipedia.org/wiki/Markovian%20arrival%20process
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In queueing theory, a discipline within the mathematical theory of probability, a Markovian arrival process (MAP or MArP) is a mathematical model for the time between job arrivals to a system. The simplest such process is a Poisson process where the time between each arrival is exponentially distributed.
The processes were first suggested by Marcel F. Neuts in 1979.
Definition
A Markov arrival process is defined by two matrices, D0 and D1 where elements of D0 represent hidden transitions and elements of D1 observable transitions. The block matrix Q below is a transition rate matrix for a continuous-time Markov chain.
The simplest example is a Poisson process where D0 = −λ and D1 = λ where there is only one possible transition, it is observable, and occurs at rate λ. For Q to be a valid transition rate matrix, the following restrictions apply to the Di
Special cases
Phase-type renewal process
The phase-type renewal process is a Markov arrival process with phase-type distributed sojourn between arrivals. For example, if an arrival process has an interarrival time distribution PH with an exit vector denoted , the arrival process has generator matrix,
Generalizations
Batch Markov arrival process
The batch Markovian arrival process (BMAP) is a generalisation of the Markovian arrival process by allowing more than one arrival at a time. The homogeneous case has rate matrix,
An arrival of size occurs every time a transition occurs in the sub-matrix . Sub-matrices have elements of , the rate of a Poisson process, such that,
and
Markov-modulated Poisson process
The Markov-modulated Poisson process or MMPP where m Poisson processes are switched between by an underlying continuous-time Markov chain. If each of the m Poisson processes has rate λi and the modulating continuous-time Markov has m × m transition rate matrix R, then the MAP representation is
Fitting
A MAP can be fitted using an expectation–maximization algorithm.
Software
KPC-toolbox a library of MATLAB scripts to fit a MAP to data.
See also
Rational arrival process
References
Queueing theory
Markov processes
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https://en.wikipedia.org/wiki/Janko%20%28disambiguation%29
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Janko is a surname and given name.
Janko may also refer to:
Janko group, in mathematics
Janko group J1
Janko group J2
Janko group J3
Janko group J4
Jankó keyboard
Jankwa (or Janko), a Newar ritual
See also
Janko Kráľ Park, a park in Bratislava's Petržalka borough
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https://en.wikipedia.org/wiki/Dynkin%20index
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In mathematics, the Dynkin index of a finite-dimensional highest-weight representation of a compact simple Lie algebra with highest weight is defined by
where is the 'defining representation', which is most often taken to be the fundamental representation if the Lie algebra under consideration is a matrix Lie algebra.
The notation is the trace form on the representation . By Schur's lemma, since the trace forms are all invariant forms, they are related by constants, so the index is well-defined.
Since the trace forms are bilinear forms, we can take traces to obtain
where the Weyl vector
is equal to half of the sum of all the positive roots of . The expression is the value of the quadratic Casimir in the representation . The index is always a positive integer.
In the particular case where is the highest root, so that is the adjoint representation, the Dynkin index is equal to the dual Coxeter number.
See also
Killing form
References
Philippe Di Francesco, Pierre Mathieu, David Sénéchal, Conformal Field Theory, 1997 Springer-Verlag New York,
Representation theory of Lie algebras
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https://en.wikipedia.org/wiki/Periodic%20continued%20fraction
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In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form
where the initial block of k + 1 partial denominators is followed by a block [ak+1, ak+2,...ak+m] of partial denominators that repeats ad infinitum. For example, can be expanded to a periodic continued fraction, namely as [1,2,2,2,...].
The partial denominators {ai} can in general be any real or complex numbers. That general case is treated in the article convergence problem. The remainder of this article is devoted to the subject of simple continued fractions that are also periodic. In other words, the remainder of this article assumes that all the partial denominators ai (i ≥ 1) are positive integers.
Purely periodic and periodic fractions
Since all the partial numerators in a regular continued fraction are equal to unity we can adopt a shorthand notation in which the continued fraction shown above is written as
where, in the second line, a vinculum marks the repeating block. Some textbooks use the notation
where the repeating block is indicated by dots over its first and last terms.
If the initial non-repeating block is not present – that is, if k = -1, a0 = am and
the regular continued fraction x is said to be purely periodic. For example, the regular continued fraction for the golden ratio φ – given by [1; 1, 1, 1, ...] – is purely periodic, while the regular continued fraction for the square root of two – [1; 2, 2, 2, ...] – is periodic, but not purely periodic.
As unimodular matrices
Such periodic fractions are in one-to-one correspondence with the real quadratic irrationals. The correspondence is explicitly provided by Minkowski's question-mark function. That article also reviews tools that make it easy to work with such continued fractions. Consider first the purely periodic part
This can, in fact, be written as
with the being integers, and satisfying Explicit values can be obtained by writing
which is termed a "shift", so that
and similarly a reflection, given by
so that . Both of these matrices are unimodular, arbitrary products remain unimodular. Then, given as above, the corresponding matrix is of the form
and one has
as the explicit form. As all of the matrix entries are integers, this matrix belongs to the modular group
Relation to quadratic irrationals
A quadratic irrational number is an irrational real root of the quadratic equation
where the coefficients a, b, and c are integers, and the discriminant, b2 − 4ac, is greater than zero. By the quadratic formula every quadratic irrational can be written in the form
where P, D, and Q are integers, D > 0 is not a perfect square (but not necessarily square-free), and Q divides the quantity P2 − D (for example (6+)/4). Such a quadratic irrational may also be written in another form with a square-root of a square-free number (for example (3+)/2) as explained for quadratic irrationals.
By considering the complete quotients of periodic continued fractio
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https://en.wikipedia.org/wiki/Lie%27s%20third%20theorem
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In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra over the real numbers is associated to a Lie group . The theorem is part of the Lie group–Lie algebra correspondence.
Historically, the third theorem referred to a different but related result. The two preceding theorems of Sophus Lie, restated in modern language, relate to the infinitesimal transformations of a group action on a smooth manifold. The third theorem on the list stated the Jacobi identity for the infinitesimal transformations of a local Lie group. Conversely, in the presence of a Lie algebra of vector fields, integration gives a local Lie group action. The result now known as the third theorem provides an intrinsic and global converse to the original theorem.
Historical notes
The equivalence between the category of simply connected real Lie groups and finite-dimensional real Lie algebras is usually called (in the literature of the second half of 20th century) Cartan's or the Cartan-Lie theorem as it was proved by Élie Cartan. Sophus Lie had previously proved the infinitesimal version: local solvability of the Maurer-Cartan equation, or the equivalence between the category of finite-dimensional Lie algebras and the category of local Lie groups.
Lie listed his results as three direct and three converse theorems. The infinitesimal variant of Cartan's theorem was essentially Lie's third converse theorem. In an influential book Jean-Pierre Serre called it the third theorem of Lie. The name is historically somewhat misleading, but often used in connection to generalizations.
Serre provided two proofs in his book: one based on Ado's theorem and another recounting the proof by Élie Cartan.
Proofs
There are several proofs of Lie's third theorem, each of them employing different algebraic and/or geometric techniques.
Algebraic proof
The classical proof is straightforward but relies on Ado's theorem, whose proof is algebraic and highly non-trivial. Ado's theorem states that any finite-dimensional Lie algebra can be represented by matrices. As a consequence, integrating such algebra of matrices via the matrix exponential yields a Lie group integrating the original Lie algebra.
Cohomological proof
A more geometric proof is due to Élie Cartan and was published by . This proof uses induction on the dimension of the center and it involves the Chevalley-Eilenberg complex.
Geometric proof
A different geometric proof was discovered in 2000 by Duistermaat and Kolk. Unlike the previous ones, it is a constructive proof: the integrating Lie group is built as the quotient of the (infinite-dimensional) Banach Lie group of paths on the Lie algebra by a suitable subgroup. This proof was influential for Lie theory since it paved the way to the generalisation of Lie third theorem for Lie groupoids and Lie algebroids.
See also
Lie group integrator
References
External links
Encyclopaedia of Mathematics (EoM) article
Lie algebras
Lie groups
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https://en.wikipedia.org/wiki/Martha%20Siegel
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Martha Jochnowitz Siegel is an American applied mathematician, probability theorist and mathematics educator who served as the editor of Mathematics Magazine from 1991 to 1996. In 2017 she won the Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service of the Mathematical Association of America for "her remarkable leadership in guiding the national conversation on undergraduate mathematics curriculum". She was a faculty member in the mathematics department of Towson University from 1971 until 2015, when she became a professor emerita.
Education and career
Siegel grew up in Brooklyn, the daughter of civil engineer Nat Jochnowitz. She became interested in mathematics through her father's interest in mathematical puzzles, and through the calculation of baseball statistics for the Brooklyn Dodgers. She did her undergraduate studies in mathematics at Russell Sage College, a small women's college in Troy, New York, while also taking classes at the nearby men-only Rensselaer Polytechnic Institute, as at that time Russell Sage had no mathematics department. At Russell Sage, she was a Kellas honor student, and president of the science club. She completed her Ph.D. in 1969 at the University of Rochester; her dissertation, On Birth and Death Processes, was supervised by Johannes Kemperman. During graduate school and until her 1971 move to Towson, she was on the faculty at Goucher College.
Contributions
At Towson, in 1981, Siegel founded an innovative and still-ongoing undergraduate applied mathematics program involving projects connected to local business and government. She is a co-author of the discrete mathematics and precalculus textbooks Finite Mathematics and Its Applications and Functioning in the Real World. She also served as chair of a committee of the Mathematical Association of America charged with producing the 2015 edition of their MAA Curriculum Guide to Undergraduate Majors in the Mathematical Sciences.
References
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
Probability theorists
Mathematics educators
Russell Sage College alumni
University of Rochester alumni
Goucher College faculty and staff
Towson University faculty
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women
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https://en.wikipedia.org/wiki/Littlemoss%20High%20School
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Littlemoss High School for Boys was a comprehensive school in Littlemoss, Droylsden, Tameside, England. It merged with Droylsden High School, Mathematics and Computing College for Girls in September 2009 to become Droylsden Academy.
Prior to the merger it educated about 550 boys and held specialist Business and Enterprise College status.
Academic standards
The school was under special measures from March 1998 until July 2002.
After their March 2004 inspection Ofsted reported that "This is a good, effective school that has been very successful in improving the areas of weakness identified in the previous report. Very good leadership, together with the significant improvements in teaching and learning and in pupils’ behaviour, has improved standards overall, although standards are still below average." They rated the school Good, point three on a seven-point scale.
Awards and mentions
Schools Achievement Award and Sportsmark in 2002.
Mentioned in Parliament as a member of the Peacemaker Consultation Programme.
Notable teachers
Sarah Joynes was awarded a distinction in the North West Guardian Award for Teacher of the Year in a Secondary School in 2007.
Woodwork teacher Harry "Bulldog" Johnson achieved national fame in 1979 when he won the jackpot on Littlewoods Pools, receiving a prize of over £750,000. He planned to retire at the end of the school term but died of a heart attack just a few weeks after his win.
In July 2008, Science teacher Chris Hilton disappeared on holiday in the French Alps. His body was discovered 7 months later in February 2009.
Former German & French teacher Geoff Rees is now part of the rock group the Badgers
Notable alumni
Howard Donald of Take That attended Littlemoss High School between 1979 and 1984.
Carl Murphy who received a Royal Humane Society Testimonial in December 2004 for saving a man's life at sea.
Dale Cregan the One eyed Manchester Cop Killer attended in the 90's
Notes
The photograph showing of the school prior to demolition is not Littlemoss School.
The photograph is of Manor Road Girls School.
Please update as this misinformation is now being shared on social media post in error.
References
External links
Official site
Defunct schools in Tameside
2009 disestablishments in England
Educational institutions disestablished in 2009
Droylsden
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https://en.wikipedia.org/wiki/Jean-Fran%C3%A7ois%20Jodar
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Jean-François Jodar (born 2 December 1949) is a retired French footballer and manager.
External links
Profile, statistics and pictures
Profile, statistics and pictures
1949 births
Living people
People from Montereau-Fault-Yonne
French men's footballers
France men's international footballers
Stade de Reims players
RC Strasbourg Alsace players
Olympique Lyonnais players
French football managers
Hassania Agadir managers
FC Montceau Bourgogne players
2008 Africa Cup of Nations managers
Men's association football defenders
Footballers from Seine-et-Marne
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https://en.wikipedia.org/wiki/Normal-gamma%20distribution
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In probability theory and statistics, the normal-gamma distribution (or Gaussian-gamma distribution) is a bivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and precision.
Definition
For a pair of random variables, (X,T), suppose that the conditional distribution of X given T is given by
meaning that the conditional distribution is a normal distribution with mean and precision — equivalently, with variance
Suppose also that the marginal distribution of T is given by
where this means that T has a gamma distribution. Here λ, α and β are parameters of the joint distribution.
Then (X,T) has a normal-gamma distribution, and this is denoted by
Properties
Probability density function
The joint probability density function of (X,T) is
Marginal distributions
By construction, the marginal distribution of is a gamma distribution, and the conditional distribution of given is a Gaussian distribution. The marginal distribution of is a three-parameter non-standardized Student's t-distribution with parameters .
Exponential family
The normal-gamma distribution is a four-parameter exponential family with natural parameters and natural statistics .
Moments of the natural statistics
The following moments can be easily computed using the moment generating function of the sufficient statistic:
where is the digamma function,
Scaling
If then for any is distributed as
Posterior distribution of the parameters
Assume that x is distributed according to a normal distribution with unknown mean and precision .
and that the prior distribution on and , , has a normal-gamma distribution
for which the density satisfies
Suppose
i.e. the components of are conditionally independent given and the conditional distribution of each of them given is normal with expected value and variance The posterior distribution of and given this dataset can be analytically determined by Bayes' theorem explicitly,
where is the likelihood of the parameters given the data.
Since the data are i.i.d, the likelihood of the entire dataset is equal to the product of the likelihoods of the individual data samples:
This expression can be simplified as follows:
where , the mean of the data samples, and , the sample variance.
The posterior distribution of the parameters is proportional to the prior times the likelihood.
The final exponential term is simplified by completing the square.
On inserting this back into the expression above,
This final expression is in exactly the same form as a Normal-Gamma distribution, i.e.,
Interpretation of parameters
The interpretation of parameters in terms of pseudo-observations is as follows:
The new mean takes a weighted average of the old pseudo-mean and the observed mean, weighted by the number of associated (pseudo-)observations.
The precision was estimated from pseudo-observations (i.e. possibly a different number of pseu
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https://en.wikipedia.org/wiki/Timothy%20A.%20Cross
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Timothy A. Cross is an American academic chemist who specializes in nuclear magnetic resonance (NMR) spectroscopy, membrane and computational biophysics, and biomathematics. He is a professor of chemistry at Florida State University and the Director of the NMR Program at the National High Magnetic Field Laboratory. His research focuses on the sets of proteins that are important for the pharmaceutical industry in the treatment of diseases such as the flu (Influenza A) and tuberculosis.
External links
National High Magnetic Field Laboratory faculty profile
FSU Faculty profile on Timothy Cross' research
Florida State University faculty
21st-century American chemists
Living people
Computational chemists
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/East%20Mallee
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East Mallee is a statistical subdivision defined under the Australian Standard Geographical Classification, and therefore used by the Australian Bureau of Statistics. It is one of three subdivisions of the Mallee statistical division of the Australian state of Victoria. It consists of four statistical local areas: Gannawarra (S), Swan Hill (RC) - Central, Swan Hill (RC) - Robinvale and Swan Hill (RC) Bal.
References
External links
Demographics of Australia
Geography of Victoria (state)
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https://en.wikipedia.org/wiki/Pooled%20variance
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In statistics, pooled variance (also known as combined variance, composite variance, or overall variance, and written ) is a method for estimating variance of several different populations when the mean of each population may be different, but one may assume that the variance of each population is the same. The numerical estimate resulting from the use of this method is also called the pooled variance.
Under the assumption of equal population variances, the pooled sample variance provides a higher precision estimate of variance than the individual sample variances. This higher precision can lead to increased statistical power when used in statistical tests that compare the populations, such as the t-test.
The square root of a pooled variance estimator is known as a pooled standard deviation (also known as combined standard deviation, composite standard deviation, or overall standard deviation).
Motivation
In statistics, many times, data are collected for a dependent variable, y, over a range of values for the independent variable, x. For example, the observation of fuel consumption might be studied as a function of engine speed while the engine load is held constant. If, in order to achieve a small variance in y, numerous repeated tests are required at each value of x, the expense of testing may become prohibitive. Reasonable estimates of variance can be determined by using the principle of pooled variance after repeating each test at a particular x only a few times.
Definition and computation
The pooled variance is an estimate of the fixed common variance underlying various populations that have different means.
We are given a set of sample variances , where the populations are indexed ,
=
Assuming uniform sample sizes, , then the pooled variance can be computed by the arithmetic mean:
If the sample sizes are non-uniform, then the pooled variance can be computed by the weighted average, using as weights the respective degrees of freedom (see also: Bessel's correction):
Variants
The unbiased least squares estimate of (as presented above),
and the biased maximum likelihood estimate below:
are used in different contexts. The former can give an unbiased to estimate when the two groups share an equal population variance. The latter one can give a more efficient to estimate , although subject to bias. Note that the quantities in the right hand sides of both equations are the unbiased estimates.
Example
Consider the following set of data for y obtained at various levels of the independent variable x.
The number of trials, mean, variance and standard deviation are presented in the next table.
These statistics represent the variance and standard deviation for each subset of data at the various levels of x. If we can assume that the same phenomena are generating random error at every level of x, the above data can be “pooled” to express a single estimate of variance and standard deviation. In a sense, this suggests finding a
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https://en.wikipedia.org/wiki/Vitali%20Volkov
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Vitali Vladimirovich Volkov (, born 22 March 1981) is a former Russian footballer.
Club career
He finished as top scorer in the UEFA Intertoto Cup 2007.
Career statistics
Notes
References
External links
Tom' Tomsk profile
Living people
Footballers from Moscow
1981 births
Russian men's footballers
Russia men's youth international footballers
Russia men's under-21 international footballers
FC Rubin Kazan players
FC Torpedo Moscow players
FC Tom Tomsk players
Russian Premier League players
Kazakhstan Premier League players
FC Volga Nizhny Novgorod players
Russian expatriate men's footballers
Expatriate men's footballers in Kazakhstan
FC Tobol players
FC Okzhetpes players
Men's association football midfielders
FC Aktobe players
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https://en.wikipedia.org/wiki/Max%20Gunzburger
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Max D. Gunzburger, Francis Eppes Distinguished Professor of Mathematics at Florida State University, is an American mathematician and computational scientist affiliated with the Florida State interdisciplinary Department of Scientific Computing. He was the 2008 winner of the SIAM W.T. and Idalia Reid Prize in Mathematics. His seminal research contributions include flow control, finite element analysis, superconductivity and Voronoi tessellations. He has also made contributions in the areas of aerodynamics, materials, acoustics, climate change, groundwater, image processing and risk assessment.
Ph.D.
After completing his BS degree at New York University in 1966, Gunzburger earned his Ph.D. degree from the same University in 1969. His thesis, titled Diffraction of shock waves by a thin wing—Symmetric and anti-symmetric problems, was written under the direction of Lu Ting.
Early career
Gunzburger began his career at New York University as a research scientist and assistant professor of mathematics, a position he held from receiving his Ph.D. until 1971. He then spent two years working as a post-doctorate at the Naval Ordnance Laboratory before transferring to the Institute for Computer Applications in Science and Engineering at NASA until 1976. He then became an associate professor and professor of mathematics at the University of Tennessee, a position he held from 1976 to 1982.
Transferring again, he moved from Carnegie Mellon University in 1981 to 1989, to Virginia Polytechnic Institute and State University from 1987 to 1997. In 1989 he completed his influential first book, "Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice and Algorithms," , which according to Google scholar has over 400 citations as of March 2009 .
Distinguished professorship
In 1995, Gunzburger took a position with Iowa State University as professor and chair of mathematics. In 2001, he was awarded distinguished professor of mathematics. At Iowa State he wrote three other books.
Gunzburger came to the Florida State University in 2002. As an Eppes professor, Gunzburger is among the university's most distinguished scholars. He currently serves as the chair of the Department of Scientific Computing.
SIAM
Gunzburger served as editor-in-chief of the SIAM Journal on Numerical Analysis from 2000 to 2007. He also served as the chairman of the board of trustees in 2003 and has held various other SIAM positions
Awards and honors
In 2007, issues 3–4 in volume 4 of the "International Journal of Numerical Analysis and Modeling" were dedicated to Gunzburger to honor the occasion of his 60th birthday. In 2008, Gunzburger was awarded the W.T. and Idalia Reid Prize in Mathematics, an award given for "research in, or other contributions to, the broadly defined areas of differential equations and control theory." The award was given based on "fundamental contributions to control of distributed parameter systems and computational mathema
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https://en.wikipedia.org/wiki/Isotropic%20quadratic%20form
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In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More explicitly, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if . A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form.
Suppose that is quadratic space and W is a subspace of V. Then W is called an isotropic subspace of V if some vector in it is isotropic, a totally isotropic subspace if all vectors in it are isotropic, and an anisotropic subspace if it does not contain any (non-zero) isotropic vectors. The of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces.
A quadratic form q on a finite-dimensional real vector space V is anisotropic if and only if q is a definite form:
either q is positive definite, i.e. for all non-zero v in V ;
or q is negative definite, i.e. for all non-zero v in V.
More generally, if the quadratic form is non-degenerate and has the signature , then its isotropy index is the minimum of a and b. An important example of an isotropic form over the reals occurs in pseudo-Euclidean space.
Hyperbolic plane
Let F be a field of characteristic not 2 and . If we consider the general element of V, then the quadratic forms and are equivalent since there is a linear transformation on V that makes q look like r, and vice versa. Evidently, and are isotropic. This example is called the hyperbolic plane in the theory of quadratic forms. A common instance has F = real numbers in which case and are hyperbolas. In particular, is the unit hyperbola. The notation has been used by Milnor and Husemoller for the hyperbolic plane as the signs of the terms of the bivariate polynomial r are exhibited.
The affine hyperbolic plane was described by Emil Artin as a quadratic space with basis satisfying , where the products represent the quadratic form.
Through the polarization identity the quadratic form is related to a symmetric bilinear form .
Two vectors u and v are orthogonal when . In the case of the hyperbolic plane, such u and v are hyperbolic-orthogonal.
Split quadratic space
A space with quadratic form is split (or metabolic) if there is a subspace which is equal to its own orthogonal complement; equivalently, the index of isotropy is equal to half the dimension. The hyperbolic plane is an example, and over a field of characteristic not equal to 2, every split space is a direct sum of hyperbolic planes.
Relation with classification of quadratic forms
From the point of view of classification of quadratic forms, anisotropic spaces are the basic building blocks for quadratic spaces of arbitrary dimensions. For a general field F, classification of anisotropic quadratic forms is a nontrivial problem. By contrast, the isotropic forms are usually much easier to handle.
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https://en.wikipedia.org/wiki/Myles%20Hollander
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Myles Hollander (born March 21, 1941) is an American academic statistician who has made research contributions to nonparametric methods, biostatistics, and reliability. He was born in Brooklyn, New York. He is Emeritus and Robert O. Lawton Distinguished Professor of Statistics at Florida State University. He is a Fellow of the American Statistical Association, the Institute of Mathematical Statistics, and the International Statistical Institute.
References
External links
Florida State University faculty profile
Florida State University faculty
Fellows of the American Statistical Association
Living people
1941 births
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https://en.wikipedia.org/wiki/Applied%20academics
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Applied academics is an approach to learning and teaching that focuses on how academic subjects (communications, mathematics, science, and basic literacy) can apply to the real world. Further, applied academics can be viewed as theoretical knowledge supporting practical applications.
Definition
Applied Academics is an approach to learning which focuses on motivating and challenging students to connect what they learn with the world they experience and with what interests them. The basic premise is that if academic content is made more relevant, participatory and concrete, students learn better, retain more and apply learning in their lives. Teaching in this model uses hands-on innovative teaching methods sometimes called contextual learning. Teachers help students understand the reasons for studying their subject matter and capitalize on students' natural learning inclinations and problem-solving approaches they can use well beyond the classroom throughout their lives.
Applied Academics is an attempt to break from disconnected learning (where students go to different classes for different subjects for specified periods of time and don't gain a sense of the interconnectedness of learning) that has become a part of traditional approaches to education. This approach attempts to reintegrate learning by doing such things (for example) as teaching math, science, writing, or speech within other contexts such as a learning experience dealing with some form of technology training.
Research in cognitive science
Research in cognitive science has shown that learners are not passive receptacles into which knowledge may be "poured". Learning occurs when the learner constructs, invents, and solves problems. Studies have shown that students taught theoretical principles, processes and skills in isolation without practice do not transfer these skills and knowledge as well to real life situations. Although "learning to know," "learning to do," and their "application" are often separated, there is no effective learning or understanding of one kind without the other two.
Relating to the SCANS Competencies
In the United States the Secretary's Commission on Achieving Necessary Skills (SCANS) states that "good jobs require people who can put knowledge to work", and listed the knowledge (academics) that it considered critical for all students to possess.
These competencies include:
Knowing how to effectively allocate time, money, materials, space, and staff
The ability to work on a team, teach, serve customers, lead, negotiate and work with people from culturally diverse backgrounds
Understanding how to effectively acquire, evaluate, and use data
Understanding and ability to design and improve social, organizational, and technological systems
The ability to effectively use technology
Relating to action research
The concept of action research is related to Applied Academics, as one tenet of action research is to discover theory through practice, and then
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https://en.wikipedia.org/wiki/Croatian%20Bureau%20of%20Statistics
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The Croatian Bureau of Statistics ( or DZS) is the Croatian national statistics bureau.
History
The bureau was formed in 1875 in Austria-Hungary as the Zemaljski statistički ured for the Kingdom of Croatia, Slavonia and Dalmatia.
In 1924, the bureau was renamed to the Statistical Office in Zagreb (Statistički ured u Zagrebu). In 1929, after royal monarchy was proclaimed in the Kingdom of Serbs, Croats and Slovenes the bureau lost its financial and technical independence.
In 1939 with the formation of the Banovina of Croatia, the office was made subject to the presidential office on the Ban's administration. In 1941 the Independent State of Croatia was formed and an Office of General State Statistics existed during this time under the control of the presidential government.
In 1945 the Statistical Office of the People's Republic of Croatia was formed. In 1951 it was renamed to the Bureau of Statistics and Evidence, in 1956 to the Bureau of Statistics of the People's Republic of Croatia and in 1963 to the Republican Bureau for Statistics of the Socialist Republic of Croatia.
The bureau was independent during this time, but was subordinated to the Yugoslavian Federal Bureau for Statistics.
Upon Croatian independence, the Central Bureau of Statistics was made the highest statistical body in the nation.
Work
The bureau collects and processes data for the Republic of Croatia. Among other things, the bureau conducts the Croatian census.
The Bureau keeps records on Croatian censuses since 1857, including the recent:
1991 Croatian census
2001 Croatian census
2011 Croatian census
References
External links
Organizations established in 1875
Government agencies of Croatia
Croatia
Donji grad, Zagreb
Organizations based in Zagreb
1875 establishments in Austria-Hungary
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https://en.wikipedia.org/wiki/James%20H.%20Davenport
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James Harold Davenport (born 26 September 1953) is a British computer scientist who works in computer algebra. Having done his PhD and early research at the Computer Laboratory, University of Cambridge, he is the Hebron and Medlock Professor of Information Technology at the University of Bath in Bath, England.
Education
Davenport was educated at Marlborough College, and was then a student at Trinity College, Cambridge. He was awarded a Bachelor of Arts degree in 1974 which was converted to a Master of Arts degree in 1978, and a Master of Mathematics in 2011. He was awarded a PhD in 1980.
Career and research
In 1969, the team that developed the automated teller machine in the United Kingdom at IBM Hursley used parts from that project to build an IBM School Computer. It was a community outreach project, and it went on tour. When it came to Marlborough College, Davenport, aged 16, discovered that, although it was ostensibly a six-digit computer, the microcode had access to a 12-digit internal register to do multiply/divide. He used this to implement Draim's algorithm from his father's book, The Higher Arithmetic, and tested eight-digit numbers for primality.
Between school and university, Davenport worked in a government laboratory for nine months, again writing and using multiword arithmetic, but also using number theory to solve a problem in hashing, which was published. He was at IBM Yorktown Heights for a year, and returned to Cambridge as a Research Fellow. He went to Grenoble for a year, before taking a post at the University of Bath in 1983.
Davenport is an author of a textbook about computer algebra and of many papers. He has been Project Chair of the European OpenMath Project and its successor Thematic Network, with responsibilities for aligning OpenMath and MathML, producing Content Dictionaries and supervised a Reduce-based OpenMath/MathML translator, and was Treasurer of the European Mathematical Trust. He was Founding Editor-in-Chief of the London Mathematical Society's Journal of Computation and Mathematics.
Awards and honours
Davenport was awarded the Honorary Degree of Doctor of Science in September 2019 by the West University of Timişoara, Romania. This was in recognition of his pioneering and ongoing work in computer algebra systems and theory of symbolic computation.
In 2014, Davenport was awarded a National Teaching Fellowship by the Higher Education Academy.
He was awarded the Bronze Medal of the University of Helsinki in 2001.
From January to June 2017 Davenport was a Fulbright CyberSecurity Scholar at New York University, and maintained a blog over the same period.
Personal life
Davenport is the son of the mathematician Harold Davenport.
References
British computer scientists
Alumni of Trinity College, Cambridge
Academics of the University of Bath
Living people
1953 births
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https://en.wikipedia.org/wiki/Ricardinho%20%28footballer%2C%20born%201975%29
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Ricardo Souza Silva or simply Ricardinho (born November 26, 1975, in São Paulo), is a Brazilian attacking midfielder.
Club statistics
Honours
São Paulo
Campeonato Paulista: 2000
Paulista
Copa do Brasil: 2005
Vitória
Campeonato Baiano: 2008
Sport
Campeonato Pernambucano: 2010
External links
CBF
1975 births
Living people
Brazilian men's footballers
Footballers from São Paulo
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
Campeonato Brasileiro Série C players
J1 League players
J2 League players
Men's association football midfielders
Nacional Atlético Clube (SP) players
Nagoya Grampus players
Shonan Bellmare players
São Paulo FC players
Kawasaki Frontale players
Coritiba Foot Ball Club players
Joinville Esporte Clube players
C.D. Nacional players
Fortaleza Esporte Clube players
Portimonense S.C. players
Paulista Futebol Clube players
Sport Club Internacional players
Sociedade Esportiva Palmeiras players
Botafogo de Futebol e Regatas players
Esporte Clube Vitória players
Guaratinguetá Futebol players
Avaí FC players
Vila Nova Futebol Clube players
Sport Club do Recife players
Grêmio Esportivo Brasil players
Bangu Atlético Clube players
Independente Futebol Clube players
Esporte Clube Água Santa players
Nacional Atlético Clube Sociedade Civil Ltda. players
Associação Atlética Portuguesa (Santos) players
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https://en.wikipedia.org/wiki/FETI
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In mathematics, in particular numerical analysis, the FETI method (finite element tearing and interconnect) is an iterative substructuring method for solving systems of linear equations from the finite element method for the solution of elliptic partial differential equations, in particular in computational mechanics In each iteration, FETI requires the solution of a Neumann problem in each substructure and the solution of a coarse problem. The simplest version of FETI with no preconditioner (or only a diagonal preconditioner) in the substructure is scalable with the number of substructures but the condition number grows polynomially with the number of elements per substructure. FETI with a (more expensive) preconditioner consisting of the solution of a Dirichlet problem in each substructure is scalable with the number of substructures and its condition number grows only polylogarithmically with the number of elements per substructure. The coarse space in FETI consists of the nullspace on each substructure.
Apart from FETI Dual-Primal (FETI-DP, see below), several extensions have been developed to solve particular physical problems, as FETI Helmholtz (FETI-H), FETI for quasi-incompressible problems, and FETI Contact (FETI-C).
See also
Balancing domain decomposition
FETI-DP
References
External links
Google Scholar search
Domain decomposition methods
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https://en.wikipedia.org/wiki/T%C3%BAlio%20%28footballer%2C%20born%201976%29
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Túlio Lustosa Seixas Pinheiro or simply Túlio (born April 25, 1976 in Brasília, Brazil), is a retired Brazilian defensive midfielder.
Club statistics
Honours
Goiás State League: 1996, 1997, 1998, 1999, 2000, 2003
Brazilian Center-West Cup: 2000
Rio de Janeiro's Cup: 2007
Contract
1 January 2008 to 31 December 2010
External links
sambafoot.com
CBF
1976 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Goiás Esporte Clube players
Botafogo de Futebol e Regatas players
Sport Club Corinthians Paulista players
Figueirense FC players
Grêmio Foot-Ball Porto Alegrense players
Oita Trinita players
Al Hilal SFC players
Expatriate men's footballers in Japan
Expatriate men's footballers in Saudi Arabia
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
J1 League players
Men's association football midfielders
Footballers from Brasília
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https://en.wikipedia.org/wiki/GRV
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GRV may refer to:
Gaussian random variable, a variable in normal distribution in statistics
Gawler Range Volcanics, a geological event in South Australia
Gravesend railway station, England
Greenville station (South Carolina), a train station
Greyhound Racing Victoria
Gross rock volume, a calculation used in hydrocarbon exploration
Groundnut rosette virus, a plant pathogen
Grozny Airport, in Chechnya, Russia
GRV, the ISO code of the Central Grebo language
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https://en.wikipedia.org/wiki/University%20of%20Minnesota%20Talented%20Youth%20Mathematics%20Program
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The University of Minnesota Talented Youth Mathematics Program (UMTYMP) is an alternative secondary mathematics education program operated by the University of Minnesota's School of Mathematics Center for Educational Programs (MathCEP). Classes are offered in St. Cloud, Rochester, Duluth, and Minneapolis, Minnesota. The Program is supported by the Minnesota state legislature. The course structure, intensity, and workload are comparable to college-level classes in rigor.
Program
UMTYMP offers a total of five years of math coursework. The program is divided into a high school component and a calculus component.
High school component
The UMTYMP high school component lasts two years and covers standard high school mathematics. The classes are taught by local secondary school teachers.
First year: Algebra I and II
Second year: Geometry and Math Analysis
Students receive high school credit for these courses. To pass each semester, a grade of at least 70% in the class must be achieved, and a grade of 70%-75% results in probation and potential ejection from the course.
Calculus component
The UMTYMP calculus component lasts three years and covers honors-level college calculus. The classes are taught by university faculty members.
Calculus 1: single-variable calculus
Calculus 2: differential equations, proof methods, set theory and linear algebra
Calculus 3: multivariable calculus
Students receive both high school and University of Minnesota credit (4 credits per year) for these courses.
Post-UMTYMP
UMTYMP graduates can take 4000- or 5000-level math classes at the University of Minnesota either as a undergraduates (if enrolled there for college) or as PSEO students (if still in high school). UMTYMP also offers an optional semester-long course known as Advanced Topics.
Enrollment
Interested students in 5th to 7th grade are eligible to take a qualifying exam in April. Students who pass the qualifying exam are invited to enroll in UMTYMP's Algebra I/II class in the fall or may defer their enrollment for a year.
Non-UMTYMP students in 7th to 10th grade may take an entrance exam to begin the calculus component.
The enrollment fee for the 2006 - 2007 school year was around $700 – $800.
Demographics
In the 2005-2006 school year, 506 students from 134 public and 28 private schools were enrolled in UMTYMP. Of them, 421 attended classes in Minneapolis, 56 in Rochester, and 29 in St. Cloud. 361 (71%) were male and 145 (29%) were female. Among those who reported their ethnicity, 277 (62%) were White, 142 (32%) were Asian/Pacific Islanders, 10 (2%) were Chicano/Latino, 3 (0.7%) were Alaskan/American Indian, 2 (0.4%) were African-American, and 14 (3%) described themselves as other.
Subreddit
On September 17, 2012, the UMTYMP subreddit (reddit.com/r/UMTYMP) was created. UMTYMP participants were slow to adopt the forum as a means of communication, favoring Moodle as the official forum. Occasional posts were made in 2017 and 2018, but the subreddit n
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https://en.wikipedia.org/wiki/Orientation%20entanglement
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In mathematics and physics, the notion of orientation entanglement is sometimes used to develop intuition relating to the geometry of spinors or alternatively as a concrete realization of the failure of the special orthogonal groups to be simply connected.
Elementary description
Spatial vectors alone are not sufficient to describe fully the properties of rotations in space.
Consider the following example. A coffee cup is suspended in a room by a pair of elastic rubber bands fixed to the walls of the room. The cup is rotated by its handle through a full twist of 360°, so that the handle is brought all the way around the central vertical axis of the cup and back to its original position.
Note that after this rotation, the cup has been returned to its original orientation, but that its orientation with respect to the walls is twisted. In other words, if we lower the coffee cup to the floor of the room, the two bands will coil around each other in one full twist of a double helix. This is an example of orientation entanglement: the new orientation of the coffee cup embedded in the room is not actually the same as the old orientation, as evidenced by the twisting of the rubber bands. Stated another way, the orientation of the coffee cup has become entangled with the orientation of the surrounding walls.
Clearly the geometry of spatial vectors alone is insufficient to express the orientation entanglement (the twist of the rubber bands). Consider drawing a vector across the cup. A full rotation will move the vector around so that the new orientation of the vector is the same as the old one. The vector alone doesn't know that the coffee cup is entangled with the walls of the room.
In fact, the coffee cup is inextricably entangled. There is no way to untwist the bands without rotating the cup. However, consider what happens instead when the cup is rotated, not through just one 360° turn, but two 360° turns for a total rotation of 720°. Then if the cup is lowered to the floor, the two rubber bands coil around each other in two full twists of a double helix. If the cup is now brought up through the center of one coil of this helix, and passed onto its other side, the twist disappears. The bands are no longer coiled about each other, even though no additional rotation had to be performed. (This experiment is more easily performed with a ribbon or belt. See below.)
Thus, whereas the orientation of the cup was twisted with respect to the walls after a rotation of only 360°, it was no longer twisted after a rotation of 720°. By only considering the vector attached to the cup, it is impossible to distinguish between these two cases, however. It is only when we attach a spinor to the cup that we can distinguish between the twisted and untwisted case.
In this situation, a spinor is a sort of polarized vector. In the adjacent diagram, a spinor can be represented as a vector whose head is a flag lying on one side of a Möbius strip, pointing i
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https://en.wikipedia.org/wiki/Schwarz%20integral%20formula
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In complex analysis, a branch of mathematics, the Schwarz integral formula, named after Hermann Schwarz, allows one to recover a holomorphic function, up to an imaginary constant, from the boundary values of its real part.
Unit disc
Let f be a function holomorphic on the closed unit disc {z ∈ C | |z| ≤ 1}. Then
for all |z| < 1.
Upper half-plane
Let f be a function holomorphic on the closed upper half-plane {z ∈ C | Im(z) ≥ 0} such that, for some α > 0, |zα f(z)| is bounded on the closed upper half-plane. Then
for all Im(z) > 0.
Note that, as compared to the version on the unit disc, this formula does not have an arbitrary constant added to the integral; this is because the additional decay condition makes the conditions for this formula more stringent.
Corollary of Poisson integral formula
The formula follows from Poisson integral formula applied to u:
This is equivalent to
By means of conformal maps, the formula can be generalized to any simply connected open set.
Notes and references
Ahlfors, Lars V. (1979), Complex Analysis, Third Edition, McGraw-Hill,
Remmert, Reinhold (1990), Theory of Complex Functions, Second Edition, Springer,
Saff, E. B., and A. D. Snider (1993), Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, Second Edition, Prentice Hall,
Theorems in complex analysis
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https://en.wikipedia.org/wiki/Hedgeley%20Dene%20Gardens
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"features": [
{
"type": "Feature",
"properties": {},
"geometry": {
"type": "Point",
"coordinates": [
145.053638,
-37.868735
]
}
}
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}
Hedgeley Dene Gardens is a public open space in the suburb of Malvern East in Melbourne, Australia. It is one of the most popular parks in the Malvern East locality. It is also significant as an example of public open space design that recreates the qualities of an informal, picturesque English garden or northern European landscape in an Australian suburb. It forms part of a network of linear open spaces in Melbourne's eastern suburbs in the local government areas of the Stonnington and Boroondara, formed along drainage easements and watercourses such as Gardiners Creek. However, Hedgeley Dene Gardens is unique due to its particular landscape character.
Bibliography
Cooper, JB (1935) The History of Malvern: From its First Settlement to a City, Melbourne: The Speciality Press
Strahan, L (1989) Public and Private Memory: A History of the City of Malvern, North Melbourne: Hargreen
References
Parks in Melbourne
Gardens in Victoria (state)
City of Stonnington
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https://en.wikipedia.org/wiki/Orthonormal%20function%20system
|
An orthonormal function system (ONS) is an orthonormal basis in a vector space of functions.
References
Linear algebra
Functional analysis
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https://en.wikipedia.org/wiki/Pseudo-Euclidean%20space
|
In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real -space together with a non-degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving
which is called the scalar square of the vector .
For Euclidean spaces, , implying that the quadratic form is positive-definite. When , is an isotropic quadratic form, otherwise it is anisotropic. Note that if , then , so that is a null vector. In a pseudo-Euclidean space with , unlike in a Euclidean space, there exist vectors with negative scalar square.
As with the term Euclidean space, the term pseudo-Euclidean space may be used to refer to an affine space or a vector space depending on the author, with the latter alternatively being referred to as a pseudo-Euclidean vector space (see point–vector distinction).
Geometry
The geometry of a pseudo-Euclidean space is consistent despite some properties of Euclidean space not applying, most notably that it is not a metric space as explained below. The affine structure is unchanged, and thus also the concepts line, plane and, generally, of an affine subspace (flat), as well as line segments.
Positive, zero, and negative scalar squares
A null vector is a vector for which the quadratic form is zero. Unlike in a Euclidean space, such a vector can be non-zero, in which case it is self-orthogonal.
If the quadratic form is indefinite, a pseudo-Euclidean space has a linear cone of null vectors given by . When the pseudo-Euclidean space provides a model for spacetime (see below), the null cone is called the light cone of the origin.
The null cone separates two open sets, respectively for which and . If , then the set of vectors for which is connected. If , then it consists of two disjoint parts, one with and another with . Similarly, if , then the set of vectors for which is connected. If , then it consists of two disjoint parts, one with and another with .
Interval
The quadratic form corresponds to the square of a vector in the Euclidean case. To define the vector norm (and distance) in an invariant manner, one has to get square roots of scalar squares, which leads to possibly imaginary distances; see square root of negative numbers. But even for a triangle with positive scalar squares of all three sides (whose square roots are real and positive), the triangle inequality does not hold in general.
Hence terms norm and distance are avoided in pseudo-Euclidean geometry, which may be replaced with scalar square and interval respectively.
Though, for a curve whose tangent vectors all have scalar squares of the same sign, the arc length is defined. It has important applications: see proper time, for example.
Rotations and spheres
The rotations group of such space is the indefinite orthogonal group , also denoted as without a reference to particular quadratic form. Such "rotations" preserve the form and, hence, the scalar square of each vector including
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https://en.wikipedia.org/wiki/LeMond%20Racing%20Cycles
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LeMond Racing Cycles is a bicycle company founded by Greg LeMond, the only American winner of the Tour de France.
LeMond initially offered bicycle frames with a geometry based on the racing bikes he used in competition, with a longer top tube and wheelbase in an otherwise traditional lightweight steel frame. This was to stretch out the rider on the bicycle, with the intent of both lowering the frontal area presented to the wind, and optimizing power and stability.
From 1995 until February 2010 LeMond did not manufacture bicycles, instead licensing the brand name to Trek. Trek believed that the cachet of the name and models offering a longer top tube than Trek's frame geometries would increase sales . In September 2013, LeMond partnered with Time to produce a limited run of 300 frames to commemorate his three Tour victories in 1986, 1989, and 1990. In August 2014, Greg LeMond launched the Washoe, a Reynolds 853 steel bike manufactured in the United States.
Greg LeMond
Greg LeMond was a pioneer in the use of carbon fiber bicycle frames in European professional road cycling, and his Tour de France win in 1986 ahead of Bernard Hinault was the first for carbon. LeMond rode a "Bernard Hinault" Signature Model Look prototype that year. LeMond also won the 1989 Tour and World's, and his final Tour de France in 1990 on carbon fiber frames, which had begun to feature "Greg LeMond" branding.
Company founding
In 1986, LeMond founded LeMond Bicycles to develop machines for himself that would also be marketed and sold to the public. In 1990, searching for an equipment edge for Team Z at the 1991 Tour de France, LeMond concluded an exclusive licensing agreement between his company and Carbonframes, Inc., to access the latter's advanced composites technology. While LeMond briefly led the 1991 Tour while riding his Carbonframes-produced "Greg LeMond" bicycle, the company faltered, something LeMond blamed on "undercapitalization" and poor management by his father, although Carbonframes and LeMond Cycles "parted amiably two years later." In 1995, LeMond reached a licensing agreement with Trek, according to which the Wisconsin-based company would manufacture and distribute bicycles designed with LeMond that would be sold under the "LeMond Bicycles" brand. LeMond would later claim that going into business with Trek "destroyed" his relationship with his father.
In 2001 the Trek deal proved painful for LeMond as he was forced by John Burke, the head of Trek, to apologize for the negative comments about Michele Ferrari, doping, and Lance Armstrong, who was by then a very important marketing force for Trek. LeMond's contract with Trek had a clause prohibiting LeMond from doing anything that would damage Trek. Burke reminded LeMond of this commitment, and strongly argued that LeMond publicly retract his statements. LeMond read a formal apology to Armstrong.
In March 2008 LeMond Cycling Inc filed a complaint against Trek for breach of contract, claiming that they
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https://en.wikipedia.org/wiki/G%26R
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G&R may refer to:
Gradshteyn and Ryzhik, aka Table of Integrals, Series, and Products, a classical book in mathematics
Guns N' Roses, an American rock band
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https://en.wikipedia.org/wiki/Conditional%20statement
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A conditional statement may refer to:
A conditional formula in logic and mathematics, which can be interpreted as:
Material conditional
Strict conditional
Variably strict conditional
Relevance conditional
A conditional sentence in natural language, including:
Indicative conditional
Counterfactual conditional
Biscuit conditional
Conditional (computer programming), a conditional statement in a computer programming language
See also
Condition (disambiguation)
Conditional (disambiguation)
Logical biconditional
Logical consequence
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https://en.wikipedia.org/wiki/Mathematics%20of%20cyclic%20redundancy%20checks
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The cyclic redundancy check (CRC) is based on division in the ring of polynomials over the finite field GF(2) (the integers modulo 2), that is, the set of polynomials where each coefficient is either zero or one, and arithmetic operations wrap around.
Any string of bits can be interpreted as the coefficients of a message polynomial of this sort, and to find the CRC, we multiply the message polynomial by and then find the remainder when dividing by the degree- generator polynomial. The coefficients of the remainder polynomial are the bits of the CRC.
Formulation
In general, computation of CRC corresponds to Euclidean division of polynomials over GF(2):
Here is the original message polynomial and is the degree- generator polynomial. The bits of are the original message with zeroes added at the end. The CRC 'checksum' is formed by the coefficients of the remainder polynomial whose degree is strictly less than . The quotient polynomial is of no interest. Using modulo operation, it can be stated that
In communication, the sender attaches the bits of R after the original message bits of M, which could be shown to be equivalent to sending out (the codeword.) The receiver, knowing and therefore , separates M from R and repeats the calculation, verifying that the received and computed R are equal. If they are, then the receiver assumes the received message bits are correct.
In practice CRC calculations most closely resemble long division in binary, except that the subtractions involved do not borrow from more significant digits, and thus become exclusive or operations.
A CRC is a checksum in a strict mathematical sense, as it can be expressed as the weighted modulo-2 sum of per-bit syndromes, but that word is generally reserved more specifically for sums computed using larger moduli, such as 10, 256, or 65535.
CRCs can also be used as part of error-correcting codes, which allow not only the detection of transmission errors, but the reconstruction of the correct message. These codes are based on closely related mathematical principles.
Polynomial arithmetic modulo 2
Since the coefficients are constrained to a single bit, any math operation on CRC polynomials must map the coefficients of the result to either zero or one. For example, in addition:
Note that is equivalent to zero in the above equation because addition of coefficients is performed modulo 2:
Polynomial addition modulo 2 is the same as bitwise XOR. Since XOR is the inverse of itself, polynominal subtraction modulo 2 is the same as bitwise XOR too.
Multiplication is similar (a carry-less product):
We can also divide polynomials mod 2 and find the quotient and remainder. For example, suppose we're dividing by . We would find that
In other words,
The division yields a quotient of x2 + 1 with a remainder of −1, which, since it is odd, has a last bit of 1.
In the above equations, represents the original message bits 111, is the generator polynomial, and
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https://en.wikipedia.org/wiki/Euclidean%20shortest%20path
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The Euclidean shortest path problem is a problem in computational geometry: given a set of polyhedral obstacles in a Euclidean space, and two points, find the shortest path between the points that does not intersect any of the obstacles.
Two dimensions
In two dimensions, the problem can be solved in polynomial time in a model of computation allowing addition and comparisons of real numbers, despite theoretical difficulties involving the numerical precision needed to perform such calculations. These algorithms are based on two different principles, either performing a shortest path algorithm such as Dijkstra's algorithm on a visibility graph derived from the obstacles or (in an approach called the continuous Dijkstra method) propagating a wavefront from one of the points until it meets the other.
Higher dimensions
In three (and higher) dimensions the problem is NP-hard in the general case, but there exist efficient approximation algorithms that run in polynomial time based on the idea of finding a suitable sample of points on the obstacle edges and performing a visibility graph calculation using these sample points.
There are many results on computing shortest paths which stays on a polyhedral surface. Given two points s and t, say on the surface
of a convex polyhedron, the problem is to compute a shortest path that never leaves the surface and connects s with t.
This is a generalization of the problem from 2-dimension but it is much easier than the 3-dimensional problem.
Variants
There are variations of this problem, where the obstacles are weighted, i.e., one can go through an obstacle, but it incurs
an extra cost to go through an obstacle. The standard problem is the special case where the obstacles have infinite weight. This is
termed as the weighted region problem in the literature.
See also
Shortest path problem, in a graph of edges and vertices
Any-angle path planning, in a grid space
Notes
References
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.
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.
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.
.
.
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External links
Implementation of Euclidean Shortest Path algorithm in Digital Geometric Kernel software
Geometric algorithms
Computational geometry
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https://en.wikipedia.org/wiki/List%20of%20United%20States%20rapid%20transit%20systems
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The following is a list of all heavy rail rapid transit systems in the United States. It does not include statistics for bus or light rail systems; see: List of United States light rail systems by ridership for light rail systems. All ridership figures represent unlinked passenger trips, so line transfers on multi-line systems register as separate trips. The data is provided by the American Public Transportation Association's Ridership Reports.
See also
List of metro systems
List of North American rapid transit systems by ridership
List of tram and light rail transit systems
List of suburban and commuter rail systems
List of United States light rail systems by ridership
List of North American light rail systems by ridership
List of United States commuter rail systems by ridership
List of United States local bus agencies by ridership
Notes
References
Rapid transit systems
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https://en.wikipedia.org/wiki/Effective%20dimension
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In mathematics, effective dimension is a modification of Hausdorff dimension and other fractal dimensions that places it in a computability theory setting. There are several variations (various notions of effective dimension) of which the most common is effective Hausdorff dimension. Dimension, in mathematics, is a particular way of describing the size of an object (contrasting with measure and other, different, notions of size). Hausdorff dimension generalizes the well-known integer dimensions assigned to points, lines, planes, etc. by allowing one to distinguish between objects of intermediate size between these integer-dimensional objects. For example, fractal subsets of the plane may have intermediate dimension between 1 and 2, as they are "larger" than lines or curves, and yet "smaller" than filled circles or rectangles. Effective dimension modifies Hausdorff dimension by requiring that objects with small effective dimension be not only small but also locatable (or partially locatable) in a computable sense. As such, objects with large Hausdorff dimension also have large effective dimension, and objects with small effective dimension have small Hausdorff dimension, but an object can have small Hausdorff but large effective dimension. An example is an algorithmically random point on a line, which has Hausdorff dimension 0 (since it is a point) but effective dimension 1 (because, roughly speaking, it can't be effectively localized any better than a small interval, which has Hausdorff dimension 1).
Rigorous definitions
This article will define effective dimension for subsets of Cantor space 2ω; closely related definitions exist for subsets of Euclidean space Rn. We will move freely between considering a set X of natural numbers, the infinite sequence given by the characteristic function of X, and the real number with binary expansion 0.X.
Martingales and other gales
A martingale on Cantor space 2ω is a function d: 2ω → R≥ 0 from Cantor space to nonnegative reals which satisfies the fairness condition:
A martingale is thought of as a betting strategy, and the function gives the capital of the better after seeing a sequence σ of 0s and 1s. The fairness condition then says that the capital after a sequence σ is the average of the capital after seeing σ0 and σ1; in other words the martingale gives a betting scheme for a bookie with 2:1 odds offered on either of two "equally likely" options, hence the name fair.
(Note that this is subtly different from the probability theory notion of martingale. That definition of martingale has a similar fairness condition, which also states that the expected value after some observation is the same as the value before the observation, given the prior history of observations. The difference is that in probability theory, the prior history of observations just refers to the capital history, whereas here the history refers to the exact sequence of 0s and 1s in the string.)
A supermartingale on Cantor sp
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https://en.wikipedia.org/wiki/Langlands%20classification
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In mathematics, the Langlands classification is a description of the irreducible representations of a reductive Lie group G, suggested by Robert Langlands (1973). There are two slightly different versions of the Langlands classification. One of these describes the irreducible admissible (g,K)-modules,
for g a Lie algebra of a reductive Lie group G, with maximal compact subgroup K, in terms of tempered representations of smaller groups. The tempered representations were in turn classified by Anthony Knapp and Gregg Zuckerman. The other version of the Langlands classification divides the irreducible representations into L-packets, and classifies the L-packets in terms of certain homomorphisms of the Weil group of R or C into the Langlands dual group.
Notation
g is the Lie algebra of a real reductive Lie group G in the Harish-Chandra class.
K is a maximal compact subgroup of G, with Lie algebra k.
ω is a Cartan involution of G, fixing K.
p is the −1 eigenspace of a Cartan involution of g.
a is a maximal abelian subspace of p.
Σ is the root system of a in g.
Δ is a set of simple roots of Σ.
Classification
The Langlands classification states that the irreducible admissible representations of (g,K) are parameterized by triples
(F, σ,λ)
where
F is a subset of Δ
Q is the standard parabolic subgroup of F, with Langlands decomposition Q = MAN
σ is an irreducible tempered representation of the semisimple Lie group M (up to isomorphism)
λ is an element of Hom(aF,C) with α(Re(λ))>0 for all simple roots α not in F.
More precisely, the irreducible admissible representation given by the data above is the irreducible quotient of a parabolically induced representation.
For an example of the Langlands classification, see the representation theory of SL2(R).
Variations
There are several minor variations of the Langlands classification. For example:
Instead of taking an irreducible quotient, one can take an irreducible submodule.
Since tempered representations are in turn given as certain representations induced from discrete series or limit of discrete series representations, one can do both inductions at once and get a Langlands classification parameterized by discrete series or limit of discrete series representations instead of tempered representations. The problem with doing this is that it is tricky to decide when two irreducible representations are the same.
References
E. P. van den Ban, Induced representations and the Langlands classification, in (T. Bailey and A. W. Knapp, eds.).
Borel, A. and Wallach, N. Continuous cohomology, discrete subgroups, and representations of reductive groups. Second edition. Mathematical Surveys and Monographs, 67. American Mathematical Society, Providence, RI, 2000. xviii+260 pp.
D. Vogan, Representations of real reductive Lie groups,
Representation theory of Lie groups
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https://en.wikipedia.org/wiki/Constructive%20Approximation
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Constructive Approximation is "an international mathematics journal dedicated to Approximations, expansions, and related research in: computation, function theory, functional analysis, interpolation spaces and interpolation of operators, numerical analysis, space of functions, special functions, and applications."
References
External links
Constructive Approximation web site
Mathematics journals
Approximation theory
English-language journals
Academic journals established in 1985
Springer Science+Business Media academic journals
Bimonthly journals
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https://en.wikipedia.org/wiki/East%20Journal%20on%20Approximations
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The East Journal on Approximations is a journal about approximation theory published in Sofia, Bulgaria.
External links
East Journal on Approximations web site
References
Mathematics journals
Academic journals established in 1995
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https://en.wikipedia.org/wiki/Exact%20category
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In mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition of such a sequence.
Definition
An exact category E is an additive category possessing a class E of "short exact sequences": triples of objects connected by arrows
satisfying the following axioms inspired by the properties of short exact sequences in an abelian category:
E is closed under isomorphisms and contains the canonical ("split exact") sequences:
Suppose occurs as the second arrow of a sequence in E (it is an admissible epimorphism) and is any arrow in E. Then their pullback exists and its projection to is also an admissible epimorphism. Dually, if occurs as the first arrow of a sequence in E (it is an admissible monomorphism) and is any arrow, then their pushout exists and its coprojection from is also an admissible monomorphism. (We say that the admissible epimorphisms are "stable under pullback", resp. the admissible monomorphisms are "stable under pushout".);
Admissible monomorphisms are kernels of their corresponding admissible epimorphisms, and dually. The composition of two admissible monomorphisms is admissible (likewise admissible epimorphisms);
Suppose is a map in E which admits a kernel in E, and suppose is any map such that the composition is an admissible epimorphism. Then so is Dually, if admits a cokernel and is such that is an admissible monomorphism, then so is
Admissible monomorphisms are generally denoted and admissible epimorphisms are denoted These axioms are not minimal; in fact, the last one has been shown by to be redundant.
One can speak of an exact functor between exact categories exactly as in the case of exact functors of abelian categories: an exact functor from an exact category D to another one E is an additive functor such that if
is exact in D, then
is exact in E. If D is a subcategory of E, it is an exact subcategory if the inclusion functor is fully faithful and exact.
Motivation
Exact categories come from abelian categories in the following way. Suppose A is abelian and let E be any strictly full additive subcategory which is closed under taking extensions in the sense that given an exact sequence
in A, then if are in E, so is . We can take the class E to be simply the sequences in E which are exact in A; that is,
is in E iff
is exact in A. Then E is an exact category in the above sense. We verify the axioms:
E is closed under isomorphisms and contains the split exact sequences: these are true by definition, since in an abelian category, any sequence isomorphic to an exact one is also exact, and since the split sequences are always exact in A.
Admissible epimorphisms (respectively, admissible monomorphisms) are stable under pullbacks (resp. pushouts): given an
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https://en.wikipedia.org/wiki/Statistical%20interference
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When two probability distributions overlap, statistical interference exists. Knowledge of the distributions can be used to determine the likelihood that one parameter exceeds another, and by how much.
This technique can be used for dimensioning of mechanical parts, determining when an applied load exceeds the strength of a structure, and in many other situations. This type of analysis can also be used to estimate the probability of failure or the frequency of failure.
Dimensional interference
Mechanical parts are usually designed to fit precisely together. For example, if a shaft is designed to have a "sliding fit" in a hole, the shaft must be a little smaller than the hole. (Traditional tolerances may suggest that all dimensions fall within those intended tolerances. A process capability study of actual production, however, may reveal normal distributions with long tails.) Both the shaft and hole sizes will usually form normal distributions with some average (arithmetic mean) and standard deviation.
With two such normal distributions, a distribution of interference can be calculated. The derived distribution will also be normal, and its average will be equal to the difference between the means of the two base distributions. The variance of the derived distribution will be the sum of the variances of the two base distributions.
This derived distribution can be used to determine how often the difference in dimensions will be less than zero (i.e., the shaft cannot fit in the hole), how often the difference will be less than the required sliding gap (the shaft fits, but too tightly), and how often the difference will be greater than the maximum acceptable gap (the shaft fits, but not tightly enough).
Physical property interference
Physical properties and the conditions of use are also inherently variable. For example, the applied load (stress) on a mechanical part may vary. The measured strength of that part (tensile strength, etc.) may also be variable. The part will break when the stress exceeds the strength.
With two normal distributions, the statistical interference may be calculated as above. (This problem is also workable for transformed units such as the log-normal distribution). With other distributions, or combinations of different distributions, a Monte Carlo method or simulation is often the most practical way to quantify the effects of statistical interference.
See also
Interference fit
Joint probability distribution
Probabilistic design
Process capability
Reliability engineering
Specification
Tolerance (engineering)
References
Paul H. Garthwaite, Byron Jones, Ian T. Jolliffe (2002) Statistical Inference.
Haugen, (1980) Probabilistic mechanical design, Wiley.
Statistical theory
Survival analysis
Reliability engineering
Probability theory
Applied probability
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https://en.wikipedia.org/wiki/Restricted%20partial%20quotients
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In mathematics, and more particularly in the analytic theory of regular continued fractions, an infinite regular continued fraction x is said to be restricted, or composed of restricted partial quotients, if the sequence of denominators of its partial quotients is bounded; that is
and there is some positive integer M such that all the (integral) partial denominators ai are less than or equal to M.
Periodic continued fractions
A regular periodic continued fraction consists of a finite initial block of partial denominators followed by a repeating block; if
then ζ is a quadratic irrational number, and its representation as a regular continued fraction is periodic. Clearly any regular periodic continued fraction consists of restricted partial quotients, since none of the partial denominators can be greater than the largest of a0 through ak+m. Historically, mathematicians studied periodic continued fractions before considering the more general concept of restricted partial quotients.
Restricted CFs and the Cantor set
The Cantor set is a set C of measure zero from which a complete interval of real numbers can be constructed by simple addition – that is, any real number from the interval can be expressed as the sum of exactly two elements of the set C. The usual proof of the existence of the Cantor set is based on the idea of punching a "hole" in the middle of an interval, then punching holes in the remaining sub-intervals, and repeating this process ad infinitum.
The process of adding one more partial quotient to a finite continued fraction is in many ways analogous to this process of "punching a hole" in an interval of real numbers. The size of the "hole" is inversely proportional to the next partial denominator chosen – if the next partial denominator is 1, the gap between successive convergents is maximized.
To make the following theorems precise we will consider CF(M), the set of restricted continued fractions whose values lie in the open interval (0, 1) and whose partial denominators are bounded by a positive integer M – that is,
By making an argument parallel to the one used to construct the Cantor set two interesting results can be obtained.
If M ≥ 4, then any real number in an interval can be constructed as the sum of two elements from CF(M), where the interval is given by
A simple argument shows that holds when M ≥ 4, and this in turn implies that if M ≥ 4, every real number can be represented in the form n + CF1 + CF2, where n is an integer, and CF1 and CF2 are elements of CF(M).
Zaremba's conjecture
Zaremba has conjectured the existence of an absolute constant A, such that the rationals with partial quotients restricted by A contain at least one for every (positive integer) denominator. The choice A = 5 is compatible with the numerical evidence. Further conjectures reduce that value, in the case of all sufficiently large denominators. Jean Bourgain and Alex Kontorovich have shown that A can be chosen so that the conclusion holds
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https://en.wikipedia.org/wiki/Luzin%20space
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In mathematics, a Luzin space (or Lusin space), named for N. N. Luzin, is an uncountable topological T1 space without isolated points in which every nowhere-dense subset is countable. There are many minor variations of this definition in use: the T1 condition can be replaced by T2 or T3, and some authors allow a countable or even arbitrary number of isolated points.
The existence of a Luzin space is independent of the axioms of ZFC. showed that the continuum hypothesis implies that a Luzin space exists.
showed that assuming Martin's axiom and the negation of the continuum hypothesis, there are no Hausdorff Luzin spaces.
In real analysis
In real analysis and descriptive set theory, a Luzin set (or Lusin set), is defined as an uncountable subset of the reals such that every uncountable subset of is nonmeager; that is, of second Baire category. Equivalently, is an uncountable set of reals that meets every first category set in only countably many points. Luzin proved that, if the continuum hypothesis holds, then every nonmeager set has a Luzin subset. Obvious properties of a Luzin set are that it must be nonmeager (otherwise the set itself is an uncountable meager subset) and of measure zero, because every set of positive measure contains a meager set that also has positive measure, and is therefore uncountable. A weakly Luzin set is an uncountable subset of a real vector space such that for any uncountable subset the set of directions between different elements of the subset is dense in the sphere of directions.
The measure-category duality provides a measure analogue of Luzin sets – sets of positive outer measure, every uncountable subset of which has positive outer measure. These sets are called Sierpiński sets, after Wacław Sierpiński. Sierpiński sets are weakly Luzin sets but are not Luzin sets.
Example of a Luzin set
Choose a collection of 2ℵ0 meager subsets of R such that every meager subset is contained in one of them. By the continuum hypothesis, it is possible to enumerate them as Sα for countable ordinals α. For each countable ordinal β choose a real number xβ that is not in any of the sets Sα for α<β, which is possible as the union of these sets is meager so is not the whole of R. Then the uncountable set X of all these real numbers xβ has only a countable number of elements in each set Sα, so is a Luzin set.
More complicated variations of this construction produce examples of Luzin sets that are subgroups, subfields or real-closed subfields of the real numbers.
References
Paper mentioning Luzin spaces
Properties of topological spaces
Descriptive set theory
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https://en.wikipedia.org/wiki/Fox%20Harbour%2C%20Newfoundland%20and%20Labrador
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Fox Harbour is a small community on the Avalon Peninsula of Newfoundland. According to Statistics Canada in 2011, the population was 270. It is surrounded by hills. It is located close to Argentia, the site of the Naval Station Argentia. According to some sources, Fox Harbour got its name from tales of foxes that came down from the surrounding hills and ate the drying fish on the flakes. As well, the community was called Little Glocester before it became officially named Fox Harbour.
History
Fox Harbour started as a fishing community in the early 19th century by the three families of Matthew, Martin, and George Spurvey. However, fisherman from England and Ireland had come overseas to fish there seasonally since the 18th century. All of them returned to England in the 1820s except for a Matthew Spurvey. Other families had settled in Fox Harbour by then with the arrival of Healey, Kelly and Dreaddy families from Ireland in 1806. The population grew over time, and peaked at 746. Fox Harbour was incorporated in 1964, and the council building opened in 1969. The council building now incorporates the fire station, the library, museum, and the council office.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Fox Harbour had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Religion
The population of Fox Harbour is predominantly Roman Catholic. The Roman Catholic Church, the only church in Fox Harbour, was built in 1890. Before the church was built, the people of Fox Harbour had to boat to Argentia to attend church service. In October, 1919 the original church was torn down and a new church built. In 1945, Fox Harbour became a parish and was called Sacred Heart Parish. Fr. Penny was the first appointed parish priest. Recently the Fox Harbour parish joined with the Placentia area to become the Placentia Area Roman Catholic Cluster.
Education
The first school established in Fox Harbour was in 1848. St. Regis was built in 1946 and expanded in 1956. High school students were initially schooled at Laval High School in Placentia. When St. Anne's Academy was opened, the high school students from St. Regis moved from Laval to St. Anne's which was closer to home. However, in the early 1990s St. Regis closed its doors and all students are now bused to St. Anne's Academy from kindergarten to grade six, and Laval from grade seven to grade twelve.
The Fox Harbour Festival
In 1993 Fox Harbour had its first festival. It has been held every year since then. The festival is held in the last weekend of July.
It commences with a cemetery mass followed by a garden party on the Sunday previous to the festival week. During the following week many events are held, including dances for adults and children, and a concert. On Saturday, there is a softball tournament; all residents over the age of 30 are invited to participate
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https://en.wikipedia.org/wiki/Tempered%20representation
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In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the Lp space
L2+ε(G)
for any ε > 0.
Formulation
This condition, as just given, is slightly weaker than the condition that the matrix coefficients are square-integrable, in other words lie in
L2(G),
which would be the definition of a discrete series representation. If G is a linear semisimple Lie group with a maximal compact subgroup K, an admissible representation ρ of G is tempered if the above condition holds for the K-finite matrix coefficients of ρ.
The definition above is also used for more general groups, such as p-adic Lie groups and finite central extensions of semisimple real algebraic groups. The definition of "tempered representation" makes sense for arbitrary unimodular locally compact groups, but on groups with infinite centers such as infinite central extensions of semisimple Lie groups it does not behave well and is usually replaced by a slightly different definition. More precisely, an irreducible representation is called tempered if it is unitary when restricted to the center Z, and the absolute values of the matrix coefficients are in L2+ε(G/Z).
Tempered representations on semisimple Lie groups were first defined and studied by Harish-Chandra (using a different but equivalent definition), who showed that they are exactly the representations needed for the Plancherel theorem. They were classified by Knapp and Zuckerman, and used by Langlands in the Langlands classification of irreducible representations of a reductive Lie group G in terms of the tempered representations of smaller groups.
History
Irreducible tempered representations were identified by Harish-Chandra in his work on harmonic analysis on a semisimple Lie group as those representations that contribute to the Plancherel measure. The original definition of a tempered representation, which has certain technical advantages, is that its Harish-Chandra character should be a "tempered distribution" (see the section about this below). It follows from Harish-Chandra's results that it is equivalent to the more elementary definition given above. Tempered representations also seem to play a fundamental role in the theory of automorphic forms. This connection was probably first realized by Satake (in the context of the Ramanujan-Petersson conjecture) and Robert Langlands and served as a motivation for Langlands to develop his classification scheme for irreducible admissible representations of real and p-adic reductive algebraic groups in terms of the tempered representations of smaller groups. The precise conjectures identifying the place of tempered representations in the automorphic spectrum were formulated later by James Arthur and constitute one of the most actively developing parts of the modern theory of automorphic forms.
Harmonic analysis
Tempered representations play an important role in the harmonic analysis on s
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https://en.wikipedia.org/wiki/Binary%20tetrahedral%20group
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In mathematics, the binary tetrahedral group, denoted 2T or , is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of order 12 by a cyclic group of order 2, and is the preimage of the tetrahedral group under the 2:1 covering homomorphism Spin(3) → SO(3) of the special orthogonal group by the spin group. It follows that the binary tetrahedral group is a discrete subgroup of Spin(3) of order 24. The complex reflection group named 3(24)3 by G.C. Shephard or 3[3]3 and by Coxeter, is isomorphic to the binary tetrahedral group.
The binary tetrahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism , where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
Elements
Explicitly, the binary tetrahedral group is given as the group of units in the ring of Hurwitz integers. There are 24 such units given by
with all possible sign combinations.
All 24 units have absolute value 1 and therefore lie in the unit quaternion group Sp(1). The convex hull of these 24 elements in 4-dimensional space form a convex regular 4-polytope called the 24-cell.
Properties
The binary tetrahedral group, denoted by 2T, fits into the short exact sequence
This sequence does not split, meaning that 2T is not a semidirect product of {±1} by T. In fact, there is no subgroup of 2T isomorphic to T.
The binary tetrahedral group is the covering group of the tetrahedral group. Thinking of the tetrahedral group as the alternating group on four letters, , we thus have the binary tetrahedral group as the covering group,
The center of 2T is the subgroup {±1}. The inner automorphism group is isomorphic to A4, and the full automorphism group is isomorphic to S4.
The binary tetrahedral group can be written as a semidirect product
where Q is the quaternion group consisting of the 8 Lipschitz units and C3 is the cyclic group of order 3 generated by . The group Z3 acts on the normal subgroup Q by conjugation. Conjugation by is the automorphism of Q that cyclically rotates , , and .
One can show that the binary tetrahedral group is isomorphic to the special linear group SL(2,3) – the group of all matrices over the finite field F3 with unit determinant, with this isomorphism covering the isomorphism of the projective special linear group PSL(2,3) with the alternating group A4.
Presentation
The group 2T has a presentation given by
or equivalently,
Generators with these relations are given by
with .
Subgroups
The quaternion group consisting of the 8 Lipschitz units forms a normal subgroup of 2T of index 3. This group and the center {±1} are the only nontrivial normal subgroups.
All other subgroups of 2T are cyclic groups generated by the various elements, with orders 3, 4, and 6.
Higher dimensions
Just as the tetrahedral group generalizes to the rotational symmetry group of
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https://en.wikipedia.org/wiki/Binary%20octahedral%20group
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In mathematics, the binary octahedral group, name as 2O or is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group O or (2,3,4) of order 24 by a cyclic group of order 2, and is the preimage of the octahedral group under the 2:1 covering homomorphism of the special orthogonal group by the spin group. It follows that the binary octahedral group is a discrete subgroup of Spin(3) of order 48.
The binary octahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
Elements
Explicitly, the binary octahedral group is given as the union of the 24 Hurwitz units
with all 24 quaternions obtained from
by a permutation of coordinates and all possible sign combinations. All 48 elements have absolute value 1 and therefore lie in the unit quaternion group Sp(1).
Properties
The binary octahedral group, denoted by 2O, fits into the short exact sequence
This sequence does not split, meaning that 2O is not a semidirect product of {±1} by O. In fact, there is no subgroup of 2O isomorphic to O.
The center of 2O is the subgroup {±1}, so that the inner automorphism group is isomorphic to O. The full automorphism group is isomorphic to O × Z2.
Presentation
The group 2O has a presentation given by
or equivalently,
Quaternion generators with these relations are given by
with
Subgroups
The binary tetrahedral group, 2T, consisting of the 24 Hurwitz units, forms a normal subgroup of index 2. The quaternion group, Q8, consisting of the 8 Lipschitz units forms a normal subgroup of 2O of index 6. The quotient group is isomorphic to S3 (the symmetric group on 3 letters). These two groups, together with the center {±1}, are the only nontrivial normal subgroups of 2O.
The generalized quaternion group, Q16, also forms a subgroup of 2O, index 3. This subgroup is self-normalizing so its conjugacy class has 3 members. There are also isomorphic copies of the binary dihedral groups Q8 and Q12 in 2O.
All other subgroups are cyclic groups generated by the various elements (with orders 3, 4, 6, and 8).
Higher dimensions
The binary octahedral group generalizes to higher dimensions: just as the octahedron generalizes to the orthoplex, the octahedral group in SO(3) generalizes to the hyperoctahedral group in SO(n), which has a binary cover under the map
See also
Binary polyhedral group
binary cyclic group, ⟨n⟩, index 2n
binary dihedral group, ⟨2,2,n⟩, index 4n
binary tetrahedral group, 2T=⟨2,3,3⟩, index 24
binary icosahedral group, 2I=⟨2,3,5⟩, index 120
References
Notes
Octahedral
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https://en.wikipedia.org/wiki/Homomorphic%20secret%20sharing
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In cryptography, homomorphic secret sharing is a type of secret sharing algorithm in which the secret is encrypted via homomorphic encryption. A homomorphism is a transformation from one algebraic structure into another of the same type so that the structure is preserved. Importantly, this means that for every kind of manipulation of the original data, there is a corresponding manipulation of the transformed data.
Technique
Homomorphic secret sharing is used to transmit a secret to several recipients as follows:
Transform the "secret" using a homomorphism. This often puts the secret into a form which is easy to manipulate or store. In particular, there may be a natural way to 'split' the new form as required by step (2).
Split the transformed secret into several parts, one for each recipient. The secret must be split in such a way that it can only be recovered when all or most of the parts are combined. (See Secret sharing.)
Distribute the parts of the secret to each of the recipients.
Combine each of the recipients' parts to recover the transformed secret, perhaps at a specified time.
Reverse the homomorphism to recover the original secret.
Examples
Suppose a community wants to perform an election, using a decentralized voting protocol, but they want to ensure that the vote-counters won't lie about the results. Using a type of homomorphic secret sharing known as Shamir's secret sharing, each member of the community can add their vote to a form that is split into pieces, each piece is then submitted to a different vote-counter. The pieces are designed so that the vote-counters can't predict how any alterations to each piece will affect the whole, thus, discouraging vote-counters from tampering with their pieces. When all votes have been received, the vote-counters combine them, allowing them to recover the aggregate election results.
In detail, suppose we have an election with:
Two possible outcomes, either yes or no. We'll represent those outcomes numerically by +1 and −1, respectively.
A number of authorities, k, who will count the votes.
A number of voters, n, who will submit votes.
In advance, each authority generates a publicly available numerical key, xk.
Each voter encodes his vote in a polynomial pn according to the following rules: The polynomial should have degree , its constant term should be either +1 or −1 (corresponding to voting "yes" or voting "no"), and its other coefficients should be randomly generated.
Each voter computes the value of his polynomial pn at each authority's public key xk.
This produces k points, one for each authority.
These k points are the "pieces" of the vote: If you know all of the points, you can figure out the polynomial pn (and hence you can figure out how the voter voted). However, if you know only some of the points, you can't figure out the polynomial. (This is because you need n points to determine a degree- polynomial. Two points determine a line, three points determine a parab
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https://en.wikipedia.org/wiki/%C5%81ukasiewicz%20logic
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In mathematics and philosophy, Łukasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued modal logic; it was later generalized to n-valued (for all finite n) as well as infinitely-many-valued (ℵ0-valued) variants, both propositional and first order. The ℵ0-valued version was published in 1930 by Łukasiewicz and Alfred Tarski; consequently it is sometimes called the ŁukasiewiczTarski logic. It belongs to the classes of t-norm fuzzy logics and substructural logics.
Łukasiewicz logic was motivated by Aristotle's suggestion that bivalent logic was not applicable to future contingents, e.g. the statement "There will be a sea battle tomorrow". In other words, statements about the future were neither true nor false, but an intermediate value could be assigned to them, to represent their possibility of becoming true in the future.
This article presents the Łukasiewicz(–Tarski) logic in its full generality, i.e. as an infinite-valued logic. For an elementary introduction to the three-valued instantiation Ł3, see three-valued logic.
Language
The propositional connectives of Łukasiewicz logic are ("implication"), and the constant ("false"). Additional connectives can be defined in terms of these:
The and connectives are called weak disjunction and conjunction, because they are non-classical, as the law of excluded middle does not hold for them. In the context of substructural logics, they are called additive connectives. They also correspond to lattice min/max connectives.
In terms of substructural logics, there are also strong or multiplicative disjunction and conjunction connectives, although these are not part of Łukasiewicz's original presentation:
There are also defined modal operators, using the Tarskian Möglichkeit:
Axioms
The original system of axioms for propositional infinite-valued Łukasiewicz logic used implication and negation as the primitive connectives, along with modus ponens:
Propositional infinite-valued Łukasiewicz logic can also be axiomatized by adding the following axioms to the axiomatic system of monoidal t-norm logic:
Divisibility
Double negation
That is, infinite-valued Łukasiewicz logic arises by adding the axiom of double negation to basic fuzzy logic (BL), or by adding the axiom of divisibility to the logic IMTL.
Finite-valued Łukasiewicz logics require additional axioms.
Proof Theory
A hypersequent calculus for three-valued Łukasiewicz logic was introduced by Arnon Avron in 1991.
Sequent calculi for finite and infinite-valued Łukasiewicz logics as an extension of linear logic were introduced by A. Prijatelj in 1994. However, these are not cut-free systems.
Hypersequent calculi for Łukasiewicz logics were introduced by A. Ciabattoni et al in 1999. However, these are not cut-free for finite-valued logics.
A labelled tableaux system was introduced by Nicola Olivetti in 2003.
Real-valued semantics
Infinit
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https://en.wikipedia.org/wiki/HGT%20%28disambiguation%29
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HGT may refer to:
Harrogate railway station, England
Holland's Got Talent, a Dutch television show
Horizontal gene transfer, non-hereditary genetic changes
Hyper geometric test, in statistics
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https://en.wikipedia.org/wiki/Strong%20partition%20cardinal
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In Zermelo–Fraenkel set theory without the axiom of choice a strong partition cardinal is an uncountable well-ordered cardinal such that every partition of the set of size subsets of into less than pieces has a homogeneous set of size .
The existence of strong partition cardinals contradicts the axiom of choice. The Axiom of determinacy implies that ℵ1 is a strong partition cardinal.
References
.
Cardinal numbers
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https://en.wikipedia.org/wiki/Harish-Chandra%20class
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In mathematics, Harish-Chandra's class is a class of Lie groups used in representation theory. Harish-Chandra's class contains all semisimple connected linear Lie groups and is closed under natural operations, most importantly, the passage to Levi subgroups. This closure property is crucial for many inductive arguments in representation theory of Lie groups, whereas the classes of semisimple or connected semisimple Lie groups are not closed in this sense.
Definition
A Lie group G with the Lie algebra g is said to be in Harish-Chandra's class if it satisfies the following conditions:
g is a reductive Lie algebra (the product of a semisimple and abelian Lie algebra).
The Lie group G has only a finite number of connected components.
The adjoint action of any element of G on g is given by an action of an element of the connected component of the Lie group of Lie algebra automorphisms of the complexification g⊗C.
The subgroup Gss of G generated by the image of the semisimple part gss=[g,g] of the Lie algebra g under the exponential map has finite center.
References
A. W. Knapp, Structure theory of semisimple Lie groups, in
Representation theory of Lie groups
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https://en.wikipedia.org/wiki/Minimal%20volume
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In mathematics, in particular in differential geometry, the minimal volume is a number that describes one aspect of a smooth manifold's topology. This diffeomorphism invariant was introduced by Mikhael Gromov.
Given a smooth Riemannian manifold , one may consider its volume and sectional curvature . The minimal volume of a smooth manifold is defined to be
Any closed manifold can be given an arbitrarily small volume by scaling any choice of a Riemannian metric. The minimal volume removes the possibility of such scaling by the constraint on sectional curvatures. So, if the minimal volume of is zero, then a certain kind of nontrivial collapsing phenomena can be exhibited by Riemannian metrics on . A trivial example, the only in which the possibility of scaling is present, is a closed flat manifold. The Berger spheres show that the minimal volume of the three-dimensional sphere is also zero. Gromov has conjectured that every closed simply connected odd-dimensional manifold has zero minimal volume.
By contrast, a positive lower bound for the minimal volume of amounts to some (usually nontrivial) geometric inequality for the volume of an arbitrary complete Riemannian metric on in terms of the size of its curvature. According to the Gauss-Bonnet theorem, if is a closed and connected two-dimensional manifold, then . The infimum in the definition of minimal volume is realized by the metrics appearing from the uniformization theorem. More generally, according to the Chern-Gauss-Bonnet formula, if is a closed and connected manifold then
Gromov, in 1982, showed that the volume of a complete Riemannian metric on a smooth manifold can always be estimated by the size of its curvature and by the simplicial volume of the manifold, via the inequality
References
Misha Gromov. Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, MA, 1999. xx+585 pp. .
Michael Gromov. Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99.
Riemannian geometry
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https://en.wikipedia.org/wiki/List%20of%20career%20achievements%20by%20LeBron%20James
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This article lists career accomplishments of the American professional basketball player LeBron James.
NBA career statistics
Correct as of the 2022–23 season.
Regular season
|-
| style="text-align:left;"|
| style="text-align:left;"| Cleveland
| 79 || 79 || 39.5 || .500|| .290 || .754 || 5.5 || 5.9 || 1.6 || .7 || 20.9
|-
| style="text-align:left;"|
| style="text-align:left;"| Cleveland
| 80 || 80 || style="background:#cfecec;"| 42.4* || .472 || .351 || .750 || 7.4 || 7.2 || 2.2 || .7 || 27.2
|-
| style="text-align:left;"|
| style="text-align:left;"| Cleveland
| 79 || 79 || 42.5 || .480 || .335 || .738 || 7.0 || 6.6 || 1.6 || .8 || 31.4
|-
| style="text-align:left;"|
| style="text-align:left;"| Cleveland
| 78 || 78 || 40.9 || .476 || .319 || .698 || 6.7 || 6.0 || 1.6 || .7|| 27.3
|-
| style="text-align:left;"|
| style="text-align:left;"| Cleveland
| 75 || 74 || 40.4 || .484 || .315 || .712 || 7.9 || 7.2 || 1.8 || 1.1 || style="background:#cfecec;" | 30.0*
|-
| style="text-align:left;"|
| style="text-align:left;"| Cleveland
| 81 || 81 || 37.7 || .489 || .344 || .780 || 7.6 || 7.2 || 1.7 || 1.1 || 28.4
|-
| style="text-align:left;"|
| style="text-align:left;"| Cleveland
| 76 || 76 || 39.0 || .503 || .333 || .767 || 7.3 || 8.6 || 1.6 || 1.0 || 29.7
|-
| style="text-align:left;"|
| style="text-align:left;"| Miami
| 79 || 79 || 38.8 || .510 || .330 || .759 || 7.5 || 7.0 || 1.6 || .6 || 26.7
|-
|style="text-align:left;background:#afe6ba;"|†
|style="text-align:left;"|Miami
|62||62||37.5||.531||.362||.771||7.9||6.2||1.9||.8||27.1
|-
|style="text-align:left;background:#afe6ba;"|†
|style="text-align:left;"|Miami
|76||76||37.9||.565||.406||.753||8.0||7.3||1.7||.9||26.8
|-
|style="text-align:left;"|
|style="text-align:left;"|Miami
|77||77||37.7||.567||.379||.750||6.9||6.4||1.6||.3||27.1
|-
|style="text-align:left;"|
|style="text-align:left;"|Cleveland
|69||69||36.1||.488||.354||.710||6.0||7.4||1.6||.7||25.3
|-
|style="text-align:left;background:#afe6ba;"|†
|style="text-align:left;"|Cleveland
|76||76||35.6||.520||.309||.731||7.4||6.8||1.4||.6||25.3
|-
|style="text-align:left;"|
|style="text-align:left;"|Cleveland
|74||74|| style="background:#cfecec;"| 37.8*||.548||.363||.674||8.6||8.7||1.2||.6||26.4
|-
|style="text-align:left;"|
|style="text-align:left;"|Cleveland
|style="background:#cfecec;"| 82*|| style="background:#cfecec;"| 82*|| style="background:#cfecec;"| 36.9*||.542||.367||.731||8.6||9.1||1.4||.9||27.5
|-
|style="text-align:left;"|
|style="text-align:left;"|L.A. Lakers
|55||55|| 35.2||.510||.339||.665||8.5||8.3||1.3||.6||27.4
|-
|style="text-align:left;background:#afe6ba;"|†
|style="text-align:left;"|L.A. Lakers
|67||67|| 34.6||.493||.348||.693||7.8||style="background:#cfecec;"| 10.2*||1.2||.5||25.3
|-
|style="text-align:left;"|
|style="text-align:left;"|L.A. Lakers
|45||45|| 33.4||.513||.365||.698||7.7|| 7.8 ||1.1||.6||25.0
|-
|style="text-align:left;"|
|style="text-align:left;"|L.A. Lakers
|56||56|| 37.2||.524||.359||.756||8.2|| 6.2 ||1.3|
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https://en.wikipedia.org/wiki/Hausdorff%20density
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In measure theory, a field of mathematics, the Hausdorff density measures how concentrated a Radon measure is at some point.
Definition
Let be a Radon measure and some point in Euclidean space. The s-dimensional upper and lower Hausdorff densities are defined to be, respectively,
and
where is the ball of radius r > 0 centered at a. Clearly, for all . In the event that the two are equal, we call their common value the s-density of at a and denote it .
Marstrand's theorem
The following theorem states that the times when the s-density exists are rather seldom.
Marstrand's theorem: Let be a Radon measure on . Suppose that the s-density exists and is positive and finite for a in a set of positive measure. Then s is an integer.
Preiss' theorem
In 1987 David Preiss proved a stronger version of Marstrand's theorem. One consequence is that sets with positive and finite density are rectifiable sets.
Preiss' theorem: Let be a Radon measure on . Suppose that m is an integer and the m-density exists and is positive and finite for almost every a in the support of . Then is m-rectifiable, i.e. ( is absolutely continuous with respect to Hausdorff measure ) and the support of is an m-rectifiable set.
External links
Density of a set at Encyclopedia of Mathematics
Rectifiable set at Encyclopedia of Mathematics
References
Pertti Mattila, Geometry of sets and measures in Euclidean spaces. Cambridge Press, 1995.
Measure theory
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https://en.wikipedia.org/wiki/Discrete%20exterior%20calculus
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In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs, finite element meshes, and lately also general polygonal meshes (non-flat and non-convex). DEC methods have proved to be very powerful in improving and analyzing finite element methods: for instance, DEC-based methods allow the use of highly non-uniform meshes to obtain accurate results. Non-uniform meshes are advantageous because they allow the use of large elements where the process to be simulated is relatively simple, as opposed to a fine resolution where the process may be complicated (e.g., near an obstruction to a fluid flow), while using less computational power than if a uniformly fine mesh were used.
The discrete exterior derivative
Stokes' theorem relates the integral of a differential (n − 1)-form ω over the boundary ∂M of an n-dimensional manifold M to the integral of dω (the exterior derivative of ω, and a differential n-form on M) over M itself:
One could think of differential k-forms as linear operators that act on k-dimensional "bits" of space, in which case one might prefer to use the bracket notation for a dual pairing. In this notation, Stokes' theorem reads as
In finite element analysis, the first stage is often the approximation of the domain of interest by a triangulation, T. For example, a curve would be approximated as a union of straight line segments; a surface would be approximated by a union of triangles, whose edges are straight line segments, which themselves terminate in points. Topologists would refer to such a construction as a simplicial complex. The boundary operator on this triangulation/simplicial complex T is defined in the usual way: for example, if L is a directed line segment from one point, a, to another, b, then the boundary ∂L of L is the formal difference b − a.
A k-form on T is a linear operator acting on k-dimensional subcomplexes of T; e.g., a 0-form assigns values to points, and extends linearly to linear combinations of points; a 1-form assigns values to line segments in a similarly linear way. If ω is a k-form on T, then the discrete exterior derivative dω of ω is the unique (k + 1)-form defined so that Stokes' theorem holds:
For every (k + 1)-dimensional subcomplex of T, S.
Other operators and operations such as the discrete wedge product, Hodge star, or Lie derivative can also be defined.
See also
Discrete differential geometry
Discrete Morse theory
Topological combinatorics
Discrete calculus
Notes
References
A simple and complete discrete exterior calculus on general polygonal meshes, Ptackova, Lenka and Velho, Luiz, Computer Aided Geometric Design, 2021, DOI: 10.1016/j.cagd.2021.102002
Discrete Calculus, Grady, Leo J., Polimeni, Jonathan R., 2010
Hirani Thesis on Discrete Exterior Calculus
A Primal-to-Primal Discretization of Exterior Calculus on Polygonal Meshes, Ptackova, L. and Velho, L., Symposium on Geometry Processing 2017, DOI: 10.2312/SGP.20171
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https://en.wikipedia.org/wiki/Finite%20element%20exterior%20calculus
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Finite element exterior calculus (FEEC) is a mathematical framework that formulates finite element methods using chain complexes. Its main application has been a comprehensive theory for finite element methods in computational electromagnetism, computational solid and fluid mechanics. FEEC was developed in the early 2000s by Douglas N. Arnold, Richard S. Falk and Ragnar Winther,
among others.
Finite element exterior calculus is sometimes called as an example of a compatible discretization technique, and bears similarities with discrete exterior calculus, although they are distinct theories.
One starts with the recognition that the used differential operators are often part of complexes: successive application results in zero. Then, the phrasing of the differential operators of relevant differential equations and relevant boundary conditions as a Hodge Laplacian. The Hodge Laplacian terms are split using the Hodge decomposition. A related variational saddle-point formulation for mixed quantities is then generated. Discretization to a mesh-related subcomplex is done requiring a collection of projection operators which commute with the differential operators. One can then prove uniqueness and optimal convergence as function of mesh density.
FEEC is of immediate relevancy for diffusion, elasticity, electromagnetism, Stokes flow.
For the important de Rham complex, pertaining to the grad, curl and div operators, suitable family of elements have been generated not only for tetrahedrons, but also for other shaped elements such as bricks. Moreover, also conforming with them, prism and pyramid shaped elements have been generated. For the latter, uniquely, the shape functions are not polynomial. The quantities are 0-forms (scalars), 1-forms (gradients), 2-forms (fluxes), and 3-forms (densities). Diffusion, electromagnetism, and elasticity, Stokes flow, general relatively, and actually all known complexes, can all be phrased in terms the de Rham complex. For Navier-Stokes, there may be possibilities too.
References
Finite element method
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https://en.wikipedia.org/wiki/Martin%20measure
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In descriptive set theory, the Martin measure is a filter on the set of Turing degrees of sets of natural numbers, named after Donald A. Martin. Under the axiom of determinacy it can be shown to be an ultrafilter.
Definition
Let be the set of Turing degrees of sets of natural numbers. Given some equivalence class , we may define the cone (or upward cone) of as the set of all Turing degrees such that ; that is, the set of Turing degrees that are "at least as complex" as under Turing reduction. In order-theoretic terms, the cone of is the upper set of .
Assuming the axiom of determinacy, the cone lemma states that if A is a set of Turing degrees, either A includes a cone or the complement of A contains a cone. It is similar to Wadge's lemma for Wadge degrees, and is important for the following result.
We say that a set of Turing degrees has measure 1 under the Martin measure exactly when contains some cone. Since it is possible, for any , to construct a game in which player I has a winning strategy exactly when contains a cone and in which player II has a winning strategy exactly when the complement of contains a cone, the axiom of determinacy implies that the measure-1 sets of Turing degrees form an ultrafilter.
Consequences
It is easy to show that a countable intersection of cones is itself a cone; the Martin measure is therefore a countably complete filter. This fact, combined with the fact that the Martin measure may be transferred to by a simple mapping, tells us that is measurable under the axiom of determinacy. This result shows part of the important connection between determinacy and large cardinals.
References
Descriptive set theory
Determinacy
Computability theory
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https://en.wikipedia.org/wiki/Accrington%20Academy
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Accrington Academy is a mixed 11-18 Academy in Accrington, Lancashire. It has designated specialisms in Sports and Mathematics. It is situated in the centre of Accrington. Accrington St Christopher's C of E High is nearby to the west.
History
The school, run by United Learning, opened on 1 September 2008 on the site of the former Accrington Moorhead Sports College, itself the successor Moorhead High School which was the successor of the one-time Accrington High School for Girls. All pupils previously at Moorhead automatically transferred to the new school, which has had a sixth form provision from September 2009.
Former schools
Accrington Grammar School had around 500 boys and 100 in the sixth form in the 1970s. Accrington High School for Girls had around 600 girls. Accrington Moorhead High School was on Cromwell Avenue off Queens Road West. The school was founded in 1895 on Blackburn Rd, Accrington as a 'Technical School' In 1968, it moved to the Moorhead site. In 1975, following the Labour government's educational reforms, it ceased to exist.
In 2008, Nosheen Iqbal wrote in The Guardian that Moorhead High School had been "failing". Her article described a "startling transformation" from 17% of children achieving 5 GCSEs at grades A*-C, to 78% of children doing so in the new school. The school's headteacher believed that the change had been brought about through the Creative Partnerships approach, an Arts Council England programme.
Notable former pupils
Accrington Moorhead Sports College
Dominic Brunt, actor, known for his part in Emmerdale as Paddy Kirk.
Accrington Grammar School
Sir Kenneth Barnes CB, Permanent Secretary from 1976 to 1982 of the Department of Employment
Jim Bowen, comedian, and former host of Bullseye
Oliver Bulleid CBE, Chief Mechanical Engineer from 1937 to 1948 of the Southern Railway
Harold Davenport FRS, mathematician
Sir James Drake CBE, civil engineer, designed the UK's first motorway
Graeme Fowler, cricketer
Harry Hill, cyclist who competed in the Olympic Games in 1936
Ron Hill, marathon runner in the 1964 Tokyo and 1972 Munich Olympics, and won the gold at the 1970 Edinburgh Commonwealth Games
Prof Leslie Howarth OBE, mathematician
Prof John Lamb CBE, James Watt Professor of Electrical Engineering from 1961 to 1991 at the University of Glasgow, President from 1970 to 1972 of the British Society of Rheology
Bryan Langton CBE, chairman and chief executive from 1990 to 1996 of Holiday Inn
Rev Fred Lord, editor from 1941 to 1956 of The Baptist Times
James Prescott CBE, FRS, agricultural scientist
Edward Slinger, cricketer, solicitor and judge
Sir John Tomlinson CBE, opera singer
Prof John Wallwork CBE FRCS FMedSci, cardiothoracic surgeon and emeritus professor who performed Europe's first successful combined heart-lung transplant in 1984
Graham Walne, theatre consultant, lighting designer, author, and lecturer
Harry Yeadon, civil engineer, worked with James Drake on the UK's first motorway
Accrin
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https://en.wikipedia.org/wiki/Hypertranscendental%20number
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A complex number is said to be hypertranscendental if it is not the value at an algebraic point of a function which is the solution of an algebraic differential equation with coefficients in and with algebraic initial conditions.
The term was introduced by D. D. Morduhai-Boltovskoi in "Hypertranscendental numbers and hypertranscendental functions" (1949).
The term is related to transcendental numbers, which are numbers which are not a solution of a non-zero polynomial equation with rational coefficients. The number is transcendental but not hypertranscendental, as it can be generated from the solution to the differential equation .
Any hypertranscendental number is also a transcendental number.
See also
Hypertranscendental function
References
Hiroshi Umemura, "On a class of numbers generated by differential equations related with algebraic groups", Nagoya Math. Journal. Volume 133 (1994), 1-55. (Downloadable from ProjectEuclid)
Transcendental numbers
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https://en.wikipedia.org/wiki/Hypertranscendental%20function
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A hypertranscendental function or transcendentally transcendental function is a transcendental analytic function which is not the solution of an algebraic differential equation with coefficients in (the integers) and with algebraic initial conditions.
History
The term 'transcendentally transcendental' was introduced by E. H. Moore in 1896; the term 'hypertranscendental' was introduced by D. D. Morduhai-Boltovskoi in 1914.
Definition
One standard definition (there are slight variants) defines solutions of differential equations of the form
,
where is a polynomial with constant coefficients, as algebraically transcendental or differentially algebraic. Transcendental functions which are not algebraically transcendental are transcendentally transcendental. Hölder's theorem shows that the gamma function is in this category.
Hypertranscendental functions usually arise as the solutions to functional equations, for example the gamma function.
Examples
Hypertranscendental functions
The zeta functions of algebraic number fields, in particular, the Riemann zeta function
The gamma function (cf. Hölder's theorem)
Transcendental but not hypertranscendental functions
The exponential function, logarithm, and the trigonometric and hyperbolic functions.
The generalized hypergeometric functions, including special cases such as Bessel functions (except some special cases which are algebraic).
Non-transcendental (algebraic) functions
All algebraic functions, in particular polynomials.
See also
Hypertranscendental number
Notes
References
Loxton,J.H., Poorten,A.J. van der, "A class of hypertranscendental functions", Aequationes Mathematicae, Periodical volume 16
Mahler,K., "Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen", Math. Z. 32 (1930) 545-585.
Analytic functions
Mathematical analysis
Types of functions
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https://en.wikipedia.org/wiki/Matrix%20coefficient
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In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data. Precisely, it is a function on a compact topological group G obtained by composing a representation of G on a vector space V with a linear map from the endomorphisms of V into V underlying field. It is also called a representative function. They arise naturally from finite-dimensional representations of G as the matrix-entry functions of the corresponding matrix representations. The Peter–Weyl theorem says that the matrix coefficients on G are dense in the Hilbert space of square-integrable functions on G.
Matrix coefficients of representations of Lie groups turned out to be intimately related with the theory of special functions, providing a unifying approach to large parts of this theory. Growth properties of matrix coefficients play a key role in the classification of irreducible representations of locally compact groups, in particular, reductive real and p-adic groups. The formalism of matrix coefficients leads to a generalization of the notion of a modular form. In a different direction, mixing properties of certain dynamical systems are controlled by the properties of suitable matrix coefficients.
Definition
A matrix coefficient (or matrix element) of a linear representation of a group on a vector space is a function on the group, of the type
where is a vector in , is a continuous linear functional on , and is an element of . This function takes scalar values on . If is a Hilbert space, then by the Riesz representation theorem, all matrix coefficients have the form
for some vectors and in .
For of finite dimension, and and taken from a standard basis, this is actually the function given by the matrix entry in a fixed place.
Applications
Finite groups
Matrix coefficients of irreducible representations of finite groups play a prominent role in representation theory of these groups, as developed by Burnside, Frobenius and Schur. They satisfy Schur orthogonality relations. The character of a representation ρ is a sum of the matrix coefficients fvi,ηi, where {vi} form a basis in the representation space of ρ, and {ηi} form the dual basis.
Finite-dimensional Lie groups and special functions
Matrix coefficients of representations of Lie groups were first considered by Élie Cartan.
Israel Gelfand realized that many classical special functions and orthogonal polynomials are expressible as the matrix coefficients of representation of Lie groups G. This description provides a uniform framework for proving many hitherto disparate properties of special functions, such as addition formulas, certain recurrence relations, orthogonality relations, integral representations, and eigenvalue properties with respect to differential operators. Special functions of mathematical physics, such as the trigonometric functions, the hypergeometric function and its generali
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https://en.wikipedia.org/wiki/Quaternion-K%C3%A4hler%20symmetric%20space
|
In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space. Any quaternion-Kähler symmetric space with positive Ricci curvature is compact and simply connected, and is a Riemannian product of quaternion-Kähler symmetric spaces associated to compact simple Lie groups.
For any compact simple Lie group G, there is a unique G/H obtained as a quotient of G by a subgroup
Here, Sp(1) is the compact form of the SL(2)-triple associated with the highest root of G, and K its centralizer in G. These are classified as follows.
The twistor spaces of quaternion-Kähler symmetric spaces are the homogeneous holomorphic contact manifolds, classified by Boothby: they are the adjoint varieties of the complex semisimple Lie groups.
These spaces can be obtained by taking a projectivization of
a minimal nilpotent orbit of the respective complex Lie group.
The holomorphic contact structure is apparent, because
the nilpotent orbits of semisimple Lie groups
are equipped with the Kirillov-Kostant holomorphic symplectic form. This argument also explains how one
can associate a unique Wolf space to each of the simple
complex Lie groups.
See also
Quaternionic discrete series representation
References
. Reprint of the 1987 edition.
.
Differential geometry
Structures on manifolds
Riemannian geometry
Homogeneous spaces
Lie groups
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https://en.wikipedia.org/wiki/Mathematics%20in%20Education%20and%20Industry
|
MEI (Mathematics in Education and Industry) is an independent educational charity and curriculum development body for mathematics education in the United Kingdom. Income generated through its work is used to support the teaching and learning of mathematics.
History
MEI was founded in 1963 with a grant from the Schools & Industry Committee of the Mathematical Association. In 1965 it produced its first exam, Additional Mathematics, then produced an A level course two years later. MEI's A-level exams were the first to include probability.
It was incorporated as a company on 18 October 1996.
Structure
Although independent, MEI works in partnership with many organisations, including the UK Government. MEI is a registered charity with a board of directors and a small professional staff.
Qualifications
GCE AS/A level Mathematics, Further Mathematics and Further Mathematics (Additional) (Published by OCR)
AS Level Statistics
GCSE Mathematics
Foundations of Advanced Mathematics (FAM) – a freestanding course
Introduction to Quantitative Methods (in association with OCR)
OCR MEI Level 3 Core Maths Qualifications (Level 3 Certificate in Quantitative Reasoning and Level 3 Certificate in Quantitative Problem Solving)
Competitions
MEI organises an annual online competition called Ritangle for teams of students of A level Mathematics, the International Baccalaureate and Scottish Highers. Questions are posted on the Integral website, with correct answers releasing a clue for the final question.
References
External links
Educational charities based in the United Kingdom
Mathematics education in the United Kingdom
Qualification awarding bodies in the United Kingdom
Organisations based in Wiltshire
Organizations established in 1963
Science and technology in Wiltshire
Trowbridge
1963 establishments in the United Kingdom
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https://en.wikipedia.org/wiki/Epsilon%20calculus
|
In logic, Hilbert's epsilon calculus is an extension of a formal language by the epsilon operator, where the epsilon operator substitutes for quantifiers in that language as a method leading to a proof of consistency for the extended formal language. The epsilon operator and epsilon substitution method are typically applied to a first-order predicate calculus, followed by a demonstration of consistency. The epsilon-extended calculus is further extended and generalized to cover those mathematical objects, classes, and categories for which there is a desire to show consistency, building on previously-shown consistency at earlier levels.
Epsilon operator
Hilbert notation
For any formal language L, extend L by adding the epsilon operator to redefine quantification:
The intended interpretation of ϵx A is some x that satisfies A, if it exists. In other words, ϵx A returns some term t such that A(t) is true, otherwise it returns some default or arbitrary term. If more than one term can satisfy A, then any one of these terms (which make A true) can be chosen, non-deterministically. Equality is required to be defined under L, and the only rules required for L extended by the epsilon operator are modus ponens and the substitution of A(t) to replace A(x) for any term t.
Bourbaki notation
In tau-square notation from N. Bourbaki's Theory of Sets, the quantifiers are defined as follows:
where A is a relation in L, x is a variable, and juxtaposes a at the front of A, replaces all instances of x with , and links them back to . Then let Y be an assembly, (Y|x)A denotes the replacement of all variables x in A with Y.
This notation is equivalent to the Hilbert notation and is read the same. It is used by Bourbaki to define cardinal assignment since they do not use the axiom of replacement.
Defining quantifiers in this way leads to great inefficiencies. For instance, the expansion of Bourbaki's original definition of the number one, using this notation, has length approximately 4.5 × 1012, and for a later edition of Bourbaki that combined this notation with the Kuratowski definition of ordered pairs, this number grows to approximately 2.4 × 1054.
Modern approaches
Hilbert's program for mathematics was to justify those formal systems as consistent in relation to constructive or semi-constructive systems. While Gödel's results on incompleteness mooted Hilbert's Program to a great extent, modern researchers find the epsilon calculus to provide alternatives for approaching proofs of systemic consistency as described in the epsilon substitution method.
Epsilon substitution method
A theory to be checked for consistency is first embedded in an appropriate epsilon calculus. Second, a process is developed for re-writing quantified theorems to be expressed in terms of epsilon operations via the epsilon substitution method. Finally, the process must be shown to normalize the re-writing process, so that the re-written theorems satisfy the axioms of the theory.
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https://en.wikipedia.org/wiki/Limit%20comparison%20test
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In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series.
Statement
Suppose that we have two series and with for all .
Then if with , then either both series converge or both series diverge.
Proof
Because we know that for every there is a positive integer such that for all we have that , or equivalently
As we can choose to be sufficiently small such that is positive.
So and by the direct comparison test, if converges then so does .
Similarly , so if diverges, again by the direct comparison test, so does .
That is, both series converge or both series diverge.
Example
We want to determine if the series converges. For this we compare it with the convergent series
As we have that the original series also converges.
One-sided version
One can state a one-sided comparison test by using limit superior. Let for all . Then if with and converges, necessarily converges.
Example
Let and for all natural numbers . Now
does not exist, so we cannot apply the standard comparison test. However,
and since converges, the one-sided comparison test implies that converges.
Converse of the one-sided comparison test
Let for all . If diverges and converges, then necessarily
, that is,
. The essential content here is that in some sense the numbers are larger than the numbers .
Example
Let be analytic in the unit disc and have image of finite area. By Parseval's formula the area of the image of is proportional to . Moreover,
diverges. Therefore, by the converse of the comparison test, we have
, that is,
.
See also
Convergence tests
Direct comparison test
References
Further reading
Rinaldo B. Schinazi: From Calculus to Analysis. Springer, 2011, , pp. 50
Michele Longo and Vincenzo Valori: The Comparison Test: Not Just for Nonnegative Series. Mathematics Magazine, Vol. 79, No. 3 (Jun., 2006), pp. 205–210 (JSTOR)
J. Marshall Ash: The Limit Comparison Test Needs Positivity. Mathematics Magazine, Vol. 85, No. 5 (December 2012), pp. 374–375 (JSTOR)
External links
Pauls Online Notes on Comparison Test
Convergence tests
Articles containing proofs
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https://en.wikipedia.org/wiki/Topological%20derivative
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The topological derivative is, conceptually, a derivative of a shape functional with respect to infinitesimal changes in its topology, such as adding an infinitesimal hole or crack. When used in higher dimensions than one, the term topological gradient is also used to name the first-order term of the topological asymptotic expansion, dealing only with infinitesimal singular domain perturbations. It has applications in shape optimization, topology optimization, image processing and mechanical modeling.
Definition
Let be an open bounded domain of , with , which is subject to a nonsmooth perturbation confined in a small region of size with an arbitrary point of and a fixed domain of . Let be a characteristic function associated to the unperturbed domain and be a characteristic function associated to the perforated domain . A given shape functional associated to the topologically perturbed domain, admits the following topological asymptotic expansion:
where is the shape functional associated to the reference domain, is a positive first order correction function of and is the remainder. The function is called the topological derivative of at .
Applications
Structural mechanics
The topological derivative can be applied to shape optimization problems in structural mechanics. The topological derivative can be considered as the singular limit of the shape derivative. It is a generalization of this classical tool in shape optimization. Shape optimization concerns itself with finding an optimal shape. That is, find to minimize some scalar-valued objective function, . The topological derivative technique can be coupled with level-set method.
In 2005, the topological asymptotic expansion for the Laplace equation with respect to the insertion of a short crack inside a plane domain had been found. It allows to detect and locate cracks for a simple model problem: the steady-state heat equation with the heat flux imposed and the temperature measured on the boundary. The topological derivative had been fully developed for a wide range of second-order differential operators and in 2011, it had been applied to Kirchhoff plate bending problem with a fourth-order operator.
Image processing
In the field of image processing, in 2006, the topological derivative has been used to perform edge detection and image restoration. The impact of an insulating crack in the domain is studied. The topological sensitivity gives information on the image edges. The presented algorithm is non-iterative and thanks to the use of spectral methods has a short computing time. Only operations are needed to detect edges, where is the number of pixels. During the following years, other problems have been considered: classification, segmentation, inpainting and super-resolution. This approach can be applied to gray-level or color images. Until 2010, isotropic diffusion was used for image reconstructions. The topological gradient is also able to provide edge orientatio
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https://en.wikipedia.org/wiki/Sharon%20Inkelas
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Sharon Inkelas is a Professor and former Chair of the Linguistics Department at the University of California, Berkeley.
Education and career
Inkelas completed her Bachelor of Arts in mathematics at Pomona College in 1984 and received her PhD in linguistics at Stanford University in 1989 with a dissertation, "Prosodic Constituency in the Lexicon," supervised by Paul Kiparsky. In 1990, she arrived at UC Berkeley as a Miller Institute for Basic Research in Science research fellow and became a faculty member at Berkeley in 1992. She was a Hellman Fellow in 1995. She was named the special faculty adviser to the chancellor on sexual violence/sexual harassment for a three-year term, beginning on July 24, 2017.
Inkelas is noted for her work on phonology interfaces and particularly the interaction between morphology and phonology. Her research interests include cophonology theory, reduplication, affix ordering, child phonology, and the analysis of Turkish.
Honors
Inkelas has long been actively involved in the Linguistic Society of America, serving on their executive committee from 2016-2018. In 2020, Inkelas was inducted as a Fellow of the Linguistic Society of America.
Personal
Inkelas is also a violinist: she played for the symphony orchestra of Stanford University and is a member of the symphony orchestra of the University of California, Davis .
Selected publications
"Reduplication", in Keith Brown, ed., Encyclopedia of Language and Linguistics, Elsevier: Oxford, pp. 417–419, 2006
"Underspecification", in Keith Brown, ed., Encyclopedia of Language and Linguistics, Elsevier: Oxford, pp. 224–226, 2006
"The architecture and the implementation of a finite state pronunciation lexicon for Turkish", with Kemal Oflazer. Computer Speech and Language, pp. 80–106, 2006
Reduplication: Doubling in Morphology, with Cheryl Zoll. Cambridge University Press. 2005. Review.
"Velar Fronting Revisited", with Yvan Rose, in Barbara Beachley, Amanda Brown & Fran Conlin (eds.), Proceedings of the 26th Annual Boston University Conference on Language Development; Somerville, MA: Cascadilla Press
"Turkish stress: a review", with C. Orhan Orgun, Phonology 20, pp. 139–161, 2003
"J's rhymes: a longitudinal case study of language play", Journal of Child Language 30, pp. 557–581, 2003
References
External links
Department Bio
Inkelas's home page
Pomona College alumni
Stanford University alumni
University of California, Berkeley College of Letters and Science faculty
Living people
Linguists from the United States
Women linguists
Year of birth missing (living people)
Fellows of the Linguistic Society of America
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https://en.wikipedia.org/wiki/Trigonometric%20moment%20problem
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In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence {α0, ... αn }, does there exist a positive Borel measure μ on the interval [0, 2π] such that
In other words, an affirmative answer to the problems means that {α0, ... αn } are the first n + 1 Fourier coefficients of some positive Borel measure μ on [0, 2π].
Characterization
The trigonometric moment problem is solvable, that is, {αk} is a sequence of Fourier coefficients, if and only if the (n + 1) × (n + 1) Toeplitz matrix
is positive semidefinite.
The "only if" part of the claims can be verified by a direct calculation.
We sketch an argument for the converse. The positive semidefinite matrix A defines a sesquilinear product on Cn + 1, resulting in a Hilbert space
of dimensional at most n + 1, a typical element of which is an equivalence class denoted by [f]. The Toeplitz structure of A means that a "truncated" shift is a partial isometry on . More specifically, let { e0, ...en } be the standard basis of Cn + 1. Let be the subspace generated by { [e0], ... [en - 1] } and be the subspace generated by { [e1], ... [en] }. Define an operator
by
Since
V can be extended to a partial isometry acting on all of . Take a minimal unitary extension U of V, on a possibly larger space (this always exists). According to the spectral theorem, there exists a Borel measure m on the unit circle T such that for all integer k
For k = 0,...,n, the left hand side is
So
Finally, parametrize the unit circle T by eit on [0, 2π] gives
for some suitable measure μ.
Parametrization of solutions
The above discussion shows that the trigonometric moment problem has infinitely many solutions if the Toeplitz matrix A is invertible. In that case, the solutions to the problem are in bijective correspondence with minimal unitary extensions of the partial isometry V.
References
N.I. Akhiezer, The Classical Moment Problem, Olivier and Boyd, 1965.
N.I. Akhiezer, M.G. Krein, Some Questions in the Theory of Moments, Amer. Math. Soc., 1962.
Probability problems
Measure theory
Functional analysis
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https://en.wikipedia.org/wiki/Discrete%20phase-type%20distribution
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The discrete phase-type distribution is a probability distribution that results from a system of one or more inter-related geometric distributions occurring in sequence, or phases. The sequence in which each of the phases occur may itself be a stochastic process. The distribution can be represented by a random variable describing the time until absorption of an absorbing Markov chain with one absorbing state. Each of the states of the Markov chain represents one of the phases.
It has continuous time equivalent in the phase-type distribution.
Definition
A terminating Markov chain is a Markov chain where all states are transient, except one which is absorbing.
Reordering the states, the transition probability matrix of a terminating Markov chain with transient states is
where is a matrix, and are column vectors with entries, and . The transition matrix is characterized entirely by its upper-left block .
Definition. A distribution on is a discrete phase-type distribution if it is the distribution of the first passage time to the absorbing state of a terminating Markov chain with finitely many states.
Characterization
Fix a terminating Markov chain. Denote the upper-left block of its transition matrix and the initial distribution.
The distribution of the first time to the absorbing state is denoted or .
Its cumulative distribution function is
for , and its density function is
for . It is assumed the probability of process starting in the absorbing state is zero. The factorial moments of the distribution function are given by,
where is the appropriate dimension identity matrix.
Special cases
Just as the continuous time distribution is a generalisation of the exponential distribution, the discrete time distribution is a generalisation of the geometric distribution, for example:
Degenerate distribution, point mass at zero or the empty phase-type distribution – 0 phases.
Geometric distribution – 1 phase.
Negative binomial distribution – 2 or more identical phases in sequence.
Mixed Geometric distribution – 2 or more non-identical phases, that each have a probability of occurring in a mutually exclusive, or parallel, manner. This is the discrete analogue of the Hyperexponential distribution, but it is not called the Hypergeometric distribution, since that name is in use for an entirely different type of discrete distribution.
See also
Phase-type distribution
Queueing model
Queueing theory
References
M. F. Neuts. Matrix-Geometric Solutions in Stochastic Models: an Algorithmic Approach, Chapter 2: Probability Distributions of Phase Type; Dover Publications Inc., 1981.
G. Latouche, V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modelling, 1st edition. Chapter 2: PH Distributions; ASA SIAM, 1999.
Discrete distributions
Types of probability distributions
Markov models
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https://en.wikipedia.org/wiki/Bindura%20University%20of%20Science%20Education
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Bindura University of Science Education is a Zimbabwean university offering courses within the fields of Science, Technology, Engineering and Mathematics, Science Education, Commerce and Social Sciences.
The main campus is located 5 km from Bindura town center, with a separate campus which houses the Faculty of Science on the Trojan Road. The Faculty of Social Science along the main library is located in the city centre.
History
The origins of the Bindura University of Science Education (BUSE) formerly Bindura University College of Science Education (BUCSE) can be traced to the Zimbabwe-Cuba Teacher Training Programme, which started in the mid-1980s. The programme used to send Zimbabwean student teachers to Cuba for training in Science Education. Known as the best University in terms of education in Zimbabwe.
The programme was relocated to Zimbabwe in 1995 for economic reasons. A decision was made to set up a college in Bindura under the auspices of the University of Zimbabwe, but which would be turned into a full-fledged university within a period of two to four years. The college admitted its first group of 125 students in March 1996.
An act of parliament, the Bindura University of Science Education Act, was passed in February 2000 conferring university status to the College becoming the fourth state university established in Zimbabwe. The first graduation ceremony was held in 2003 where the Chancellor and Vice-Chancellor of the university were installed as well as capping a group of 140 graduands.
Zimbabwe opened its first-ever School of Optometry at the Bindura University of Science Education in 2018.
In 2019, a BUSE alumnus, Tatenda Magetsi received a prestigious Rhodes scholarship to study at the Oxford University in the United Kingdom.
Location
The university is located in Bindura, which is about 87 km north east of Harare. Bindura is the provincial capital of Mashonaland Central Province and is a small town of about 40,000 inhabitants. It offers a quiet semi-urban environment.
Politics
The university has experienced problems in Zimbabwe mainly because of its location in Mashonaland Central which is considered a stronghold of the ruling party ZANU PF. In 2002 a student belonging to the MDC party was brutally assaulted by suspected ZANU-PF supporters and the university was briefly closed.
Vice Chancellors
Professor Cowdeng Chikomba 1996-2002 (Pro-Vice Chancellor) late
Professor Sam Abel Tswana 2002 - 2008 (late)
Professor Eddie Mwenje 2010–present
Notable faculty
Christopher Chetsanga - Harvard - discovered two enzymes involved in DNA repair
Sports
Bindura University houses the National Sports Academy, which facilitates the development of identified talent with the aim of achieving international success. It is also set to promote research in high-performance sport development and any such research that can improve decision making in sport as well as the establishment of partnerships with international organizations and don
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https://en.wikipedia.org/wiki/Albert%20Gemmrich
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Albert Gemmrich (born 13 February 1955) is a French former professional footballer who played as a striker. He obtained five caps scoring twice for the France national team.
Career statistics
Scores and results list France's goal tally first, score column indicates score after each Gemmrich goal.
Honours
French championship in 1979 with RC Strasbourg
References
External links
French Football Federation Profile
Stats
1955 births
Living people
People from Haguenau
Footballers from Bas-Rhin
French men's footballers
France men's international footballers
French people of German descent
Men's association football forwards
Ligue 1 players
Ligue 2 players
AS Mutzig players
RC Strasbourg Alsace players
FC Girondins de Bordeaux players
Lille OSC players
OGC Nice players
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https://en.wikipedia.org/wiki/Population%20change
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Population change is simply the change in the number of people in a specified area during a specific time period. Demographics (or demography) is the study of population statistics, their variation and its causes. These statistics include birth rates, death rates (and hence life expectancy), migration rates and sex ratios. All of these statistics are investigated by censuses and surveys conducted over a period of time. Some demographic information can also be obtained from historical maps, and aerial photographs.
A major purpose of demography is to inform government and business planning of the resources that will be required as a result of population changes.
Population trends
The change in total population over a period is equal to the number of births, minus the number of deaths, plus or minus the net amount of migration in a population. The number of births can be projected as the number of females at each relevant age multiplied by the assumed fertility rate. The number of deaths can be projected as the sum of the numbers of each age and sex in the population multiplied by their respective mortality rates. For many centuries, the overall population of the world changed relatively slowly: very broadly, the numbers of births were balanced by numbers of deaths (including high rates of infant immortality). Infant mortality was high for various reasons such as ignorance, insufficient health facilities, and sometimes lack of food. Occasionally, farmers were unable to produce enough food for the population, resulting in death from starvation. However more recently, and especially in the 20th and 21st centuries, due to growth in technology, education, and medical care, the world population has increased rapidly, as many more people have survived to child-bearing age. Natural resources that we were once scarce are now being mass-produced. Because of this increase, some countries have adopted policies to try to control population growth. These policies include active measures to reduce the numbers of births (e.g. "one-child policy") as well as education. In many countries, fertility rates have declined, due to better education, better available birth control, better pension provision reducing economic dependence on one's children in old age, and in response to lower infant mortality. Those who wait until they are older before starting a family may find it more difficult to do so as fertility declines with age. One of the biological reasons for this is abnormal chromosome segregation during cell division in older eggs. In some parts of society there are also now more women formally employed in the workforce. Recent studies show that there has been a decline in fertility from the ages 25 to 29. In general, fertility rates have relatively decreased at ages under 30.
One way to visualize population change is to examine population pyramids. These display graphically how many people of each gender there are in each age bracket in a given population. (A
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https://en.wikipedia.org/wiki/K-finite
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In mathematics, a K-finite function is a type of generalized trigonometric polynomial. Here K is some compact group, and the generalization is from the circle group T.
From an abstract point of view, the characterization of trigonometric polynomials amongst other functions F, in the harmonic analysis of the circle, is that for functions F in any of the typical function spaces, F is a trigonometric polynomial if and only if its Fourier coefficients
an
vanish for |n| large enough, and that this in turn is equivalent to the statement that all the translates
F(t + θ)
by a fixed angle θ lie in a finite-dimensional subspace. One implication here is trivial, and the other, starting from a finite-dimensional invariant subspace, follows from complete reducibility of representations of T.
From this formulation, the general definition can be seen: for a representation ρ of K on a vector space V, a K-finite vector v in V is one for which the
ρ(k).v
for k in K span a finite-dimensional subspace. The union of all finite-dimension K-invariant subspaces is itself a subspace, and K-invariant, and consists of all the K-finite vectors. When all v are K-finite, the representation ρ itself is called K-finite.
References
Representation theory of groups
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https://en.wikipedia.org/wiki/Admissible%20representation
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In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra.
Real or complex reductive Lie groups
Let G be a connected reductive (real or complex) Lie group. Let K be a maximal compact subgroup. A continuous representation (π, V) of G on a complex Hilbert space V is called admissible if π restricted to K is unitary and each irreducible unitary representation of K occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation of G.
An admissible representation π induces a -module which is easier to deal with as it is an algebraic object. Two admissible representations are said to be infinitesimally equivalent if their associated -modules are isomorphic. Though for general admissible representations, this notion is different than the usual equivalence, it is an important result that the two notions of equivalence agree for unitary (admissible) representations. Additionally, there is a notion of unitarity of -modules. This reduces the study of the equivalence classes of irreducible unitary representations of G to the study of infinitesimal equivalence classes of admissible representations and the determination of which of these classes are infinitesimally unitary. The problem of parameterizing the infinitesimal equivalence classes of admissible representations was fully solved by Robert Langlands and is called the Langlands classification.
Totally disconnected groups
Let G be a locally compact totally disconnected group (such as a reductive algebraic group over a nonarchimedean local field or over the finite adeles of a global field). A representation (π, V) of G on a complex vector space V is called smooth if the subgroup of G fixing any vector of V is open. If, in addition, the space of vectors fixed by any compact open subgroup is finite dimensional then π is called admissible. Admissible representations of p-adic groups admit more algebraic description through the action of the Hecke algebra of locally constant functions on G.
Deep studies of admissible representations of p-adic reductive groups were undertaken by Casselman and by Bernstein and Zelevinsky in the 1970s. Progress was made more recently by Howe, Moy, Gopal Prasad and Bushnell and Kutzko, who developed a theory of types and classified the admissible dual (i.e. the set of equivalence classes of irreducible admissible representations) in many cases.
Notes
References
Chapter VIII of
Representation theory
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https://en.wikipedia.org/wiki/Joel%20B.%20Wolowelsky
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Joel Blumenthal Wolowelsky (b. 1946) is a Modern Orthodox thinker and author. He is the dean of faculty at the Yeshivah of Flatbush high school, where he teaches Ethics and mathematics. He has written extensively on topics pertaining to the role of women in Judaism and Jewish medical ethics. He served as Associate Editor of Tradition, the Journal of Jewish Thought, and The Young One, published by the Rabbinical Council of America, the Tora u-Madda Journal published by Yeshiva University, and MeOtzer HoRav: Selected Writings of Rabbi Joseph B. Soloveitchik.
Education and career
Wolowelsky earned his Bachelor of Science degree at Yeshiva University in 1969 and his doctorate in philosophy at New York University Steinhardt School of Culture, Education and Human Development in 1979.
He served as chairman of advanced placement studies at Yeshivah of Flatbush.
Wolowelsky is on the advisory boards of the Lookstein Center for Jewish Education at Bar-Ilan University, the Boston Initiative for Excellence in Jewish Day Schools, and the Pardes Educators Program in Jerusalem.
Awards
Yeshiva University Lifetime Achievement Award in Jewish Education (2010)
Selected bibliography
Books
(ed. with Emanuel Feldman)
(ed.)
(ed. with Lawrence H. Schiffman)
(ed. with David Shatz)
(ed. with Emanuel Feldman)
(ed. with Emanuel Feldman)
MeOtzer HoRav series
(ed. with David Shatz and Reuven Ziegler)
(ed. with Eli D. Clark and Reuven Ziegler)
(ed. with Reuven Ziegler)
(ed. with David Shatz and Reuven Ziegler)
(ed. with David Shatz)
References
American Modern Orthodox Jews
1946 births
Living people
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https://en.wikipedia.org/wiki/Kelvin%20functions
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In applied mathematics, the Kelvin functions berν(x) and beiν(x) are the real and imaginary parts, respectively, of
where x is real, and , is the νth order Bessel function of the first kind. Similarly, the functions kerν(x) and keiν(x) are the real and imaginary parts, respectively, of
where is the νth order modified Bessel function of the second kind.
These functions are named after William Thomson, 1st Baron Kelvin.
While the Kelvin functions are defined as the real and imaginary parts of Bessel functions with x taken to be real, the functions can be analytically continued for complex arguments With the exception of bern(x) and bein(x) for integral n, the Kelvin functions have a branch point at x = 0.
Below, is the gamma function and is the digamma function.
ber(x)
For integers n, bern(x) has the series expansion
where is the gamma function. The special case ber0(x), commonly denoted as just ber(x), has the series expansion
and asymptotic series
,
where
bei(x)
For integers n, bein(x) has the series expansion
The special case bei0(x), commonly denoted as just bei(x), has the series expansion
and asymptotic series
where α, , and are defined as for ber(x).
ker(x)
For integers n, kern(x) has the (complicated) series expansion
The special case ker0(x), commonly denoted as just ker(x), has the series expansion
and the asymptotic series
where
kei(x)
For integer n, kein(x) has the series expansion
The special case kei0(x), commonly denoted as just kei(x), has the series expansion
and the asymptotic series
where β, f2(x), and g2(x) are defined as for ker(x).
See also
Bessel function
References
External links
Weisstein, Eric W. "Kelvin Functions." From MathWorld—A Wolfram Web Resource.
GPL-licensed C/C++ source code for calculating Kelvin functions at codecogs.com:
Special hypergeometric functions
Functions
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https://en.wikipedia.org/wiki/George%20White%20%28preacher%29
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Rev. George White (March 12, 1802 – April 30, 1887) was an Episcopalian preacher, amateur historian, and archaeologist in Georgia, United States. In 1849 he published Statistics of the State of Georgia: Including an Account of Its Natural, Civil, and Ecclesiastical History Together with a Particular Description of Each County, Notices of the Manners and Customs of Its Aboriginal Tribes, and a Correct Map of the State. His book entitled Historical Collections of Georgia: Containing the Most Interesting Facts, Traditions, Biographical Sketches, Etc., Relating to Its History and Antiquities, from Its First Settlement to the Present Time, has been widely referenced by scholars working with Georgia history since its publication in 1854.
References
Coulter, E. Merton (1936) "What the South Has Done About Its History," Journal of Southern History 2(1):3-28.
1802 births
1887 deaths
19th-century American Episcopalians
American archaeologists
American Episcopal priests
19th-century American historians
19th-century American male writers
People from Georgia (U.S. state)
Place of birth missing
Place of death missing
19th-century American clergy
American male non-fiction writers
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https://en.wikipedia.org/wiki/Linear%20relation
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In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution.
More precisely, if are elements of a (left) module over a ring (the case of a vector space over a field is a special case), a relation between is a sequence of elements of such that
The relations between form a module. One is generally interested in the case where is a generating set of a finitely generated module , in which case the module of the relations is often called a syzygy module of . The syzygy module depends on the choice of a generating set, but it is unique up to the direct sum with a free module. That is, if and are syzygy modules corresponding to two generating sets of the same module, then they are stably isomorphic, which means that there exist two free modules and such that and are isomorphic.
Higher order syzygy modules are defined recursively: a first syzygy module of a module is simply its syzygy module. For , a th syzygy module of is a syzygy module of a -th syzygy module. Hilbert's syzygy theorem states that, if is a polynomial ring in indeterminates over a field, then every th syzygy module is free. The case is the fact that every finite dimensional vector space has a basis, and the case is the fact that is a principal ideal domain and that every submodule of a finitely generated free module is also free.
The construction of higher order syzygy modules is generalized as the definition of free resolutions, which allows restating Hilbert's syzygy theorem as a polynomial ring in indeterminates over a field has global homological dimension .
If and are two elements of the commutative ring , then is a relation that is said trivial. The module of trivial relations of an ideal is the submodule of the first syzygy module of the ideal that is generated by the trivial relations between the elements of a generating set of an ideal. The concept of trivial relations can be generalized to higher order syzygy modules, and this leads to the concept of the Koszul complex of an ideal, which provides information on the non-trivial relations between the generators of an ideal.
Basic definitions
Let be a ring, and be a left -module. A linear relation, or simply a relation between elements of is a sequence of elements of such that
If is a generating set of , the relation is often called a syzygy of . It makes sense to call it a syzygy of without regard to because, although the syzygy module depends on the chosen generating set, most of its properties are independent; see , below.
If the ring is Noetherian, or, at least coherent, and if is finitely generated, then the syzygy module is also finitely generated. A syzygy module of this syzygy module is a second syzygy module of . Continuing this way one can define a th syzygy module for every positive integer .
Hilbert's syzygy theorem asserts that, if is a finitely generated module o
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https://en.wikipedia.org/wiki/Prismatic%20uniform%20polyhedron
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In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in two infinite families, the uniform prisms and the uniform antiprisms. All have their vertices in parallel planes and are therefore prismatoids.
Vertex configuration and symmetry groups
Because they are isogonal (vertex-transitive), their vertex arrangement uniquely corresponds to a symmetry group.
The difference between the prismatic and antiprismatic symmetry groups is that Dph has the vertices lined up in both planes, which gives it a reflection plane perpendicular to its p-fold axis (parallel to the {p/q} polygon); while Dpd has the vertices twisted relative to the other plane, which gives it a rotatory reflection. Each has p reflection planes which contain the p-fold axis.
The Dph symmetry group contains inversion if and only if p is even, while Dpd contains inversion symmetry if and only if p is odd.
Enumeration
There are:
prisms, for each rational number p/q > 2, with symmetry group Dph;
antiprisms, for each rational number p/q > 3/2, with symmetry group Dpd if q is odd, Dph if q is even.
If p/q is an integer, i.e. if q = 1, the prism or antiprism is convex. (The fraction is always assumed to be stated in lowest terms.)
An antiprism with p/q < 2 is crossed or retrograde; its vertex figure resembles a bowtie. If p/q < 3/2 no uniform antiprism can exist, as its vertex figure would have to violate the triangle inequality. If p/q = 3/2 the uniform antiprism is degenerate (has zero height).
Forms by symmetry
Note: The tetrahedron, cube, and octahedron are listed here with dihedral symmetry (as a digonal antiprism, square prism and triangular antiprism respectively), although if uniformly colored, the tetrahedron also has tetrahedral symmetry and the cube and octahedron also have octahedral symmetry.
See also
Uniform polyhedron
Prism (geometry)
Antiprism
References
Cromwell, P.; Polyhedra, CUP, Hbk. 1997, . Pbk. (1999), . p.175
.
External links
Prisms and Antiprisms George W. Hart
Prismatoid polyhedra
Uniform polyhedra
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https://en.wikipedia.org/wiki/Darboux%20frame
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In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a surface embedded in Euclidean space. It is named after French mathematician Jean Gaston Darboux.
Darboux frame of an embedded curve
Let S be an oriented surface in three-dimensional Euclidean space E3. The construction of Darboux frames on S first considers frames moving along a curve in S, and then specializes when the curves move in the direction of the principal curvatures.
Definition
At each point of an oriented surface, one may attach a unit normal vector in a unique way, as soon as an orientation has been chosen for the normal at any particular fixed point. If is a curve in , parametrized by arc length, then the Darboux frame of is defined by
(the unit tangent)
(the unit normal)
(the tangent normal)
The triple defines a positively oriented orthonormal basis attached to each point of the curve: a natural moving frame along the embedded curve.
Geodesic curvature, normal curvature, and relative torsion
Note that a Darboux frame for a curve does not yield a natural moving frame on the surface, since it still depends on an initial choice of tangent vector. To obtain a moving frame on the surface, we first compare the Darboux frame of γ with its Frenet–Serret frame. Let
(the unit tangent, as above)
(the Frenet normal vector)
(the Frenet binormal vector).
Since the tangent vectors are the same in both cases, there is a unique angle α such that a rotation in the plane of N and B produces the pair t and u:
Taking a differential, and applying the Frenet–Serret formulas yields
where:
κg is the geodesic curvature of the curve,
κn is the normal curvature of the curve, and
τr is the relative torsion (also called geodesic torsion) of the curve.
Darboux frame on a surface
This section specializes the case of the Darboux frame on a curve to the case when the curve is a principal curve of the surface (a line of curvature). In that case, since the principal curves are canonically associated to a surface at all non-umbilic points, the Darboux frame is a canonical moving frame.
The trihedron
The introduction of the trihedron (or trièdre), an invention of Darboux, allows for a conceptual simplification of the problem of moving frames on curves and surfaces by treating the coordinates of the point on the curve and the frame vectors in a uniform manner. A trihedron consists of a point P in Euclidean space, and three orthonormal vectors e1, e2, and e3 based at the point P. A moving trihedron is a trihedron whose components depend on one or more parameters. For example, a trihedron moves along a curve if the point P depends on a single parameter s, and P(s) traces out the curve. Similarly, if P(s,t) depends on a pair of parameters, then this traces out a surface.
A trihed
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