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https://en.wikipedia.org/wiki/Condensation%20lemma | In set theory, a branch of mathematics, the condensation lemma is a result about sets in the
constructible universe.
It states that if X is a transitive set and is an elementary submodel of some level of the constructible hierarchy Lα, that is, , then in fact there is some ordinal such that .
More can be said: If X is not transitive, then its transitive collapse is equal to some , and the hypothesis of elementarity can be weakened to elementarity only for formulas which are in the Lévy hierarchy. Also, the assumption that X be transitive automatically holds when .
The lemma was formulated and proved by Kurt Gödel in his proof that the axiom of constructibility implies GCH.
References
(theorem II.5.2 and lemma II.5.10)
Inline citations
Constructible universe
Lemmas in set theory |
https://en.wikipedia.org/wiki/FriCAS | FriCAS is a general purpose computer algebra system with a strong focus on mathematical research and development of new algorithms. It comprises an interpreter, a compiler and a still-growing library
of more than 1,000 domains and categories.
FriCAS provides a strongly typed high-level programming language called SPAD and a similar interactive language
that uses type-inferencing for convenience. Aldor was intentionally developed being the
next generation compiler for the Axiom CAS and its forks. FriCAS (optionally) allows running Aldor programs. Both languages
share a similar syntax and a sophisticated (dependent) type system.
FriCAS is comprehensively documented and available as source code and as a binary distribution for the most common
platforms. Compiling the sources requires besides other prerequisites a Common Lisp environment (whereby
many of the major implementations are supported and freely available as open source).
FriCAS runs on many POSIX platforms such as Linux, macOS, Unix,
BSD as well as under Cygwin and
Microsoft Windows (WSL).
History
FriCAS is a descendant of Axiom
which itself has its origin in Scratchpad, a project that started in 1965 by James Griesmer
at IBM laboratories.
For more details see Axiom/History.
Examples
FriCAS has a largely complete implementation of the
Risch–Bronstein–Trager algorithm.
Another useful feature is stream:
)set stream calculate 5
exp_series := series(exp x, x=0)
Type: UnivariatePuiseuxSeries(Expression(Integer),x,0)
So any coefficient may be retrieved, for instance :
coefficient(exp_series,40)
Type: Expression(Integer)
See also
List of computer algebra systems
References
External links
FriCAS Homepage
FriCAS Wiki
SandBox: Try FriCAS online
Documentation at fricas.github.io
FriCAS API (sphinx)
Source code repositories: SourceForge, GitHub
Rosetta stone (pdf)
Rosetta stone (some other formats)
Rosetta Stone (html)
rosettacode.org, Category:SPAD
Forum: fricas-devel
FriCAS Jupyter Kernel (jfricas/src)
FriCAS Jupyter (pypi)
Related:
Axiom
OpenAxiom
A# programming language
Aldor programming language
Common Lisp (programming language) software
Computer algebra system software for Linux
Free computer algebra systems
Free software programmed in Lisp
Software using the BSD license |
https://en.wikipedia.org/wiki/MWX | MWX may refer to:
Morley–Wang–Xu element, a canonical construction in applied mathematics
Muan International Airport (IATA: MWX), South Jeolla Province, South Korea |
https://en.wikipedia.org/wiki/Brumer%E2%80%93Stark%20conjecture | The Brumer–Stark conjecture is a conjecture in algebraic number theory giving a rough generalization of both the analytic class number formula for Dedekind zeta functions, and also of Stickelberger's theorem about the factorization of Gauss sums. It is named after Armand Brumer and Harold Stark.
It arises as a special case (abelian and first-order) of Stark's conjecture, when the place that splits completely in the extension is finite. There are very few cases where the conjecture is known to be valid. Its importance arises, for instance, from its connection with Hilbert's twelfth problem.
Statement of the conjecture
Let be an abelian extension of global fields, and let be a set of places of containing the Archimedean places and the prime ideals that ramify in . The -imprimitive equivariant Artin L-function is obtained from the usual equivariant Artin L-function by removing the Euler factors corresponding to the primes in from the Artin L-functions from which the equivariant function is built. It is a function on the complex numbers taking values in the complex group ring where is the Galois group of . It is analytic on the entire plane, excepting a lone simple pole at .
Let be the group of roots of unity in . The group acts on ; let be the annihilator of as a -module. An important theorem, first proved by C. L. Siegel and later independently by Takuro Shintani, states that is actually in . A deeper theorem, proved independently by Pierre Deligne and Ken Ribet, Daniel Barsky, and Pierrette Cassou-Noguès, states that is in . In particular, is in , where is the cardinality of .
The ideal class group of is a -module. From the above discussion, we can let act on it. The Brumer–Stark conjecture says the following:
Brumer–Stark Conjecture. For each nonzero fractional ideal of , there is an "anti-unit" such that
The extension is abelian.
The first part of this conjecture is due to Armand Brumer, and Harold Stark originally suggested that the second condition might hold. The conjecture was first stated in published form by John Tate.
The term "anti-unit" refers to the condition that is required to be 1 for each Archimedean place .
Progress
The Brumer Stark conjecture is known to be true for extensions where
is cyclotomic: this follows from Stickelberger's theorem
is abelian over
is a quadratic extension
is a biquadratic extension
In 2020, Dasgupta and Kakde proved the Brumer–Stark conjecture away from the prime 2. In 2023, a full proof of the conjecture has been announced.
Function field analogue
The analogous statement in the function field case is known to be true, having been proved by John Tate and Pierre Deligne in 1984, with a different proof by David Hayes in 1985.
References
Conjectures
Unsolved problems in number theory
Algebraic number theory
Zeta and L-functions |
https://en.wikipedia.org/wiki/Torashi%20Shimazu | is a former Japanese football player.
Club statistics
References
External links
1978 births
Living people
Toin University of Yokohama alumni
Japanese men's footballers
J1 League players
J2 League players
Ventforet Kofu players
Tokushima Vortis players
JEF United Chiba players
Iwate Grulla Morioka players
Men's association football goalkeepers
Association football people from Hamamatsu |
https://en.wikipedia.org/wiki/Deutscher%20Sportclub%20f%C3%BCr%20Fu%C3%9Fballstatistiken | The Deutscher Sportclub für Fußballstatistiken e.V., (English: German sports club for football statistics) short DSFS is an association dedicated to collecting and publishing German football statistics, similar to the RSSSF, and is a member of the German Olympic Society.
The club used to be best known for its annual publication, the Deutscher Fussball-Almanach, a yearbook on German football. Unlike other yearbooks, it does not so much focus on professional football, but rather covers the higher amateur leagues.
History
The DSFS was formed on 1 July 1971 but was only registered in 1979. It was formed by six football enthusiasts after Helmut Druwen posted an add in the kicker sport magazine looking for people interested in football statistics. Having grown to a membership of 50, another add in the kicker in 1978 pushed the membership drive ahead and in 1979 the club finally became properly registered, joining the German Olympic Society as well.
From 1980, the club started publishing brochures and a club magazine. But after 1983, the DSFS went into decline for the first time, recovering only in 1986 from several years of inactivity.
With the re-establishment of the Regionalliga in German football in 1994, a reform of the club's annual publication was started. Year by year until 2004 the contents was expanded until covering all divisions of German football on the top five levels including the top youth and women's leagues. Consequently, the name ofthis annual publication was changed to Deutschlands Fussball in Zahlen (English: Germany's football by numbers).
In 1998, it was declared Gemeinnützig, similar to a Non-profit organization, which entitles German clubs to certain tax benefits. In 2004, the number of members was at its peak with almost 430 members. Since then the club lost several of them and now has approximately 378 members (2023 census).
As of September 2022, the club's president is Michael Diepenbrock.
See also
International Federation of Football History & Statistics
RSSSF
References
External links
Association football organizations
Football mass media in Germany
1971 establishments in Germany
Sports organizations established in 1971 |
https://en.wikipedia.org/wiki/2009%20Brisbane%20Lions%20season | This article covers the 2009 AFL season results for AFL team, the Brisbane Lions.
Squad for 2009
Statistics are correct as of start of 2009 season.
Player Changes
In
Out
Ladder
Results
NAB Cup
Home and Away Season
Finals series
Week 1
Week 2
Statistics
Leading Goalkickers
Milestones
Awards
References
External links
Official Website of the Brisbane Lions Football Club
Official Website of the Australian Football League
Brisbane Lions
2009
Brisbane Lions |
https://en.wikipedia.org/wiki/Christophe%20Gadbled | Christophe Gadbled (1734 – 11 October 1782) was a mathematics professor at the University of Caen Normandy.
Gadbled was born in Saint-Martin-le-Bouillant. He is known to have been the mentor of Pierre-Simon Laplace. He died in Caen.
Books by Gadbled
Exposé des quelques unes des vérités rigoureusement démontrées par les géomètres et rejetées par l'auteur du "Compendium de physique", Caen, 1775
Exercice sur la théorie de la navigation, Caen, 1779
References
Édouard Frère, Manuel du bibliographe normand, t. II, Rouen, Le Brument, 1860, p. 1
Annuaire du département de la Manche, p. 307, édition de 1829 (Saint-Lô, imprimerie J. Elie)
1734 births
1782 deaths
Academic staff of the University of Caen Normandy
18th-century French mathematicians |
https://en.wikipedia.org/wiki/Reiichi%20Ikegami | is a former Japanese football player.
Ikegami previously played for FC Tokyo in the J1 League.
Club statistics
References
External links
1983 births
Living people
Sendai University alumni
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
FC Tokyo players
Thespakusatsu Gunma players
FC Kariya players
FC Gifu players
Men's association football midfielders
Association football people from Chiba (city) |
https://en.wikipedia.org/wiki/Satake%20diagram | In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify simple Lie algebras over the field of real numbers. The Satake diagrams associated to a Dynkin diagram classify real forms of the complex Lie algebra corresponding to the Dynkin diagram.
More generally, the Tits index or Satake–Tits diagram of a reductive algebraic group over a field is a generalization of the Satake diagram to arbitrary fields, introduced by , that reduces the classification of reductive algebraic groups to that of anisotropic reductive algebraic groups.
Satake diagrams are not the same as Vogan diagrams of a Lie group, although they look similar.
Definition
A Satake diagram is obtained from a Dynkin diagram by blackening some vertices, and connecting other vertices in pairs by arrows, according to certain rules.
Suppose that G is an algebraic group defined over a field k, such as the reals. We let S be a maximal split torus in G, and take T to be a maximal torus containing S defined over the separable algebraic closure K of k. Then G(K) has a Dynkin diagram with respect to some choice of positive roots of T. This Dynkin diagram has a natural action of the Galois group of K/k. Also some of the simple roots vanish on S. The Satake–Tits diagram is given by the Dynkin diagram D, together with the action of the Galois group, with the simple roots vanishing on S colored black. In the case when k is the field of real numbers, the absolute Galois group has order 2, and its action on D is represented by drawing conjugate points of the Dynkin diagram near each other, and the Satake–Tits diagram is called a Satake diagram.
Examples
Compact Lie algebras correspond to the Satake diagram with all vertices blackened.
Split Lie algebras correspond to the Satake diagram with only white (i.e., non blackened) and unpaired vertices.
A table can be found at .
Differences between Satake and Vogan diagrams
Both Satake and Vogan diagrams are used to classify semisimple Lie groups or algebras (or algebraic groups) over the reals and both consist of Dynkin diagrams enriched by blackening a subset of the nodes and connecting some pairs of vertices by arrows. Satake diagrams, however, can be generalized to any field (see above) and fall under the general paradigm of Galois cohomology, whereas Vogan diagrams are defined specifically over the reals. Generally speaking, the structure of a real semisimple Lie algebra is encoded in a more transparent way in its Satake diagram, but Vogan diagrams are simpler to classify.
The essential difference is that the Satake diagram of a real semisimple Lie algebra with Cartan involution θ and associated Cartan pair (the +1 and −1 eigenspaces of θ) is defined by starting from a maximally noncompact θ-stable Cartan subalgebra , that is, one for which and is as small as possible (in the presentation above, appears as the Lie algebra of the maxim |
https://en.wikipedia.org/wiki/Daiki%20Sato%20%28footballer%2C%20born%201988%29 | was a Japanese football player.
Death
On 19 July 2010, Daiki Sato died cause of heart attack.
Club statistics
References
External links
1988 births
2010 deaths
Association football people from Chiba Prefecture
Japanese men's footballers
J2 League players
Thespakusatsu Gunma players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Akira%20Ishigame | is a former Japanese football player.
Club statistics
References
External links
1985 births
Living people
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
Omiya Ardija players
Thespakusatsu Gunma players
Men's association football defenders |
https://en.wikipedia.org/wiki/Kazuma%20Kita | is a former Japanese football player.
Club statistics
References
External links
1981 births
Living people
Kokushikan University alumni
Association football people from Ishikawa Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Thespakusatsu Gunma players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Satoshi%20Tokizawa | is a Japanese football player.
Club statistics
Updated to 23 February 2020.
References
External links
1985 births
Living people
Association football people from Gunma Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Tokyo Verdy players
Thespakusatsu Gunma players
FC Tokyo players
Montedio Yamagata players
FC Gifu players
FC Maruyasu Okazaki players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Dimension%20theory%20%28algebra%29 | In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a theory for such an apparently simple notion results from the existence of many definitions of dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety). A large part of dimension theory consists in studying the conditions under which several dimensions are equal, and many important classes of commutative rings may be defined as the rings such that two dimensions are equal; for example, a regular ring is a commutative ring such that the homological dimension is equal to the Krull dimension.
The theory is simpler for commutative rings that are finitely generated algebras over a field, which are also quotient rings of polynomial rings in a finite number of indeterminates over a field. In this case, which is the algebraic counterpart of the case of affine algebraic sets, most of the definitions of the dimension are equivalent. For general commutative rings, the lack of geometric interpretation is an obstacle to the development of the theory; in particular, very little is known for non-noetherian rings. (Kaplansky's Commutative rings gives a good account of the non-noetherian case.)
Throughout the article, denotes Krull dimension of a ring and the height of a prime ideal (i.e., the Krull dimension of the localization at that prime ideal.) Rings are assumed to be commutative except in the last section on dimensions of non-commutative rings.
Basic results
Let R be a noetherian ring or valuation ring. Then
If R is noetherian, this follows from the fundamental theorem below (in particular, Krull's principal ideal theorem), but it is also a consequence of a more precise result. For any prime ideal in R,
for any prime ideal in that contracts to .
This can be shown within basic ring theory (cf. Kaplansky, commutative rings). In addition, in each fiber of , one cannot have a chain of primes ideals of length .
Since an artinian ring (e.g., a field) has dimension zero, by induction one gets a formula: for an artinian ring R,
Local rings
Fundamental theorem
Let be a noetherian local ring and I a -primary ideal (i.e., it sits between some power of and ). Let be the Poincaré series of the associated graded ring . That is,
where refers to the length of a module (over an artinian ring ). If generate I, then their image in have degree 1 and generate as -algebra. By the Hilbert–Serre theorem, F is a rational function with exactly one pole at of order . Since
we find that the coefficient of in is of the form
That is to say, is a polynomial in n of degree . P is called the Hilbert polynomial of .
We set . We also set to be the minimum number of elements of R that can generate an -primary ideal of R. Our ambition is to prove the fundamental theorem:
Since we can take s to be , we already have from the above. Next we pro |
https://en.wikipedia.org/wiki/Damiano%20Brigo | Damiano Brigo (born Venice, Italy 1966) is a mathematician known for research in mathematical finance, filtering theory, stochastic analysis with differential geometry, probability theory and statistics, authoring more than 130 research publications and three monographs.
From 2012 he serves as full professor with a chair in mathematical finance at the Department of Mathematics of Imperial College London, where he headed the Mathematical Finance group in 2012–2019. He is also a well known quantitative finance researcher, manager and advisor in the industry. His research has been cited and published also in mainstream industry publications, including Risk Magazine, where he has been the most cited author in the twenty years 1998–2017. He is often requested as a plenary or invited speaker both at academic and industry international events.
Brigo's research has also been used in court as support for legal proceedings.
Brigo holds a Ph.D. in stochastic nonlinear filtering with differential geometric methods from the Free University of Amsterdam, following a laurea degree in mathematics from the University of Padua.
Education and career
Brigo studied for a laurea degree in mathematics at the University of Padua, where he graduated cum laude with a dissertation on the nonlinear filtering problem under the supervision of Prof. Giovanni Battista Di Masi. Brigo continued his studies with a Ph.D. under the primary supervision of Bernard Hanzon at the Free University of Amsterdam, with periods under the supervision of Francois Le Gland at IRISA/INRIA in Rennes, France, with the oversight of Jan van Schuppen at CWI in Amsterdam, with a dissertation that introduced and studied the projection filters.
After his PhD, Brigo pursued a career in the financial industry with several subsequent roles, first as a quantitative analyst in Banca Intesa in Milan, then as head of credit models in Banca IMI in London, and finally as a managing director with Fitch Ratings in London.
While in the industry, Brigo had been appointed as external fixed income professor at Bocconi University and as a visiting professor at the Department of Mathematics at Imperial College London. By then a well known researcher and manager in the financial industry, Brigo moved to a full time academic career, starting with the Gilbart Chair full professorship in Financial Mathematics at King's College London (2010-2012), where he headed the financial mathematics group. In 2012 Brigo moved to a full professor position at the Department of Mathematics of Imperial College London, where he headed the group in 2012-2019 and where he still serves as chair in mathematical finance, while continuing advisory work in the financial industry, serving in the academic advisory board of several financial institutions, and as director of two industry research institutes in two subsequent periods in 2012–2017, being often invited as a speaker both at academic events and at events organized by the industry, with |
https://en.wikipedia.org/wiki/Henri%20Bacry | Henri Bacry (1928–2010) was Professor Emeritus at the Université de la Méditerranée. Henri Bacry was assistant of physics at the Faculté des Sciences d'Alger and then Professor of mathematics at Lycée Bugeaud, before becoming, in 1969, Professor at the Faculté des Sciences de Luminy. He was a visiting scholar at the Institute for Advanced Study in Princeton in 1966-6 and a researcher at CERN. He is the founder in 1972 of the International Colloquium of Group Theoretical Methods in Physics. He has numerous publications on theoretical physics, problems of symmetry in various fields ranging from relativity to particle physics, optics, physics of sound and statistical mechanics and some work in mathematics.
Selected publications
with Jean-Marc Lévy-Leblond Possible Kinematics, J. Math. Phys., vol. 9, 1969, pp. 1605–1614 (discussed by Freeman Dyson his 1972 Gibbs Lecture Missed opportunities)
Constellations and projective classical groups, Comm. Math. Pays., vol. 72, 1980, pp. 119–130
Group theory and constellations. Editions Publibook, 2004.
References
External links
Henri Bacry biography at www.canal-u.fr
Henri Bacry (1928–2010)
French physicists
Institute for Advanced Study visiting scholars
1928 births
2010 deaths
People associated with CERN |
https://en.wikipedia.org/wiki/Jean-Marc%20L%C3%A9vy-Leblond | Jean-Marc Lévy-Leblond (born 1940) is a physicist and essayist.
Biography
After high school in Cannes, Lévy-Leblond studied mathematics at the Lycée Janson-de-Sailly (Paris), then entered the École Normale Supérieure in 1958. A member of the Union of Communist Students (UEC) from 1956, then of the Communist Party, he left in 1968 to become one of the leaders of the movement of radical political criticism of science (see the journal Impasciences). After a doctorate (1962), then a doctorate in physical sciences (theoretical physics) at the University of Orsay in 1965, he was successively research fellow at the CNRS, lecturer at the University of Nice Sophia Antipolis, professor at the University of Paris 7, and at the University of Nice, where he taught in the departments of physics, philosophy and communication. He is professor emeritus at the University of Nice and was programme director at the Collège international de philosophie from 2001 to 2007.
He has published numerous articles on his research work, which mainly concerns theoretical and mathematical physics and epistemology.
He founded and directs the journal Alliage (culture, science, technique), directs the Science Ouverte and Points (science series) collections at Seuil, and works more generally to "(re )bringing science into culture”.
For a long time, Jean-Marc Lévy-Leblond has been sounding the alarm on the need for a public science intelligence, where knowledge, research, culture and politics would be tied together […]. In order to preserve authentic scientific discourse and avoid a gap of misunderstanding between specialists and the general public, but also to cultivate the need for a history of science, against the illusion of a universality of scientific knowledge, against the presentism and the fantasies of absolute contemporaneity, against the submission of science to industrial imperatives, against the planetary standardization that the domination of technosciences installs.
According to him
If these enemy brothers, scientism and irrationalism, prosper today, it is because uncultivated science becomes cult or occult with the same ease,
and the divorce between science and culture sometimes seems dangerously consummated. He has developed a discourse on the need for "science criticism", which he compares to art criticism, and calls for "a much higher level of collective consciousness on the part of society as a whole as to what scientific activity is”.
Educational books
Quantique (with Françoise Balibar), original textbook of quantum physics, published by Masson/CNRS. Volume 1 published in 1984, volume 2 unfinished, available online (archives-ouvertes.fr (CEL/CNRS)
Physics in question, general physics exercises, in 2 volumes: 1. Mechanics, 2. Electricity and magnetism (Vuibert)
Matter – relativistic, quantum, interactive, Paris, Seuil, 2006
Richard Feynman, The Nature of Physics, trans. Jean-Marc Lévy-Leblond, Françoise Balibar, Paris, Seuil, 1980 (Points Sciences collec |
https://en.wikipedia.org/wiki/Shinichiro%20Kuwada | is a former Japanese football player.
Club statistics
Updated to 23 February 2014.
References
External links
1986 births
Living people
Association football people from Hiroshima Prefecture
Japanese men's footballers
J1 League players
J2 League players
Sanfrecce Hiroshima players
Fagiano Okayama players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Stanislav%20Smrek | Stanislav Smrek (born 1 December 1986 in Košice) is a professional Slovak football defender who currently plays for the 2. liga club MFK Zemplín Michalovce.
Career statistics
Last updated: 4 January 2010
External links
Player profile MFK Košice profile
Eurofotbal profile
References
Living people
1986 births
Slovak men's footballers
MFK Zemplín Michalovce players
FC VSS Košice players
FK Bodva Moldava nad Bodvou players
FC Silon Táborsko players
Slovak First Football League players
Expatriate men's footballers in the Czech Republic
Footballers from Košice
Men's association football fullbacks |
https://en.wikipedia.org/wiki/Endoscopic%20group | In mathematics, endoscopic groups of reductive algebraic groups were introduced by in his work on the stable trace formula.
Roughly speaking, an endoscopic group H of G is a quasi-split group whose L-group is the connected component of the centralizer of a semisimple element of the L-group of G.
In the stable trace formula, unstable orbital integrals on a group G correspond to stable orbital integrals on its endoscopic groups H. The relation between them is given by the fundamental lemma.
References
Automorphic forms
Langlands program |
https://en.wikipedia.org/wiki/List%20of%20probabilistic%20proofs%20of%20non-probabilistic%20theorems | Probability theory routinely uses results from other fields of mathematics (mostly, analysis). The opposite cases, collected below, are relatively rare; however, probability theory is used systematically in combinatorics via the probabilistic method. They are particularly used for non-constructive proofs.
Analysis
Normal numbers exist. Moreover, computable normal numbers exist. These non-probabilistic existence theorems follow from probabilistic results: (a) a number chosen at random (uniformly on (0,1)) is normal almost surely (which follows easily from the strong law of large numbers); (b) some probabilistic inequalities behind the strong law. The existence of a normal number follows from (a) immediately. The proof of the existence of computable normal numbers, based on (b), involves additional arguments. All known proofs use probabilistic arguments.
Dvoretzky's theorem which states that high-dimensional convex bodies have ball-like slices is proved probabilistically. No deterministic construction is known, even for many specific bodies.
The diameter of the Banach–Mazur compactum was calculated using a probabilistic construction. No deterministic construction is known.
The original proof that the Hausdorff–Young inequality cannot be extended to is probabilistic. The proof of the de Leeuw–Kahane–Katznelson theorem (which is a stronger claim) is partially probabilistic.
The first construction of a Salem set was probabilistic. Only in 1981 did Kaufman give a deterministic construction.
Every continuous function on a compact interval can be uniformly approximated by polynomials, which is the Weierstrass approximation theorem. A probabilistic proof uses the weak law of large numbers. Non-probabilistic proofs were available earlier.
Existence of a nowhere differentiable continuous function follows easily from properties of Wiener process. A non-probabilistic proof was available earlier.
Stirling's formula was first discovered by Abraham de Moivre in his `The Doctrine of Chances' (with a constant identified later by Stirling) in order to be used in probability theory. Several probabilistic proofs of Stirling's formula (and related results) were found in the 20th century.
The only bounded harmonic functions defined on the whole plane are constant functions by Liouville's theorem. A probabilistic proof via n-dimensional Brownian motion is well known. Non-probabilistic proofs were available earlier.
Non-tangential boundary values of an analytic or harmonic function exist at almost all boundary points of non-tangential boundedness. This result (Privalov's theorem), and several results of this kind, are deduced from martingale convergence. Non-probabilistic proofs were available earlier.
The boundary Harnack principle is proved using Brownian motion (see also). Non-probabilistic proofs were available earlier.
Euler's Basel sum, can be demonstrated by considering the expected exit time of planar Brownian motion from an infinite strip. A numb |
https://en.wikipedia.org/wiki/Tsubasa%20Yokotake | is a Japanese football player and plays for Kochi United SC of the Japan Football League.
Club statistics
Updated to 20 February 2020.
References
External links
Profile at Kochi United SC
1989 births
Living people
Association football people from Hiroshima Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Sanfrecce Hiroshima players
Gainare Tottori players
Kochi United SC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Issei%20Takayanagi | is a Japanese football player and he is the currently first-team coach of Japan Football League club Okinawa SV..
Playing career
Takayanagi joined Okinawa SV in February 2019.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Renofa Yamaguchi FC
1986 births
Living people
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
Sanfrecce Hiroshima players
Hokkaido Consadole Sapporo players
Vissel Kobe players
Roasso Kumamoto players
Renofa Yamaguchi FC players
Okinawa SV players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Tomoyasu%20Hirose | is a former Japanese football player.
Club statistics
References
External links
1989 births
Living people
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
Montedio Yamagata players
Tokushima Vortis players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Tetsuro%20Ota | is a retired Japanese football player who last featured for ReinMeer Aomori.
Career statistics
Updated to 1 January 2020.
References
External links
Profile at Sagan Tosu
1989 births
Living people
Association football people from Ibaraki Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Montedio Yamagata players
Kashiwa Reysol players
Sagan Tosu players
ReinMeer Aomori players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Shogo%20Sakai | is a Japanese former football player.
Club statistics
References
External links
1988 births
Living people
Association football people from Mie Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Montedio Yamagata players
Veertien Mie players
Men's association football forwards |
https://en.wikipedia.org/wiki/Takuya%20Sonoda | is a Japanese footballer who plays for FC Imabari, as a defender.
Career
FC Imabari
On 16 December 2018, FC Imabari announced the signing of Sonoda.
Career statistics
Updated to 23 December 2018.
References
External links
Profile at Roasso Kumamoto
1984 births
Living people
Chuo University alumni
Association football people from Miyazaki Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Montedio Yamagata players
Ehime FC players
Roasso Kumamoto players
FC Imabari players
Men's association football defenders |
https://en.wikipedia.org/wiki/Kenta%20Kifuji | is a former Japanese football player.
Club statistics
References
External links
1981 births
Living people
Kindai University alumni
Association football people from Nagasaki Prefecture
Japanese men's footballers
J1 League players
J2 League players
Avispa Fukuoka players
Montedio Yamagata players
Men's association football defenders |
https://en.wikipedia.org/wiki/Kentaro%20Sato%20%28footballer%29 | is a Japanese football player currently playing for Renofa Yamaguchi FC.
Club statistics
Updated to end of 2018 season.
References
External links
Profile at Renofa Yamaguchi
Profile at Kyoto Sanga
1984 births
Living people
Juntendo University alumni
Association football people from Mie Prefecture
Japanese men's footballers
J1 League players
J2 League players
Montedio Yamagata players
JEF United Chiba players
Kyoto Sanga FC players
Renofa Yamaguchi FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Yu%20Hasegawa | is a Japanese footballer who last played for Nankatsu SC.
Club statistics
Updated to 1 March 2019.
References
External links
Profile at Shimizu S-Pulse
1987 births
Living people
Association football people from Yamanashi Prefecture
Japanese men's footballers
Japanese expatriate men's footballers
J1 League players
J2 League players
Kashiwa Reysol players
FC Gifu players
Avispa Fukuoka players
Montedio Yamagata players
Omiya Ardija players
Tokushima Vortis players
Shimizu S-Pulse players
V-Varen Nagasaki players
Sydney Olympic FC players
Men's association football forwards
Japanese expatriate sportspeople in Australia
Expatriate men's soccer players in Australia
Wollongong United FC players |
https://en.wikipedia.org/wiki/Takuya%20Miyamoto%20%28footballer%2C%20born%201983%29 | was a Japanese footballer who played as a defender.
Career statistics
References
External links
1983 births
2022 deaths
Osaka University of Commerce alumni
Association football people from Hiroshima Prefecture
Japanese men's footballers
J1 League players
J2 League players
Cerezo Osaka players
Montedio Yamagata players
Avispa Fukuoka players
Men's association football defenders
Sportspeople from Hiroshima Prefecture |
https://en.wikipedia.org/wiki/Daishi%20Hiramatsu | is a former Japanese football player.
Club statistics
References
External links
1983 births
Living people
Chuo University alumni
Association football people from Tochigi Prefecture
Japanese men's footballers
J1 League players
J2 League players
Mito HollyHock players
FC Tokyo players
Men's association football defenders |
https://en.wikipedia.org/wiki/Kento%20Hori | is a former Japanese football player.
Club statistics
References
External links
j-league
1982 births
Living people
Juntendo University alumni
Japanese men's footballers
J2 League players
Japan Football League players
Sagawa Shiga FC players
Mito HollyHock players
Verspah Oita players
Men's association football midfielders
Universiade medalists in football
FISU World University Games gold medalists for Japan
Association football people from Hiroshima |
https://en.wikipedia.org/wiki/Listwise%20deletion | In statistics, listwise deletion is a method for handling missing data. In this method, an entire record is excluded from analysis if any single value is missing.
Example
For example, consider the following questionnaire, as answered by 10 subjects:
A researcher is hoping to model income (dependent variable) based on age and gender (independent variables). Using listwise deletion, the researcher would remove subjects 3, 4, and 8 from the sample before performing any further analysis.
Problems with listwise deletion
Listwise deletion affects statistical power of the tests conducted. Statistical power relies in part on high sample size. Because listwise deletion excludes data with missing values, it reduces the sample which is being statistically analysed.
Listwise deletion is also problematic when the reason for missing data may not be random (i.e., questions in questionnaires aiming to extract sensitive information. Due to the method, much of the subjects' data will be excluded from analysis, leaving a bias in data findings. For instance, a questionnaire may include questions about respondents drug use history, current earnings, or sexual persuasions. Many of the subjects in the sample may not answer due to the intrusive nature of the questions, but may answer all other items. Listwise deletion will exclude these respondents from analysis. This may create a bias as participants who do divulge this information may have different characteristics than participants who do not. Multiple imputation is an alternate technique for dealing with missing data that attempts to eliminate this bias.
Compared to other methods
While listwise deletion does have its problems, it is preferable to many other methods for handling missing data. In some cases, it may even be the least problematic method. The following table provides some comparisons of listwise deletions to other methods:
References
Missing data |
https://en.wikipedia.org/wiki/Tero%20Koskela | Tero Koskela (born 13 October 1976) is a Finnish footballer who plays for Vaasan Palloseura.
References
Hønefoss BK season statistics
Tero Koskela at Vaasan Palloseura web site
1976 births
Living people
Sportspeople from Kokkola
Finnish men's footballers
Veikkausliiga players
Eliteserien players
FC Jokerit players
FC Honka players
Tampere United players
Fredrikstad FK players
Hønefoss BK players
Vaasan Palloseura players
Kokkolan Palloveikot players
Finnish expatriate men's footballers
Finnish expatriate sportspeople in Norway
Expatriate men's footballers in Norway
Men's association football midfielders |
https://en.wikipedia.org/wiki/Matrix%20%28mathematics%29 | In mathematics, a matrix (: matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension .
Matrices are used to represent linear maps and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents the composition of linear maps.
Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. This article focuses on matrices related to linear algebra, and, unless otherwise specified, all matrices represent linear maps or may be viewed as such.
Square matrices, matrices with the same number of rows and columns, play a major role in matrix theory. Square matrices of a given dimension form a noncommutative ring, which is one of the most common examples of a noncommutative ring. The determinant of a square matrix is a number associated to the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is invertible if and only if it has a nonzero determinant, and the eigenvalues of a square matrix are the roots of a polynomial determinant.
In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimension. Matrices are used in most areas of mathematics and most scientific fields, either directly, or through their use in geometry and numerical analysis.
Matrix theory is the branch of mathematics that focuses on the study of matrices. It was initially a sub-branch of linear algebra, but soon grew to include subjects related to graph theory, algebra, combinatorics and statistics.
Definition
A matrix is a rectangular array of numbers (or other mathematical objects), called the entries of the matrix. Matrices are subject to standard operations such as addition and multiplication. Most commonly, a matrix over a field F is a rectangular array of elements of F. A real matrix and a complex matrix are matrices whose entries are respectively real numbers or complex numbers. More general types of entries are discussed below. For instance, this is a real matrix:
The numbers, symbols, or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.
Size
The size of a matrix is defined by the number of rows and columns it cont |
https://en.wikipedia.org/wiki/Skerdilaid%20Curri | Skerdilaid Curri (born 6 October 1975) is a retired Albanian footballer who last played for FC Erzgebirge Aue. He also holds German citizenship.
Career statistics
International
National team statistics
Honours
Partizani Tirana
Albanian Cup: 1996–97
References
External links
1975 births
Living people
Footballers from Kavajë
Albanian men's footballers
Men's association football midfielders
Men's association football forwards
KF Besa Kavajë players
FK Partizani Tirana players
SpVgg Unterhaching players
FC Erzgebirge Aue players
2. Bundesliga players
3. Liga players
Albanian expatriate men's footballers
Expatriate men's footballers in Germany
Albanian expatriate sportspeople in Germany |
https://en.wikipedia.org/wiki/Equivariant%20L-function | In algebraic number theory, an equivariant Artin L-function is a function associated to a finite Galois extension of global fields created by packaging together the various Artin L-functions associated with the extension. Each extension has many traditional Artin L-functions associated with it, corresponding to the characters of representations of the Galois group. By contrast, each extension has a unique corresponding equivariant L-function.
Equivariant L-functions have become increasingly important as a wide range of conjectures and theorems in number theory have been developed around them. Among these are the Brumer–Stark conjecture, the Coates-Sinnott conjecture, and a recently developed equivariant version of the main conjecture in Iwasawa theory.
References
Algebraic number theory
Zeta and L-functions |
https://en.wikipedia.org/wiki/SPICE%20%28observation%20geometry%20system%29 | SPICE (Spacecraft Planet Instrument C-matrix Events) is a NASA ancillary information system used to compute geometric information used in planning and analyzing science observations obtained from robotic spacecraft. It is also used in planning missions and conducting numerous engineering functions needed to carry out those missions.
SPICE was developed at NASA's Navigation and Ancillary Information Facility (NAIF), located at the Jet Propulsion Laboratory. It has become the de facto standard for handling much of the so-called observation geometry information on NASA's planetary missions, and it is now widely used in support of science data analysis on planetary missions of other space agencies as well. Some SPICE capabilities are also used on a variety of astrophysics, solar physics and earth science missions.
Data
SPICE data files are usually referred to as "kernels." These files provide information such as spacecraft trajectory and orientation; target body ephemeris, size and shape; instrument field-of-view size, shape and orientation; specifications for reference frames; and tabulations of time system conversion coefficients.
SPICE data are archived in a national archive center such as the NASA Planetary Data System archives.
Kernels
There are five major kernels and five other kernels.
SPK: Space vehicle or target body trajectory (ephemeris)
PcK: Target body size, shape and orientation
IK: Instrument field-of-view size, shape and orientation
CK: Orientation of space vehicle or any articulating structure on it
EK: Events information
ESP: Science Plan
ESQ: Sequence of events
ENB: Experimenter’s Notebook
Others:
FK: Reference frame specifications
LSK: Leapseconds tabulation
SCLK: Spacecraft clock coefficients
DSK: Digital shape models
MK: Meta-kernel
Software
The SPICE system includes software referred to as The SPICE Toolkit, used for reading the SPICE data files and computing geometric parameters based on data from those files. These tools are provided as subroutine libraries in four programming languages: C, FORTRAN, IDL, MATLAB and Java Native Interface. Third parties offer Python and Ruby interfaces to the C-language Toolkit. The Toolkits also include a number of utility and application programs. The SPICE Toolkits are available for most popular computing platforms, operating systems and compilers. Extensive documentation accompanies each Toolkit.
Those unable to write their own SPICE-based program may try using WebGeocalc, a browser interface to a SPICE-based geometry engine running on the NAIF server. Using WebGeocalc is much easier than writing your own program, but it still requires considerable knowledge about SPICE data and solar system geometry, and it doesn't offer the full range of computations available when using Toolkit software in your own program.
The NAIF Group also offers a 3-D mission visualization program named SPICE-Enhanced Cosmographia. This program runs in the OSX, Windows and Linux environments. Visual repre |
https://en.wikipedia.org/wiki/Guadalupe%20Victoria%2C%20Puebla | Guadalupe Victoria Municipality is a municipality in the Mexican state of Puebla. According to the National Statistics Institute (INEGI), it had a population of 15,041 inhabitants in the 2005 census. Its total area is 239.83 km². It is named after Guadalupe Victoria, the first president of Mexico.
Its geographical coordinates are 19° 17′ North, and 97° 20′ West. Its average altitude is above sea level. Its highest elevation is the rhyolitic twin dome volcano Las Derrumbadas (3480 m).
External links
https://web.archive.org/web/20070324000550/http://www.e-local.gob.mx/work/templates/enciclo/puebla/Mpios/21067a.htm
Municipalities of Puebla |
https://en.wikipedia.org/wiki/Early%20numeracy | Early numeracy is a branch of numeracy that aims to enhance numeracy learning for younger learners, particularly those at-risk in the area of mathematics. Usually the mathematical learning begins with simply learning the first digits, 1 through 10. This is done because it acts as an entry way to the expansion of counting. One can keep track of the digits using any of the fingers.
See also
Primary education
References
Mathematics education |
https://en.wikipedia.org/wiki/David%20A.%20Freedman | David Amiel Freedman (5 March 1938 – 17 October 2008) was Professor of Statistics at the University of California, Berkeley. He was a distinguished mathematical statistician whose wide-ranging research included the analysis of martingale inequalities, Markov processes, de Finetti's theorem, consistency of Bayes estimators, sampling, the bootstrap, and procedures for testing and evaluating models. He published extensively on methods for causal inference and the behavior of standard statistical models under non-standard conditions – for example, how regression models behave when fitted to data from randomized experiments. Freedman also wrote widely on the application—and misapplication—of statistics in the social sciences, including epidemiology, public policy, and law.
Biography and awards
Freedman was a fellow of the Institute of Mathematical Statistics and the American Statistical Association and an elected fellow of the American Academy of Arts and Sciences. He won the 2003 John J. Carty Award for the Advancement of Science from the National Academy of Sciences "for his profound contributions to the theory and practice of statistics, including rigorous foundations for Bayesian inference and trenchant analysis of census adjustment." He was a Fellow at the Miller Institute for Basic Research in Science in 1990, an Alfred P. Sloan Foundation Fellow in 1964–66, and a Canada Council Fellow at Imperial College London in 1960–61.
Freedman was born in Montreal, Quebec, Canada, on 5 March 1938. He received a B.Sc. from McGill University in 1958 and a M.A. and a Ph.D. from Princeton University in 1959 and 1960, respectively. He joined the University of California, Berkeley Department of Statistics in 1961 as a lecturer and was appointed to the research faculty in 1962. He remained at Berkeley his entire career. He started his professional life as a probabilist and mathematical statistician with Bayesian leanings but became one of the world's leading applied statisticians and a circumspect frequentist.
Freedman was a consulting or testifying expert on statistics in disputes involving employment discrimination, fair loan practices, voting rights, duplicate signatures on petitions, railroad taxation, ecological inference, flight patterns of golf balls, price scanner errors, bovine spongiform encephalopathy (mad cow disease), and sampling. He consulted for the Bank of Canada, the Carnegie Commission, the City of San Francisco, the County of Los Angeles, and the Federal Reserve, as well as the U.S. departments of energy, treasury, justice, and commerce. Freedman and his colleague Kenneth Wachter testified to the United States Congress and the courts against adjusting the 1980 and 1990 censuses using estimates of differential undercounts. A 1990 lawsuit that sought to compel the United States Department of Commerce to adjust the census was heard on appeal by the U.S. Supreme Court, which ruled unanimously in favor of the Commerce Department and Freedman |
https://en.wikipedia.org/wiki/Arley%20Dinas | José Arley Dinas Rodríguez, or simply known as Arley Dinas (born May 16, 1974) is a former Colombian football player.
Club statistics
National team statistics
External links
1974 births
Living people
Colombian men's footballers
Colombian expatriate men's footballers
América de Cali footballers
Deportivo Cali footballers
Shonan Bellmare players
Millonarios F.C. players
Deportes Tolima footballers
Boca Juniors footballers
Expatriate men's footballers in Japan
Expatriate men's footballers in Argentina
Categoría Primera A players
J2 League players
Colombia men's under-20 international footballers
Colombia men's international footballers
2000 CONCACAF Gold Cup players
2004 Copa América players
Men's association football defenders
Footballers from Cauca Department
Footballers at the 1995 Pan American Games
Pan American Games bronze medalists for Colombia
Pan American Games medalists in football
Medalists at the 1995 Pan American Games |
https://en.wikipedia.org/wiki/Takuro%20Kikuoka | is a Japanese football player who plays for ReinMeer Aomori.
Club statistics
Updated to 23 February 2020.
References
External links
Profile at Consadole Sapporo
Profile at SC Sagamihara
1985 births
Living people
Hosei University alumni
Association football people from Shizuoka Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Mito HollyHock players
Tokyo Verdy players
Tochigi SC players
Hokkaido Consadole Sapporo players
SC Sagamihara players
ReinMeer Aomori players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Function%20application | In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range. In this sense, function application can be thought of as the opposite of function abstraction.
Representation
Function application is usually depicted by juxtaposing the variable representing the function with its argument encompassed in parentheses. For example, the following expression represents the application of the function ƒ to its argument x.
In some instances, a different notation is used where the parentheses aren't required, and function application can be expressed just by juxtaposition. For example, the following expression can be considered the same as the previous one:
The latter notation is especially useful in combination with the currying isomorphism. Given a function , its application is represented as by the former notation and (or with the argument written with the less common angle brackets) by the latter. However, functions in curried form can be represented by juxtaposing their arguments: , rather than . This relies on function application being left-associative.
As an operator
Function application can be trivially defined as an operator, called apply or , by the following definition:
The operator may also be denoted by a backtick (`).
If the operator is understood to be of low precedence and right-associative, the application operator can be used to cut down on the number of parentheses needed in an expression. For example;
can be rewritten as:
However, this is perhaps more clearly expressed by using function composition instead:
or even:
if one considers to be a constant function returning .
Other instances
Function application in the lambda calculus is expressed by β-reduction.
The Curry–Howard correspondence relates function application to the logical rule of modus ponens.
See also
Polish notation
Functions and mappings |
https://en.wikipedia.org/wiki/Mitsuru%20Mansho | is a former Japanese football player who last played as a striker for J3 League side Fujieda MYFC.
Club statistics
Updated to 23 February 2017.
References
External links
Profile at Fujieda MYFC
1989 births
Living people
Osaka Sangyo University alumni
Association football people from Osaka Prefecture
Japanese men's footballers
J2 League players
J3 League players
Mito HollyHock players
Fujieda MYFC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Takuya%20Shiihara | is a former Japanese football player.
Club statistics
References
External links
1980 births
Living people
Association football people from Kagoshima Prefecture
Japanese men's footballers
J1 League players
J2 League players
Osaka Gas SC players
JEF United Chiba players
Mito HollyHock players
Men's association football midfielders
Sportspeople from Kagoshima |
https://en.wikipedia.org/wiki/Yoichi%20Akiba | is a former Japanese football player.
He played for Yokohama and Mito HollyHock.
Club statistics
References
External links
1983 births
Living people
University of Tsukuba alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
Yokohama FC players
Mito HollyHock players
Men's association football defenders |
https://en.wikipedia.org/wiki/Willoughby%20D.%20Miller | Willoughby Dayton Miller (1853–1907) was an American dentist and the first oral microbiologist.
Biography
Willoughy D. Miller was born in Alexandria, Ohio, and studied mathematics and physics at the University of Michigan. He traveled to Edinburgh to continue his studies, but financial problems caused him to move to Berlin where he was assisted by an American dentist Frank Abbot. Miller later married Abbot's daughter, Caroline. Becoming interested in his father-in-law's profession, Miller returned to the United States to train as a dentist at the Pennsylvania Dental College. This college merged with the University of Pennsylvania Department of Dentistry in 1878, and Miller was one of the members of the first graduating class in 1879. After graduating, Miller returned to Berlin where he worked at first in Abbot's dental office and pursued his interest in the emerging science of microbiology. In his later years, he was appointed Dean of the University of Michigan School of Dentistry in 1906, but he died in 1907 following an operation for appendicitis, prior to assuming the position.
Miller worked during the golden age of microbiology. Pasteur had discovered that bacteria can ferment sugars into lactic acid, and another Frenchman, Emil Magitot, showed that fermentation of sugars could dissolve teeth in the laboratory. Bacteria had been observed inside carious dentin by Underwood and Miles in 1881, and these researchers also proposed that bacterial acids were necessary for removing the mineral of teeth. It was against this background that Miller developed his oral microbiological research, soon becoming appointed Professor of Operative Dentistry at the University of Berlin. He worked in the microbiological laboratory of Robert Koch in Berlin and began numerous research projects that introduced modern biological principles to dentistry. In 1890 Miller formulated the chemo-parasitic theory of caries (tooth decay). This theory held that caries is caused by acids produced by oral bacteria following fermentation of sugars. The principles of the chemo-parasitic theory were bolstered by the descriptions of bacterial plaque on tooth surfaces independently by GV Black and by JL Williams in 1898. The biomass of plaque helps localize acids at the tooth surface and prevent dilution by saliva. Miller thought that no single species of bacteria could cause caries. This idea was supplanted in the 1950s when the role of Streptococcus mutans as a primary pathogen in caries was established. More recent examination of the microbiology of carious lesions using 16S rRNA sequencing and high throughput DNA sequencing indicates that communities of diverse organisms may be more important than individual species.
A second major contribution of WD Miller was the focal infection theory. Miller proposed that oral microorganisms or their products have a role in the development of a variety of diseases in sites removed from the oral cavity, including brain abscess |
https://en.wikipedia.org/wiki/Hiromasa%20Kanazawa | is a former Japanese football player.
Club statistics
References
External links
1983 births
Living people
Tokyo Gakugei University alumni
Association football people from Shizuoka Prefecture
Japanese men's footballers
J2 League players
Yokohama FC players
Mito HollyHock players
SC Sagamihara players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kohei%20Nishino | is a former Japanese football player. His brother is Makoto Nishino.
He played for Japanese club Oita Trinita, Mito HollyHock and Fagiano Okayama.
Club Statistics
References
External links
Kohei Nishino at Yahoo! Japan
1982 births
Living people
Nippon Bunri University alumni
Association football people from Hyōgo Prefecture
Japanese men's footballers
J1 League players
J2 League players
Oita Trinita players
Mito HollyHock players
Fagiano Okayama players
Men's association football forwards |
https://en.wikipedia.org/wiki/Takashi%20Kuramoto | is a former Japanese football player.
Club statistics
References
External links
1984 births
Living people
Association football people from Ōita Prefecture
Japanese men's footballers
J1 League players
J2 League players
Oita Trinita players
Mito HollyHock players
Men's association football defenders |
https://en.wikipedia.org/wiki/Kenichi%20Mori | is a former Japanese football player.
Club statistics
References
External links
1984 births
Living people
Meiji University alumni
Association football people from Gunma Prefecture
Japanese men's footballers
J2 League players
Mito HollyHock players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Keisuke%20Endo | is a Japanese footballer who plays for Nara Club.
Career
On 22 July 2019, Endo joined Nara Club.
Club statistics
Updated to 23 February 2020.
References
External links
Profile at Fujieda MYFC
Profile at Kataller Toyama
1989 births
Living people
People from Ichihara, Chiba
Association football people from Chiba Prefecture
Japanese men's footballers
J2 League players
J3 League players
Mito HollyHock players
Thespakusatsu Gunma players
FC Machida Zelvia players
Fujieda MYFC players
Kataller Toyama players
Nara Club players
Vonds Ichihara players
Men's association football forwards |
https://en.wikipedia.org/wiki/Masahiro%20Takahashi | is a former Japanese football player.
Takahashi previously played for Mito Hollyhock in the J2 League.
Club statistics
References
External links
1985 births
Living people
Chukyo University alumni
Association football people from Ibaraki Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Mito HollyHock players
Reilac Shiga FC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Yu%20Kobayashi%20%28footballer%29 | is a Japanese football player currently playing for Kawasaki Frontale and the Japan national team.
Club statistics
1=Japanese Super Cup appearances
National team statistics
International goals
Scores and results list Japan's goal tally first.
Honours
Club
Kawasaki Frontale
J1 League (4): 2017, 2018, 2020, 2021
Emperor's Cup: 2020
J.League Cup: 2019
Japanese Super Cup (2): 2019, 2021
Individual
J.League MVP Award: 2017
J.League Top Scorer: 2017
J.League Best XI (2): 2016, 2017
References
External links
Profile at Kawasaki Frontale
1987 births
Living people
Takushoku University alumni
Association football people from Aomori Prefecture
Japanese men's footballers
Japan men's international footballers
J1 League players
J2 League players
Mito HollyHock players
Kawasaki Frontale players
2015 AFC Asian Cup players
Men's association football forwards
21st-century Japanese people |
https://en.wikipedia.org/wiki/Yusuke%20Hayashi | is a Japanese footballer who plays as a midfielder for Philippines Football League side Stallion Laguna.
Career statistics
Updated to 2 February 2018.
References
External links
Profile at Grulla Morioka
1990 births
Living people
Association football people from Iwate Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Urawa Red Diamonds players
Thespakusatsu Gunma players
Iwate Grulla Morioka players
Men's association football midfielders
Naturalized citizens of Japan
Japanese people of Korean descent
Expatriate men's footballers in the Philippines
Philippines Football League players |
https://en.wikipedia.org/wiki/Chartered%20Statistician | Chartered Statistician (CStat) is a professional qualification in Statistics awarded to practising professional statisticians by the Royal Statistical Society in the United Kingdom. A Chartered Statistician may use the post-nominal letters CStat.
Chartered Statistician is the Royal Statistical Society's highest professional qualification; achieving it is done through a rigorous peer-reviewed process. It provides formal recognition of a member's qualifications in Statistics, professional practice of Statistics at an advanced level, technical standing, and commitment to remain at the forefront of Statistical theory and practise throughout one's professional career.
The required standard for Chartered Statistician registration is typically an accredited UK MMath degree in Mathematics and Statistics, at least five years of peer-reviewed professional practice of advanced Statistics, attainment of a senior-level of technical standing, and an ongoing commitment to Continuing Professional Development.
The Royal Statistical Society's Chartered Statistician qualification is equal in status to the Accredited Professional Statistician (PStat) qualification awarded by the American Statistical Association. This formal mutual recognition entitles Chartered Statisticians and Accredited Professional Statisticians to be automatically eligible for each other's designations, should they wish to apply.
See also
Royal Statistical Society
American Statistical Association
References
External links
Royal Statistical Society website
Royal Statistical Society
Statistician
Statistics education |
https://en.wikipedia.org/wiki/Giannis%20Kanotidis | Giannis Kanotidis (; born 2 May 1979) is a Greek footballer. He currently plays for Panthrakikos F.C. in Beta Ethniki.
Career
Career statistics
Last update: 21 July 2011
References
External links
Panthracrocs
Guardian
1979 births
Living people
Greek men's footballers
Athlitiki Enosi Larissa F.C. players
Kastoria 1980 F.C. players
Kavala F.C. players
Olympiacos Volos F.C. players
Kalamata F.C. players
Panthrakikos F.C. players
Men's association football defenders |
https://en.wikipedia.org/wiki/Daniel%20Spielman | Daniel Alan Spielman (born March 1970 in Philadelphia, Pennsylvania) has been a professor of applied mathematics and computer science at Yale University since 2006. As of 2018, he is the Sterling Professor of Computer Science at Yale. He is also the Co-Director of the Yale Institute for Network Science, since its founding, and chair of the newly established Department of Statistics and Data Science.
Education
Daniel Spielman attended The Philadelphia School, and Germantown Friends School. He received his bachelor of arts degree in mathematics and computer science from Yale University in 1992 and a PhD in applied mathematics from MIT in 1995 (his dissertation was called "Computationally Efficient Error-Correcting Codes and Holographic Proofs"). He taught in the Mathematics Department at MIT from 1996 to 2005.
Awards
Spielman and his collaborator Shang-Hua Teng have jointly won the Gödel Prize twice: in 2008 for their work on smoothed analysis of algorithms and in 2015 for their work on nearly-linear-time Laplacian solvers.
In 2010 he was awarded the Nevanlinna Prize "for smoothed analysis of Linear Programming, algorithms for graph-based codes and applications of graph theory to Numerical Computing" and the same year he was named a Fellow of the Association for Computing Machinery.
In 2012 he was part of the inaugural class of Simons Investigators providing $660,000 for five years for curiosity driven research.
In October 2012, he was named a recipient of the MacArthur Fellowship.
In 2013, together with Adam Marcus and Nikhil Srivastava, he provided a positive solution to the Kadison–Singer problem, a result that was awarded the 2014 Pólya Prize.
He gave a plenary lecture at the International Congress of Mathematicians in 2010.
In 2017 he was elected to the National Academy of Sciences.
In 2022 he won the Breakthrough Prize in Mathematics "for breakthrough contributions to theoretical computer science and mathematics, including to spectral graph theory, the Kadison-Singer problem, numerical linear algebra, optimization, and coding theory.".
References
External links
Yale faculty homepage
1970 births
Living people
Mathematicians from Philadelphia
American computer scientists
Researchers in geometric algorithms
MacArthur Fellows
Gödel Prize laureates
Nevanlinna Prize laureates
Fellows of the Association for Computing Machinery
Scientists from Pennsylvania
Massachusetts Institute of Technology School of Science faculty
Yale University faculty
Yale Sterling Professors
Massachusetts Institute of Technology School of Science alumni
Yale University alumni
Jewish American scientists
Members of the United States National Academy of Sciences
Simons Investigator
Germantown Friends School alumni
21st-century American Jews
Theoretical computer scientists |
https://en.wikipedia.org/wiki/James%20Henry%20Taylor | James Henry Taylor (February 21, 1893 – March 30, 1972) was a professor of mathematics at George Washington University from 1929–1958, and professor emeritus from 1959 until his death.
Early life
Born on February 21, 1893, in Sharon, Pennsylvania, Taylor died of cancer on March 30, 1972, at the age of 79. In addition to the title of professor, Taylor was also referred to as an emeritus of mathematics in Residence from 1958 until his death.
Taylor was a graduate of three different universities. The first was the University of Nebraska at Omaha. Secondly he enrolled in the University of Chicago. Lastly, he attended Princeton University where he was a National Research Fellow.
Military service
Before becoming a professor, Taylor was a Second Lieutenant in the United States Army in World War I from August 1917 to August 1918. As a Second Lieutenant, the entry-level rank for most commissioned officers, Taylor led a platoon-size element. He was then promoted to First Lieutenant and saw duty in France in the 351st Infantry, a position he held until June 1919. (Washington Post, Dr. James H. Taylor, Mathematician at GW).
During World War II, he was a mathematical advisor at Fort Belvoir in Virginia and at the Department of Terrestrial Magnetism at the Carnegie Institute. According to DTM, "scientists bring the perspective of several disciplines to broad questions about nature."
After World War I, he was a National Research Fellow in Mathematics from September 1924 until September 1925. During the summer of 1919, he was a "boilermaker’s helper," at the Chicago, Burlington and Quincy Railroad Shops in Havelock, Nebraska. He worked with the Nebraska Department of Public Works computing road coasts during the summers of 1921 and 1922.
Teaching career
Taylor began his teaching career as an assistant in mathematics at the University of Nebraska at Lincoln in 1919, where he taught for a year until becoming an instructor from 1920 to 1922. He then transferred to Northwestern University in Chicago in 1923, where he was a part-time math instructor for a year. In 1924, Taylor received his Ph.D. in mathematics from the University of Chicago. He furthered his profession in 1925 when he became an assistant professor of mathematics at Lehigh University in Bethlehem, Pennsylvania. After working for a year at Lehigh, Taylor began working at the University of Wisconsin, again as an assistant professor of mathematics from 1926 until 1929. Then, in 1929, he started his career at the George Washington University where he was a full-time professor of mathematics for the first time.
During the time that Taylor taught at the George Washington University from 1929–1958 the mathematics department was relatively basic. He taught classes in advanced analytics, geometry, and tensor analysis. In 1950–1951 the department expanded a little, offering 34 classes ranging from college algebra to analytic geometry to plane trigonometry. Taylor taught continued to teach classes |
https://en.wikipedia.org/wiki/Mountain%20climbing%20problem | In mathematics, the mountain climbing problem is a mathematical problem that considers a two-dimensional mountain range (represented as a continuous function), and asks whether it is possible for two mountain climbers starting at sea level on the left and right sides of the mountain to meet at the summit, while maintaining equal altitudes at all times. This problem was named and posed in this form by , but its history goes back to , who solved a version of it. The problem has been repeatedly rediscovered and solved independently in different contexts by a number of people (see references below).
Since the 1990s, the problem was shown to be connected to the weak Fréchet distance of curves in the plane, various planar motion planning problems in computational geometry, the inscribed square problem, semigroup of polynomials, etc. The problem was popularized in the article by , which received the Mathematical Association of America's Lester R. Ford Award in 1990.
Analysis
The problem can be rephrased as asking whether, for a given pair of continuous functions with (corresponding to rescaled versions of the left and right faces of the mountain), it is possible to find another pair of functions with (the climbers' horizontal positions at time ) such that the function compositions and (the climbers' altitudes at time ) are the same function.
Finite number of peaks and valleys
When have only a finite number of peaks and valleys (local maxima and local minima) it is always possible to coordinate the climbers' movements. This can be shown by drawing out a sort of game tree: an undirected graph with one vertex labeled whenever and either or is a local maximum or minimum. Two vertices will be connected by an edge if and only if one node is immediately reachable from the other; the degree of a vertex will be greater than one only when the climbers have a non-trivial choice to make from that position.
At the vertex , the degree is one: the only possible direction for both climbers to go is onto the mountain. Similarly, at the degree is one, because both climbers can only return down the mountain.
At a vertex where one climber is at a peak or a valley and the other one is not, then the degree is two: the climber at the peak or valley has two choices of which way to go, and the other climber can only go one way.
At a vertex where both climbers are at peaks or both climbers are at valleys, the degree is four: both climbers may choose independently of each other which direction to go.
At a vertex where one climber is at a peak and the other is at a valley, the degree is zero: such positions are unreachable. (That is, if such a vertex exists, then the graph is not connected.)
According to the handshaking lemma, every connected component of an undirected graph has an even number of odd-degree vertices. Since the only odd-degree vertices in all of are and , these two vertices must belong to the same connected component. That is, must contain |
https://en.wikipedia.org/wiki/Noriyoshi%20Omichi | is a Nippon Professional Baseball player for the Yomiuri Giants of Japan's Central League. Before playing for the Giants, he was a member of Fukuoka SoftBank Hawks.
External links
Career statistics - NPB.jp
75 Noriyoshi Omichi PLAYERS2021 - Fukuoka SoftBank Hawks Official site
Living people
1969 births
Baseball people from Mie Prefecture
Japanese baseball players
Nankai Hawks players
Fukuoka Daiei Hawks players
Fukuoka SoftBank Hawks players
Yomiuri Giants players
Nippon Professional Baseball first basemen
Nippon Professional Baseball outfielders
Nippon Professional Baseball coaches
Japanese baseball coaches |
https://en.wikipedia.org/wiki/Polynomial%20identity%20testing | In mathematics, polynomial identity testing (PIT) is the problem of efficiently determining whether two multivariate polynomials are identical. More formally, a PIT algorithm is given an arithmetic circuit that computes a polynomial p in a field, and decides whether p is the zero polynomial. Determining the computational complexity required for polynomial identity testing, in particular finding deterministic algorithms for PIT, is one of the most important open problems in algebraic computing complexity.
Description
The question "Does equal " is a question about whether two polynomials are identical. As with any polynomial identity testing question, it can be trivially transformed into the question "Is a certain polynomial equal to 0?"; in this case we can ask "Does "? If we are given the polynomial as an algebraic expression (rather than as a black-box), we can confirm that the equality holds through brute-force multiplication and addition, but the time complexity of the brute-force approach grows as , where is the number of variables (here, : is the first and is the second), and is the degree of the polynomial (here, ). If and are both large, grows exponentially.
PIT concerns whether a polynomial is identical to the zero polynomial, rather than whether the function implemented by the polynomial always evaluates to zero in the given domain. For example, the field with two elements, GF(2), contains only the elements 0 and 1. In GF(2), always evaluates to zero; despite this, PIT does not consider to be equal to the zero polynomial.
Determining the computational complexity required for polynomial identity testing is one of the most important open problems in the mathematical subfield known as "algebraic computing complexity". The study of PIT is a building-block to many other areas of computational complexity, such as the proof that IP=PSPACE. In addition, PIT has applications to Tutte matrices and also to primality testing, where PIT techniques led to the AKS primality test, the first deterministic (though impractical) polynomial time algorithm for primality testing.
Formal problem statement
Given an arithmetic circuit that computes a polynomial in a field, determine whether the polynomial is equal to the zero polynomial (that is, the polynomial with no nonzero terms).
Solutions
In some cases, the specification of the arithmetic circuit is not given to the PIT solver, and the PIT solver can only input values into a "black box" that implements the circuit, and then analyze the output. Note that the solutions below assume that any operation (such as multiplication) in the given field takes constant time; further, all black-box algorithms below assume the size of the field is larger than the degree of the polynomial.
The Schwartz–Zippel algorithm provides a practical probabilistic solution, by simply randomly testing inputs and checking whether the output is zero. It was the first randomized polynomial time PIT algorithm to be prov |
https://en.wikipedia.org/wiki/Jens%20H%C3%A4rtel | Jens Härtel (born 7 June 1969) is a German professional football manager, who last managed Eintracht Braunschweig and former player who last managed Hansa Rostock.
Managerial statistics
Honours
Manager
Individual
3. Liga Manager of the Season: 2017–18
References
External links
1969 births
Living people
People from Rochlitz
Men's association football defenders
German men's footballers
1. FC Lokomotive Leipzig players
FSV Zwickau players
FC Sachsen Leipzig players
1. FC Union Berlin players
SV Babelsberg 03 players
SV Germania Schöneiche players
2. Bundesliga players
3. Liga managers
1. FC Magdeburg managers
FC Hansa Rostock managers
Eintracht Braunschweig managers
German football managers
Footballers from Saxony
2. Bundesliga managers
Footballers from Lower Saxony |
https://en.wikipedia.org/wiki/Moshe%20Milevsky | Moshe Arye Milevsky is a professor of finance at the Schulich School of Business at York University, and a member of the Graduate Faculty in the Department of Mathematics & Statistics, in Toronto, Canada, where he has been based and teaching for over 25 years. He earned a B.A. in mathematics and physics from Yeshiva University in 1990, an M.A. in mathematics and statistics from York University in 1992 and a Ph.D. in business finance from York University in 1996.
His area of expertise is in mathematical financial economics, pensions, insurance, actuarial science and history of financial products. He has done extensive research on exotic option pricing, quantitative personal financial planning (focusing on investment strategies for retiring individuals), insurance derivatives, pensions, annuities, tontines and stochastic mortality models.
He is also the Executive Director of the Individual Finance and Insurance Decisions Centre (IFID), a non-profit corporation dedicated to generating advanced research at the intersection of wealth management, personal finance, and insurance. For his contributions to the Fields Institute and to the Canadian mathematical community, Moshe was inducted as a Fields Institute Fellow in 2002.
Moshe A. Milevsky is the author of 17 books, including the popular Are You a Stock or a Bond, and The 7 Most Important Equations for Your Retirement and the more advanced The Calculus of Retirement Income, which summarizes much of the research that Milevsky has done on quantitative retirement income planning. His recent books include King William's Tontine: Why the Retirement Annuity of the Future Should Resemble Its Past (Cambridge 2015) and The Day the King Defaulted: Financial Lessons from the Stop of the Exchequer in 1672.
Selected books
References
Academic staff of York University
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Fast%20and%20Realistic%20OpenGL%20Displayer | The Fast & Realistic OpenGL Displayer (FROG) is a generic framework dedicated to visualize events in a given geometry. It has been written in C++ and use OpenGL cross-platform libraries. Its first application is the visualization of events in the Compact Muon Solenoid detector.
Introduction
FROG has been firstly written by Loïc Quertenmont and Vincent Roberfroid in order to view, in a matter of seconds, events in the Compact Muon Solenoid detector. This detector is located at CERN and is dedicated to measure emitted particles produced by the collisions of high energetic protons accelerated by the Large Hadron Collider.
Two other tools already exists in order to view events in the CMS detector : IGUANA and Fireworks.
However, FROG has the advantage to have been implemented as such that any particular physics experiments or detector designs can be visualized.
Moreover, in comparison with the 2 other event displayers, FROG is very light and very fast and can run on various Operating System (Windows, Linux, Mac OS). In addition, FROG is self-consistent and does not require installation of big libraries generally used by High Energy physic experiments such as ROOT.
The article describes the principle of the algorithm and its many functionalities such as : 3D and 2D visualization, graphical user interface, mouse interface, configuration files, production of pictures of various format, integration of personal objects... Finally the application of FROG to the CMS experiment will be described.
References
External links
FROG on hepforge
ROOT : An object oriented analysis framework
CERN software |
https://en.wikipedia.org/wiki/Hilbert%20space | In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.
Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear subspace or a subspace (the analog of "dropping the altitude" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis, in analogy with Cartesian coordinates in classical geometry. When this basis is countably infinite, it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space.
Definition and illustration
Motivating example: Euclidean vector space
One of the most familiar examples of a Hilbert space is the Euclidean vector space consisting of three-dimensional vectors, denoted by , and equipped with the dot product. The dot product takes two vectors and , and produces a real number . If and are represented in Cartesian coordinates, then the dot product is defined by
The dot product satisfies the properties
It is symmetric in and : .
It is linear in its first argument: for any scalars , , and vectors , , and .
It is positive definite: for all vectors , , with equality if and only if .
An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner pro |
https://en.wikipedia.org/wiki/Basic%20statistical%20unit%20%28Norway%29 | The basic statistical unit () is a type of statistical unit used by Statistics Norway to provide stable and coherent geographical units for regional statistics in Norway. Basic statistical units are subdivisions of municipalities (they never include land in more than one municipality), and cover generally homogeneous areas. Most basic statistical units include a few hundred inhabitants, but as their borders are near constant, this can vary widely over time.
References
Demographics of Norway
Subdivisions of Norway |
https://en.wikipedia.org/wiki/Gerald%20Folland | Gerald Budge Folland is an American mathematician and a professor of mathematics at the University of Washington.
He is the author of several textbooks on mathematical analysis. His areas of interest include harmonic analysis (on both Euclidean space and Lie groups), differential equations, and mathematical physics. The title of his doctoral dissertation at Princeton University (1971) is "The Tangential Cauchy-Riemann Complex on Spheres".
In 2012 he became a fellow of the American Mathematical Society.
Publications and books
A Guide to Advanced Real Analysis, Washington, D.C. : Mathematical Association of America, 2009.
Quantum Field Theory : A Tourist Guide for Mathematicians, Providence, R.I. : American Mathematical Society, 2008.
Advanced Calculus, Prentice-Hall, 2002.
Real Analysis: Modern Techniques and their Applications (2nd ed.), John Wiley, 1999, .
"The uncertainty principle: a mathematical survey", J. Fourier Anal. Appl. 4 (1997), 207–238 (with A. Sitaram).
Introduction to Partial Differential Equations (2nd ed.), Princeton University Press, 1995.
A Course in Abstract Harmonic Analysis, CRC Press, 1995.
Fourier Analysis and Its Applications, Pacific Grove, Calif. : Wadsworth & Brooks/Cole Advanced Books & Software, 1992.
Harmonic Analysis in Phase Space, Princeton University Press, 1989.
Lectures on Partial Differential Equations : lectures delivered at the Indian Institute of Science, Bangalore, Springer, 1983.
Hardy Spaces on Homogeneous Groups (with Elias M. Stein), Princeton University Press, 1982.
"Estimates for the ∂b complex and analysis on the Heisenberg group", Comm. Pure Appl. Math. 27 (1974), 429–522 (with E. M. Stein)
References
.
External links
Jerry Folland's Personal Homepage
Jerry Folland's Official Homepage
20th-century American mathematicians
21st-century American mathematicians
Fellows of the American Mathematical Society
Living people
1947 births
Princeton University alumni
University of Washington faculty |
https://en.wikipedia.org/wiki/Jacquet%E2%80%93Langlands%20correspondence | In mathematics, the Jacquet–Langlands correspondence is a correspondence between automorphic forms on GL2 and its twisted forms, proved by in their book Automorphic Forms on GL(2) using the Selberg trace formula. It was one of the first examples of the Langlands philosophy that maps between L-groups should induce maps between automorphic representations. There are generalized versions of the Jacquet–Langlands correspondence relating automorphic representations of GLr(D) and GLdr(F), where D is a division algebra of degree d2 over the local or global field F.
Suppose that G is an inner twist of the algebraic group GL2, in other words the multiplicative group of a quaternion algebra. The Jacquet–Langlands correspondence is bijection between
Automorphic representations of G of dimension greater than 1
Cuspidal automorphic representations of GL2 that are square integrable (modulo the center) at each ramified place of G.
Corresponding representations have the same local components at all unramified places of G.
and extended the Jacquet–Langlands correspondence to division algebras of higher dimension.
References
Automorphic forms
Theorems in harmonic analysis |
https://en.wikipedia.org/wiki/James%20Howard%20Gore | James Howard Gore (September 18, 1856 – June 10, 1939) was professor of mathematics at The Corcoran Scientific School (which became the George Washington University School of Engineering and Applied Science). In 1905, Gore was the head of the mathematics department and taught a majority of the undergraduate and graduate courses. He is the author of nearly 20 books covering topics including mathematics, geodesy, European politics, travel and art. Much of his writing was influenced by his time spent in Berlin, Leyden, and Brussels, where he completed his post-graduate studies. Gore was also a member of several European royal societies such as Commandeur of the Order of Leopold (Belgium), a member of the Order of the Crown (Belgium) and a member of the Order of Orange-Nassau. Each of these societies are related to the royal parties of Belgium and the Netherlands and are generally bestowed upon individuals who have demonstrated dedication and commitment towards these respective areas.
History
James Howard Gore was born to Mahlon Gore and Sydney Cather in Frederick County, Virginia. Growing up, he lived on a farm with his family. At the age of four, his father died. From this point on, his mother took on the responsibility of raising Gore and his two brothers, with whom he shared farm chores. From an early age, Gore displayed interest in the sciences and mathematics and often read scientific literature. Gore took his education very seriously and chose to attend the Hamilton Academy, Richmond College, and finally Columbian University (now George Washington University). Gore also took time abroad and performed his post graduate studies in Berlin, Leyden and Brussels from 1894 through 1897. However, it was in 1877 that Gore graduated from Columbian University but remained there as a tutor. Thus began his profession of teaching. During 1878–1881 he served as a tutor at Columbian University in mathematics; during 1881–1883 Gore served as an adjunct professor of mathematics at Columbian University; in 1884–1887 he was professor of mathematics and geodesy at the Corcoran Scientific School at the George Washington University; during 1884–1909 Gore was a respected professor at the Corcoran Scientific school until his retirement; in 1888 he received an honorary doctorate from Columbian University; and during 1909–1939 Gore was a professor emeritus until his death in 1939. Throughout his time teaching at George Washington University, Gore lived not too far away in a neighborhood frequented by today's students in Friendship Heights, Maryland. On July 20, 1889, Gore married Lillian van Sparrendahl and they had one child. He died on June 10, 1939, at the age of 82. George Washington University continues to preserve materials from his days as a professor.
Publications
Outside of his teaching career, Gore wrote several publications, including Geodesy; Elements of Geodesy; History of Geodesy; Physical Geography; Parliamentary Law; German Science Reader; Editor of |
https://en.wikipedia.org/wiki/Masashi%20Owada | is a former Japanese football player. he is the current first-team coach J2 League club of JEF United Chiba.
Club statistics
References
External links
1981 births
Living people
Meiji University alumni
Association football people from Ibaraki Prefecture
Japanese men's footballers
J2 League players
Mito HollyHock players
Tochigi SC players
Men's association football defenders |
https://en.wikipedia.org/wiki/Yoshinobu%20Harada | is a Japanese footballer who currently plays for Tochigi Uva F.C. in the Kantō Soccer League.
Club statistics
Updated to 23 February 2016.
References
External links
1986 births
Living people
Association football people from Saitama Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Mito HollyHock players
Tochigi City FC players
V-Varen Nagasaki players
Zweigen Kanazawa players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Hideyuki%20Nakamura | is a former Japanese football player.
Club statistics
References
External links
1984 births
Living people
Juntendo University alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
Mito HollyHock players
Thespakusatsu Gunma players
FC Gifu players
Montedio Yamagata players
Men's association football defenders |
https://en.wikipedia.org/wiki/Kazuhiko%20Shingyoji | is a former Japanese football player.
Club statistics
References
External links
1986 births
Living people
Association football people from Tokyo
Japanese men's footballers
J2 League players
Japan Football League players
Mito HollyHock players
Blaublitz Akita players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Jun%20Tanaka%20%28footballer%29 | is a former Japanese football player.
Club statistics
References
External links
1983 births
Living people
Shobi University alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J2 League players
Thespakusatsu Gunma players
Men's association football defenders |
https://en.wikipedia.org/wiki/Kazuki%20Sakurada | is a former Japanese football player.
Club statistics
References
External links
1982 births
Living people
Shizuoka Sangyo University alumni
Association football people from Shizuoka Prefecture
Japanese men's footballers
J2 League players
Thespakusatsu Gunma players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Charles%20Epstein | Charles L. Epstein is a Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, Philadelphia.
Research interests
Epstein is an analyst and applied mathematician. His interests include microlocal analysis and index theory; boundary value problems; nuclear magnetic resonance and medical imaging; and mathematical biology.
Education and work
Epstein was an undergraduate in mathematics at the Massachusetts Institute of Technology and graduate student at the Courant Institute, New York University, where he received his Ph.D. in 1983 under the direction of Peter Lax.
He did a postdoc with William Thurston before moving to the University of Pennsylvania, where he has been since. Epstein won a Sloan Research Fellowship in 1988.
He is currently Thomas A. Scott Professor of Mathematics and serves as graduate chair of Applied Mathematics and Computational Science from 2008 to June 2019.
Awards and honors
In 2014, he became a Fellow of the American Mathematical Society "for contributions to analysis, geometry, and applied mathematics including medical imaging, as well as for service to the profession".
Books
C L Epstein, Introduction to the mathematics of medical imaging. Second edition. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. xxxiv+761 pp.
C L Epstein, The spectral theory of geometrically periodic hyperbolic 3-manifolds. Mem. Amer. Math. Soc. 58 (1985), no. 335, ix+161 pp.
Publications
C L Epstein, R B Melrose, G A Mendoza, Resolvent of the Laplacian on strictly pseudoconvex domains. Acta Mathematica 167 (1991), no. 1–2, 1–106.
C L Epstein, The hyperbolic Gauss map and quasiconformal reflections. Journal für die Reine und Angewandte Mathematik 372 (1986), 96–135.
C L Epstein, R Melrose, Contact degree and the index of Fourier integral operators. Math. Res. Lett. 5 (1998), no. 3, 363–381.
C L Epstein, Embeddable CR-structures and deformations of pseudoconvex surfaces. I. Formal deformations. J. Algebraic Geom. 5 (1996), no. 2, 277–368.
C L Epstein, CR-structures on three-dimensional circle bundles. Invent. Math. 109 (1992), no. 2, 351–403.
D M Burns, C L Epstein, Embeddability for three-dimensional CR-manifolds. J. Amer. Math. Soc. 3 (1990), no. 4, 809–841.
C L Epstein A relative index on the space of embeddable CR-structures. I. Annals of Mathematics (2) 147 (1998), no. 1, 1–59.
C L Epstein, Asymptotics for closed geodesics in a homology class, the finite volume case. Duke Math. J. 55 (1987), no. 4, 717–757.
C L Epstein; G M Henkin, Stability of embeddings for pseudoconcave surfaces and their boundaries. Acta Mathematica 185 (2000), no. 2, 161–237.
C L Epstein, A relative index on the space of embeddable CR-structures. II. Annals of Mathematics (2) 147 (1998), no. 1, 61–91.
D Burns, C L Epstein, Characteristic numbers of bounded domains. Acta Mathematica 164 (1990), no. 1–2, 29–71.
C L Epstein, M Gage, The curve shortening flow. Wave motion: theory, modelling, and computation (Berkeley, Ca |
https://en.wikipedia.org/wiki/Daisuke%20Fujii | is a former Japanese football player.
Club statistics
References
External links
1986 births
Living people
Association football people from Osaka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Albirex Niigata players
Thespakusatsu Gunma players
V-Varen Nagasaki players
Kamatamare Sanuki players
Men's association football defenders
People from Ibaraki, Osaka |
https://en.wikipedia.org/wiki/Shota%20Iwata | is a former Japanese football player.
Club statistics
References
External links
1988 births
Living people
Association football people from Chiba Prefecture
Japanese men's footballers
J2 League players
Thespakusatsu Gunma players
Men's association football forwards |
https://en.wikipedia.org/wiki/Nobuhide%20Akiba | is a Japanese football player. He plays for Vonds Ichihara.
Club statistics
References
External links
1985 births
Living people
Shizuoka Sangyo University alumni
Association football people from Chiba Prefecture
Japanese men's footballers
J2 League players
Thespakusatsu Gunma players
Vonds Ichihara players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kenta%20Togawa | is a former Japanese football player.
Club statistics
References
External links
1981 births
Living people
Meiji University alumni
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Tokyo Verdy players
Yokohama FC players
Gainare Tottori players
Fukushima United FC players
Men's association football defenders
FISU World University Games gold medalists for Japan
Universiade medalists in football |
https://en.wikipedia.org/wiki/Anton%20Ehmann | Anton Ehmann (born 17 December 1972) is a retired Austrian football player.
National team statistics
Honours
Austrian Football Bundesliga: 2003–2004
Austrian Cup: 1999–2000, 2001–2002, 2003–2004
References
External links
1972 births
Living people
Austrian men's footballers
Austria men's international footballers
LASK players
SV Ried players
Grazer AK players
Men's association football defenders |
https://en.wikipedia.org/wiki/Eduard%20Glieder | Eduard "Edi" Glieder (born 28 January 1969) is an Austrian former professional footballer played as a forward.
Career statistics
Club
International
Honours
Austria Salzburg
Austrian Bundesliga winner: 1994–95, 1996–97
Austrian Supercup winner: 1997
Austrian Bundesliga top scorer: 1998–99 (22 goals)
Tirol Innsbruck
Austrian Bundesliga winner: 1999–2000, 2000–01, 2001–02
Background
His youth club St. Margarethen renamed its stadium to "Edi-Glieder Stadion" in June 2001.
References
1969 births
Living people
Austrian men's footballers
Men's association football forwards
Austria men's international footballers
Grazer AK players
FC Red Bull Salzburg players
FC Schalke 04 players
FC Kärnten players
Austrian Football Bundesliga players
2. Liga (Austria) players
Bundesliga players
Austrian expatriate men's footballers
Austrian expatriate sportspeople in Germany
Expatriate men's footballers in Germany
SK Vorwärts Steyr managers
FC Tirol Innsbruck players |
https://en.wikipedia.org/wiki/Michael%20Wagner%20%28footballer%2C%20born%201975%29 | Michael Wagner (born 18 December 1975) is an Austrian former professional footballer who played as a midfielder. He made ten appearances for the Austria national team.
Career statistics
References
Living people
1975 births
Austrian men's footballers
Men's association football midfielders
Austria men's international footballers
FK Austria Wien players
SC Freiburg players
SK Rapid Wien players
FC Admira Wacker Mödling players
Austrian Football Bundesliga players
Bundesliga players
Austrian expatriate men's footballers
Expatriate men's footballers in Germany |
https://en.wikipedia.org/wiki/Kabanga%2C%20Kagera%20Region | Kabanga is a ward in the Ngara District of the Kagera Region in Tanzania near the Burundian border. Wahagaza are the indigenous of Ngara. In 2016 the Tanzania National Bureau of Statistics report there were 24,979 people in the ward, from 22,010 in 2012.
Villages
The ward has 25 villages.
Mukafigiri
Mukitangaro
Mkisagara
Muntamba
Ibuga Na. 1
Ibuga Centre
Mukihahe
Kichacha A
Kichacha B
Mukigoti
Mndarangavye
Djululigwa centre
Murukoli
Kabanga ya juu
Kumushiha
Nzaza A
Nzaza B
Mkagobero
Murukukumbo chini
Mundimanga
Muchuya
Mukitamo
Murutete
Mukirehe
Murulango
Transport
In 2021, a proposed railway to Rwanda would pass through this town.
See also
Kabanga Nickel Project
References
Populated places in Kagera Region |
https://en.wikipedia.org/wiki/Three-point%20estimation | The three-point estimation technique is used in management and information systems applications for the construction of an approximate probability distribution representing the outcome of future events, based on very limited information. While the distribution used for the approximation might be a normal distribution, this is not always so. For example, a triangular distribution might be used, depending on the application.
In three-point estimation, three figures are produced initially for every distribution that is required, based on prior experience or best-guesses:
a = the best-case estimate
m = the most likely estimate
b = the worst-case estimate
These are then combined to yield either a full probability distribution, for later combination with distributions obtained similarly for other variables, or summary descriptors of the distribution, such as the mean, standard deviation or percentage points of the distribution. The accuracy attributed to the results derived can be no better than the accuracy inherent in the three initial points, and there are clear dangers in using an assumed form for an underlying distribution that itself has little basis.
Estimation
Based on the assumption that a PERT distribution governs the data, several estimates are possible. These values are used to calculate an E value for the estimate and a standard deviation (SD) as L-estimators, where:
E = (a + 4m + b) / 6
SD = (b − a) / 6
E is a weighted average which takes into account both the most optimistic and most pessimistic estimates provided. SD measures the variability or uncertainty in the estimate.
In Program Evaluation and Review Techniques (PERT) the three values are used to fit a PERT distribution for Monte Carlo simulations.
The triangular distribution is also commonly used. It differs from the double-triangular by its simple triangular shape and by the property that the mode does not have to coincide with the median. The mean (expected value) is then:
E = (a + m + b) / 3.
In some applications, the triangular distribution is used directly as an estimated probability distribution, rather than for the derivation of estimated statistics.
Project management
To produce a project estimate the project manager:
Decomposes the project into a list of estimable tasks, i.e. a work breakdown structure
Estimates the expected value E(task) and the standard deviation SD(task) of this estimate for each task time
Calculates the expected value for the total project work time as
Calculates the value SD(project) for the standard error of the estimated total project work time as: under the assumption that the project work time estimates are uncorrelated
The E and SD values are then used to convert the project time estimates to confidence intervals as follows:
The 68% confidence interval for the true project work time is approximately E(project) ± SD(project)
The 90% confidence interval for the true project work time is approximately E(project) ± 1.645 × S |
https://en.wikipedia.org/wiki/Jan%20H.%20van%20Schuppen | Jan Hendrik van Schuppen (born 6 October 1947) is a Dutch mathematician and Professor at the Department of Mathematics of the Vrije Universiteit, known for his contributions in the field of systems theory, particularly on control theory and system identification, on probability, and on a number of related practical applications.
Biography
Van Schuppen obtained a PhD at the University of California, Berkeley, in 1973, where his PhD supervisor was Pravin Varaiya.
Van Schuppen works as a full professor at the Department of Mathematics of the Free University of Amsterdam and as a research leader at the CWI research institute in Amsterdam. He has been coordinating several European Union funded research networks such as the European Research Network System Identification, for which he has been the Netherlands leader. The lists among the PhD students who worked under van Schuppen's supervision Hendrik (Henk) Nijmeijer, Jan Willem Polderman, Peter Spreij and Damiano Brigo.
Van Schuppen is Editor in Chief of Mathematics of Control, Signals, and Systems, has been Departmental Editor of the Journal of Discrete Event Dynamic Systems in 1990–2000, and has been Associate Editor-at-Large of the prestigious and leading journal IEEE Transactions on Automatic Control in 1999–2001.
Work
Van Schuppen's research interest are in the areas of systems theory and probability. These include system identification, and realization theory, and the area of control theory, with control of discrete-event systems, control of hybrid systems, control and system theory of positive systems, control of stochastic systems, and adaptive control.
He worked also on the filtering problem, on dynamic games and team problems, on probability and stochastic processes, and on applications of the theory including control and system theory of biochemical reaction networks, control of communication systems and networks, and control of motorway traffic in a consultancy for the Dutch administration.
Publications
Van Schuppen has authored more than one hundred publications in the field and is a universally recognized and respected authority in the area. A selection, obtained by Jan van Schuppen's web site, is as follows.
Realization theory
J.H. van Schuppen, System theory for system identification, J. Econometrics 188 (2004), 313–339.
J.M. van den Hof, J.H. van Schuppen, Positive matrix factorization via extremal polyhedral cones, Linear Algebra and its Appl. 293(1999), 171–186.
J.H. van Schuppen, Equivalences of discrete-event systems and of hybrid systems, Open problems of mathematical systems and control theory, V.D. Blondel, E.D. Sontag, M. Vidyasagar, J.C. Willems, Springer Verlag, London, 1998, 251–257.
G. Picci, J.M. van den Hof, J.H. van Schuppen, Primes in several classes of the positive matrices, Linear Algebra and its Applications 277 (1998), 149–185.
J.H. van Schuppen, Stochastic realization of a Gaussian stochastic control system, Acta Applicandae Mathematicae 35(1994), |
https://en.wikipedia.org/wiki/Cohomology%20with%20compact%20support | In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.
Singular cohomology with compact support
Let be a topological space. Then
This is also naturally isomorphic to the cohomology of the sub–chain complex consisting of all singular cochains that have compact support in the sense that there exists some compact such that vanishes on all chains in .
Functorial definition
Let be a topological space and the map to the point. Using the direct image and direct image with compact support functors , one can define cohomology and cohomology with compact support of a sheaf of abelian groups on as
Taking for the constant sheaf with coefficients in a ring recovers the previous definition.
de Rham cohomology with compact support for smooth manifolds
Given a manifold X, let be the real vector space of k-forms on X with compact support, and d be the standard exterior derivative. Then the de Rham cohomology groups with compact support are the homology of the chain complex :
i.e., is the vector space of closed q-forms modulo that of exact q-forms.
Despite their definition as the homology of an ascending complex, the de Rham groups with compact support demonstrate covariant behavior; for example, given the inclusion mapping j for an open set U of X, extension of forms on U to X (by defining them to be 0 on X–U) is a map inducing a map
.
They also demonstrate contravariant behavior with respect to proper maps - that is, maps such that the inverse image of every compact set is compact. Let f: Y → X be such a map; then the pullback
induces a map
.
If Z is a submanifold of X and U = X–Z is the complementary open set, there is a long exact sequence
called the long exact sequence of cohomology with compact support. It has numerous applications, such as the Jordan curve theorem, which is obtained for X = R² and Z a simple closed curve in X.
De Rham cohomology with compact support satisfies a covariant Mayer–Vietoris sequence: if U and V are open sets covering X, then
where all maps are induced by extension by zero is also exact.
See also
Borel–Moore homology
Poincaré duality
Constructible sheaf
Derived category
References
Cohomology theories |
https://en.wikipedia.org/wiki/Junki%20Koike | is a Japanese football player currently playing for Tokyo Verdy.
Club statistics
Updated to 19 July 2022.
References
External links
Profile at Ehime FC
Profile at JEF United Chiba
1987 births
Living people
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
Urawa Red Diamonds players
Thespakusatsu Gunma players
Mito HollyHock players
Tokyo Verdy players
Yokohama FC players
JEF United Chiba players
Ehime FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Tetsushi%20Kondo | is a former Japanese football player.
Club statistics
References
External links
1986 births
Living people
Japanese men's footballers
J1 League players
J2 League players
Urawa Red Diamonds players
Ehime FC players
Fagiano Okayama players
Men's association football defenders
Association football people from Fukuoka (city) |
https://en.wikipedia.org/wiki/Hiroyuki%20Takasaki | is a Japanese football player currently playing for Ventforet Kofu.
Club statistics
Updated to 24 February 2019.
1Includes J2 Playoffs.
References
External links
Profile at Matsumoto Yamaga
Profile at Urawa Reds
Hiroyuki Takasaki at Yahoo! Japan sports
1986 births
Living people
Komazawa University alumni
Association football people from Ibaraki Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Urawa Red Diamonds players
Mito HollyHock players
Ventforet Kofu players
Tokushima Vortis players
Kashima Antlers players
Montedio Yamagata players
Matsumoto Yamaga FC players
FC Gifu players
Men's association football forwards |
https://en.wikipedia.org/wiki/Yoshiya%20Nishizawa | is a Japanese football player currently playing for Okinawa SV.
Career statistics
Updated to 23 February 2020.
References
External links
Profile at Tochigi SC
1987 births
Living people
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Urawa Red Diamonds players
Thespakusatsu Gunma players
Tochigi SC players
Okinawa SV players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Abuse%20of%20language | Abuse of language may refer to:
Abuse of terminology, in mathematics, a use of terminology in a way that is not formally correct but that simplifies exposition or suggests the correct intuition
Misnomer, a term which suggests an interpretation that is known to be untrue
Barbarism (linguistics), use of non-standard language forms |
https://en.wikipedia.org/wiki/Spectral%20geometry | Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other Laplace operators in differential geometry have also been examined. The field concerns itself with two kinds of questions: direct problems and inverse problems.
Inverse problems seek to identify features of the geometry from information about the eigenvalues of the Laplacian. One of the earliest results of this kind was due to Hermann Weyl who used David Hilbert's theory of integral equation in 1911 to show that the volume of a bounded domain in Euclidean space can be determined from the asymptotic behavior of the eigenvalues for the Dirichlet boundary value problem of the Laplace operator. This question is usually expressed as "Can one hear the shape of a drum?", the popular phrase due to Mark Kac. A refinement of Weyl's asymptotic formula obtained by Pleijel and Minakshisundaram produces a series of local spectral invariants involving covariant differentiations of the curvature tensor, which can be used to establish spectral rigidity for a special class of manifolds. However as the example given by John Milnor tells us, the information of eigenvalues is not enough to determine the isometry class of a manifold (see isospectral). A general and systematic method due to Toshikazu Sunada gave rise to a veritable cottage industry of such examples which clarifies the phenomenon of isospectral manifolds.
Direct problems attempt to infer the behavior of the eigenvalues of a Riemannian manifold from knowledge of the geometry. The solutions to direct problems are typified by the Cheeger inequality which gives a relation between the first positive eigenvalue and an isoperimetric constant (the Cheeger constant). Many versions of the inequality have been established since Cheeger's work (by R. Brooks and P. Buser for instance).
See also
Isospectral
Hearing the shape of a drum
References
.
.
Differential geometry
Spectral theory
Riemannian geometry |
https://en.wikipedia.org/wiki/2008%20Queensland%20Reds%20season | This article covers the 2008 Super 14 season results and statistics of Super Rugby side, the Reds.
Regular season
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Week 9
! align=centre colspan=100| Bye
|- bgcolor="#FFFFFF"
| Chiefs
| Reds
Week 10
Week 11
Week 12
Week 13
Week 14
Table
Table notes
Pos = Table Position
Pld = Played
W = Win (Worth 4 points)
D = Draw (Worth 2 points)
L = Loss (Worth 0 points)
F = For (Total points scored)
A = Against (Total points scored against)
+/- = Points difference (The total of For minus Against points)
BP = Bonus Point
Teams can score up to two additional bonus points in each regular season match. One bonus point will be awarded to any team that scores four tries or more in a single game, regardless of win/loss/draw. A bonus point will also be awarded to the losing side if the margin of loss is 7 points or less. Only the losing side can achieve the maximum 2 bonus points.
Pts = Progressive points tally
References
2008
2008 in Australian rugby union
2008 Super 14 season by team |
https://en.wikipedia.org/wiki/Surface%20plot | Surface plot may refer to:
Surface plot (mathematics), a graph of a function of two variables
Surface plot (radar) |
https://en.wikipedia.org/wiki/Mutual%20coherence | Mutual coherence can refer to:
Mutual coherence (physics), sinusoidal waves which exhibit a constant phase relationship
Mutual coherence (linear algebra), a property of a matrix describing the maximum correlation between its columns
See also
Coherence (disambiguation) |
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