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https://en.wikipedia.org/wiki/Kakutani%27s%20theorem | In mathematics, Kakutani's theorem may refer to:
the Kakutani fixed-point theorem, a fixed-point theorem for set-valued functions;
Kakutani's theorem (geometry): the result that every convex body in 3-dimensional space has a circumscribed cube;
Kakutani's theorem (measure theory): a result on the mutual equivalence or singularity of infinite product measures
the result that a Banach space is reflexive if and only if its closed unit ball is compact in the weak topology: see Reflexive space#Properties.
the Birkhoff-Kakutani theorem: the result that for a topological group, metrizability, first countability, and the existence of a compatible left-invariant metric are all equivalent. |
https://en.wikipedia.org/wiki/Edward%20Bierstone | Edward Bierstone (born ) is a Canadian mathematician at the University of Toronto who specializes in singularity theory, analytic geometry, and differential analysis.
Education and career
He received his B.Sc. from the University of Toronto and his Ph.D. at Brandeis University in 1972. He was a visiting scholar at the Institute for Advanced Study in the summer of 1973. He served as the Director of the Fields Institute from 2009 to 2013.
Recognition
Bierstone was elected a member of the Royal Society of Canada in 1992 and, together with Pierre Milman, received the Jefferey Williams Prize in 2005. In 2012 he became a fellow of the American Mathematical Society. In 2018 the Canadian Mathematical Society listed him in their inaugural class of fellows.
Notes
External links
personal webpage
Living people
Canadian mathematicians
Fellows of the Canadian Mathematical Society
Fellows of the Royal Society of Canada
Academic staff of the University of Toronto
Institute for Advanced Study visiting scholars
Fellows of the American Mathematical Society
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/G%C3%BCnter%20Heimbeck | Günter Heimbeck (born June 23, 1946 in Gunzenhausen, Germany) is a German–Namibian retired professor of mathematics. His particular research interest is geometry; the Heimbeck Planes are named for him. Heimbeck probably is the first and only Namibian scholar to have a scientific sub-discipline carry his name.
Heimbeck studied mathematics at University of Würzburg from 1965. He completed his PhD in 1974, and his habilitation in 1981. He then became lecturer at his alma mater. In 1985 Heimbeck emigrated to South Africa, where he taught at the University of the Witwatersrand in Johannesburg. In 1987 he took up a professorship for mathematics at the University of Namibia. Heimbeck is advisor to the Namibian Ministry of Education.
References
1946 births
Living people
20th-century German mathematicians
21st-century German mathematicians
Namibian mathematicians
People from Gunzenhausen |
https://en.wikipedia.org/wiki/1977%20S%C3%A3o%20Paulo%20FC%20season | The 1977 football season was São Paulo's 48th season since club's existence.
Statistics
Scorers
Overall
{|class="wikitable"
|-
|Games played || 74 (21 Campeonato Paulista, 46 Campeonato Brasileiro, 9 Friendly match)
|-
|Games won || 40 (13 Campeonato Paulista, 23 Campeonato Brasileiro, 4 Friendly match)
|-
|Games drawn || 20 (4 Campeonato Paulista, 13 Campeonato Brasileiro, 3 Friendly match)
|-
|Games lost || 16 (4 Campeonato Paulista, 10 Campeonato Brasileiro, 2 Friendly match)
|-
|Goals scored || 121
|-
|Goals conceded || 58
|-
|Goal difference || +59
|-
|Best result || 6–1 (H) v Marília - Campeonato Paulista - 1977.07.13
|-
|Worst result || 0–3 (A) v Portuguesa - Campeonato Paulista - 1977.03.27
|-
|Most appearances ||
|-
|Top scorer || Serginho (50)
|-
Friendlies
Torneio Triangular Luiz Lamejo
Official competitions
Campeonato Pauista
Record
Campeonato Brasileiro
Record
External links
official website
Association football clubs 1977 season
1977
1977 in Brazilian football |
https://en.wikipedia.org/wiki/M%C3%BCnster%20Journal%20of%20Mathematics | The Münster Journal of Mathematics is a peer-reviewed mathematics journal covering research in all fields of pure and applied mathematics. It is published by the Mathematical Institutes of the University of Münster. It was established in 1948 as the Schriftenreihe des Mathematischen Institutes Münster by Heinrich Behnke and continued by Reinhold Remmert, G. Maltese and Christopher Deninger. The journal obtained its current title in 2008 with volume numbering restarted at 1. The current editor-in-chief is Linus Kramer.
The journal appears both in print and electronically. Free full-text PDF versions of all articles are available on-line. According to the Mathematical Reviews, the journal has a Mathematical Citation Quotient for 2012 of 0.88.
References
External links
Mathematical Institute Münster
Mathematics journals
Academic journals established in 1948
Multilingual journals
University of Münster |
https://en.wikipedia.org/wiki/Allelomimetic%20behavior | Allelomimetic behavior or allomimetic behavior is a range of activities in which the performance of a behavior increases the probability of that behavior being performed by other nearby animals. Allelomimetic behavior is sometimes called contagious behavior and has strong components of social facilitation, imitation, and group coordination. It is usually considered to occur between members of the same species. An alternate definition is that allelomimetic behavior is a more intense or more frequent response or the initiation of an already known response, when others around the individual are engaged in the same behavior. It is often referred to as synchronous behavior, mimetic behavior, imitative behavior, and social facilitation.
Allelomimetic behavior is displayed in all animals and can occur in any stage of life, but usually starts at a young age. This behavior will continue throughout life, especially when an individual is living in a large group that emphasizes group cohesion. Cohesion is seen as a prerequisite for group living, with synchronous activity being crucial for social cohesion. However, animals in large cohesive groups face trade-offs when allelomimetic behavior is adopted. If the behavior is adopted then the risk of predation or capture decreases significantly but the inter-individual competition for immediate resources, such as food, mates, and space, will increase when cohesion is still stressed. Many collective group decisions in animals are the result of allelomimetism and can be explained by allelomimetic behaviors. Some examples are the cockroaches choosing a single aggregation site, schooling behaviors in fishes, and pheromone-based path selection in ants that allows all the workers to go down the same path to a specific food source. Allelomimetic behavior can also be seen as an animal welfare indicator. For example, if cattle do not have enough room to all lie down simultaneously then it indicates that there are not enough resources present and this can result in lameness of the animals that are forced to stand. Allomimicry is affected by circadian rhythms and circadian cycles of activity within groups which can give the overall appearance of poor animal welfare, if allomimetic behavior were to be used as a welfare indicator then it must be measured several times throughout the course of a day. Most mechanisms involved in performing allelomimetic behavior do not require circadian rhythms to function. Decisions at the individual level are, more often than not, enough to encourage allelomimetism. Patterns of allelomimetic behavior can vary from species to species and can possibly explain other behaviors seen in the animal kingdom.
Function
Group cohesion: Social animals often benefit by behaving in a similar manner to others within their group. This means that when animals switch behaviors, e.g. from lying to grazing, a degree of synchrony is beneficial. Sometimes this synchrony can be provided by environmental cues, at |
https://en.wikipedia.org/wiki/Microcontinuity | In nonstandard analysis, a discipline within classical mathematics, microcontinuity (or S-continuity) of an internal function f at a point a is defined as follows:
for all x infinitely close to a, the value f(x) is infinitely close to f(a).
Here x runs through the domain of f. In formulas, this can be expressed as follows:
if then .
For a function f defined on , the definition can be expressed in terms of the halo as follows: f is microcontinuous at if and only if , where the natural extension of f to the hyperreals is still denoted f. Alternatively, the property of microcontinuity at c can be expressed by stating that the composition is constant on the halo of c, where "st" is the standard part function.
History
The modern property of continuity of a function was first defined by Bolzano in 1817. However, Bolzano's work was not noticed by the larger mathematical community until its rediscovery in Heine in the 1860s. Meanwhile, Cauchy's textbook Cours d'Analyse defined continuity in 1821 using infinitesimals as above.
Continuity and uniform continuity
The property of microcontinuity is typically applied to the natural extension f* of a real function f. Thus, f defined on a real interval I is continuous if and only if f* is microcontinuous at every point of I. Meanwhile, f is uniformly continuous on I if and only if f* is microcontinuous at every point (standard and nonstandard) of the natural extension I* of its domain I (see Davis, 1977, p. 96).
Example 1
The real function on the open interval (0,1) is not uniformly continuous because the natural extension f* of f fails to be microcontinuous at an infinitesimal . Indeed, for such an a, the values a and 2a are infinitely close, but the values of f*, namely and are not infinitely close.
Example 2
The function on is not uniformly continuous because f* fails to be microcontinuous at an infinite point . Namely, setting and K = H + e, one easily sees that H and K are infinitely close but f*(H) and f*(K) are not infinitely close.
Uniform convergence
Uniform convergence similarly admits a simplified definition in a hyperreal setting. Thus, a sequence converges to f uniformly if for all x in the domain of f* and all infinite n, is infinitely close to .
See also
Standard part function
Bibliography
Martin Davis (1977) Applied nonstandard analysis. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney. xii+181 pp.
Gordon, E. I.; Kusraev, A. G.; Kutateladze, S. S.: Infinitesimal analysis. Updated and revised translation of the 2001 Russian original. Translated by Kutateladze. Mathematics and its Applications, 544. Kluwer Academic Publishers, Dordrecht, 2002.
References
Nonstandard analysis |
https://en.wikipedia.org/wiki/George%20A.%20Elliott | George Arthur Elliott (born 1945) is a Canadian mathematician specializing in operator algebras, K-theory, and non-commutative geometry. He is a professor at the University of Toronto Department of Mathematics, and holds a Canada Research Chair. He is best known for his work on classifying C*-algebras, both for initiating their classification and highlighting the importance of K-theory in this respect.
He was an invited speaker at the International Congress of Mathematicians, Zurich–1994.
Awards and honours
1982 Elected as Fellow of Royal Society of Canada.
1996 CRM/Fields Institute Prize
1996-1998 Killam Research Fellow.
1998 Jeffery–Williams Prize.
1999 John L. Synge Award.
2012 Fellow of the American Mathematical Society.
2019 Fellow of the Canadian Mathematical Society
References
External links
Living people
1945 births
Canadian mathematicians
Academic staff of the University of Toronto
Fellows of the Royal Society of Canada
Canada Research Chairs
Fellows of the American Mathematical Society
Fellows of the Canadian Mathematical Society
University of Toronto alumni
Queen's University at Kingston alumni |
https://en.wikipedia.org/wiki/Denjoy%E2%80%93Wolff%20theorem | In mathematics, the Denjoy–Wolff theorem is a theorem in complex analysis and dynamical systems concerning fixed points and iterations of holomorphic mappings of the unit disc in the complex numbers into itself. The result was proved independently in 1926 by the French mathematician Arnaud Denjoy and the Dutch mathematician Julius Wolff.
Statement
Theorem. Let D be the open unit disk in C and let f be a holomorphic function mapping D into D which is not an automorphism of D (i.e. a Möbius transformation). Then there is a unique point z in the closure of D such that the iterates of f tend to z uniformly on compact subsets of D. If z lies in D, it is the unique fixed point of f. The mapping f leaves invariant hyperbolic disks centered on z, if z lies in D, and disks tangent to the unit circle at z, if z lies on the boundary of D.
When the fixed point is at z = 0, the hyperbolic disks centred at z are just the Euclidean disks with centre 0. Otherwise f can be conjugated by a Möbius transformation so that the fixed point is zero. An elementary proof of the theorem is given below, taken from Shapiro (1993) and Burckel (1981). Two other short proofs can be found in Carleson & Gamelin (1993).
Proof of theorem
Fixed point in the disk
If f has a fixed point z in D then, after conjugating by a Möbius transformation, it can be assumed that z = 0. Let M(r) be the maximum modulus of f on |z| = r < 1. By the Schwarz lemma
for |z| ≤ r, where
It follows by iteration that
for |z| ≤ r. These two inequalities imply the result in this case.
No fixed points
When f acts in D without fixed points, Wolff showed that there is a point z on the boundary such that the iterates of f leave invariant each disk tangent to the boundary at that point.
Take a sequence increasing to 1 and set
By applying Rouché's theorem to and , has exactly one zero in D.
Passing to a subsequence if necessary, it can be assumed that The point z cannot lie in D, because,
by passing to the limit, z would have to be a fixed point. The result for the case of fixed points implies that the maps leave invariant all Euclidean disks whose hyperbolic center is located at . Explicit computations show that, as k increases, one can choose such disks so that they tend to any given disk tangent to the boundary at z. By continuity, f leaves each such disk Δ invariant.
To see that converges uniformly on compacta to the constant z, it is enough to show that the same is true for any subsequence , convergent in the same sense to g, say. Such limits exist by Montel's theorem, and if
g is non-constant, it can also be assumed that has a limit, h say. But then
for w in D.
Since h is holomorphic and g(D) open,
for all w.
Setting , it can also be assumed that is convergent to F say.
But then f(F(w)) = w = f(F(w)), contradicting the fact that f is not an automorphism.
Hence every subsequence tends to some constant uniformly on compacta in D.
The invariance of Δ implies each such constant lies |
https://en.wikipedia.org/wiki/Domestic%20violence%20in%20Norway | Domestic violence in Norway is officially referred to as vold i nære relasjoner (). It is defined as:
Extent
According to Norwegian police statistics, 5,284 cases of domestic violence were reported in 2008. These cases ranged from serious acts of violence such as murder and attempted murder to physical assault. The number of reported cases of domestic violence increased by 500 percent from 2005 to 2011. It is argued that the majority of cases go unrecorded. A 2011 study claimed that one in four women will experience domestic violence in their lifetime.
Government measures
In 2004 the Government established the Norwegian Centre for Violence and Traumatic Stress Studies, a research centre affiliated with the University of Oslo and with the national responsibility for violence research in Norway, including domestic violence.
Domestic violence was also addressed through the 2007 Handlingsplan mot vold i nære relasjoner (). This plan was drafted as a collaboration between the Ministry of Children and Family Affairs, the Ministry of Health, the Ministry of Justice and the Ministry of Social Affairs. The stated goal of the plan was expressed as follows:
See also
Crime in Norway
Notes
Violence against women in Norway
Women's rights in Norway
Norway
Family in Norway |
https://en.wikipedia.org/wiki/Maillet%27s%20determinant | In mathematics, Maillet's determinant Dp is the determinant of the matrix introduced by whose entries are R(s/r) for s,r = 1, 2, ..., (p – 1)/2 ∈ Z/pZ for an odd prime p, where and R(a) is the least positive residue of a mod p . calculated the determinant Dp for p = 3, 5, 7, 11, 13 and found that in these cases it is given by (–p)(p – 3)/2, and conjectured that it is given by this formula in general. showed that this conjecture is incorrect; the determinant in general is given by Dp = (–p)(p – 3)/2h−, where h− is the first factor of the class number of the cyclotomic field generated by pth roots of 1, which happens to be 1 for p less than 23. In particular, this verifies Maillet's conjecture that the determinant is always non-zero. Chowla and Weil had previously found the same formula but did not publish it.
Their results have been extended to all non-prime odd numbers by K. Wang(1982).
References
Algebraic number theory
Determinants |
https://en.wikipedia.org/wiki/1936%20Suomensarja%20%E2%80%93%20Finnish%20League%20Division%202 | These are statistics for the first season of the Suomensarja held in 1936.
Overview
The 1936 Suomensarja was contested by 13 teams divided into 2 regional sections. The 2 top teams from each section then participated in a promotion play-off group with VIFK Vaasa and UL Turku eventually gaining promotion with the former finishing as champions.
League tables
Itäsarja (Eastern League)
Länsisarja (Western League)
Nousukarsinnat (Promotion Playoffs)
See also
Mestaruussarja (Tier 1)
References
Suomensarja
2
Fin
Fin |
https://en.wikipedia.org/wiki/1937%20Suomensarja%20%E2%80%93%20Finnish%20League%20Division%202 | These are statistics for the first season of the Suomensarja held in 1937.
Overview
The 1937 Suomensarja was contested by 13 teams divided into 2 regional sections. The top teams from each section then participated in a promotion play-offs with KPT Kuopio and VPS Vaasa eventually gaining promotion with the former finishing as champions.
League tables
Itäsarja (Eastern League)
Länsisarja (Western League)
Nousukarsinnat (Promotion Playoffs)
KPT Kuopio were promoted and VPS Vaasa were required to undertake a further round of playoffs.
Mestaruussarja/Suomensarja promotion/relegation playoffs
VPS Vaasa were promoted to the Mestaruussarja and VIFK Vaasa relegated.
See also
Mestaruussarja (Tier 1)
References
Suomensarja
2
Fin
Fin |
https://en.wikipedia.org/wiki/1976%20S%C3%A3o%20Paulo%20FC%20season | The 1976 football season was São Paulo's 47th season since club's existence.
Statistics
Scorers
Overall
{|class="wikitable"
|-
|Games played || 63 (28 Campeonato Paulista, 13 Campeonato Brasileiro, 22 Friendly match)
|-
|Games won || 27 (12 Campeonato Paulista, 4 Campeonato Brasileiro, 11 Friendly match)
|-
|Games drawn || 21 (10 Campeonato Paulista, 4 Campeonato Brasileiro, 7 Friendly match)
|-
|Games lost || 15 (6 Campeonato Paulista, 5 Campeonato Brasileiro, 4 Friendly match)
|-
|Goals scored || 89
|-
|Goals conceded || 49
|-
|Goal difference || +40
|-
|Best result || 5–0 (H) v Noroeste - Campeonato Paulista - 1976.08.21
|-
|Worst result || 0–2 (A) v Portuguesa - Friendly match - 1976.02.170–2 (A) v Botafogo-SP - Campeonato Brasileiro - 1976.09.120–2 (H) v Atlético Paranaense - Campeonato Brasileiro - 1976.09.15
|-
|Most appearances ||
|-
|Top scorer || Pedro Rocha and Serginho (15)
|-
Friendlies
II Copa São Paulo (Taça Governador Laudo Natel)
Taça Governador do Estado de São Paulo
Taça Cidade de Maringá
Taça Cidade de Guaíra
Torneio Triangular Piracicabano
Torneio Nunes Freire
Official competitions
Campeonato Paulista
Record
Campeonato Brasileiro
Record
External links
official website
Association football clubs 1976 season
1976
1976 in Brazilian football |
https://en.wikipedia.org/wiki/Weyl%20law | In mathematics, especially spectral theory, Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator. This description was discovered in 1911 (in the case) by Hermann Weyl for eigenvalues for the Laplace–Beltrami operator acting on functions that vanish at the boundary of a bounded domain . In particular, he proved that the number, , of Dirichlet eigenvalues (counting their multiplicities) less than or equal to satisfies
where is a volume of the unit ball in . In 1912 he provided a new proof based on variational methods.
Generalizations
The Weyl law has been extended to more general domains and operators. For the Schrödinger operator
it was extended to
as tending to or to a bottom of essential spectrum and/or .
Here is the number of eigenvalues of below unless there is essential spectrum below in which case .
In the development of spectral asymptotics, the crucial role was played by variational methods and microlocal analysis.
Counter-examples
The extended Weyl law fails in certain situations. In particular, the extended Weyl law "claims" that there is no essential spectrum if and only if the right-hand expression is finite for all .
If one considers domains with cusps (i.e. "shrinking exits to infinity") then the (extended) Weyl law claims that there is no essential spectrum if and only if the volume is finite. However for the Dirichlet Laplacian there is no essential spectrum even if the volume is infinite as long as cusps shrinks at infinity (so the finiteness of the volume is not necessary).
On the other hand, for the Neumann Laplacian there is an essential spectrum unless cusps shrinks at infinity faster than the negative exponent (so the finiteness of the volume is not sufficient).
Weyl conjecture
Weyl conjectured that
where the remainder term is negative for Dirichlet boundary conditions and positive for Neumann.
The remainder estimate was improved upon by many mathematicians.
In 1922, Richard Courant proved a bound of .
In 1952, Boris Levitan proved the tighter bound of for compact closed manifolds. Robert Seeley extended this to include certain Euclidean domains in 1978.
In 1975, Hans Duistermaat and Victor Guillemin proved the bound of
when the set of periodic bicharacteristics has measure 0. This was finally generalized by Victor Ivrii in 1980. This generalization assumes that the set of periodic trajectories of a billiard in has measure 0, which Ivrii conjectured is fulfilled for all bounded Euclidean domains with smooth boundaries. Since then, similar results have been obtained for wider classes of operators.
References
Partial differential equations
Spectral theory |
https://en.wikipedia.org/wiki/Koenigs%20function | In mathematics, the Koenigs function is a function arising in complex analysis and dynamical systems. Introduced in 1884 by the French mathematician Gabriel Koenigs, it gives a canonical representation as dilations of a univalent holomorphic mapping, or a semigroup of mappings, of the unit disk in the complex numbers into itself.
Existence and uniqueness of Koenigs function
Let D be the unit disk in the complex numbers. Let be a holomorphic function mapping D into itself, fixing the point 0, with not identically 0 and not an automorphism of D, i.e. a Möbius transformation defined by a matrix in SU(1,1).
By the Denjoy-Wolff theorem, leaves invariant each disk |z | < r and the iterates of converge uniformly on compacta to 0: in fact for 0 < < 1,
for |z | ≤ r with M(r ) < 1. Moreover '(0) = with 0 < || < 1.
proved that there is a unique holomorphic function h defined on D, called the Koenigs function,
such that (0) = 0, '(0) = 1 and Schröder's equation is satisfied,
The function h is the uniform limit on compacta of the normalized iterates, .
Moreover, if is univalent, so is .
As a consequence, when (and hence ) are univalent, can be identified with the open domain . Under this conformal identification, the mapping becomes multiplication by , a dilation on .
Proof
Uniqueness. If is another solution then, by analyticity, it suffices to show that k = h near 0. Let
near 0. Thus H(0) =0, H'''(0)=1 and, for |z | small,
Substituting into the power series for , it follows that near 0. Hence near 0.Existence. If then by the Schwarz lemma
On the other hand,
Hence gn converges uniformly for |z| ≤ r by the Weierstrass M-test sinceUnivalence. By Hurwitz's theorem, since each gn is univalent and normalized, i.e. fixes 0 and has derivative 1 there , their limit is also univalent.
Koenigs function of a semigroup
Let be a semigroup of holomorphic univalent mappings of into itself fixing 0 defined
for such that
is not an automorphism for > 0
is jointly continuous in and
Each with > 0 has the same Koenigs function, cf. iterated function. In fact, if h is the Koenigs function of
, then satisfies Schroeder's equation and hence is proportion to h.
Taking derivatives gives
Hence is the Koenigs function of .
Structure of univalent semigroups
On the domain , the maps become multiplication by , a continuous semigroup.
So where is a uniquely determined solution of with Re < 0. It follows that the semigroup is differentiable at 0. Let
a holomorphic function on with v(0) = 0 and {{math|v'(0)}} = .
Then
so that
and
the flow equation for a vector field.
Restricting to the case with 0 < λ < 1, the h(D'') must be starlike so that
Since the same result holds for the reciprocal,
so that satisfies the conditions of
Conversely, reversing the above steps, any holomorphic vector field satisfying these conditions is associated to a semigroup , with
Notes
References
ASIN: B0006BTAC2
Complex analys |
https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3%20Filep | László Filep (6 December 1941 Császló – 19 November 2004 Budapest) was a Hungarian mathematician who specialized in history of mathematics. His Ph.D. advisors at the University of Debrecen were Barna Szénássy and Lajos Tamássy.
Selected publications
Books
Queen of the sciences (Development of mathematics)
Game theory
The history of number writing
References
External links
In memoriam László Filep
1941 births
2004 deaths
University of Debrecen alumni
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Historians of mathematics
People from Szabolcs-Szatmár-Bereg County |
https://en.wikipedia.org/wiki/120-cell%20honeycomb | In the geometry of hyperbolic 4-space, the 120-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol {5,3,3,3}, it has three 120-cells around each face. Its dual is the order-5 5-cell honeycomb, {3,3,3,5}.
Related honeycombs
It is related to the order-4 120-cell honeycomb, {5,3,3,4}, and order-5 120-cell honeycomb, {5,3,3,5}.
It is topologically similar to the finite 5-cube, {4,3,3,3}, and 5-simplex, {3,3,3,3}.
It is analogous to the 120-cell, {5,3,3}, and dodecahedron, {5,3}.
See also
List of regular polytopes
References
Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)
Honeycombs (geometry) |
https://en.wikipedia.org/wiki/Order-5%205-cell%20honeycomb | In the geometry of hyperbolic 4-space, the order-5 5-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol {3,3,3,5}, it has five 5-cells around each face. Its dual is the 120-cell honeycomb, {5,3,3,3}.
Related honeycombs
It is related to the order-5 tesseractic honeycomb, {4,3,3,5}, and order-5 120-cell honeycomb, {5,3,3,5}.
It is topologically similar to the finite 5-orthoplex, {3,3,3,4}, and 5-simplex, {3,3,3,3}.
It is analogous to the 600-cell, {3,3,5}, and icosahedron, {3,5}.
See also
List of regular polytopes
References
Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
Honeycombs (geometry) |
https://en.wikipedia.org/wiki/1938%20Suomensarja%20%E2%80%93%20Finnish%20League%20Division%202 | These are statistics for the first season of the Suomensarja held in 1938.
Overview
The 1938 Suomensarja was contested by 26 teams divided into 4 regional sections. The top teams from each section then participated in a promotion play-offs with Reipas Viipuri eventually gaining promotion andfinishing as champions.
League tables
Itäsarja, Eteläinen lohko (Eastern League, Southern Section)
Itäsarja, Pohjoinen lohko (Eastern League, Northern Section)
Länsisarja, Eteläinen lohko (Western League, Southern Section)
Länsisarja, Pohjoinen lohko (Western League, Northern Section)
Nousukarsinnat (Promotion Playoffs)
Semi-finals
Finals
Reipas Viipuri were promoted and KIF Helsinki were required to undertake a further round of playoffs.
Mestaruussarja/Suomensarja promotion/relegation playoffs
KIF Helsinki remained in the Suomensarja and KPT Kuopio were not relegated from the Mestaruussarja.
See also
Mestaruussarja (Tier 1)
References
Suomensarja
2
Fin
Fin |
https://en.wikipedia.org/wiki/Godofredo%20Garc%C3%ADa | Godofredo García (born in Lima, Peru, November 8, 1888 - July 16, 1970) was a Peruvian mathematician and engineer. He was the author of more than 80 publications covering mathematics, physics, astronomy, astrophysics, and engineering.
Background
He studied at the Colegio de Lima, under Pedro A. Labarthe. In 1906 he entered the Faculty of Sciences of the National University of San Marcos, where he received a bachelor's degree (1909) and later his doctorate degree in Mathematical Sciences (1912), with his thesis on "Singular points of flat curves" and "Resistance of Columns of reinforced concrete", respectively. Simultaneously, he studied at the School of Engineers of Peru, now called the National University of Engineering (1908-1910), graduating from Civil engineer in 1911.
From 1912, he taught at the Chorrillos Military School, where he was in charge of the courses of Flat, Descriptive and Analytical Geometry, Infinitesimal Calculus, Rational Mechanics and Exterior Ballistics. He was also professor of Rational Mechanics in the Faculty of Sciences of the University of San Marcos beginning in 1919 and later served as dean (1928-1940). He became Rector (1941-1943). He was also a professor at the School of Engineers corresponding with Albert Einstein.
In the 1920s he worked with the Polish mathematician Alfred Rosenblatt in San Marcos. In 1938, together with Rosenblatt and other San Marcos mathematicians, he founded the National Academy of Exact, Physical and Natural Sciences of Peru, an institution that he presided over from 1960 until he died in 1970. He also directed the publication Actas de la Academia".
García was elected to the American Philosophical Society in 1943. He was awarded the national prize for scientific research, in recognition of his contributions in the field of mathematical sciences and his "Exact equations and exact solutions to the movement and stresses of viscous fluids" (1948).
He organized conferences in Lima with the participation of Tullio Levi-Civita, Arthur Compton and Garret Birkhoff, among others. In each conference Godofredo García presented a review of the work of these scientists.
He married Alicia Rendón (Ecuadorian) and fathered four children.
Bibliography
Lessons of Rational Mechanics, UNMSM; 1937.
On a New Cosmogonic Theory, 1940.
Algebraic Analysis, Ed. Sanmarti, 1955.
References
Carranza, César. “La Matemática en el Perú”. Discurso pronunciado en XXIII Coloquio Nacional de Matemática, Lambayeque, 2004.
Samamé Boggio, Mario: “Godofredo García Díaz”, pg.71-81 de Hacer ciencia en el Perú. Biografías de ocho científicos. Lima, 1990.
Tauro del Pino, Alberto (2001): Enciclopedia Ilustrada del Perú. Tercera Edición. Tomo 7. FER/GUZ. Lima, PEISA.
People from Lima
National University of San Marcos alumni
Academic staff of the National University of San Marcos
1970 deaths
1888 births
Peruvian mathematicians
Members of the American Philosophical Society |
https://en.wikipedia.org/wiki/Grunsky%27s%20theorem | In mathematics, Grunsky's theorem, due to the German mathematician Helmut Grunsky, is a result in complex analysis concerning holomorphic univalent functions defined on the unit disk in the complex numbers. The theorem states that a univalent function defined on the unit disc, fixing the point 0, maps every disk |z| < r onto a starlike domain for r ≤ tanh π/4. The largest r for which this is true is called the radius of starlikeness of the function.
Statement
Let f be a univalent holomorphic function on the unit disc D such that f(0) = 0. Then for all r ≤ tanh π/4, the image of the disc |z| < r is starlike with respect to 0, , i.e. it is invariant under multiplication by real numbers in (0,1).
An inequality of Grunsky
If f(z) is univalent on D with f(0) = 0, then
Taking the real and imaginary parts of the logarithm, this implies the two inequalities
and
For fixed z, both these equalities are attained by suitable Koebe functions
where |w| = 1.
Proof
originally proved these inequalities based on extremal techniques of Ludwig Bieberbach. Subsequent proofs, outlined in , relied on the Loewner equation. More elementary proofs were subsequently given based on Goluzin's inequalities, an equivalent form of Grunsky's inequalities (1939) for the Grunsky matrix.
For a univalent function g in z > 1 with an expansion
Goluzin's inequalities state that
where the zi are distinct points with |zi| > 1 and λi are arbitrary complex numbers.
Taking n = 2. with λ1 = – λ2 = λ, the inequality implies
If g is an odd function and η = – ζ, this yields
Finally if f is any normalized univalent function in D, the required inequality for f follows by taking
with
Proof of the theorem
Let f be a univalent function on D with f(0) = 0. By Nevanlinna's criterion, f is starlike on |z| < r if and only if
for |z| < r. Equivalently
On the other hand by the inequality of Grunsky above,
Thus if
the inequality holds at z. This condition is equivalent to
and hence f is starlike on any disk |z| < r with r ≤ tanh π/4.
References
(in Russian)
(in German)
(in German)
Theorems in complex analysis |
https://en.wikipedia.org/wiki/Applied%20Maths | Applied Maths NV, a bioMérieux company headquartered in Sint-Martens-Latem, Belgium, is a bioinformatics company developing software for the biosciences.
History
Applied Maths was founded in 1992 and gained worldwide recognition with the software GelCompar, used as a standard tool for the normalization and comparative analysis of electrophoresis patterns (PFGE, AFLP, RAPD, REP-PCR and variants, etc.).
GelCompar II was released in 1998 to deal with the ever growing amounts of information following the success and expansion of electrophoresis and other fingerprinting techniques in various application fields in microbiology, virology and mycology. Following the introduction of the concepts of polyphasic taxonomy and the growing need to combine genotypic, phenotypic, electrophoresis and sequence information, Applied Maths released in 1996 the software package BIONUMERICS which still today is a platform for the management, storage and (statistical) analysis of all types of biological data. BIONUMERICS and GelCompar II are used by several networks around the globe, such as PulseNet and CaliciNet, to share and identify strain information.
In January 2016, Applied Maths was acquired by bioMérieux. In October 2021, the company announced the phase out of its flagship software BIONUMERICS by the end 2024. Applied Maths merged with bioMérieux in June 2023.
Products
BIONUMERICS: BIONUMERICS is a commercial suite of 4 configurations used for the analysis of all major applications in (microbial) bioinformatics: 1D electrophoresis gels, chromatographic and spectrometric profiles, phenotype characters, microarrays, nucleic acid sequences, whole genome sequences etc.
GelCompar II: GelCompar II is a suite of 5 modules developed for the analysis of fingerprint patterns, covering the normalization, import into a relational database and the comparative analysis.
BNServer: BNserver is the web-based platform generally installed between a centrally maintained database and distributed clients using BIONUMERICS, GelCompar II or a web browser to exchange biological information and analysis results. BNServer has been used since the nineties in Food outbreak detection.
EPISEQ CS: a cloud-based software for the analysis of clinical microbial outbreaks.
EPISEQ 16S: a cloud-based software for microbiome profiling studies based on the 16S rRNA gene.
EPISEQ SARS-CoV-2: a cloud-based software for the identification of SARS-CoV-2 variants.
Reception
Over 20,000 peer-reviewed research articles mention the use of Applied Maths software packages BIONUMERICS or Gelcompar II.
References
External links
Biotechnology companies of Belgium
Computational science
Bioinformatics companies
Software companies established in 1992
Biotechnology companies established in 1992
1992 establishments in Belgium
Software companies of Belgium
Companies based in East Flanders |
https://en.wikipedia.org/wiki/Nearly%20completely%20decomposable%20Markov%20chain | In probability theory, a nearly completely decomposable (NCD) Markov chain is a Markov chain where the state space can be partitioned in such a way that movement within a partition occurs much more frequently than movement between partitions. Particularly efficient algorithms exist to compute the stationary distribution of Markov chains with this property.
Definition
Ando and Fisher define a completely decomposable matrix as one where "an identical rearrangement of rows and columns leaves a set of square submatrices on the principal diagonal and zeros everywhere else." A nearly completely decomposable matrix is one where an identical rearrangement of rows and columns leaves a set of square submatrices on the principal diagonal and small nonzeros everywhere else.
Example
A Markov chain with transition matrix
is nearly completely decomposable if ε is small (say 0.1).
Stationary distribution algorithms
Special-purpose iterative algorithms have been designed for NCD Markov chains though the multi–level algorithm, a general purpose algorithm, has been shown experimentally to be competitive and in some cases significantly faster.
See also
Lumpability
References
Markov processes |
https://en.wikipedia.org/wiki/Christian%20Wiener | Ludwig Christian Wiener (7 December 1826 Darmstadt – 31 July 1896 Karlsruhe) was a German mathematician who specialized in descriptive geometry. Wiener was also a physicist and philosopher. In 1863, he was the first person to identify qualitatively the internal molecular cause of Brownian motion.
Wiener was the son of a judge and studied architecture and engineering in Giessen. After the state examination in 1848, he became a teacher at the "Höhere Gewerbeschule" in Darmstadt, today the Technische Universität Darmstadt.
The mathematician Hermann Wiener was his son.
Selected publications
Lehrbuch der darstellenden Geometrie, 2 Bände, Teubner, Leipzig 1884, 1887, online at archiv.org: ,
Die ersten Sätze der Erkenntniß, insbesondere das Gesetz der Ursächlichkeit und die Wirklichkeit der Außenwelt, Berlin, Lüderitz 1874
Die Freiheit des Willens, Darmstadt, Brill 1894
Die Grundzüge der Weltordnung, Leipzig, Winter 1863,
Über Vielecke und Vielflache, Teubner 1864
References
Thompson, D W., 1992. On Growth and Form. Cambridge Univ. Press. Abridged edition by John Tyler Bonner, p. 45. , .Online in Google Books
Otto Wiener, "Christian Wiener zum 100. Geburtstag", Naturwissenschaften Bd.15, 1927, Issue 4.
1826 births
1896 deaths
University of Giessen alumni
Academic staff of the Karlsruhe Institute of Technology
19th-century German mathematicians
Geometers
Technische Universität Darmstadt alumni |
https://en.wikipedia.org/wiki/M/G/k%20queue | In queueing theory, a discipline within the mathematical theory of probability, an M/G/k queue is a queue model where arrivals are Markovian (modulated by a Poisson process), service times have a General distribution and there are k servers. The model name is written in Kendall's notation, and is an extension of the M/M/c queue, where service times must be exponentially distributed and of the M/G/1 queue with a single server. Most performance metrics for this queueing system are not known and remain an open problem.
Model definition
A queue represented by a M/G/k queue is a stochastic process whose state space is the set {0,1,2,3...}, where the value corresponds to the number of customers in the queue, including any being served. Transitions from state i to i + 1 represent the arrival of a new customer: the times between such arrivals have an exponential distribution with parameter λ. Transitions from state i to i − 1 represent the departure of a customer who has just finished being served: the length of time required for serving an individual customer has a general distribution function. The lengths of times between arrivals and of service periods are random variables which are assumed to be statistically independent.
Steady state distribution
Tijms et al. believe it is "not likely that computationally tractable methods can be developed to compute the exact numerical values of the steady-state probability in the M/G/k queue."
Various approximations for the average queue size, stationary distribution and approximation by a reflected Brownian motion have been offered by different authors. Recently a new approximate approach based on Laplace transform for steady state probabilities has been proposed by Hamzeh Khazaei et al.. This new approach is yet accurate enough in cases of large number of servers and when the distribution of service time has a Coefficient of variation more than one.
Average delay/waiting time
There are numerous approximations for the average delay a job experiences. The first such was given in 1959 using a factor to adjust the mean waiting time in an M/M/c queue This result is sometimes known as Kingman's law of congestion.
where C is the coefficient of variation of the service time distribution. Ward Whitt described this approximation as “usually an excellent approximation, even given extra information about the service-time distribution."
However, it is known that no approximation using only the first two moments can be accurate in all cases.
A Markov–Krein characterization has been shown to produce tight bounds on the mean waiting time.
Inter-departure times
It is conjectured that the times between departures, given a departure leaves n customers in a queue, has a mean which as n tends to infinity is different from the intuitive 1/μ result.
Two servers
For an M/G/2 queue (the model with two servers) the problem of determining marginal probabilities can be reduced to solving a pair of integral equations or the L |
https://en.wikipedia.org/wiki/End%20%28graph%20theory%29 | In the mathematics of infinite graphs, an end of a graph represents, intuitively, a direction in which the graph extends to infinity. Ends may be formalized mathematically as equivalence classes of infinite paths, as havens describing strategies for pursuit–evasion games on the graph, or (in the case of locally finite graphs) as topological ends of topological spaces associated with the graph.
Ends of graphs may be used (via Cayley graphs) to define ends of finitely generated groups. Finitely generated infinite groups have one, two, or infinitely many ends, and the Stallings theorem about ends of groups provides a decomposition for groups with more than one end.
Definition and characterization
Ends of graphs were defined by in terms of equivalence classes of infinite paths. A in an infinite graph is a semi-infinite simple path; that is, it is an infinite sequence of vertices in which each vertex appears at most once in the sequence and each two consecutive vertices in the sequence are the two endpoints of an edge in the graph. According to Halin's definition, two rays and are equivalent if there is a ray (which may equal one of the two given rays) that contains infinitely many of the vertices in each of and . This is an equivalence relation: each ray is equivalent to itself, the definition is symmetric with regard to the ordering of the two rays, and it can be shown to be transitive. Therefore, it partitions the set of all rays into equivalence classes, and Halin defined an end as one of these equivalence classes.
An alternative definition of the same equivalence relation has also been used: two rays and are equivalent if there is no finite set of vertices that separates infinitely many vertices of from infinitely many vertices of . This is equivalent to Halin's definition: if the ray from Halin's definition exists, then any separator must contain infinitely many points of and therefore cannot be finite, and conversely if does not exist then a path that alternates as many times as possible between and must form the desired finite separator.
Ends also have a more concrete characterization in terms of havens, functions that describe evasion strategies for pursuit–evasion games on a graph . In the game in question, a robber is trying to evade a set of policemen by moving from vertex to vertex along the edges of . The police have helicopters and therefore do not need to follow the edges; however the robber can see the police coming and can choose where to move next before the helicopters land. A haven is a function that maps each set of police locations to one of the connected components of the subgraph formed by deleting ; a robber can evade the police by moving in each round of the game to a vertex within this component. Havens must satisfy a consistency property (corresponding to the requirement that the robber cannot move through vertices on which police have already landed): if is a subset of , and both and are valid se |
https://en.wikipedia.org/wiki/Andrew%20G.%20Reid | Andrew Graham Reid (May 24, 1878 – July 6, 1941) was an American football player, coach, and official, athletics administrator, professor of mathematics, businessman, and lawyer. He played football for the University of Michigan's 1901 "Point-a-Minute" team. He was the head football coach and athletic director at Monmouth College in Illinois from 1907 to 1909. He also served as a football official in Big Ten Conference football games.
Early years and playing career
Reid was born in Warren County, Illinois in 1878. He grew up in Iowa and attended the public schools there before enrolling at Simpson College in Indianola, Iowa. He received Ph.B. degree there in 1901.
Michigan
After graduating from Simpson, Reid enrolled in the law department at the University of Michigan. While attending Michigan, he was a "star athlete." During his first year of law school, Reid played football for the 1901 Michigan Wolverines football team. At five feet, eight inches, and 158 pounds, Reid was a reserve on the 1901 team. He appeared as a substitute fullback in Michigan's 50–0 victory over Albion and the team's 33–0 victory over Indiana. In its "Football Year-Book," The Michigan Daily-News said of Reid, "He has developed rapidly in the fullback position." The 1901 Michigan team was the first of Fielding H. Yost's "Point-a-Minute" teams. The team compiled a record of 11–0 and outscored its opponents 550 to 0.
Reid was also a member of the varsity track team, competing in the hammer throw, during his first year at Michigan. In May 1902, The Michigan Alumnus noted: "Reid, the sub-fullback on the Varsity, has been showing up remarkably well with the hammer, and the first of this week, threw it 126 feet, within less than four feet of the Intercollegiate record." Reid received his law degree from Michigan in 1906.
Coaching career
Simpson
Reid coached at Simpson College in Indianola, Iowa from 1902 to 1905.
Monmouth
After graduating from Michigan, Reid became an assistant professor of mathematics at Monmouth College in Illinois. In 1907, he was also hired by Monmouth College as its athletic director and coach. He stayed at Monmouth for three years, resigning in 1910. In Monmouth College's 1911 yearbook, Reid was remembered as follows
Coming to take charge of the work in the spring of 1907 when the prospects were none too bright, he has developed and trained teams for the last three years that are a credit to him and to the school, and through them his hard work and excellent coaching is clearly shown. In his coaching and training he has not only given especial attention to each individual and to keeping the entire team in the best possible condition, but also has always insisted on team work so that his teams have all worked together like well lubricated machines. Too much cannot be said of Reid's ability as a coach and trainer of athletes.
In three seasons as the head football coach at Monmouth, he compiled a record of 8–10–1.
Later years
In 1910, Reid moved |
https://en.wikipedia.org/wiki/Daniel%20Gigante | Daniel Carvalho da Silva (born March 10, 1981), known as Daniel Gigante, is a Brazilian football player.
Career
Daniel Gigante has played professionally for Thespa Kusatsu in Japan.
Club statistics
References
External links
j-league
jsgoal
1981 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
J2 League players
Thespakusatsu Gunma players
Expatriate men's footballers in Japan
Men's association football defenders |
https://en.wikipedia.org/wiki/Daisuke%20Ito%20%28footballer%29 | Daisuke Ito (伊藤 大介, born April 18, 1987) is a Japanese footballer who plays as a central midfielder for Criacao Shinjuku. He is known as a free kick specialist.
Club statistics
Updated to 23 February 2018.
References
External links
1987 births
Living people
Juntendo University alumni
Association football people from Chiba Prefecture
Japanese men's footballers
J2 League players
J3 League players
JEF United Chiba players
Oita Trinita players
Fagiano Okayama players
SC Sagamihara players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Ilias%20Miniatis | Ilias Miniatis () (1669 at Lixouri – 1714 at Patras) was a Greek clergyman, writer and preacher. At the Flanginian School he learned Ancient Greek and Latin and became interested in mathematics and philology. He was ordained very early. He preached God's word at his home island, Cephalonia, at Zakynthos, at Corfu and at Constantinople. His preachings are considered exemplars for modern ecclesiastical rhetoric. As of his language, it is simple Modern Greek and his style has something dramatical and hymnographic. His eloquent preachings are collected into the book "Διδαχαί" (Teachings), first published at Venice on 1725. An older book is "Η Πέτρα του Σκανδάλου" (The Start of the Scandal) about the Photian schism. Many historians consider him a student of Frangiscos Scoufos and others an imitator of Paolo Segneri. With his speeches he helped the development of ecclesiastical rhetoric and the configuration of Modern Greek language.
References
Τσουράκης, Διονύσιος (1997). Βιογραφίες Ελλήνων Λογοτεχνών και Συγγραφέων, pp. 246–247
1669 births
1714 deaths
18th-century Greek people
Greek religious writers
People from Paliki
Greek bishops
17th-century Greek writers
17th-century Greek educators
17th-century Greek scientists
18th-century Greek writers
18th-century Greek educators
18th-century Greek scientists |
https://en.wikipedia.org/wiki/1975%20S%C3%A3o%20Paulo%20FC%20season | The 1975 football season was São Paulo's 46th season since club's existence.
Statistics
Scorers
Overall
{|class="wikitable"
|-
|Games played || 72 (35 Campeonato Paulista, 28 Campeonato Brasileiro, 9 Friendly match)
|-
|Games won || 42 (26 Campeonato Paulista, 11 Campeonato Brasileiro, 5 Friendly match)
|-
|Games drawn || 24 (7 Campeonato Paulista, 14 Campeonato Brasileiro, 3 Friendly match)
|-
|Games lost || 5 (2 Campeonato Paulista, 3 Campeonato Brasileiro, 1 Friendly match)
|-
|Goals scored || 104
|-
|Goals conceded || 40
|-
|Goal difference || +64
|-
|Best result || 5–0 (H) v Noroeste - Campeonato Paulista - 1975.07.10
|-
|Worst result || 1–2 (A) v Santos - Campeonato Paulista - 1975.08.071–2 (H) v Portuguesa - Campeonato Brasileiro - 1975.11.231–2 (H) v Grêmio - Campeonato Brasileiro - 1975.11.29
|-
|Most appearances || Waldir Peres (67)
|-
|Top scorer || Serginho (35)
|-
Friendlies
Torneio Laudo Natel
I Copa Internacional de São Paulo
Official competitions
Campeonato Paulista
Record
Campeonato Brasileiro
Record
External links
official website
References
Association football clubs 1975 season
1975
1975 in Brazilian football |
https://en.wikipedia.org/wiki/Edward%20Barbeau | Edward Barbeau is a Canadian mathematician and a Canadian Mathematical Educator. He is a Professor Emeritus at the University of Toronto Department of Mathematics.
Awards
Fellowship of the Ontario Institute for Studies in Education.
David Hilbert Award from the World Federation of National Mathematics Competitions.
Adrien Pouliot Award from the Canadian Mathematical Society.
Inaugural fellow of the Canadian Mathematical Society, 2018
References
External links
Edward J. Barbeau archival papers held at the University of Toronto Archives and Records Management Services
Living people
Canadian mathematicians
Mathematics educators
Mathematics popularizers
Academic staff of the University of Toronto
Year of birth missing (living people)
Fellows of the Canadian Mathematical Society
University of Toronto alumni |
https://en.wikipedia.org/wiki/Nevanlinna%27s%20criterion | In mathematics, Nevanlinna's criterion in complex analysis, proved in 1920 by the Finnish mathematician Rolf Nevanlinna, characterizes holomorphic univalent functions on the unit disk which are starlike. Nevanlinna used this criterion to prove the Bieberbach conjecture for starlike univalent functions.
Statement of criterion
A univalent function h on the unit disk satisfying h(0) = 0 and h(0) = 1 is starlike, i.e. has image invariant under multiplication by real numbers in [0,1], if and only if has positive real part for |z| < 1 and takes the value 1 at 0.
Note that, by applying the result to a•h(rz), the criterion applies on any disc |z| < r with only the requirement that f(0) = 0 and f'''(0) ≠ 0.
Proof of criterion
Let h(z) be a starlike univalent function on |z| < 1 with h(0) = 0 and h(0) = 1.
For t < 0, define
a semigroup of holomorphic mappings of D into itself fixing 0.
Moreover h is the Koenigs function for the semigroup ft.
By the Schwarz lemma, |ft(z)| decreases as t increases.
Hence
But, setting w = ft(z),
where
Hence
and so, dividing by |w|2,
Taking reciprocals and letting t go to 0 gives
for all |z| < 1. Since the left hand side is a harmonic function, the maximum principle implies the inequality is strict.
Conversely if
has positive real part and g(0) = 1, then h can vanish only at 0, where it must have a simple zero.
Now
Thus as z traces the circle , the argument of the image increases strictly. By the argument principle, since has a simple zero at 0,
it circles the origin just once. The interior of the region bounded by the curve it traces is therefore starlike. If a is a point in the interior then the number of solutions N(a) of h(z) = a with |z| < r is given by
Since this is an integer, depends continuously on a and N(0) = 1, it is identically 1. So h is univalent and starlike in each disk |z| < r and hence everywhere.
Application to Bieberbach conjecture
Carathéodory's lemma
Constantin Carathéodory proved in 1907 that if
is a holomorphic function on the unit disk D with positive real part, then
In fact it suffices to show the result with g
replaced by gr(z) = g(rz) for any r < 1 and then pass to the limit r = 1.
In that case g extends to a continuous function on the closed disc with positive real part and by Schwarz formula
Using the identity
it follows that
,
so defines a probability measure, and
Hence
Proof for starlike functions
Let
be a univalent starlike function in |z| < 1. proved that
In fact by Nevanlinna's criterion
has positive real part for |z|<1. So by Carathéodory's lemma
On the other hand
gives the recurrence relation
where a''1 = 1. Thus
so it follows by induction that
Notes
References
Analytic functions |
https://en.wikipedia.org/wiki/Takuya%20Ito%20%28footballer%2C%20born%201974%29 | is a former Japanese football player.
Club statistics
References
External links
j-league
1974 births
Living people
Senshu University alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J2 League players
Japan Football League (1992–1998) players
Japan Football League players
Omiya Ardija players
Saitama SC players
Sagawa Shiga FC players
Fagiano Okayama players
Giravanz Kitakyushu players
Men's association football defenders |
https://en.wikipedia.org/wiki/Bojan%20Mohar | Bojan Mohar (born September 21, 1956) is a Slovenian and Canadian mathematician, working in graph theory. He is a professor of mathematics at the University of Ljubljana and the holder of a Canada Research Chair in graph theory at Simon Fraser University in Vancouver, British Columbia, Canada.
Education
Mohar received his PhD from the University of Ljubljana in 1986, under the supervision of Tomo Pisanski.
Research
Mohar's research concerns topological graph theory, algebraic graph theory, graph minors, and graph coloring.
With Carsten Thomassen he is the co-author of the book Graphs on Surfaces (Johns Hopkins University Press, 2001).
Books
Awards and honors
Mohar was a Fulbright visiting scholar at Ohio State University in 1988, and won the Boris Kidrič prize of the Socialist Republic of Slovenia in 1990. He has been a member of the Slovenian Academy of Engineering since 1999.
He was named a SIAM Fellow in 2018.
He was elected as a Fellow of the American Mathematical Society in the 2020 Class, for "contributions to topological graph theory, including the theory of graph embedding algorithms, graph coloring and crossing numbers, and for service to the profession".
References
External links
Home page at U. Ljubljana
Living people
21st-century Slovenian mathematicians
20th-century Slovenian mathematicians
Graph theorists
Canada Research Chairs
University of Ljubljana alumni
Academic staff of Simon Fraser University
Academic staff of the University of Ljubljana
Fellows of the American Mathematical Society
Fellows of the Society for Industrial and Applied Mathematics
1956 births |
https://en.wikipedia.org/wiki/Elastic%20net%20regularization | In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the lasso and ridge methods.
Specification
The elastic net method overcomes the limitations of the LASSO (least absolute shrinkage and selection operator) method which uses a penalty function based on
Use of this penalty function has several limitations. For example, in the "large p, small n" case (high-dimensional data with few examples), the LASSO selects at most n variables before it saturates. Also if there is a group of highly correlated variables, then the LASSO tends to select one variable from a group and ignore the others. To overcome these limitations, the elastic net adds a quadratic part () to the penalty, which when used alone is ridge regression (known also as Tikhonov regularization).
The estimates from the elastic net method are defined by
The quadratic penalty term makes the loss function strongly convex, and it therefore has a unique minimum. The elastic net method includes the LASSO and ridge regression: in other words, each of them is a special case where or . Meanwhile, the naive version of elastic net method finds an estimator in a two-stage procedure : first for each fixed it finds the ridge regression coefficients, and then does a LASSO type shrinkage. This kind of estimation incurs a double amount of shrinkage, which leads to increased bias and poor predictions. To improve the prediction performance, sometimes the coefficients of the naive version of elastic net is rescaled by multiplying the estimated coefficients by .
Examples of where the elastic net method has been applied are:
Support vector machine
Metric learning
Portfolio optimization
Cancer prognosis
Reduction to support vector machine
In late 2014, it was proven that the elastic net can be reduced to the linear support vector machine.
A similar reduction was previously proven for the LASSO in 2014.
The authors showed that for every instance of the elastic net, an artificial binary classification problem can be constructed such that the hyper-plane solution of a linear support vector machine (SVM) is identical to the solution (after re-scaling). The reduction immediately enables the use of highly optimized SVM solvers for elastic net problems. It also enables the use of GPU acceleration, which is often already used for large-scale SVM solvers. The reduction is a simple transformation of the original data and regularization constants
into new artificial data instances and a regularization constant that specify a binary classification problem and the SVM regularization constant
Here, consists of binary labels . When it is typically faster to solve the linear SVM in the primal, whereas otherwise the dual formulation is faster.
Some authors have referred to the transformation as Support Vector Elastic Net (SVEN), and provided the following MATLAB pseud |
https://en.wikipedia.org/wiki/Erich%20Neuwirth | Erich Neuwirth (born October 17, 1948) is a professor emeritus of statistics and computer science at the University of Vienna.
Research
Neuwirth studied Mathematics and Statistics at the University of Vienna and received the doctorate in 1974. He started teaching at the University of Vienna in 1969 and was promoted to professor in 1987. He was employed at the Department for Statistics and Decision Support Systems at the faculty of mathematics and the faculty of computer science.
He was a visiting professor at the Northeastern University (Boston) and the National Institute of Multimedia Education in Makuhari (Japan).
The main areas of his research are election analysis and forecasts, combinatorics, mathematics and music and spreadsheets as tools for mathematical understanding. Another important research area is the PISA studies. He investigated the statistical analysis of the PISA studies 2003 and recognized significant errors. Due to his findings the results of the studies were corrected.
Other
Erich Neuwirth developed the statconn Server and R package RExcel. with Thomas Baier. The statconn server combines R and Scilab with Microsoft Excel or OpenOffice. RExcel is an add-in for Microsoft Excel, which allows the use of R within Excel.
Awards
Neuwirth won the European Academic Software Award 1996 for STIMM.
This project later became the book (+ Multimedia CD) Musical Temperament mentioned in the Books section.
Books
Erich Neuwirth: Musical Temperaments, Springer, Wien (January 1998),
Erich Neuwirth, Ivo Ponocny und Wilfried Grossmann: PISA 2000 und PISA 2003: Vertiefende Analysen und Beiträge zur Methodik, Leykam (June 2006),
Erich Neuwirth, Deane Arganbright: The Active Modeler: Mathematical Modeling with Microsoft Excel, Duxbury,
Erich Neuwirth, Richard M. Heiberger: R Through Excel: A Spreadsheet Interface for Statistics, Data Analysis, and Graphics, Springer,
References
External links
Sunsite
Homepage of Erich Neuwirth
Neuwirth - Faculty of Computer Science
Date of birth missing (living people)
1948 births
Living people
Austrian statisticians
Academic staff of the University of Vienna
R (programming language) people |
https://en.wikipedia.org/wiki/Countably%20generated%20module | In mathematics, a module over a (not necessarily commutative) ring is countably generated if it is generated as a module by a countable subset. The importance of the notion comes from Kaplansky's theorem (Kaplansky 1958), which states that a projective module is a direct sum of countably generated modules.
More generally, a module over a possibly non-commutative ring is projective if and only if (i) it is flat, (ii) it is a direct sum of countably generated modules and (iii) it is a Mittag-Leffler module. (Bazzoni–Stovicek)
References
Module theory |
https://en.wikipedia.org/wiki/D%C3%B3ra%20Ivanics | Dóra Ivanics (born 29 June 1994 in Balassagyarmat) is a Hungarian handballer who plays for Vasas SC.
Achievements
Nemzeti Bajnokság I/B:
Winner: 2018
References
External links
Career statistics at Worldhandball
BUDAÖRS KC (2014/2015)
1994 births
Living people
People from Balassagyarmat
Hungarian female handball players
Sportspeople from Nógrád County |
https://en.wikipedia.org/wiki/Krisztina%20Gyetv%C3%A1n | Krisztina Gyetván (born 20 December 1979 in Vác) is a former Hungarian handballer.
References
External links
Career statistics at Worldhandball
1979 births
Living people
Sportspeople from Vác
Hungarian female handball players |
https://en.wikipedia.org/wiki/Homogeneous%20graph | In mathematics, a k-ultrahomogeneous graph is a graph in which every isomorphism between two of its induced subgraphs of at most k vertices can be extended to an automorphism of the whole graph. A k-homogeneous graph obeys a weakened version of the same property in which every isomorphism between two induced subgraphs implies the existence of an automorphism of the whole graph that maps one subgraph to the other (but does not necessarily extend the given isomorphism).
A homogeneous graph is a graph that is k-homogeneous for every k, or equivalently k-ultrahomogeneous for every k.
Classification
The only finite homogeneous graphs are the cluster graphs mKn formed from the disjoint unions of isomorphic complete graphs, the Turán graphs formed as the complement graphs of mKn, the 3 × 3 rook's graph, and the 5-cycle.
The only countably infinite homogeneous graphs are the disjoint unions of isomorphic complete graphs (with the size of each complete graph, the number of complete graphs, or both numbers countably infinite), their complement graphs, the Henson graphs together with their complement graphs, and the Rado graph.
If a graph is 5-ultrahomogeneous, then it is ultrahomogeneous for every k.
There are only two connected graphs that are 4-ultrahomogeneous but not 5-ultrahomogeneous: the Schläfli graph and its complement. The proof relies on the classification of finite simple groups.
Variations
A graph is connected-homogeneous if every isomorphism between two connected induced subgraphs can be extended to an automorphism of the whole graph.
In addition to the homogeneous graphs, the finite connected-homogeneous graphs include all cycle graphs, all square rook's graphs, the Petersen graph, and the 5-regular Clebsch graph.
Notes
References
. As cited by .
. As cited by .
.
.
.
.
.
.
Graph families |
https://en.wikipedia.org/wiki/Littlewood%20subordination%20theorem | In mathematics, the Littlewood subordination theorem, proved by J. E. Littlewood in 1925, is a theorem in operator theory and complex analysis. It states that any holomorphic univalent self-mapping of the unit disk in the complex numbers that fixes 0 induces a contractive composition operator on various function spaces of holomorphic functions on the disk. These spaces include the Hardy spaces, the Bergman spaces and Dirichlet space.
Subordination theorem
Let h be a holomorphic univalent mapping of the unit disk D into itself such that h(0) = 0. Then the composition operator Ch defined on holomorphic functions f on D by
defines a linear operator with operator norm less than 1 on the Hardy spaces , the Bergman spaces .
(1 ≤ p < ∞) and the Dirichlet space .
The norms on these spaces are defined by:
Littlewood's inequalities
Let f be a holomorphic function on the unit disk D and let h be a holomorphic univalent mapping of D into itself with h(0) = 0. Then
if 0 < r < 1 and 1 ≤ p < ∞
This inequality also holds for 0 < p < 1, although in this case there is no operator interpretation.
Proofs
Case p = 2
To prove the result for H2 it suffices to show that for f a polynomial
Let U be the unilateral shift defined by
This has adjoint U* given by
Since f(0) = a0, this gives
and hence
Thus
Since U*f has degree less than f, it follows by induction that
and hence
The same method of proof works for A2 and
General Hardy spaces
If f is in Hardy space Hp, then it has a factorization
with fi an inner function and fo an outer function.
Then
Inequalities
Taking 0 < r < 1, Littlewood's inequalities follow by applying the Hardy space inequalities to the function
The inequalities can also be deduced, following , using subharmonic functions. The inequaties in turn immediately imply the subordination theorem for general Bergman spaces.
Notes
References
Operator theory
Theorems in complex analysis |
https://en.wikipedia.org/wiki/1974%20S%C3%A3o%20Paulo%20FC%20season | The 1974 football season was São Paulo's 45th season since club's existence.
Statistics
Scorers
Overall
{|class="wikitable"
|-
|Games played || 67 (26 Campeonato Paulista, 13 Copa Libertadores, 24 Campeonato Brasileiro, 4 Friendly match)
|-
|Games won || 31 (12 Campeonato Paulista, 8 Copa Libertadores, 8 Campeonato Brasileiro, 3 Friendly match)
|-
|Games drawn || 27 (10 Campeonato Paulista, 3 Copa Libertadores, 13 Campeonato Brasileiro, 1 Friendly match)
|-
|Games lost || 9 (4 Campeonato Paulista, 2 Copa Libertadores, 3 Campeonato Brasileiro, 0 Friendly match)
|-
|Goals scored || 87
|-
|Goals conceded || 42
|-
|Goal difference || +45
|-
|Best result || 5–0 (H) v Jorge Wilstermann - Copa Libertadores - 1974.05.08
|-
|Worst result || 0–3 (A) v Saad - Campeonato Paulista - 1974.09.21
|-
|Most appearances ||
|-
|Top scorer || Mirandinha (16)
|-
Friendlies
Official competitions
Campeonato Brasileiro
Record
Copa Libertadores
Record
Campeonato Paulista
Record
External links
official website
Association football clubs 1974 season
1974
1974 in Brazilian football |
https://en.wikipedia.org/wiki/Yoo%20Ji-hoon | Yoo Ji-hoon (; born 9 June 1988) is a South Korean footballer who plays as a defender for Gyeongnam FC.
Club career statistics
External links
1988 births
Living people
People from Namyangju
South Korean men's footballers
Men's association football defenders
Gyeongnam FC players
Busan IPark players
Gimcheon Sangmu FC players
Seoul E-Land FC players
K League 1 players
K League 2 players
Hanyang University alumni
Footballers from Gyeonggi Province |
https://en.wikipedia.org/wiki/Dirichlet%20space | In mathematics, the Dirichlet space on the domain (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space , for which the Dirichlet integral, defined by
is finite (here dA denotes the area Lebesgue measure on the complex plane ). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichlet integral defines a seminorm on . It is not a norm in general, since whenever f is a constant function.
For , we define
This is a semi-inner product, and clearly . We may equip with an inner product given by
where is the usual inner product on The corresponding norm is given by
Note that this definition is not unique, another common choice is to take , for some fixed .
The Dirichlet space is not an algebra, but the space is a Banach algebra, with respect to the norm
We usually have (the unit disk of the complex plane ), in that case , and if
then
and
Clearly, contains all the polynomials and, more generally, all functions , holomorphic on such that is bounded on .
The reproducing kernel of at is given by
See also
Banach space
Bergman space
Hardy space
Hilbert space
References
Complex analysis
Functional analysis |
https://en.wikipedia.org/wiki/Exterior%20space | In mathematics, the notion of externology in a topological space X generalizes the basic properties of the family
εXcc = {E ⊆ X : X\E is a closed compact subset of X}
of complements of the closed compact subspaces of X, which are used to construct its Alexandroff compactification. An externology permits to introduce a notion of end point, to study the divergence of nets in terms of convergence to end points and it is a useful tool for the study and classification of some families of non compact topological spaces. It can also be used to approach a topological space as the limit of other topological spaces: the externologies are very useful when a compact metric space embedded in a Hilbert space is approached by its open neighbourhoods.
Definition
Let (X,τ) be a topological space. An externology on (X,τ) is a non-empty collection ε of open subsets satisfying:
If E1, E2 ∈ ε, then E1 ∩ E2 ∈ ε;
if E ∈ ε, U ∈ τ and E ⊆ U, then U ∈ ε.
An exterior space (X,τ,ε) consists of a topological space (X,τ) together with an externology ε. An open E which is in ε is said to be an exterior-open subset. A map f:(X,τ,ε) → (X',τ',ε') is said to be an exterior map if it is continuous and f−1(E) ∈ ε, for all E ∈ ε'.
The category of exterior spaces and exterior maps will be denoted by E. It is remarkable that E is a complete and cocomplete category.
Some examples of exterior spaces
For a space (X,τ) one can always consider the trivial externology εtr={X}, and, on the other hand, the total externology εtot=τ. Note that an externology ε is a topology if and only if the empty set is a member of ε if and only if ε=τ.
Given a space (X,τ), the externology εXcc of the complements of closed compact subsets of X permits a connection with the theory of proper maps.
Given a space (X,τ) and a subset A⊆X the family ε(X,A)={U⊆X:A⊆U,U∈τ} is an externology in X. Two particular cases with important applications on shape theory and on dynamical systems, respectively, are the following:
If A is a closed subspace of the Hilbert cube X=Q the externology εA=ε(Q,A) is a resolution of A in the sense of the shape theory.
Let X be a continuous dynamical system and P the subset of periodic points; we can consider the externology ε(X,P). More generally, if A is an invariant subset the externology ε(X,A) is useful to study the dynamical properties of the flow.
Applications of exterior spaces
Proper homotopy theory: A continuous map f:X→Y between topological spaces is said to be proper if for every closed compact subset K of Y, f−1(K) is a compact subset of X. The category of spaces and proper maps will be denoted by P. This category and the corresponding proper homotopy category are very useful for the study of non compact spaces. Nevertheless, one has the problem that this category does not have enough limits and colimits and then we can not develop the usual homotopy constructions like loops, homotopy limits and colimits, etc. An answer to this problem is the category of exter |
https://en.wikipedia.org/wiki/1973%20S%C3%A3o%20Paulo%20FC%20season | The 1973 football season was São Paulo's 44th season since club's existence.
Statistics
Scorers
Overall
{|class="wikitable"
|-
|Games played || 81 (22 Campeonato Paulista, 40 Campeonato Brasileiro, 19 Friendly match)
|-
|Games won || 29 (6 Campeonato Paulista, 17 Campeonato Brasileiro, 6 Friendly match)
|-
|Games drawn || 36 (9 Campeonato Paulista, 18 Campeonato Brasileiro, 9 Friendly match)
|-
|Games lost || 16 (7 Campeonato Paulista, 5 Campeonato Brasileiro, 4 Friendly match)
|-
|Goals scored || 86
|-
|Goals conceded || 59
|-
|Goal difference || +27
|-
|Best result || 1–3 (A) v Botafogo - Campeonato Paulista - 1973.0.22
|-
|Worst result || 4–1 (A) v Portuguesa Santista - Friendly match - 1973.03.01 4–1 (A) v Moto Club - Campeonato Brasileiro - 1973.10.03 4–1 (H) v Internacional - Campeonato Brasileiro - 1974.02.13
|-
|Most appearances ||
|-
|Top scorer || Pedro Rocha (21)
|-
Friendlies
Torneio Laudo Natel
Taça Estado de São Paulo
Official competitions
Campeonato Paulista
Record
Campeonato Brasileiro
Record
External links
official website
Association football clubs 1973 season
1973
1973 in Brazilian football |
https://en.wikipedia.org/wiki/Simplectic%20honeycomb | In geometry, the simplectic honeycomb (or -simplex honeycomb) is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of nodes with one node ringed. It is composed of -simplex facets, along with all rectified -simplices. It can be thought of as an -dimensional hypercubic honeycomb that has been subdivided along all hyperplanes , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an -simplex honeycomb is an expanded -simplex.
In 2 dimensions, the honeycomb represents the triangular tiling, with Coxeter graph filling the plane with alternately colored triangles. In 3 dimensions it represents the tetrahedral-octahedral honeycomb, with Coxeter graph filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph , with 5-cell and rectified 5-cell facets. In 5 dimensions it is called the 5-simplex honeycomb, with Coxeter graph , filling space by 5-simplex, rectified 5-simplex, and birectified 5-simplex facets. In 6 dimensions it is called the 6-simplex honeycomb, with Coxeter graph , filling space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets.
By dimension
Projection by folding
The (2n-1)-simplex honeycombs and 2n-simplex honeycombs can be projected into the n-dimensional hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
Kissing number
These honeycombs, seen as tangent n-spheres located at the center of each honeycomb vertex have a fixed number of contacting spheres and correspond to the number of vertices in the vertex figure. This represents the highest kissing number for 2 and 3 dimensions, but falls short on higher dimensions. In 2-dimensions, the triangular tiling defines a circle packing of 6 tangent spheres arranged in a regular hexagon, and for 3 dimensions there are 12 tangent spheres arranged in a cuboctahedral configuration. For 4 to 8 dimensions, the kissing numbers are 20, 30, 42, 56, and 72 spheres, while the greatest solutions are 24, 40, 72, 126, and 240 spheres respectively.
See also
Hypercubic honeycomb
Alternated hypercubic honeycomb
Quarter hypercubic honeycomb
Truncated simplectic honeycomb
Omnitruncated simplectic honeycomb
References
George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
Norman Johnson Uniform Polytopes, Manuscript (1991)
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition,
Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscien |
https://en.wikipedia.org/wiki/Duflo%20isomorphism | In mathematics, the Duflo isomorphism is an isomorphism between the center of the universal enveloping algebra of a finite-dimensional Lie algebra and the invariants of its symmetric algebra. It was introduced by and later generalized to arbitrary finite-dimensional Lie algebras by Kontsevich.
The Poincaré-Birkoff-Witt theorem gives for any Lie algebra a vector space isomorphism from the polynomial algebra to the universal enveloping algebra . This map is not an algebra homomorphism. It is equivariant with respect to the natural representation of on these spaces, so it restricts to a vector space isomorphism
where the superscript indicates the subspace annihilated by the action of . Both and are commutative subalgebras, indeed is the center of , but is still not an algebra homomorphism. However, Duflo proved that in some cases we can compose with a map
to get an algebra isomorphism
Later, using the Kontsevich formality theorem, Kontsevich showed that this works for all finite-dimensional Lie algebras.
Following Calaque and Rossi, the map can be defined as follows. The adjoint action of is the map
sending to the operation on . We can treat map as an element of
or, for that matter, an element of the larger space , since . Call this element
Both and are algebras so their tensor product is as well. Thus, we can take powers of , say
Going further, we can apply any formal power series to and obtain an element of , where denotes the algebra of formal power series on . Working with formal power series, we thus obtain an element
Since the dimension of is finite, one can think of as , hence is and by applying the determinant map, we obtain an element
which is related to the Todd class in algebraic topology.
Now, acts as derivations on since any element of gives a translation-invariant vector field on . As a result, the algebra acts on
as differential operators on , and this extends to an action of on . We can thus define a linear map
by
and since the whole construction was invariant, restricts to the desired linear map
Properties
For a nilpotent Lie algebra the Duflo isomorphism coincides with the symmetrization map from symmetric algebra to universal enveloping algebra. For a semisimple Lie algebra the Duflo isomorphism is compatible in a natural way with the Harish-Chandra isomorphism.
References
Lie algebras |
https://en.wikipedia.org/wiki/Cyclotruncated%20simplectic%20honeycomb | In geometry, the cyclotruncated simplectic honeycomb (or cyclotruncated n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the symmetry of the affine Coxeter group. It is given a Schläfli symbol t0,1{3[n+1]}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of n+1 nodes with two adjacent nodes ringed. It is composed of n-simplex facets, along with all truncated n-simplices.
It is also called a Kagome lattice in two and three dimensions, although it is not a lattice.
In n-dimensions, each can be seen as a set of n+1 sets of parallel hyperplanes that divide space. Each hyperplane contains the same honeycomb of one dimension lower.
In 1-dimension, the honeycomb represents an apeirogon, with alternately colored line segments. In 2-dimensions, the honeycomb represents the trihexagonal tiling, with Coxeter graph . In 3-dimensions it represents the quarter cubic honeycomb, with Coxeter graph filling space with alternately tetrahedral and truncated tetrahedral cells. In 4-dimensions it is called a cyclotruncated 5-cell honeycomb, with Coxeter graph , with 5-cell, truncated 5-cell, and bitruncated 5-cell facets. In 5-dimensions it is called a cyclotruncated 5-simplex honeycomb, with Coxeter graph , filling space by 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets. In 6-dimensions it is called a cyclotruncated 6-simplex honeycomb, with Coxeter graph , filling space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets.
Projection by folding
The cyclotruncated (2n+1)- and 2n-simplex honeycombs and (2n-1)-simplex honeycombs can be projected into the n-dimensional hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
See also
Hypercubic honeycomb
Alternated hypercubic honeycomb
Quarter hypercubic honeycomb
Simplectic honeycomb
Omnitruncated simplectic honeycomb
References
George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
Norman Johnson Uniform Polytopes, Manuscript (1991)
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition,
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Honeycombs (geometry)
Polytopes
Truncated tilings |
https://en.wikipedia.org/wiki/Shahril%20Saa%27ri | Mohd Shahril bin Saa'ri (born 7 March 1990) is a Malaysian footballer who plays as a goalkeeper for Malaysia Super League club Kelantan United.
Career statistics
Club
Notes
References
External links
1990 births
Living people
Malaysian men's footballers
Footballers from Terengganu
Terengganu FC players
Sabah F.C. (Malaysia) players
Terengganu F.C. II players
Kedah Darul Aman F.C. players
Malaysia Super League players
Men's association football goalkeepers
Malaysian people of Malay descent |
https://en.wikipedia.org/wiki/Akito%20Fukumori | Akito Fukumori (福森 晃斗, born December 16, 1992) is a Japanese football player for Hokkaido Consadole Sapporo.
Club statistics
Updated to 5 November 2022.
References
External links
Profile at Consadole Sapporo
1992 births
Living people
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
Kawasaki Frontale players
Hokkaido Consadole Sapporo players
Men's association football defenders |
https://en.wikipedia.org/wiki/Yudai%20Tanaka%20%28footballer%2C%20born%201988%29 | is a Japanese former football player who last played for Blaublitz Akita.
Career
Blaublitz Akita
Tanaka signed with Blaublitz Akita for the 2019 season.
Club statistics
Updated to 31 December 2020.
Honours
Blaublitz Akita
J3 League (1): 2020
References
External links
Profile at Consadole Sapporo
1988 births
Living people
Kansai University alumni
Association football people from Shiga Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Kawasaki Frontale players
Tochigi SC players
Gainare Tottori players
Mito HollyHock players
Vissel Kobe players
Hokkaido Consadole Sapporo players
Blaublitz Akita players
Men's association football defenders
Universiade bronze medalists for Japan
Universiade medalists in football
Medalists at the 2009 Summer Universiade |
https://en.wikipedia.org/wiki/Kentaro%20Moriya | is a Japanese footballer who plays as a attacking midfielder for J1 League club Sagan Tosu.
Club statistics
Updated to 19 July 2022.
References
External links
Profile at Kawasaki Frontale
1988 births
Living people
University of Tsukuba alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
Yokohama F. Marinos players
Kawasaki Frontale players
Júbilo Iwata players
Ehime FC players
Sagan Tosu players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Shigeto%20Masuda | is a Japanese football player who plays for NPL Victoria club Heidelberg United.
Club statistics
Updated to 25 November 2022.
1Includes J2/J3 Playoffs.
References
External links
Profile at Gunma
Profile at Oita
Profile at Machida Zelvia
Profile at Okayama
Profile at Fujieda
1992 births
Living people
People from Kamagaya
Association football people from Chiba Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Albirex Niigata players
Thespakusatsu Gunma players
Oita Trinita players
FC Machida Zelvia players
Fagiano Okayama players
Fujieda MYFC players
Blaublitz Akita players
Men's association football defenders |
https://en.wikipedia.org/wiki/Proto-value%20function | In applied mathematics, proto-value functions (PVFs) are automatically learned basis functions that are useful in approximating task-specific value functions, providing a compact representation of the powers of transition matrices. They provide a novel framework for solving the credit assignment problem. The framework introduces a novel approach to solving Markov decision processes (MDP) and reinforcement learning problems, using multiscale spectral and manifold learning methods. Proto-value functions are generated by spectral analysis of a graph, using the graph Laplacian.
Proto-value functions were first introduced in the context of reinforcement learning by Sridhar Mahadevan in his paper, Proto-Value Functions: Developmental Reinforcement Learning at ICML 2005.
Motivation
Value function approximation is a critical component to solving Markov decision processes (MDPs) defined over a continuous state space. A good function approximator allows a reinforcement learning (RL) agent to accurately represent the value of any state it has experienced, without explicitly storing its value. Linear function approximation using basis functions is a common way of constructing a value function approximation, like radial basis functions, polynomial state encodings, and CMACs. However, parameters associated with these basis functions often require significant domain-specific hand-engineering. Proto-value functions attempts to solve this required hand-engineering by accounting for the underlying manifold structure of the problem domain.
Overview
Proto-value functions are task-independent global basis functions that collectively span the entire space of possible value functions for a given state space. They incorporate geometric constraints intrinsic to the environment. For example, states close in Euclidean distance (such as states on opposite sides of a wall) may be far apart in manifold space. Previous approaches to this nonlinearity problem lacked a broad theoretical framework, and consequently have only been explored in the context of discrete MDPs.
Proto-value functions arise from reformulating the problem of value function approximation as real-valued function approximation on a graph or manifold. This results in broader applicability of the learned bases and enables a new class of learning algorithms, which learn representations and policies at the same time.
Basis functions from graph Laplacian
This approach constructs the basis functions by spectral analysis of the graph Laplacian, a self-adjoint (or symmetric) operator on the space of functions on the graph, closely related to the random walk operator.
For the sake of simplicity, assume that the underlying state space can be represented as an undirected unweighted graph The combinatorial Laplacian is defined as the operator , where is a diagonal matrix called the degree matrix and is the adjacency matrix.
The spectral analysis of the Laplace operator on a graph consists of finding t |
https://en.wikipedia.org/wiki/Matrix%20Chernoff%20bound | For certain applications in linear algebra, it is useful to know properties of the probability distribution of the largest eigenvalue of a finite sum of random matrices. Suppose is a finite sequence of random matrices. Analogous to the well-known Chernoff bound for sums of scalars, a bound on the following is sought for a given parameter t:
The following theorems answer this general question under various assumptions; these assumptions are named below by analogy to their classical, scalar counterparts. All of these theorems can be found in , as the specific application of a general result which is derived below. A summary of related works is given.
Matrix Gaussian and Rademacher series
Self-adjoint matrices case
Consider a finite sequence of fixed,
self-adjoint matrices with dimension , and let be a finite sequence of independent standard normal or independent Rademacher random variables.
Then, for all ,
where
Rectangular case
Consider a finite sequence of fixed, self-adjoint matrices with dimension , and let be a finite sequence of independent standard normal or independent Rademacher random variables.
Define the variance parameter
Then, for all ,
Matrix Chernoff inequalities
The classical Chernoff bounds concern the sum of independent, nonnegative, and uniformly bounded random variables.
In the matrix setting, the analogous theorem concerns a sum of positive-semidefinite random matrices subjected to a uniform eigenvalue bound.
Matrix Chernoff I
Consider a finite sequence of independent, random, self-adjoint matrices with dimension .
Assume that each random matrix satisfies
almost surely.
Define
Then
Matrix Chernoff II
Consider a sequence of independent, random, self-adjoint matrices that satisfy
almost surely.
Compute the minimum and maximum eigenvalues of the average expectation,
Then
The binary information divergence is defined as
for .
Matrix Bennett and Bernstein inequalities
In the scalar setting, Bennett and Bernstein inequalities describe the upper tail of a sum of independent, zero-mean random variables that are either bounded or subexponential. In the matrix
case, the analogous results concern a sum of zero-mean random matrices.
Bounded case
Consider a finite sequence of independent, random, self-adjoint matrices with dimension .
Assume that each random matrix satisfies
almost surely.
Compute the norm of the total variance,
Then, the following chain of inequalities holds for all :
The function is defined as for .
Subexponential case
Consider a finite sequence of independent, random, self-adjoint matrices with dimension .
Assume that
for .
Compute the variance parameter,
Then, the following chain of inequalities holds for all :
Rectangular case
Consider a finite sequence of independent, random, matrices with dimension .
Assume that each random matrix satisfies
almost surely.
Define the variance parameter
Then, for all
holds.
Matrix Azuma, Hoeffding, and McDiarmid inequalities
Matrix Az |
https://en.wikipedia.org/wiki/Naoya%20Okane | Naoya Okane (岡根 直哉, born April 19, 1988) is a Japanese football player who playing as Centre-back and currently play for Okinawa SV.
Career statistics
Club
Updated to 20 February 2023.
Honours
Okinawa SV
Kyushu Soccer League: 2019, 2021, 2022
References
External links
Profile at FC Gifu
Profile at SC Sagamihara
1988 births
Living people
Waseda University alumni
Association football people from Osaka Prefecture
People from Kishiwada, Osaka
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Shimizu S-Pulse players
Montedio Yamagata players
Tochigi SC players
FC Gifu players
SC Sagamihara players
Okinawa SV players
Men's association football defenders |
https://en.wikipedia.org/wiki/Atomu%20Nabeta | is a former Japanese football player.
Club statistics
References
External links
j-league
1991 births
Living people
Association football people from Shizuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Shimizu S-Pulse players
Avispa Fukuoka players
Men's association football forwards |
https://en.wikipedia.org/wiki/Yuki%20Kobayashi%20%28footballer%2C%20born%201988%29 | is a Japanese footballer who plays as a defensive midfielder for Oita Trinita.
Club statistics
Updated to 1 August 2022.
1Includes Suruga Bank Championship and J2 Play-offs.
References
External links
Profile at Nagoya Grampus
1988 births
Living people
Meiji University alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
Júbilo Iwata players
Albirex Niigata players
Nagoya Grampus players
Oita Trinita players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Makito%20Yoshida | is a Japanese football player for Ehime FC.
Career statistics
Updated to 6 January 2018.
1Includes Emperor's Cup.
2Includes J. League Cup.
3Includes AFC Champions League.
References
External links
Profile at JEF United Chiba
1992 births
Living people
Association football people from Chiba Prefecture
Japanese men's footballers
J1 League players
J2 League players
Nagoya Grampus players
Matsumoto Yamaga FC players
Mito HollyHock players
JEF United Chiba players
FC Machida Zelvia players
Ehime FC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Teruki%20Tanaka | is a Japanese football player who lastly played as midfielder for V-Varen Nagasaki.
Career statistics
Updated to 23 February 2017.
References
External links
Profile at V-Varen Nagasaki
1992 births
Living people
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
Nagoya Grampus players
Oita Trinita players
V-Varen Nagasaki players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kotaro%20Omori | is a Japanese professional footballer who plays as an attacking midfielder or a winger for J.League club Júbilo Iwata.
Club statistics
Last update: 2 December 2018.
1 includes J. League Championship, Japanese Super Cup and Suruga Bank Championship appearances.
Reserves performance
Last Update:25 February 2019
Honors
J. League Division 1 - 2014
J. League Division 2 - 2013
Emperor's Cup - 2014, 2015
J. League Cup - 2014
Japanese Super Cup - 2015
References
External links
Profile at FC Tokyo
1992 births
Living people
Association football people from Osaka
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Gamba Osaka players
Gamba Osaka U-23 players
Vissel Kobe players
FC Tokyo players
Júbilo Iwata players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kazuya%20Murata%20%28footballer%29 | Kazuya Murata (村田 和哉, born 7 October 1988) is a Japanese football player.
Career statistics
Club
Updated to end of 2018 season.
1Includes Emperor's Cup.
2Includes J. League Cup.
3Includes AFC Champions League.
References
External links
Profile at Shimizu S-Pulse
Profile at Cerezo Osaka
1988 births
Living people
Osaka University of Health and Sport Sciences alumni
Association football people from Shiga Prefecture
Japanese men's footballers
J1 League players
J2 League players
Cerezo Osaka players
Shimizu S-Pulse players
Kashiwa Reysol players
Avispa Fukuoka players
Renofa Yamaguchi FC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Kohei%20Mishima | is a Japanese football player for SC Sagamihara.
Club statistics
Updated to 23 February 2020.
References
External links
Profile at Matsumoto Yamaga
1987 births
Living people
Komazawa University alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Vissel Kobe players
Mito HollyHock players
Matsumoto Yamaga FC players
Roasso Kumamoto players
SC Sagamihara players
Men's association football forwards
Universiade bronze medalists for Japan
Universiade medalists in football
Medalists at the 2009 Summer Universiade |
https://en.wikipedia.org/wiki/Keisuke%20Hayashi | is a Japanese football player. He plays for the Australian club Sutherland Sharks.
Club statistics
Updated to 20 February 2017.
References
External links
Profile at Gainare Tottori
Profile at Nara Club
1988 births
Living people
Doshisha University alumni
Association football people from Hyōgo Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Vissel Kobe players
Gainare Tottori players
Nara Club players
Men's association football defenders |
https://en.wikipedia.org/wiki/Bae%20Chun-suk | Bae Chun-Suk (born April 27, 1990) is a South Korean football player who plays for Jeonnam Dragons as a striker.
Club statistics
Statistics accurate as of 6 December 2015
1Includes Emperor's Cup.
2Includes J.League Cup.
References
External links
1990 births
Living people
Men's association football forwards
South Korean men's footballers
South Korean expatriate men's footballers
Vissel Kobe players
Pohang Steelers players
Jeonnam Dragons players
J1 League players
K League 1 players
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan
Busan IPark players
Footballers from Daegu |
https://en.wikipedia.org/wiki/Tate%20group | In mathematics, a Tate group, named for John Tate, may refer to:
Barsotti–Tate group
Mumford–Tate group
Tate cohomology group
Tate–Shafarevich group |
https://en.wikipedia.org/wiki/Morrey%E2%80%93Campanato%20space | In mathematics, the Morrey–Campanato spaces (named after Charles B. Morrey, Jr. and Sergio Campanato) are Banach spaces which extend the notion of functions of bounded mean oscillation, describing situations where the oscillation of the function in a ball is proportional to some power of the radius other than the dimension. They are used in the theory of elliptic partial differential equations, since for certain values of , elements of the space are Hölder continuous functions over the domain .
The seminorm of the Morrey spaces is given by
When , the Morrey space is the same as the usual space. When , the spatial dimension, the Morrey space is equivalent to , due to the Lebesgue differentiation theorem. When , the space contains only the 0 function.
Note that this is a norm for .
The seminorm of the Campanato space is given by
where
It is known that the Morrey spaces with are equivalent to the Campanato spaces with the same value of when is a sufficiently regular domain, that is to say, when there is a constant A such that for every and .
When , the Campanato space is the space of functions of bounded mean oscillation. When , the Campanato space is the space of Hölder continuous functions with . For , the space contains only constant functions.
References
Function spaces |
https://en.wikipedia.org/wiki/Dynamic%20topic%20model | Within statistics, Dynamic topic models''' are generative models that can be used to analyze the evolution of (unobserved) topics of a collection of documents over time. This family of models was proposed by David Blei and John Lafferty and is an extension to Latent Dirichlet Allocation (LDA) that can handle sequential documents.
In LDA, both the order the words appear in a document and the order the documents appear in the corpus are oblivious to the model. Whereas words are still assumed to be exchangeable, in a dynamic topic model the order of the documents plays a fundamental role. More precisely, the documents are grouped by time slice (e.g.: years) and it is assumed that the documents of each group come from a set of topics that evolved from the set of the previous slice.
Topics
Similarly to LDA and pLSA, in a dynamic topic model, each document is viewed as a mixture of unobserved topics. Furthermore, each topic defines a multinomial distribution over a set of terms. Thus, for each word of each document, a topic is drawn from the mixture and a term is subsequently drawn from the multinomial distribution corresponding to that topic.
The topics, however, evolve over time. For instance, the two most likely terms of a topic at time could be "network" and "Zipf" (in descending order) while the most likely ones at time could be "Zipf" and "percolation" (in descending order).
Model
Define
as the per-document topic distribution at time t.
as the word distribution of topic k at time t.
as the topic distribution for document d in time t,
as the topic for the nth word in document d in time t, and
as the specific word.
In this model, the multinomial distributions and are generated from and , respectively.
Even though multinomial distributions are usually written in terms of the mean parameters, representing them in terms of the natural parameters is better in the context of dynamic topic models.
The former representation has some disadvantages due to the fact that the parameters are constrained to be non-negative and sum to one. When defining the evolution of these distributions, one would need to assure that such constraints were satisfied. Since both distributions are in the exponential family, one solution to this problem is to represent them in terms of the natural parameters, that can assume any real value and can be individually changed.
Using the natural parameterization, the dynamics of the topic model are given by
and
.
The generative process at time slice 't' is therefore:
Draw topics
Draw mixture model
For each document:
Draw
For each word:
Draw topic
Draw word
where is a mapping from the natural parameterization x'' to the mean parameterization, namely
.
Inference
In the dynamic topic model, only is observable. Learning the other parameters constitutes an inference problem. Blei and Lafferty argue that applying Gibbs sampling to do inference in this model is more difficult than in static models, due |
https://en.wikipedia.org/wiki/Approximate%20entropy | In statistics, an approximate entropy (ApEn) is a technique used to quantify the amount of regularity and the unpredictability of fluctuations over time-series data. For example, consider two series of data:
Series A: (0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...), which alternates 0 and 1.
Series B: (0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, ...), which has either a value of 0 or 1, chosen randomly, each with probability 1/2.
Moment statistics, such as mean and variance, will not distinguish between these two series. Nor will rank order statistics distinguish between these series. Yet series A is perfectly regular: knowing a term has the value of 1 enables one to predict with certainty that the next term will have the value of 0. In contrast, series B is randomly valued: knowing a term has the value of 1 gives no insight into what value the next term will have.
Regularity was originally measured by exact regularity statistics, which has mainly centered on various entropy measures.
However, accurate entropy calculation requires vast amounts of data, and the results will be greatly influenced by system noise, therefore it is not practical to apply these methods to experimental data. ApEn was developed by Steve M. Pincus to handle these limitations by modifying an exact regularity statistic, Kolmogorov–Sinai entropy. ApEn was initially developed to analyze medical data, such as heart rate, and later spread its applications in finance, physiology, human factors engineering, and climate sciences.
Algorithm
A comprehensive step-by-step tutorial with an explanation of the theoretical foundations of Approximate Entropy is available. The algorithm is:
Step 1 Assume a time series of data . These are raw data values from measurements equally spaced in time.
Step 2 Let be a positive integer, with , which represents the length of a run of data (essentially a window).Let be a positive real number, which specifies a filtering level.Let .
Step 3 Define for each where . In other words, is an -dimensional vector that contains the run of data starting with .Define the distance between two vectors and as the maximum of the distances between their respective components, given by
for .
Step 4 Define a count as
for each where . Note that since takes on all values between 1 and , the match will be counted when (i.e. when the test subsequence, , is matched against itself, ).
Step 5 Define
where is the natural logarithm, and for a fixed , , and as set in Step 2.
Step 6 Define approximate entropy () as
Parameter selection Typically, choose or , whereas depends greatly on the application.
An implementation on Physionet, which is based on Pincus, use instead of in Step 4. While a concern for artificially constructed examples, it is usually not a concern in practice.
Example
Consider a sequence of samples of heart rate equally spaced in time:
Note the sequence is periodic with a period of 3. Let's choose and (the v |
https://en.wikipedia.org/wiki/Lee%20Ho-seung | Lee Ho-Seung (; born December 21, 1989) is a South Korean football player who plays for Jeonnam Dragons.
Club statistics
References
External links
1989 births
Living people
Men's association football goalkeepers
South Korean men's footballers
South Korean expatriate men's footballers
J1 League players
J2 League players
K League 1 players
K League 2 players
Hokkaido Consadole Sapporo players
Shonan Bellmare players
Jeonnam Dragons players
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan |
https://en.wikipedia.org/wiki/Tatsuki%20Nara | (born 19 September 1993) is a Japanese professional footballer who plays as a Centre-back for J1 League club Avispa Fukuoka.
Club career statistics
Updated to 5 November 2022.
1Includes Emperor's Cup.
2Includes J. League Cup.
3Includes Japanese Super Cup.
Honours
Club
J1 League (2) : 2017, 2018
Japanese Super Cup (1) : 2019
References
External links
Profile at Avispa Fukuoka
1993 births
Living people
Association football people from Hokkaido
Japanese men's footballers
Japan men's youth international footballers
J1 League players
J2 League players
J3 League players
Hokkaido Consadole Sapporo players
FC Tokyo players
Kawasaki Frontale players
J.League U-22 Selection players
Kashima Antlers players
Avispa Fukuoka players
Men's association football defenders
People from Kitami, Hokkaido |
https://en.wikipedia.org/wiki/Masaya%20Suzuki | is a Japanese football player for Toho Titanium.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Azul Claro Numazu
j-league
1988 births
Living people
Kanagawa University alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Mito HollyHock players
SC Sagamihara players
Azul Claro Numazu players
Men's association football midfielders
Sportspeople from Yokosuka, Kanagawa |
https://en.wikipedia.org/wiki/Sho%20Kamimura | is a former Japanese football player.
Club statistics
References
External links
j-league
1989 births
Living people
Senshu University alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Mito HollyHock players
FC Machida Zelvia players
Men's association football forwards
Sportspeople from Sagamihara |
https://en.wikipedia.org/wiki/Jumpei%20Obata | is a Japanese football player for ReinMeer Aomori.
Club statistics
Updated to 1 February 2020.
References
External links
Profile at ReinMeer Aomori
1988 births
Living people
Senshu University alumni
Association football people from Tokyo
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Mito HollyHock players
FC Ryukyu players
ReinMeer Aomori players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Hironobu%20Ono | is a former Japanese football player.
Club statistics
References
External links
j-league
1987 births
Living people
Chuo University alumni
Association football people from Yamanashi Prefecture
Japanese men's footballers
J2 League players
Mito HollyHock players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/McKay%20graph | In mathematics, the McKay graph of a finite-dimensional representation of a finite group is a weighted quiver encoding the structure of the representation theory of . Each node represents an irreducible representation of . If are irreducible representations of , then there is an arrow from to if and only if is a constituent of the tensor product Then the weight of the arrow is the number of times this constituent appears in For finite subgroups of the McKay graph of is the McKay graph of the defining 2-dimensional representation of .
If has irreducible characters, then the Cartan matrix of the representation of dimension is defined by where is the Kronecker delta. A result by states that if is a representative of a conjugacy class of , then the vectors are the eigenvectors of to the eigenvalues where is the character of the representation .
The McKay correspondence , named after John McKay, states that there is a one-to-one correspondence between the McKay graphs of the finite subgroups of and the extended Dynkin diagrams, which appear in the ADE classification of the simple Lie algebras.
Definition
Let be a finite group, be a representation of and be its character. Let be the irreducible representations of . If
then define the McKay graph of , relative to , as follows:
Each irreducible representation of corresponds to a node in .
If , there is an arrow from to of weight , written as or sometimes as unlabeled arrows.
If we denote the two opposite arrows between as an undirected edge of weight . Moreover, if we omit the weight label.
We can calculate the value of using inner product on characters:
The McKay graph of a finite subgroup of is defined to be the McKay graph of its canonical representation.
For finite subgroups of the canonical representation on is self-dual, so for all . Thus, the McKay graph of finite subgroups of is undirected.
In fact, by the McKay correspondence, there is a one-to-one correspondence between the finite subgroups of and the extended Coxeter-Dynkin diagrams of type A-D-E.
We define the Cartan matrix of as follows:
where is the Kronecker delta.
Some results
If the representation is faithful, then every irreducible representation is contained in some tensor power and the McKay graph of is connected.
The McKay graph of a finite subgroup of has no self-loops, that is, for all .
The arrows of the McKay graph of a finite subgroup of are all of weight one.
Examples
Suppose , and there are canonical irreducible representations of respectively. If , are the irreducible representations of and , are the irreducible representations of , then
are the irreducible representations of , where In this case, we have
Therefore, there is an arrow in the McKay graph of between and if and only if there is an arrow in the McKay graph of between and there is an arrow in the McKay graph of between . In this case, the weight on the arrow in the |
https://en.wikipedia.org/wiki/Identity%20group | Identity group may refer to:
in group theory |
https://en.wikipedia.org/wiki/2007%E2%80%9308%20FK%20Partizan%20season | The 2007–08 season was FK Partizan's 2nd season in Serbian SuperLiga. This article shows player statistics and all matches (official and friendly) that the club played during the 2007–08 season.
Tournaments
Players
Squad information
Competitions
Overview
Serbian SuperLiga
League table
Matches
Serbian Cup
UEFA Cup
UEFA expelled Partizan from the 2007–08 UEFA Cup due to crowd trouble at their away tie in Mostar, which forced the match to be interrupted for 10 minutes. UEFA adjudged travelling Partizan fans to have been the culprits of the trouble, but Partizan were allowed to play the return leg while the appeal was being processed. However, Partizan's appeal was rejected so Zrinjski Mostar qualified.
Trofeo Santiago Bernabéu
References
External links
Official website
Partizanopedia 2007-2008 (in Serbian)
FK Partizan seasons
Partizan
Serbian football championship-winning seasons |
https://en.wikipedia.org/wiki/1972%20S%C3%A3o%20Paulo%20FC%20season | The 1972 football season was São Paulo's 43rd season since club's existence.
Statistics
Scorers
Overall
{|class="wikitable"
|-
|Games played || 71 (10 Copa Libertadores, 22 Campeonato Paulista, 28 Campeonato Brasileiro, 11 Friendly match)
|-
|Games won || 37 (4 Copa Libertadores, 14 Campeonato Paulista, 13 Campeonato Brasileiro, 6 Friendly match)
|-
|Games drawn || 21 (4 Copa Libertadores, 8 Campeonato Paulista, 6 Campeonato Brasileiro, 3 Friendly match)
|-
|Games lost || 13 (2 Copa Libertadores, 0 Campeonato Paulista, 9 Campeonato Brasileiro, 2 Friendly match)
|-
|Goals scored || 116
|-
|Goals conceded || 56
|-
|Goal difference || +60
|-
|Best result || 6–0 (A) v Náutico - Campeonato Brasileiro - 1972.11.06
|-
|Worst result || 0–4 (A) v Coritiba - Campeonato Brasileiro - 1972.10.15
|-
|Most appearances || Gilberto Sorriso (68)
|-
|Top scorer || Toninho Guerreiro (28)
|-
Friendlies
Torneio Laudo Natel
Official competitions
Copa Libertadores
Record
Campeonato Paulista
Record
Campeonato Brasileiro
Record
External links
official website
Association football clubs 1972 season
1972
1972 in Brazilian football |
https://en.wikipedia.org/wiki/Maciej%20Zworski | Maciej Zworski is a Polish-Canadian mathematician, currently a professor of mathematics at the University of California, Berkeley. His mathematical interests include microlocal analysis, scattering theory, and partial differential equations.
He was an invited speaker at International Congress of Mathematicians in Beijing in 2002, and a plenary speaker at the conference Dynamics, Equations and Applications in Kraków in 2019.
Selected publications
Articles
Books
with Richard Melrose and Antônio Sá Barreto: Semi-linear diffraction of conormal waves, Astérisque, vol. 240, Societé Mathématique de France, 1996 abstract
Semiclassical analysis, American Mathematical Society 2012
as editor with Plamen Stefanov and András Vasy:
with Semyon Dyatlov:
References
External links
Professor Zworski's webpage
Living people
Polish emigrants to Canada
Canadian mathematicians
20th-century American mathematicians
21st-century American mathematicians
University of California, Berkeley College of Letters and Science faculty
1963 births
Fellows of the American Academy of Arts and Sciences
Massachusetts Institute of Technology alumni |
https://en.wikipedia.org/wiki/Stella%20octangula%20number | In mathematics, a stella octangula number is a figurate number based on the stella octangula, of the form .
The sequence of stella octangula numbers is
0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, ...
Only two of these numbers are square.
Ljunggren's equation
There are only two positive square stella octangula numbers, and , corresponding to and respectively. The elliptic curve describing the square stella octangula numbers,
may be placed in the equivalent Weierstrass form
by the change of variables , . Because the two factors and of the square number are relatively prime, they must each be squares themselves, and the second change of variables and leads to Ljunggren's equation
A theorem of Siegel states that every elliptic curve has only finitely many integer solutions, and found a difficult proof that the only integer solutions to his equation were and , corresponding to the two square stella octangula numbers. Louis J. Mordell conjectured that the proof could be simplified, and several later authors published simplifications.
Additional applications
The stella octangula numbers arise in a parametric family of instances to the crossed ladders problem in which the lengths and heights of the ladders and the height of their crossing point are all integers. In these instances, the ratio between the heights of the two ladders is a stella octangula number.
References
External links
Figurate numbers |
https://en.wikipedia.org/wiki/Sendov%27s%20conjecture | In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable. It is named after Blagovest Sendov.
The conjecture states that for a polynomial
with all roots r1, ..., rn inside the closed unit disk |z| ≤ 1, each of the n roots is at a distance no more than 1 from at least one critical point.
The Gauss–Lucas theorem says that all of the critical points lie within the convex hull of the roots. It follows that the critical points must be within the unit disk, since the roots are.
The conjecture has been proven for n < 9 by Brown-Xiang and for n sufficiently large by Tao.
History
The conjecture was first proposed by Blagovest Sendov in 1959; he described the conjecture to his colleague Nikola Obreshkov. In 1967 the conjecture was misattributed to Ljubomir Iliev by Walter Hayman. In 1969 Meir and Sharma proved the conjecture for polynomials with n < 6. In 1991 Brown proved the conjecture for n < 7. Borcea extended the proof to n < 8 in 1996. Brown and Xiang proved the conjecture for n < 9 in 1999. Terence Tao proved the conjecture for sufficiently large n in 2020.
References
G. Schmeisser, "The Conjectures of Sendov and Smale," Approximation Theory: A Volume Dedicated to Blagovest Sendov (B. Bojoanov, ed.), Sofia: DARBA, 2002 pp. 353–369.
External links
Sendov's Conjecture by Bruce Torrence with contributions from Paul Abbott at The Wolfram Demonstrations Project
Complex analysis
Conjectures
Unsolved problems in mathematics |
https://en.wikipedia.org/wiki/Wilhelm%20Ljunggren | Wilhelm Ljunggren (7 October 1905 – 25 January 1973) was a Norwegian mathematician, specializing in number theory.
Career
Ljunggren was born in Kristiania and finished his secondary education in 1925. He studied at the University of Oslo, earning a master's degree in 1931 under the supervision of Thoralf Skolem, and found employment as a secondary school mathematics teacher in Bergen, following Skolem who had moved in 1930 to the Chr. Michelsen Institute there. While in Bergen, Ljunggren continued his studies, earning a dr.philos. from the University of Oslo in 1937.
In 1938 he moved to work as a teacher at Hegdehaugen in Oslo. In 1943 he became a fellow of the Norwegian Academy of Science and Letters, and he also joined the Selskapet til Vitenskapenes Fremme. He was appointed as a docent at the University of Oslo in 1948, but in 1949 he returned to Bergen as a professor at the recently founded University of Bergen. He moved back to the University of Oslo again in 1956, where he served until his death in 1973 in Oslo.
Research
Ljunggren's research concerned number theory, and in particular Diophantine equations. He showed that Ljunggren's equation,
X2 = 2Y4 − 1.
has only the two integer solutions (1,1) and (239,13); however, his proof was complicated, and after Louis J. Mordell conjectured that it could be simplified, simpler proofs were published by several other authors.
Ljunggren also posed the question of finding the integer solutions to the Ramanujan–Nagell equation
2n − 7 = x2
(or equivalently, of finding triangular Mersenne numbers) in 1943, independently of Srinivasa Ramanujan who had asked the same question in 1913.
Ljunggren's publications are collected in a book edited by Paulo Ribenboim.
References
1905 births
1973 deaths
Number theorists
University of Oslo alumni
Academic staff of the University of Bergen
Academic staff of the University of Oslo
Norwegian schoolteachers
Members of the Norwegian Academy of Science and Letters
20th-century Norwegian mathematicians |
https://en.wikipedia.org/wiki/Maximal%20information%20coefficient | In statistics, the maximal information coefficient (MIC) is a measure of the strength of the linear or non-linear association between two variables X and Y.
The MIC belongs to the maximal information-based nonparametric exploration (MINE) class of statistics. In a simulation study, MIC outperformed some selected low power tests, however concerns have been raised regarding reduced statistical power in detecting some associations in settings with low sample size when compared to powerful methods such as distance correlation and Heller–Heller–Gorfine (HHG). Comparisons with these methods, in which MIC was outperformed, were made in Simon and Tibshirani and in Gorfine, Heller, and Heller. It is claimed that MIC approximately satisfies a property called equitability which is illustrated by selected simulation studies. It was later proved that no non-trivial coefficient can exactly satisfy the equitability property as defined by Reshef et al., although this result has been challenged. Some criticisms of MIC are addressed by Reshef et al. in further studies published on arXiv.
Overview
The maximal information coefficient uses binning as a means to apply mutual information on continuous random variables. Binning has been used for some time as a way of applying mutual information to continuous distributions; what MIC contributes in addition is a methodology for selecting the number of bins and picking a maximum over many possible grids.
The rationale is that the bins for both variables should be chosen in such a way that the mutual information between the variables be maximal. That is achieved whenever . Thus, when the mutual information is maximal over a binning of the data, we should expect that the following two properties hold, as much as made possible by the own nature of the data. First, the bins would have roughly the same size, because the entropies and are maximized by equal-sized binning. And second, each bin of X will roughly correspond to a bin in Y.
Because the variables X and Y are real numbers, it is almost always possible to create exactly one bin for each (x,y) datapoint, and that would yield a very high value of the MI. To avoid forming this kind of trivial partitioning, the authors of the paper propose taking a number of bins for X and whose product is relatively small compared with the size N of the data sample. Concretely, they propose:
In some cases it is possible to achieve a good correspondence between and with numbers as low as and , while in other cases the number of bins required may be higher. The maximum for is determined by H(X), which is in turn determined by the number of bins in each axis, therefore, the mutual information value will be dependent on the number of bins selected for each variable. In order to compare mutual information values obtained with partitions of different sizes, the mutual information value is normalized by dividing by the maximum achieveable value for the given partition size. It i |
https://en.wikipedia.org/wiki/List%20of%20Mumbai%20FC%20managers | This is a list of Mumbai Football Club's managers and their records, from 2007, when the first ever manager was appointed, to the present day.
Statistics
Information correct as of 19 December 2011. Only competitive matches are counted. Wins, losses and draws are results at the final whistle; the results of penalty shoot-outs are not counted.
Mumbai |
https://en.wikipedia.org/wiki/Kenji%20Dai | Kenji Dai (代 健司, born March 27, 1989) is a Japanese football player who currently plays as a defender for Tegevajaro Miyazaki.
Club statistics
Updated to 20 February 2020.
References
External links
Profile at Kataller Toyama
Profile at Renofa Yamaguchi
1989 births
Living people
Fukuoka University alumni
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Mito HollyHock players
Ehime FC players
Renofa Yamaguchi FC players
Kataller Toyama players
Tegevajaro Miyazaki players
Men's association football defenders
Association football people from Hiroshima |
https://en.wikipedia.org/wiki/Eita%20Kasagawa | is a Japanese football player. He plays for Albirex Niigata Singapore.
Club statistics
References
External links
1990 births
Living people
Japanese men's footballers
J1 League players
J2 League players
Avispa Fukuoka players
Men's association football goalkeepers
Association football people from Fukuoka (city) |
https://en.wikipedia.org/wiki/Freudenthal%20spectral%20theorem | In mathematics, the Freudenthal spectral theorem is a result in Riesz space theory proved by Hans Freudenthal in 1936. It roughly states that any element dominated by a positive element in a Riesz space with the principal projection property can in a sense be approximated uniformly by simple functions.
Numerous well-known results may be derived from the Freudenthal spectral theorem. The well-known Radon–Nikodym theorem, the validity of the Poisson formula and the spectral theorem from the theory of normal operators can all be shown to follow as special cases of the Freudenthal spectral theorem.
Statement
Let e be any positive element in a Riesz space E. A positive element of p in E is called a component of e if . If are pairwise disjoint components of e, any real linear combination of is called an e-simple function.
The Freudenthal spectral theorem states: Let E be any Riesz space with the principal projection property and e any positive element in E. Then for any element f in the principal ideal generated by e, there exist sequences and of e-simple functions, such that is monotone increasing and converges e-uniformly to f, and is monotone decreasing and converges e-uniformly to f.
Relation to the Radon–Nikodym theorem
Let be a measure space and the real space of signed -additive measures on . It can be shown that is a Dedekind complete Banach Lattice with the total variation norm, and hence has the principal projection property. For any positive measure , -simple functions (as defined above) can be shown to correspond exactly to -measurable simple functions on (in the usual sense). Moreover, since by the Freudenthal spectral theorem, any measure in the band generated by can be monotonously approximated from below by -measurable simple functions on , by Lebesgue's monotone convergence theorem can be shown to correspond to an function and establishes an isometric lattice isomorphism between the band generated by and the Banach Lattice .
See also
Radon–Nikodym theorem
References
Theorems in functional analysis |
https://en.wikipedia.org/wiki/Leon%20Mirsky | Leonid Mirsky (19 December 1918 – 1 December 1983) was a Russian-British mathematician who worked in number theory, linear algebra, and combinatorics. Mirsky's theorem is named after him.
Biography
Mirsky was born in Russia on 19 December 1918 to a medical family, but his parents sent him to live with his aunt and uncle, a wool merchant in Germany, when he was eight. His uncle's family moved to Bradford, England in 1933, bringing Mirsky with them. He studied at Herne Bay High School and King's College, London, graduating in 1940. Because of the evacuation of London during the Blitz, students at King's College were moved to Bristol University, where Mirsky earned a master's degree. He took a short-term faculty position at Sheffield University in 1942, and then a similar position in Manchester; he returned to Sheffield in 1945, where (except for a term as visiting faculty at Bristol) he would stay for the rest of his career. He became a lecturer in 1947, earned a Ph.D. from Sheffield in 1949, became senior lecturer in 1958, reader in 1961, and was given a personal chair in 1971.
In 1953 Mirsky married Aileen Guilding who was, at that time, a lecturer in Biblical History and Literature at Sheffield but later became a professor and Head of Department.
He retired in September 1983, and died on 1 December 1983.
Mirsky was an editor of the Journal of Linear Algebra and its Applications, the Journal of Mathematical Analysis and Applications, and Mathematical Spectrum.
Research
Number theory
Mirsky's early research concerned number theory. He was particularly interested in the r-free numbers, a generalization of the square-free integers consisting of the numbers not divisible by any rth power. These numbers are a superset of the prime numbers, and Mirsky proved theorems for them analogous to Vinogradov's theorem, Goldbach's conjecture, and the twin prime conjecture for prime numbers.
With Paul Erdős in 1952, Mirsky proved strong asymptotic bounds on the number of distinct values taken by the divisor function d(n) counting the number of divisors of the number n. If D(n) denotes the number of distinct values of d(m) for m ≤ n, then
The Mirsky–Newman theorem concerns partitions of the integers into arithmetic progressions, and states that any such partition must have two progressions with the same difference. That is, there cannot be a covering system that covers every integer exactly once and has distinct differences. This result is a special case of the Herzog–Schönheim conjecture in group theory; it was conjectured in 1950 by Paul Erdős and proved soon thereafter by Mirsky and Donald J. Newman. However, Mirsky and Newman never published their proof. The same proof was also found independently by Harold Davenport and Richard Rado.
Linear algebra
In 1947, Mirsky was asked to teach a course in linear algebra. He soon after wrote a textbook on the subject, An introduction to linear algebra (Oxford University Press, 1955), as well as writing a number |
https://en.wikipedia.org/wiki/William%20O.%20Aydelotte | William Osgood Aydelotte (September 1, 1910 – January 17, 1996) was an American historian focused on the British Parliament, a pioneer in applying the statistics to historical research.
Aydelotte was one of the first historians elected to the National Academy of Sciences. The New York Times called him "an authority on British history".
The National Academies Press called him "A leading figure in the development of social science history in the United States". Aydelotte served as the chairman of the University of Iowa history department.
Early life and education
Aydelotte was born in Bloomington, Indiana, the only child of Marie Jeanette Osgood and Franklin Ridgeway Aydelotte. He graduated from Harvard College in 1931, and a received doctoral degree from Cambridge University in 1934.
Career
Aydelotte was the chairman of the University of Iowa history department from 1947 to 1959 and from 1965 to 1968. He retired in 1978. He was married to Myrtle Aydelotte, former nursing school dean at Iowa, from 1956 until his death.
Notable works
Bismarck and British Colonial Policy: The Problem of South West Africa, 1883-1885 (University of Pennsylvania Press, 1937; 2d edition, Russell & Russell, 1970)
The History of Parliamentary Behavior (Princeton University Press, 1977.)
Quantification in History (Addison-Wesley, 1971)
References
External links
Allan G. Bogue and Gilbert White, "William Osgood Aydelotte", Biographical Memoirs of the National Academy of Sciences (1998)
The William O. Aydelotte Papers are housed at the University of Iowa Special Collections & University Archives.
1910 births
1996 deaths
Harvard College alumni
Honorary Members of the Order of the British Empire
Members of the United States National Academy of Sciences
Writers from Bloomington, Indiana
Writers from Iowa City, Iowa
Princeton University faculty
Smith College faculty
Trinity College (Connecticut) faculty
University of Iowa faculty
20th-century American historians
American male non-fiction writers
American expatriates in the United Kingdom
20th-century American male writers
Historians from Iowa |
https://en.wikipedia.org/wiki/Tate%20pairing | In mathematics, Tate pairing is any of several closely related bilinear pairings involving elliptic curves or abelian varieties, usually over local or finite fields, based on the Tate duality pairings introduced by and extended by . applied the Tate pairing over finite fields to cryptography.
See also
Weil pairing
References
Pairing-based cryptography
Elliptic curve cryptography
Elliptic curves |
https://en.wikipedia.org/wiki/Loewner%20differential%20equation | In mathematics, the Loewner differential equation, or Loewner equation, is an ordinary differential equation discovered by Charles Loewner in 1923 in complex analysis and geometric function theory. Originally introduced for studying slit mappings (conformal mappings of the open disk onto the complex plane with a curve joining 0 to ∞ removed), Loewner's method was later developed in 1943 by the Russian mathematician Pavel Parfenevich Kufarev (1909–1968). Any family of domains in the complex plane that expands continuously in the sense of Carathéodory to the whole plane leads to a one parameter family of conformal mappings, called a Loewner chain, as well as a two parameter family of holomorphic univalent self-mappings of the unit disk, called a Loewner semigroup. This semigroup corresponds to a time dependent holomorphic vector field on the disk given by a one parameter family of holomorphic functions on the disk with positive real part. The Loewner semigroup generalizes the notion of a univalent semigroup.
The Loewner differential equation has led to inequalities for univalent functions that played an important role in the solution of the Bieberbach conjecture by Louis de Branges in 1985. Loewner himself used his techniques in 1923 for proving the conjecture for the third coefficient. The Schramm–Loewner equation, a stochastic generalization of the Loewner differential equation discovered by Oded Schramm in the late 1990s, has been extensively developed in probability theory and conformal field theory.
Subordinate univalent functions
Let and be holomorphic univalent functions on the unit disk , , with .
is said to be subordinate to if and only if there is a univalent mapping of into itself fixing such that
for .
A necessary and sufficient condition for the existence of such a mapping is that
Necessity is immediate.
Conversely must be defined by
By definition φ is a univalent holomorphic self-mapping of with .
Since such a map satisfies and takes each disk , with , into itself, it follows that
and
Loewner chain
For let be a family of open connected and simply connected subsets of containing , such that
if ,
and
Thus if ,
in the sense of the Carathéodory kernel theorem.
If denotes the unit disk in , this theorem implies that the unique univalent maps
given by the Riemann mapping theorem are uniformly continuous on compact subsets
of .
Moreover, the function is positive, continuous, strictly increasing and continuous.
By a reparametrization it can be assumed that
Hence
The univalent mappings are called a Loewner chain.
The Koebe distortion theorem shows that knowledge of the chain is equivalent to the properties of the open sets .
Loewner semigroup
If is a Loewner chain, then
for so that there is a unique univalent self mapping of the disk fixing such that
By uniqueness the mappings have the following semigroup property:
for .
They constitute a Loewner semigroup.
The self-mappings depend continuous |
https://en.wikipedia.org/wiki/Paide%20Linnameeskond%20U21 | Paide Linnameeskond U21 is an Estonian football club, the reserve team of Paide Linnameeskond. Until 2013 they played under the name Paide Kumake.
Statistics
League and Cup
Players
Current squad
''As of 19 May 2017.
References
Sport in Paide
Estonian reserve football teams
Paide Linnameeskond
2008 establishments in Estonia
Association football clubs established in 2009 |
https://en.wikipedia.org/wiki/Reward-based%20selection | Reward-based selection is a technique used in evolutionary algorithms for selecting potentially useful solutions for recombination.
The probability of being selected for an individual is proportional to the cumulative reward obtained by the individual. The cumulative reward can be computed as a sum of the individual reward and the reward inherited from parents.
Description
Reward-based selection can be used within Multi-armed bandit framework for Multi-objective optimization to obtain a better approximation of the Pareto front.
The newborn and its parents receive a reward , if was selected for new population , otherwise the reward is zero.
Several reward definitions are possible:
1. , if the newborn individual was selected for new population .
2. , where is the rank of newly inserted individual in the population of individuals. Rank can be computed using a well-known non-dominated sorting procedure.
3. , where is the hypervolume indicator contribution of the individual to the population . The reward if the newly inserted individual improves the quality of the population, which is measured as its hypervolume contribution in the objective space.
4. A relaxation of the above reward, involving a rank-based penalization for points for -th dominated Pareto front:
Reward-based selection can quickly identify the most fruitful directions of search by maximizing the cumulative reward of individuals.
See also
Fitness proportionate selection
Selection (genetic algorithm)
Stochastic universal sampling
Tournament selection
References
Evolutionary algorithms
Genetic algorithms |
https://en.wikipedia.org/wiki/Totative | In number theory, a totative of a given positive integer is an integer such that and is coprime to . Euler's totient function φ(n) counts the number of totatives of n. The totatives under multiplication modulo n form the multiplicative group of integers modulo n.
Distribution
The distribution of totatives has been a subject of study. Paul Erdős conjectured that, writing the totatives of n as
the mean square gap satisfies
for some constant C, and this was proven by Bob Vaughan and Hugh Montgomery.
See also
Reduced residue system
References
Further reading
External links
Modular arithmetic |
https://en.wikipedia.org/wiki/Higgs%20bundle | In mathematics, a Higgs bundle is a pair consisting of a holomorphic vector bundle E and a Higgs field , a holomorphic 1-form taking values in the bundle of endomorphisms of E such that . Such pairs were introduced by , who named the field after Peter Higgs because of an analogy with Higgs bosons. The term 'Higgs bundle', and the condition (which is vacuous in Hitchin's original set-up on Riemann surfaces) was introduced later by Carlos Simpson.
A Higgs bundle can be thought of as a "simplified version" of a flat holomorphic connection on a holomorphic vector bundle, where the derivative is scaled to zero. The nonabelian Hodge correspondence says that, under suitable stability conditions, the category of flat holomorphic connections on a smooth projective complex algebraic variety, the category of representations of the fundamental group of the variety, and the category of Higgs bundles over this variety are actually equivalent. Therefore, one can deduce results about gauge theory with flat connections by working with the simpler Higgs bundles.
History
Higgs bundles were first introduced by Hitchin in 1987, for the specific case where the holomorphic vector bundle E is over a compact Riemann surface. Further, Hitchin's paper mostly discusses the case where the vector bundle is rank 2 (that is, the fiber is a 2-dimensional vector space). The rank 2 vector bundle arises as the solution space to Hitchin's equations for a principal SU(2) bundle.
The theory on Riemann surfaces was generalized by Carlos Simpson to the case where the base manifold is compact and Kähler. Restricting to the dimension one case recovers Hitchin's theory.
Stability of a Higgs bundle
Of particular interest in the theory of Higgs bundles is the notion of a stable Higgs bundle. To do so, -invariant subbundles must first be defined.
In Hitchin's original discussion, a rank-1 subbundle labelled L is -invariant if with the canonical bundle over the Riemann surface M. Then a Higgs bundle is stable if, for each invariant subbundle of ,
with being the usual notion of degree for a complex vector bundle over a Riemann surface.
See also
Hitchin system
References
Vector bundles
Complex manifolds |
https://en.wikipedia.org/wiki/Carath%C3%A9odory%20kernel%20theorem | In mathematics, the Carathéodory kernel theorem is a result in complex analysis and geometric function theory established by the Greek mathematician Constantin Carathéodory in 1912. The uniform convergence on compact sets of a sequence of holomorphic univalent functions, defined on the unit disk in the complex plane and fixing 0, can be formulated purely geometrically in terms of the limiting behaviour of the images of the functions. The kernel theorem has wide application in the theory of univalent functions and in particular provides the geometric basis for the Loewner differential equation.
Kernel of a sequence of open sets
Let Un be a sequence of open sets in C containing 0. Let Vn be the connected component of the interior of containing 0. The kernel of the sequence is defined to be the union of the Vn's, provided it is non-empty; otherwise it is defined to be . Thus the kernel is either a connected open set containing 0 or the one point set . The sequence is said to converge to a kernel if each subsequence has the same kernel.
Examples
If Un is an increasing sequence of connected open sets containing 0, then the kernel is just the union.
If Un is a decreasing sequence of connected open sets containing 0, then, if 0 is an interior point of U1 ∩ U2 ∩ ..., the sequence converges to the component of the interior containing 0. Otherwise, if 0 is not an interior point, the sequence converges to .
Kernel theorem
Let fn(z) be a sequence of holomorphic univalent functions on the unit disk D, normalised so that fn(0) = 0 and f 'n (0) > 0. Then fn converges uniformly on compacts in D to a function f if and only if Un = fn(D) converges to its kernel and this kernel is not C. If the kernel is , then f = 0. Otherwise the kernel is a connected open set U, f is univalent on D and f(D) = U.
Proof
Using Hurwitz's theorem and Montel's theorem, it is straightforward to check that if fn tends uniformly on compacta to f then each subsequence of Un has kernel U = f(D).
Conversely if Un converges to a kernel not equal to C, then by the Koebe quarter theorem Un contains the disk of radius f 'n(0) / 4 with centre 0. The assumption that U ≠ C implies that these radii are uniformly bounded. By the Koebe distortion theorem
Hence the sequence fn is uniformly bounded on compact sets. If two subsequences converge to holomorphic limits f and g, then f(0) = g(0) and with f'(0), g'''(0) ≥ 0. By the first part and the assumptions it follows that f(D) = g(D). Uniqueness in the Riemann mapping theorem forces f = g, so the original sequence f''n is uniformly convergent on compact sets.
References
Theorems in complex analysis |
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