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https://en.wikipedia.org/wiki/Digermane | Digermane is an inorganic compound with the chemical formula . One of the few hydrides of germanium, it is a colourless liquid. Its molecular geometry is similar to ethane.
Synthesis
Digermane was first synthesized and examined in 1924 by Dennis, Corey, and Moore. Their method involves the hydrolysis of magnesium germanide using hydrochloric acid. Many of the properties of digermane and trigermane were determined in the following decade using electron diffraction studies. Further considerations of the compound involved examinations of various reactions such as pyrolysis and oxidation.
Digermane is produced together with germane by the reduction of germanium dioxide with sodium borohydride. Although the major product is germane, a quantifiable amount of digermane is produced in addition to traces of trigermane. It also arises by the hydrolysis of magnesium-germanium alloys.
Reactions
The reactions of digermane exhibit some differences between analogous compounds of the Group 14 elements carbon and silicon. However, there are still some similarities seen, especially in regards to pyrolysis reactions.
The oxidation of digermane takes place at lower temperatures than monogermane. The product of the reaction, germanium oxide, has been shown to act in turn as a catalyst of the reaction. This exemplifies a fundamental difference between germanium and the other Group 14 elements carbon and silicon (carbon dioxide and silicon dioxide do not exhibit the same catalytic properties).
In liquid ammonia, digermane undergoes disproportionation. Ammonia acts as a weakly basic catalyst. Products of the reaction are hydrogen, germane, and a solid polymeric germanium hydride.
Pyrolysis of digermane is proposed to follow multiple steps:
This pyrolysis has been found to be more endothermic than the pyrolysis of disilane. This difference is attributed to the greater strength of the Ge-H bond vs the Si-H bond. As seen in the last reaction of the mechanism above, pyrolysis of digermane may induce polymerization of the group, where acts as a chain propagator and molecular hydrogen gas is released. The dehydrogenation of digermane on gold leads to the formation of germanium nanowires.
Digermane is a precursor to , where E is either sulfur or selenium. These trifluoromethylthio () and trifluoromethylseleno () derivatives possess a markedly higher thermal stability than digermane itself.
Applications
Digermane has a limited number of applications; germane itself is the preferred volatile germanium hydride. Generally, digermane is primarily used a precursor to germanium for use in various applications. Digermane can be used to deposit Ge-containing semiconductors via chemical vapor deposition.
References
Germanium compounds
Metal hydrides
Substances discovered in the 1920s
Chemical compounds containing metal–metal bonds |
https://en.wikipedia.org/wiki/Jean-Louis%20Loday | Jean-Louis Loday (12 January 1946 – 6 June 2012) was a French mathematician who worked on cyclic homology and who introduced Leibniz algebras (sometimes called Loday algebras) and Zinbiel algebras.
He occasionally used the pseudonym Guillaume William Zinbiel, formed by reversing the last name of Gottfried Wilhelm Leibniz.
Education and career
Loday studied at Lycée Louis-le-Grand and at École Normale Supérieure in Paris. He completed his Ph.D. at the University of Strasbourg in 1975 under the supervision of Max Karoubi, with a dissertation titled K-Théorie algébrique et représentations de groupes. He went on to become a senior scientist at CNRS and a member of the Institute for Advanced Mathematical Research (IRMA) at the University of Strasbourg.
Publications
See also
Associahedron
Blakers–Massey theorem
Loday functor
References
Obituary
Home page
External links
Loday's biography of Guillaume William Zinbiel
1946 births
2012 deaths
20th-century French mathematicians
21st-century French mathematicians
Lycée Louis-le-Grand alumni
École Normale Supérieure alumni
University of Strasbourg alumni
Academic staff of the University of Strasbourg
Topologists
Algebraists |
https://en.wikipedia.org/wiki/Uniformly%20bounded%20representation | In mathematics, a uniformly bounded representation of a locally compact group on a Hilbert space is a homomorphism into the bounded invertible operators which is continuous for the strong operator topology, and such that is finite. In 1947 Béla Szőkefalvi-Nagy established that any uniformly bounded representation of the integers or the real numbers is unitarizable, i.e. conjugate by an invertible operator to a unitary representation. For the integers this gives a criterion for an invertible operator to be similar to a unitary operator: the operator norms of all the positive and negative powers must be uniformly bounded. The result on unitarizability of uniformly bounded representations was extended in 1950 by Dixmier, Day and Nakamura-Takeda to all locally compact amenable groups, following essentially the method of proof of Sz-Nagy. The result is known to fail for non-amenable groups such as SL(2,R) and the free group on two generators. conjectured that a locally compact group is amenable if and only if every uniformly bounded representation is unitarizable.
Statement
Let G be a locally compact amenable group and let Tg be a homomorphism of G into GL(H), the group of an invertible operators on a Hilbert space such that
for every x in H the vector-valued gx on G is continuous;
the operator norms of the operators Tg are uniformly bounded.
Then there is a positive invertible operator S on H such that S Tg S−1 is unitary for every g in G.
As a consequence, if T is an invertible operator with all its positive and negative powers uniformly bounded in operator norm, then T is conjugate by a positive invertible operator to a unitary.
Proof
By assumption the continuous functions
generate a separable unital C* subalgebra A of the uniformly bounded continuous functions on G. By construction the algebra is invariant under left translation. By amenability there is an invariant state φ on A. It follows that
is a new inner product on H satisfying
where
So there is a positive invertible operator P such that
By construction
Let S be the unique positive square root of P. Then
Applying S−1 to x and y, it follows that
Since the operators
are invertible, it follows that they are unitary.
Examples of non-unitarizable representations
SL(2,R)
The complementary series of irreducible unitary representations of SL(2,R) was introduced by . These representations can be realized on functions on the circle or on the real line: the Cayley transform provides the unitary equivalence between the two realizations.
In fact for 0 < σ < 1/2 and f, g continuous functions on the circle define
where
Since the function kσ is integrable, this integral converges. In fact
where the norms are the usual L2 norms.
The functions
are orthogonal with
Since these quantities are positive, (f,g)σ defines an inner product. The Hilbert space completion is denoted by Hσ.
For F, G continuous functions of compact support on R, define
Since, regarded as distributions, the |
https://en.wikipedia.org/wiki/Tyson%20Marsh | Tyson Marsh (born June 20, 1984) is a Canadian former professional ice hockey defenceman who last played for the Cardiff Devils in the Elite Ice Hockey League in the United Kingdom.
Career statistics
External links
1984 births
Living people
Alaska Aces (ECHL) players
Canadian ice hockey defencemen
Cardiff Devils players
Chicago Wolves players
Columbia Inferno players
HC Alleghe players
Ice hockey people from British Columbia
Reading Royals players
Rockford IceHogs (AHL) players
Pensacola Ice Pilots players
SC Riessersee players
St. John's Maple Leafs players
Toronto Marlies players
Vancouver Giants players
Canadian expatriate ice hockey players in Wales |
https://en.wikipedia.org/wiki/Earle%E2%80%93Hamilton%20fixed-point%20theorem | In mathematics, the Earle–Hamilton fixed point theorem is a result in geometric function theory giving sufficient conditions for a holomorphic mapping of an open domain in a complex Banach space into itself to have a fixed point. The result was proved in 1968 by Clifford Earle and Richard S. Hamilton by showing that, with respect to the Carathéodory metric on the domain, the holomorphic mapping becomes a contraction mapping to which the Banach fixed-point theorem can be applied.
Statement
Let D be a connected open subset of a complex Banach space X and let f be a holomorphic mapping of D into itself such that:
the image f(D) is bounded in norm;
the distance between points f(D) and points in the exterior of D is bounded below by a positive constant.
Then the mapping f has a unique fixed point x in D and if y is any point in D, the iterates fn(y) converge to x.
Proof
Replacing D by an ε-neighbourhood of f(D), it can be assumed that D is itself bounded in norm.
For z in D and v in X, set
where the supremum is taken over all holomorphic functions g on D with |g(z)| < 1.
Define the α-length of a piecewise differentiable curve γ:[0,1] D by
The Carathéodory metric is defined by
for x and y in D. It is a continuous function on D x D for the norm topology.
If the diameter of D is less than R then, by taking suitable holomorphic functions g of the form
with a in X* and b in C, it follows that
and hence that
In particular d defines a metric on D.
The chain rule
implies that
and hence f satisfies the following generalization of the Schwarz-Pick inequality:
For δ sufficiently small and y fixed in D, the same inequality can be applied to the holomorphic mapping
and yields the improved estimate:
The Banach fixed-point theorem can be applied to the restriction of f to the closure of f(D) on which d defines a complete metric, defining the same
topology as the norm.
Other holomorphic fixed point theorems
In finite dimensions the existence of a fixed point can often be deduced from the Brouwer fixed point theorem without any appeal to holomorphicity of the mapping. In the case of bounded symmetric domains with the Bergman metric, and showed that the same scheme of proof as that used in the Earle-Hamilton theorem applies. The bounded symmetric domain D = G / K is a complete metric space for the Bergman metric. The open semigroup of the complexification Gc taking the closure of D into D acts by contraction mappings, so again the Banach fixed-point theorem can be applied. Neretin extended this argument by continuity to some infinite-dimensional bounded symmetric domains, in particular the Siegel generalized disk of symmetric Hilbert-Schmidt operators with operator norm less than 1. The Earle-Hamilton theorem applies equally well in this case.
References
Theorems in complex analysis
Fixed-point theorems |
https://en.wikipedia.org/wiki/Gossard%20perspector | In geometry the Gossard perspector (also called the Zeeman–Gossard perspector) is a special point associated with a plane triangle. It is a triangle center and it is designated as X(402) in Clark Kimberling's Encyclopedia of Triangle Centers. The point was named Gossard perspector by John Conway in 1998 in honour of Harry Clinton Gossard who discovered its existence in 1916. Later it was learned
that the point had appeared in an article by Christopher Zeeman published during 1899 – 1902. From 2003 onwards the Encyclopedia of Triangle Centers has been referring to this point as Zeeman–Gossard perspector.
Definition
Gossard triangle
Let ABC be any triangle. Let the Euler line of triangle ABC meet the sidelines BC, CA and AB of triangle ABC at D, E and F respectively. Let AgBgCg be the triangle formed by the Euler lines of the triangles AEF, BFD and CDE, the vertex Ag being the intersection of the Euler lines of the triangles BFD and CDE, and similarly for the other two vertices.
The triangle AgBgCg is called the Gossard triangle of triangle ABC.
Gossard perspector
Let ABC be any triangle and let AgBgCg be its Gossard triangle. Then the lines AAg, BBg and CCg are concurrent. The point of concurrence is called the Gossard perspector of triangle ABC.
Properties
Let AgBgCg be the Gossard triangle of triangle ABC. The lines BgCg, CgAg and AgBg are respectively parallel to the lines BC, CA and AB.
Any triangle and its Gossard triangle are congruent.
Any triangle and its Gossard triangle have the same Euler line.
The Gossard triangle of triangle ABC is the reflection of triangle ABC in the Gossard perspector.
Trilinear coordinates
The trilinear coordinates of the Gossard perspector of triangle ABC are
( f ( a, b, c ) : f ( b, c, a ) : f ( c, a, b ) )
where
f ( a, b, c ) = p ( a, b, c ) y ( a, b, c ) / a
where
p ( a, b, c ) = 2a4 − a2b2 − a2c2 − ( b2 − c2 )2
and
y ( a, b, c ) = a8 − a6 ( b2 + c2 ) + a4 ( 2b2 − c2 ) ( 2c2 − b2 ) + ( b2 − c2 )2 [ 3a2 ( b2 + c2 ) − b4 − c4 − 3b2c2 ]
Generalisations
The construction yielding the Gossard triangle of a triangle ABC can be generalised to produce triangles A'B'C' which are congruent to triangle ABC and whose sidelines are parallel to the sidelines of triangle ABC.
Generalisation 1
This result is due to Christopher Zeeman.
Let l be any line parallel to the Euler line of triangle ABC. Let l intersect the sidelines BC, CA, AB of triangle ABC at X, Y, Z respectively. Let A'B'C' be the triangle formed by the Euler lines of the triangles AYZ, BZX and CXY. Then triangle A'B'C' is congruent to triangle ABC and its sidelines are parallel to the sidelines of triangle ABC.
Generalisation 2
This generalisation is due to Paul Yiu.
Let P be any point in the plane of the triangle ABC different from its centroid G.
Let the line PG meet the sidelines BC, CA and AB at X, Y and Z respectively.
Let the centroids of the triangles AYZ, BZX and CXY be Ga, Gb and Gc respectively.
Let Pa be a point such tha |
https://en.wikipedia.org/wiki/Fractal%20derivative | In applied mathematics and mathematical analysis, the fractal derivative or Hausdorff derivative is a non-Newtonian generalization of the derivative dealing with the measurement of fractals, defined in fractal geometry. Fractal derivatives were created for the study of anomalous diffusion, by which traditional approaches fail to factor in the fractal nature of the media. A fractal measure t is scaled according to tα. Such a derivative is local, in contrast to the similarly applied fractional derivative. Fractal calculus is formulated as a generalized of standard calculus
Physical background
Porous media, aquifers, turbulence, and other media usually exhibit fractal properties. Classical diffusion or dispersion laws based on random walks in free space (essentially the same result variously known as Fick's laws of diffusion, Darcy's law, and Fourier's law) are not applicable to fractal media. To address this, concepts such as distance and velocity must be redefined for fractal media; in particular, scales for space and time are to be transformed according to (xβ, tα). Elementary physical concepts such as velocity are redefined as follows for fractal spacetime (xβ, tα):
,
where Sα,β represents the fractal spacetime with scaling indices α and β. The traditional definition of velocity makes no sense in the non-differentiable fractal spacetime.
Definition
Based on above discussion, the concept of the fractal derivative of a function u(t) with respect to a fractal measure t has been introduced as follows:
,
A more general definition is given by
.
For a function y(t) on -perfect fractal set F the fractal derivative or -derivative of at t, is defined by
.
Motivation
The derivatives of a function f can be defined in terms of the coefficients ak in the Taylor series expansion:
From this approach one can directly obtain:
This can be generalized approximating f with functions (xα-(x0)α)k:
note: the lowest order coefficient still has to be b0=f(x0), since it's still the constant approximation of the function f at x0.
Again one can directly obtain:
The Fractal Maclaurin series of f(t) with fractal support F is as follows:
Properties
Expansion coefficients
Just like in the Taylor series expansion, the coefficients bk can be expressed in terms of the fractal derivatives of order k of f:
Proof idea: assuming exists, bk can be written as
one can now use and since
Connection with Derivative
If for a given function f both the derivative Df and the fractal derivative Dαf exists, one can find an analog to the chain rule:
The last step is motivated by the Implicit function theorem which, under appropriate conditions, gives us dx/dxα = (dxα/dx)−1
Similarly for the more general definition:
Application in anomalous diffusion
As an alternative modeling approach to the classical Fick's second law, the fractal derivative is used to derive a linear anomalous transport-diffusion equation underlying anomalous diffusion process,
where 0 < α < 2, 0 |
https://en.wikipedia.org/wiki/Misleading%20graph | In statistics, a misleading graph, also known as a distorted graph, is a graph that misrepresents data, constituting a misuse of statistics and with the result that an incorrect conclusion may be derived from it.
Graphs may be misleading by being excessively complex or poorly constructed. Even when constructed to display the characteristics of their data accurately, graphs can be subject to different interpretations, or unintended kinds of data can seemingly and ultimately erroneously be derived.
Misleading graphs may be created intentionally to hinder the proper interpretation of data or accidentally due to unfamiliarity with graphing software, misinterpretation of data, or because data cannot be accurately conveyed. Misleading graphs are often used in false advertising. One of the first authors to write about misleading graphs was Darrell Huff, publisher of the 1954 book How to Lie with Statistics.
The field of data visualization describes ways to present information that avoids creating misleading graphs.
Misleading graph methods
There are numerous ways in which a misleading graph may be constructed.
Excessive usage
The use of graphs where they are not needed can lead to unnecessary confusion/interpretation. Generally, the more explanation a graph needs, the less the graph itself is needed. Graphs do not always convey information better than tables.
Biased labeling
The use of biased or loaded words in the graph's title, axis labels, or caption may inappropriately prime the reader.
Fabricated trends
Similarly, attempting to draw trend lines through uncorrelated data may mislead the reader into believing a trend exists where there is none. This can be both the result of intentionally attempting to mislead the reader or due to the phenomenon of illusory correlation.
Pie chart
Comparing pie charts of different sizes could be misleading as people cannot accurately read the comparative area of circles.
The usage of thin slices, which are hard to discern, may be difficult to interpret.
The usage of percentages as labels on a pie chart can be misleading when the sample size is small.
Making a pie chart 3D or adding a slant will make interpretation difficult due to distorted effect of perspective. Bar-charted pie graphs in which the height of the slices is varied may confuse the reader.
Comparing pie charts
Comparing data on barcharts is generally much easier. In the image below, it is very hard to tell where the blue sector is bigger than the green sector on the piecharts.
3D Pie chart slice perspective
A perspective (3D) pie chart is used to give the chart a 3D look. Often used for aesthetic reasons, the third dimension does not improve the reading of the data; on the contrary, these plots are difficult to interpret because of the distorted effect of perspective associated with the third dimension. The use of superfluous dimensions not used to display the data of interest is discouraged for charts in general, not only for pie charts. In |
https://en.wikipedia.org/wiki/Gerad%20Adams | Gerad Adams (born May 3, 1978) is a Canadian professional ice hockey defenceman who was previously the coach of the Sheffield Steelers of the Elite Ice Hockey League.
Career statistics
External links
1978 births
Living people
Canadian expatriate ice hockey players in England
Canadian expatriate ice hockey players in Scotland
Canadian expatriate ice hockey players in the United States
Canadian expatriate ice hockey players in Wales
Canadian ice hockey defencemen
Cardiff Devils players
Edinburgh Capitals players
Hampton Roads Admirals players
Ice hockey player-coaches
Kelowna Rockets players
London Knights (UK) players
Portland Pirates players
Regina Pats players
Richmond Renegades players
Sheffield Steelers players
Ice hockey people from Regina, Saskatchewan |
https://en.wikipedia.org/wiki/Ehrenpreis%20conjecture | In mathematics, the Ehrenpreis conjecture of Leon Ehrenpreis states that for any K greater than 1, any two closed Riemann surfaces of genus at least 2 have finite-degree covers which are K-quasiconformal: that is, the covers are arbitrarily close in the Teichmüller metric.
A proof was announced by Jeremy Kahn and Vladimir Markovic in January 2011, using their proof of the Surface subgroup conjecture and a newly developed "good pants homology" theory. In June 2012, Kahn and Markovic were given the Clay Research Awards for their work on these two problems by the Clay Mathematics Institute at a ceremony at Oxford University.
See also
Surface subgroup conjecture
Virtually Haken conjecture
Virtually fibered conjecture
References
3-manifolds
Conjectures that have been proved
Theorems in topology |
https://en.wikipedia.org/wiki/Non-wellfounded%20mereology | In philosophy, specifically metaphysics, mereology is the study of parthood relationships. In mathematics and formal logic, wellfoundedness prohibits for any x.
Thus non-wellfounded mereology treats topologically circular, cyclical, repetitive, or other eventual self-containment.
More formally, non-wellfounded partial orders may exhibit for some x whereas well-founded orders prohibit that.
See also
Aczel's anti-foundation axiom
Peter Aczel
John Barwise
Steve Awodey
Dana Scott
External links
Mereology
Mathematical logic |
https://en.wikipedia.org/wiki/Yutaka%20Nishiyama | is a Japanese mathematician and professor at the Osaka University of Economics, where he teaches mathematics and information. He is known as the "boomerang professor". He has written nine books about the mathematics in daily life. The most recent one, The mystery of five in nature, investigates, amongst other things, why many flowers have five petals.
Biography
1967-1971: Faculty of Mathematics, Department of Science, Kyoto University
1971-1985: IBM Japan as a Systems Engineer
1985: Lecturer of Information Mathematics at Osaka University of Economics
1995–present: Professor at Osaka University of Economics
2005-2006: Visiting fellow at University of Cambridge, UK, joined for MMP.
Books
50 Visions of Mathematics, Oxford University Press, May 2014,
The Mysterious Number 6174: One of 30 Mathematical Topics in Daily Life, Gendai Sugakusha, July 2013,
Papers
General Solution for Multiple Foldings of Hexaflexagons IJPAM, Vol. 58, No. 1, (2010). 113-124. "19 faces of Flexagons"
Fixed Points in Similarity Transformations IJPAM, Vol. 56, No. 3, (2009). 429-438.
Articles for Plus Magazine
A bright idea, Plus Magazine, issue 36, University of Cambridge, September 2005.
Mysterious Number 6174, Plus Magazine, issue 38, University of Cambridge, March 2006.
Winning Odds, with Steve Humble, Plus Magazine, issue 55, University of Cambridge, June 2010.
Having fun with unit fractions, Plus Magazine, University of Cambridge, Feb 2012.
Circles rolling on circles, Plus Magazine, University of Cambridge, May 2014.
References
External links
Homepage of Yutaka Nishiyama
1948 births
20th-century Japanese mathematicians
21st-century Japanese mathematicians
Kyoto University alumni
Living people |
https://en.wikipedia.org/wiki/Markov%E2%80%93Kakutani%20fixed-point%20theorem | In mathematics, the Markov–Kakutani fixed-point theorem, named after Andrey Markov and Shizuo Kakutani, states that a commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space has a common fixed point. This theorem is a key tool in one of the quickest proofs of amenability of abelian groups.
Statement
Let be a locally convex topological vector space, with a compact convex subset .
Let be a family of continuous mappings of to itself which commute and are affine, meaning that for all in and in . Then the mappings in share a fixed point.
Proof for a single affine self-mapping
Let be a continuous affine self-mapping of .
For in define a net in by
Since is compact, there is a convergent subnet in :
To prove that is a fixed point, it suffices to show that for every in the dual of . (The dual separates points by the Hahn-Banach theorem; this is where the assumption of local convexity is used.)
Since is compact, is bounded on by a positive constant . On the other hand
Taking and passing to the limit as goes to infinity, it follows that
Hence
Proof of theorem
The set of fixed points of a single affine mapping is a non-empty compact convex set by the result for a single mapping. The other mappings in the family commute with so leave invariant. Applying the result for a single mapping successively, it follows that any finite subset of has a non-empty fixed point set given as the intersection of the compact convex sets as ranges over the subset. From the compactness of it follows that the set
is non-empty (and compact and convex).
Citations
References
Theorems in functional analysis
Topological vector spaces
Fixed-point theorems |
https://en.wikipedia.org/wiki/Fischer%20group%20Fi24 | {{DISPLAYTITLE:Fischer group Fi24}}
In the area of modern algebra known as group theory, the Fischer group Fi24 or F24′ is a sporadic simple group of order
22131652731113172329
= 1255205709190661721292800
≈ 1.
History and properties
Fi24 is one of the 26 sporadic groups and is the largest of the three Fischer groups introduced by while investigating 3-transposition groups. It is the 3rd largest of the sporadic groups (after the Monster group and Baby Monster group).
The outer automorphism group has order 2, and the Schur multiplier has order 3. The automorphism group is a 3-transposition group Fi24, containing the simple group with index 2.
The centralizer of an element of order 3 in the monster group is a triple cover of the sporadic simple group Fi24, as a result of which the prime 3 plays a special role in its theory.
Representations
The centralizer of an element of order 3 in the monster group is a triple cover of the Fischer group, as a result of which the prime 3 plays a special role in its theory. In particular it acts on a vertex operator algebra over the field with 3 elements.
The simple Fischer group has a rank 3 action on a graph of 306936 (=23.33.72.29) vertices corresponding to the 3-transpositions of Fi24, with point stabilizer the Fischer group Fi23.
The triple cover has a complex representation of dimension 783. When reduced modulo 3 this has 1-dimensional invariant subspaces and quotient spaces, giving an irreducible representation of dimension 781 over the field with 3 elements.
Generalized Monstrous Moonshine
Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For Fi24 (as well as Fi23), the relevant McKay-Thompson series is where one can set the constant term a(0) = 42 (),
Maximal subgroups
found the 22 conjugacy classes of maximal subgroups of Fi24 as follows:
Fi23 Centralizes a 3-transposition in the automorphism group Fi24.
2.Fi22:2
(3 x O(3):3):2
O(2)
37.O7(3)
31+10:U5(2):2
211.M24
22.U6(2):S3
21+12:3.U4(3).2
32+4+8.(A5 x 2A4).2
(A4 x O(2):3):2
He:2 (Two classes, fused by an outer automorphism)
23+12.(L3(2) x A6)
26+8.(S3 x A8)
(G2(3) x 32:2).2
(A9 x A5):2
A7 x 7:6
[313]:(L3(3) x 2)
L2(8):3 x A6
U3(3):2 (Two classes, fused by an outer automorphism)
L2(13):2 (Two classes, fused by an outer automorphism)
29:14
References
contains a complete proof of Fischer's theorem.
This is the first part of Fischer's preprint on the construction of his groups. The remainder of the paper is unpublished (as of 2010).
Wilson, R. A. ATLAS of Finite Group Representation.
External links
MathWorld: Fischer Groups
Atlas of Finite Group Representations: Fi24
Sporadic groups |
https://en.wikipedia.org/wiki/Etsuji%20Fujita | is a Japanese former water polo player who competed in the 1984 Summer Olympics.
See also
Japan men's Olympic water polo team records and statistics
List of men's Olympic water polo tournament goalkeepers
References
External links
1961 births
Living people
Japanese male water polo players
Water polo goalkeepers
Olympic water polo players for Japan
Water polo players at the 1984 Summer Olympics
Asian Games medalists in water polo
Water polo players at the 1982 Asian Games
Asian Games silver medalists for Japan
Medalists at the 1982 Asian Games |
https://en.wikipedia.org/wiki/Yukiharu%20Oshita | is a Japanese former water polo player who competed in the 1972 Summer Olympics.
See also
Japan men's Olympic water polo team records and statistics
List of men's Olympic water polo tournament goalkeepers
References
External links
1949 births
Living people
Japanese male water polo players
Water polo goalkeepers
Olympic water polo players for Japan
Water polo players at the 1972 Summer Olympics
Asian Games medalists in water polo
Water polo players at the 1970 Asian Games
Asian Games gold medalists for Japan
Medalists at the 1970 Asian Games
20th-century Japanese people
21st-century Japanese people |
https://en.wikipedia.org/wiki/Tetsunosuke%20Ishii | is a Japanese former water polo player who competed in the 1968 Summer Olympics.
See also
Japan men's Olympic water polo team records and statistics
List of men's Olympic water polo tournament goalkeepers
References
External links
1944 births
Living people
Japanese male water polo players
Water polo goalkeepers
Olympic water polo players for Japan
Water polo players at the 1968 Summer Olympics
Asian Games medalists in water polo
Water polo players at the 1966 Asian Games
Asian Games gold medalists for Japan
Medalists at the 1966 Asian Games
20th-century Japanese people |
https://en.wikipedia.org/wiki/Fischer%20group%20Fi23 | {{DISPLAYTITLE:Fischer group Fi23}}
In the area of modern algebra known as group theory, the Fischer group Fi23 is a sporadic simple group of order
21831352711131723
= 4089470473293004800
≈ 4.
History
Fi23 is one of the 26 sporadic groups and is one of the three Fischer groups introduced by while investigating 3-transposition groups.
The Schur multiplier and the outer automorphism group are both trivial.
Representations
The Fischer group Fi23 has a rank 3 action on a graph of 31671 vertices corresponding to 3-transpositions, with point stabilizer the double cover of the Fischer group Fi22. It has a second rank-3 action on 137632 points
The smallest faithful complex representation has dimension . The group has an irreducible representation of dimension 253 over the field with 3 elements.
Generalized Monstrous Moonshine
Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For Fi23, the relevant McKay-Thompson series is where one can set the constant term a(0) = 42 (),
and η(τ) is the Dedekind eta function.
Maximal subgroups
found the 14 conjugacy classes of maximal subgroups of Fi23 as follows:
2.Fi22
O8+(3):S3
22.U6(2).2
S8(2)
O7(3) × S3
211.M23
31+8.21+6.31+2.2S4
[310].(L3(3) × 2)
S12
(22 × 21+8).(3 × U4(2)).2
26+8:(A7 × S3)
S6(2) × S4
S4(4):4
L2(23)
References
contains a complete proof of Fischer's theorem.
This is the first part of Fischer's preprint on the construction of his groups. The remainder of the paper is unpublished (as of 2010).
Wilson, R. A. ATLAS of Finite Group Representations.
External links
MathWorld: Fischer Groups
Atlas of Finite Group Representations: Fi23
Sporadic groups |
https://en.wikipedia.org/wiki/Fischer%20group%20Fi22 | {{DISPLAYTITLE:Fischer group Fi22}}
In the area of modern algebra known as group theory, the Fischer group Fi22 is a sporadic simple group of order
217395271113
= 64561751654400
≈ 6.
History
Fi22 is one of the 26 sporadic groups and is the smallest of the three Fischer groups. It was introduced by while investigating 3-transposition groups.
The outer automorphism group has order 2, and the Schur multiplier has order 6.
Representations
The Fischer group Fi22 has a rank 3 action on a graph of 3510 vertices corresponding to its 3-transpositions, with point stabilizer the double cover of the group PSU6(2). It also has two rank 3 actions on 14080 points, exchanged by an outer automorphism.
Fi22 has an irreducible real representation of dimension 78. Reducing an integral form of this mod 3 gives a representation of Fi22 over the field with 3 elements, whose quotient by the 1-dimensional space of fixed vectors is a 77-dimensional irreducible representation.
The perfect triple cover of Fi22 has an irreducible representation of dimension 27 over the field with 4 elements. This arises from the fact that Fi22 is a subgroup of 2E6(22).
All the ordinary and modular character tables of Fi22 have been computed. found the 5-modular character table, and found the 2- and 3-modular character tables.
The automorphism group of Fi22 centralizes an element of order 3 in the baby monster group.
Generalized Monstrous Moonshine
Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For Fi22, the McKay-Thompson series is where one can set a(0) = 10 (),
and η(τ) is the Dedekind eta function.
Maximal subgroups
found the 12 conjugacy classes of maximal subgroups of Fi22 as follows:
2·U6(2)
O7(3) (Two classes, fused by an outer automorphism)
O(2):S3
210:M22
26:S6(2)
(2 × 21+8):(U4(2):2)
U4(3):2 × S3
2F4(2)' (This is the Tits group)
25+8:(S3 × A6)
31+6:23+4:32:2
S10 (Two classes, fused by an outer automorphism)
M12
References
contains a complete proof of Fischer's theorem.
This is the first part of Fischer's preprint on the construction of his groups. The remainder of the paper is unpublished (as of 2010).
Wilson, R. A. ATLAS of Finite Group Representations.
External links
MathWorld: Fischer Groups
Atlas of Finite Group Representations: Fi22
Sporadic groups |
https://en.wikipedia.org/wiki/Regression-kriging | In applied statistics and geostatistics, regression-kriging (RK) is a spatial prediction technique that combines a regression of the dependent variable on auxiliary variables (such as parameters derived from digital elevation modelling, remote sensing/imagery, and thematic maps) with interpolation (kriging) of the regression residuals. It is mathematically equivalent to the interpolation method variously called universal kriging and kriging with external drift, where auxiliary predictors are used directly to solve the kriging weights.
BLUP for spatial data
Regression-kriging is an implementation of the best linear unbiased predictor (BLUP) for spatial data, i.e. the best linear interpolator assuming the universal model of spatial variation. Matheron (1969) proposed that a value of a target variable at some location can be modeled as a sum of the deterministic and stochastic components:
which he termed universal model of spatial variation. Both deterministic and stochastic components of spatial variation can be modeled separately. By combining the two approaches, we obtain:
where is the fitted deterministic part, is the interpolated residual, are estimated deterministic model coefficients ( is the estimated intercept), are kriging weights determined by the spatial dependence structure of the residual and where is the residual at location . The regression coefficients can be estimated from the sample by some fitting method, e.g. ordinary least squares (OLS) or, optimally, using generalized least squares (GLS):
where is the vector of estimated regression coefficients, is the covariance matrix of the residuals, is a matrix of predictors at the sampling locations and is the vector of measured values of the target variable. The GLS estimation of regression coefficients is, in fact, a special case of the geographically weighted regression. In the case, the weights are determined objectively to account for the spatial auto-correlation between the residuals.
Once the deterministic part of variation has been estimated (regression-part), the residual can be interpolated with kriging and added to the estimated trend. The estimation of the residuals is an iterative process: first the deterministic part of variation is estimated using OLS, then the covariance function of the residuals is used to obtain the GLS coefficients. Next, these are used to re-compute the residuals, from which an updated covariance function is computed, and so on. Although this is by many geostatisticians recommended as the proper procedure, Kitanidis (1994) showed that use of the covariance function derived from the OLS residuals (i.e. a single iteration) is often satisfactory, because it is not different enough from the function derived after several iterations; i.e. it does not affect much the final predictions. Minasny and McBratney (2007) report similar results—it seems that using more higher quality data is more important than to use more sophisticated statistical |
https://en.wikipedia.org/wiki/Khosrov%20Forest%20State%20Reserve | {
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Khosrov Forest State Reserve (), is a nature reserve in Ararat Province of Armenia. The reserve is one of the oldest protected areas in the world having a history of about 1700 years. It was founded in the 4th century (334–338) by the order of Khosrov Kotak, King of Armenia who gave it his name. It was founded to improve the natural climatic conditions of adjacent territories of Artashat – the capital city of Armenia of the given period and the newly established city of Dvin to ensure conservation and enrichment of flora and fauna species; serve as a ground for royal hunting, military exercises and entertainments.
This area was designated as a state reserve in September 1958 and covers around at elevations from 700 to above sea level. The Khosrov reserve protects juniper (Juniperus polycarpos) and oak (Quercus macranthera) forests from Tertiary Period, arid associations of semi-desert and phrygana landscapes and other Mediterranean relict plant ecosystems as well as the genetic fund of rare animals and plants adapted to the reserve conditions. It also includes wetlands of international importance. Khosrov Forest State Reserve, thanks to its numerous peculiarities, is unique not only in Armenia but also in the whole Caucasus ecoregion.
Since 2002, administrative, scientific, practical and organizational activities in the reserve have been implemented by the Khosrov Forest State Reserve State Non-Commercial Organization (SNCO) of the Bioresources Management Agency under the aegis of the Ministry of Nature Protection of Armenia. According to the reserved charter (RA Governmental decision N 925 from 30 May 2002, Annex 1), the main goals of the reserve establishment are to ensure natural development of water and terrestrial ecosystems of the rivers Azat and Vedi basins, to protect the landscapes and biological diversity including the genetic fund of rare and endangered plants and animals as well as their habitats, to implement scientific research and to create prerequisites for the development of scientific and educational tourism, environmental education and responsibility.
The following protected areas are under the subordination of the Khosrov Forest State Reserve: SNCO:
Khosrov Forest state reserve, which at present is divided into four districts- Garni (4253ha), Kaqavaberd (4745ha), Khosrov (6860.8ha), and Khachadzor (7354.7ha).
Goravan Sands state sanctuary (95.99ha)
Khor Virap State Sanctuary (50.28ha)
Realizing the importance of the reserve, on 26 August 2013 Khosrov Forest State Reserve SNCO was awarded with European Diploma of Protected Areas.
The reserve has wonderful landscapes, rich biological diversity, a huge variety of interesting and important untouched forests that are the result of long-term preservation, proper manage |
https://en.wikipedia.org/wiki/Generalized%20beta%20distribution | In probability and statistics, the generalized beta distribution is a continuous probability distribution with four shape parameters (however it's customary to make explicit the scale parameter as a fifth parameter, while the location parameter is usually left implicit), including more than thirty named distributions as limiting or special cases. It has been used in the modeling of income distribution, stock returns, as well as in regression analysis. The exponential generalized beta (EGB) distribution follows directly from the GB and generalizes other common distributions.
Definition
A generalized beta random variable, Y, is defined by the following probability density function:
and zero otherwise. Here the parameters satisfy , and , , and positive. The function B(p,q) is the beta function. The parameter is the scale parameter and can thus be set to without loss of generality, but it is usually made explicit as in the function above (while the location parameter is usually left implicit and set to as in the function above).
Properties
Moments
It can be shown that the hth moment can be expressed as follows:
where denotes the hypergeometric series (which converges for all h if c<1, or for all h/a<q if c=1 ).
Related distributions
The generalized beta encompasses many distributions as limiting or special cases. These are depicted in the GB distribution tree shown above. Listed below are its three direct descendants, or sub-families.
Generalized beta of first kind (GB1)
The generalized beta of the first kind is defined by the following pdf:
for where , , and are positive. It is easily verified that
The moments of the GB1 are given by
The GB1 includes the beta of the first kind (B1), generalized gamma(GG), and Pareto as special cases:
Generalized beta of the second kind (GB2)
The GB2 is defined by the following pdf:
for and zero otherwise. One can verify that
The moments of the GB2 are given by
The GB2 is also known as the Generalized Beta Prime (Patil, Boswell, Ratnaparkhi (1984)), the transformed beta (Venter, 1983), the generalized F (Kalfleisch and Prentice, 1980), and is a special case (μ≡0) of the Feller-Pareto (Arnold, 1983) distribution. The GB2 nests common distributions such as the generalized gamma (GG), Burr type 3, Burr type 12, Dagum, lognormal, Weibull, gamma, Lomax, F statistic, Fisk or Rayleigh, chi-square, half-normal, half-Student's t, exponential, asymmetric log-Laplace, log-Laplace, power function, and the log-logistic.
Beta
The beta family of distributions (B) is defined by:
for and zero otherwise. Its relation to the GB is seen below:
The beta family includes the beta of the first and second kind (B1 and B2, where the B2 is also referred to as the Beta prime), which correspond to c = 0 and c = 1, respectively. Setting , yields the standard two-parameter beta distribution.
Generalized Gamma
The generalized gamma distribution (GG) is a limiting case of the GB2. Its PDF is defined by:
with the |
https://en.wikipedia.org/wiki/Shyamaprasad%20Mukherjee | Shyamaprasad Mukherjee, FNASc, known as S. P. Mukherjee (born 16 June 1938), is an Indian statistician and the former Centenary Professor of Statistics at the University of Calcutta. He is currently a visiting professor at University of Calcutta after retiring formally as the Centenary Professor of Statistics in 2004. He is currently serving as the Chairman of Expert Group conducting All-India surveys under the Labour Bureau.
Early life and education
He was born in Kidderpore, Kolkata, India. In 1954 he appeared in his school final (10th grade) examination and was ranked second in the state of West Bengal. He received an M.Sc. in Statistics in 1960 and Ph.D. in Statistics in 1967 from University of Calcutta. After a short stint of teaching in the Presidency College, Calcutta, he joined the University of Calcutta as a lecturer in Statistics in 1964.
Academic career
Prof Mukherjee has done extensive research work in Applied Probability, Parametric and Bayesian Estimation, Reliability Analysis, Quality Management, and Operations Research; published more than sixty research papers and review and expository articles; and supervised twenty one Ph. D. students.
He was the President of the Operational Research Society of India as well as of the Association of Asian-Pacific Operational Research Societies (APORS). He was also a Vice-President of the International Federation of Operational Research Societies (IFORS). He has been the President of the Calcutta Statistical Association. He has served as the Chairman of the Committee on Statistical Methods for Quality and Reliability of the Bureau of Indian Standards. He is the president of the Indian Association for Productivity, Quality and Reliability (IAPQR) as well as the editor of IAPQR Transactions and is associated with the editorial work of several other journals.
Areas of research contributions
Applied probability
Parametric and Bayesian estimation
Reliability
Quality management
Operations research
Books authored
Graduate Employment and Higher Education in West Bengal
Frontiers in Probability and Statistics
Quality: Domains and Dimensions
Statistical Methods in Social Science Research<ref></ref>
A Guide to Research Methodology: An Overview of Research Problems, Tasks and Methods Decision-making: Concepts, Methods and Techniques''
Awards and medals
Prof P V Sukhatme Award for Senior Statisticians June 2012
P C Mahalanobis Birth Centenary Award January 2000
Fellow of ISPS 2012
References
Indian statisticians
1939 births
Living people
Presidency University, Kolkata alumni
University of Calcutta alumni
20th-century Indian mathematicians
Bengali mathematicians
Academic staff of the University of Calcutta
Scientists from Kolkata |
https://en.wikipedia.org/wiki/Weak%20form%20and%20strong%20form | Weak form and strong form may refer to:
Weaker and stronger versions of a hypothesis, theorem or physical law
Weak formulations and strong formulations of differential equations in mathematics
Differing pronunciations of words depending on emphasis; see Weak and strong forms in English
Weak and strong pronouns
See also
Weakened weak form (mathematics)
Clitic (linguistics)
Weak inflection (linguistics)
Strong (disambiguation)
Weak (disambiguation) |
https://en.wikipedia.org/wiki/Holonomic%20basis | In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold is a set of basis vector fields defined at every point of a region of the manifold as
where is the displacement vector between the point and a nearby point
whose coordinate separation from is along the coordinate curve (i.e. the curve on the manifold through for which the local coordinate varies and all other coordinates are constant).
It is possible to make an association between such a basis and directional derivative operators. Given a parameterized curve on the manifold defined by with the tangent vector , where , and a function defined in a neighbourhood of , the variation of along can be written as
Since we have that , the identification is often made between a coordinate basis vector and the partial derivative operator , under the interpretation of vectors as operators acting on functions.
A local condition for a basis to be holonomic is that all mutual Lie derivatives vanish:
A basis that is not holonomic is called an anholonomic, non-holonomic or non-coordinate basis.
Given a metric tensor on a manifold , it is in general not possible to find a coordinate basis that is orthonormal in any open region of . An obvious exception is when is the real coordinate space considered as a manifold with being the Euclidean metric at every point.
References
See also
Jet bundle
Tetrad formalism
Ricci calculus
Differential geometry
Mathematical physics |
https://en.wikipedia.org/wiki/Mathematical%20Social%20Sciences | Mathematical Social Sciences is a peer-reviewed mathematics journal in the field of social science, in particular economics. The journal covers research on mathematical modelling in fields such as economics, psychology, political science, and other social sciences, including individual decision making and preferences, decisions under risk, collective choice, voting, theories of measurement, and game theory.
It was established in 1980 and is published by Elsevier. The editors-in-chief have been Ki Hang Kim (1980-1983), Hervé Moulin (1983-2004), Jean-François Laslier (2005-2016), Simon Grant, Christopher Chambers (2009-2020), Yusufcan Masatlioglu (2020-2021), Juan Moreno-Ternero (2017-) and Emel Filiz-Ozbay (2021-).
See also
List of scholarly journals in economics
External links
Economics journals
Elsevier academic journals
Academic journals established in 1980
English-language journals
Bimonthly journals
Mathematics journals
Multidisciplinary social science journals |
https://en.wikipedia.org/wiki/Wilhelm%20Ahrens | Wilhelm Ahrens (3 March 1872 – 23 May 1927) was a German mathematician and writer on recreational mathematics.
Biography
Ahrens was born in Lübz at the Elde in Mecklenburg and studied from 1890 to 1897 at the University of Rostock, Humboldt University of Berlin, and the University of Freiburg. In 1895 at the University of Rostock he received his Promotion (Ph.D.), summa cum laude, under the supervision of Otto Staude with dissertation entitled Über eine Gattung n-fach periodischer Functionen von n reellen Veränderlichen. From 1895 to 1896 he taught at the German school in Antwerp and then studied another semester under Sophus Lie in Leipzig. In 1897 Ahrens was a teacher in Magdeburg at the Baugewerkeschule, from 1901 at the engineering school. Inspired by Sophus Lie, he wrote "On transformation groups, all of whose subgroups are invariant" (Hamburger Math Society Vol 4, 1902).
He worked a lot on the history of mathematics and mathematical games (recreational mathematics), about which he wrote a great work and also contributed to the Encyclopedia of mathematical sciences His predecessors were the great Jacques Ozanam in France, where the number theorist Édouard Lucas (1842–1891) in the 19th century wrote similar books, and Walter William Rouse Ball (1850–1925) in England (Mathematical recreations and essays 1892), Sam Loyd (1841–1901) in the U.S. and Henry Dudeney (1857–1930) in England. In this sense Martin Gardner (1914-2010) and Ian Stewart, the editor of the math column in Scientific American, might be regarded as his successors. He also wrote a book of quotations and anecdotes about mathematicians. He was the author of numerous journal articles.
Scherz und Ernst in der Mathematik
According to R. C. Archibald:
Bibliography
Mathematische Unterhaltungen und Spiele [Mathematical Recreations and Games], 1901
Mathematische Spiele [Mathematical Games], 1902
Scherz und Ernst in der Mathematik; geflügelte und ungeflügelte Worte [Fun and seriousness in mathematics: well-known and less well-known words], 1904. ; 2002 Auflage
Gelehrten-Anekdoten [Scholarly anecdotes], 1911
Mathematiker-Anekdoten [Anecdotes of Mathematicians], 1916; Zweite, stark veränderte Auflage (2nd revised edition) 1920
References
External links
Wilhelm Ernst Martin Georg Ahrens at the Mathematics Genealogy Project
1872 births
1927 deaths
People from Lübz
Military personnel from the Grand Duchy of Mecklenburg-Schwerin
19th-century German mathematicians
Recreational mathematicians
Mathematics popularizers
20th-century German mathematicians
Mathematicians from the German Empire |
https://en.wikipedia.org/wiki/Kneser%27s%20theorem%20%28combinatorics%29 | In the branch of mathematics known as additive combinatorics, Kneser's theorem can refer to one of several related theorems regarding the sizes of certain sumsets in abelian groups. These are named after Martin Kneser, who published them in 1953 and 1956. They may be regarded as extensions of the Cauchy–Davenport theorem, which also concerns sumsets in groups but is restricted to groups whose order is a prime number.
The first three statements deal with sumsets whose size (in various senses) is strictly smaller than the sum of the size of the summands. The last statement deals with the case of equality for Haar measure in connected compact abelian groups.
Strict inequality
If is an abelian group and is a subset of , the group is the stabilizer of .
Cardinality
Let be an abelian group. If and are nonempty finite subsets of satisfying and is the stabilizer of , then
This statement is a corollary of the statement for LCA groups below, obtained by specializing to the case where the ambient group is discrete. A self-contained proof is provided in Nathanson's textbook.
Lower asymptotic density in the natural numbers
The main result of Kneser's 1953 article is a variant of Mann's theorem on Schnirelmann density.
If is a subset of , the lower asymptotic density of is the number . Kneser's theorem for lower asymptotic density states that if and are subsets of satisfying , then there is a natural number such that satisfies the following two conditions:
is finite,
and
Note that , since .
Haar measure in locally compact abelian (LCA) groups
Let be an LCA group with Haar measure and let denote the inner measure induced by (we also assume is Hausdorff, as usual). We are forced to consider inner Haar measure, as the sumset of two -measurable sets can fail to be -measurable. Satz 1 of Kneser's 1956 article can be stated as follows:
If and are nonempty -measurable subsets of satisfying , then the stabilizer is compact and open. Thus is compact and open (and therefore -measurable), being a union of finitely many cosets of . Furthermore,
Equality in connected compact abelian groups
Because connected groups have no proper open subgroups, the preceding statement immediately implies that if is connected, then for all -measurable sets and . Examples where
can be found when is the torus and and are intervals. Satz 2 of Kneser's 1956 article says that all examples of sets satisfying equation () with non-null summands are obvious modifications of these. To be precise: if is a connected compact abelian group with Haar measure and are -measurable subsets of satisfying , and equation (), then there is a continuous surjective homomorphism and there are closed intervals , in such that , , , and .
Notes
References
Theorems in combinatorics
Sumsets |
https://en.wikipedia.org/wiki/Conway%20group%20Co2 | {{DISPLAYTITLE:Conway group Co2}}
In the area of modern algebra known as group theory, the Conway group Co2 is a sporadic simple group of order
218365371123
= 42305421312000
≈ 4.
History and properties
Co2 is one of the 26 sporadic groups and was discovered by as the group of automorphisms of the Leech lattice Λ fixing a lattice vector of type 2. It is thus a subgroup of Co0. It is isomorphic to a subgroup of Co1. The direct product 2×Co2 is maximal in Co0.
The Schur multiplier and the outer automorphism group are both trivial.
Representations
Co2 acts as a rank 3 permutation group on 2300 points. These points can be identified with planar hexagons in the Leech lattice having 6 type 2 vertices.
Co2 acts on the 23-dimensional even integral lattice with no roots of determinant 4, given as a sublattice of the Leech lattice orthogonal to a norm 4 vector. Over the field with 2 elements it has a 22-dimensional faithful representation; this is the smallest faithful representation over any field.
showed that if a finite group has an absolutely irreducible faithful rational representation of dimension 23 and has no subgroups of index 23 or 24 then it is contained in either Z/2Z × Co2 or Z/2Z × Co3.
The Mathieu group M23 is isomorphic to a maximal subgroup of Co2 and one representation, in permutation matrices, fixes the type 2 vector u = (-3,123). A block sum ζ of the involution η =
and 5 copies of -η also fixes the same vector. Hence Co2 has a convenient matrix representation inside the standard representation of Co0. The trace of ζ is -8, while the involutions in M23 have trace 8.
A 24-dimensional block sum of η and -η is in Co0 if and only if the number of copies of η is odd.
Another representation fixes the vector v = (4,-4,022). A monomial and maximal subgroup includes a representation of M22:2, where any α interchanging the first 2 co-ordinates restores v by then negating the vector. Also included are diagonal involutions corresponding to octads (trace 8), 16-sets (trace -8), and dodecads (trace 0). It can be shown that Co2 has just 3 conjugacy classes of involutions. η leaves (4,-4,0,0) unchanged; the block sum ζ provides a non-monomial generator completing this representation of Co2.
There is an alternate way to construct the stabilizer of v. Now u and u+v = (1,-3,122) are vertices of a 2-2-2 triangle (vide infra). Then u, u+v, v, and their negatives form a coplanar hexagon fixed by ζ and M22; these generate a group Fi21 ≈ U6(2). α (vide supra) extends this to Fi21:2, which is maximal in Co2. Lastly, Co0 is transitive on type 2 points, so that a 23-cycle fixing u has a conjugate fixing v, and the generation is completed.
Maximal subgroups
Some maximal subgroups fix or reflect 2-dimensional sublattices of the Leech lattice. It is usual to define these planes by h-k-l triangles: triangles including the origin as a vertex, with edges (differences of vertices) being vectors of types h, k, and l.
found the 11 conjugacy classes |
https://en.wikipedia.org/wiki/Conway%20group%20Co3 | {{DISPLAYTITLE:Conway group Co3}}
In the area of modern algebra known as group theory, the Conway group is a sporadic simple group of order
210375371123
= 495766656000
≈ 5.
History and properties
is one of the 26 sporadic groups and was discovered by as the group of automorphisms of the Leech lattice fixing a lattice vector of type 3, thus length . It is thus a subgroup of . It is isomorphic to a subgroup of . The direct product is maximal in .
The Schur multiplier and the outer automorphism group are both trivial.
Representations
Co3 acts on the unique 23-dimensional even lattice of determinant 4 with no roots, given by the orthogonal complement of a norm 4 vector of the Leech lattice. This gives 23-dimensional representations over any field; over fields of characteristic 2 or 3 this can be reduced to a 22-dimensional faithful representation.
Co3 has a doubly transitive permutation representation on 276 points.
showed that if a finite group has an absolutely irreducible faithful rational representation of dimension 23 and has no subgroups of index 23 or 24 then it is contained in either or .
Maximal subgroups
Some maximal subgroups fix or reflect 2-dimensional sublattices of the Leech lattice. It is usual to define these planes by h-k-l triangles: triangles including the origin as a vertex, with edges (differences of vertices) being vectors of types h, k, and l.
found the 14 conjugacy classes of maximal subgroups of as follows:
McL:2 – McL fixes a 2-2-3 triangle. The maximal subgroup also includes reflections of the triangle. has a doubly transitive permutation representation on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by .
HS – fixes a 2-3-3 triangle.
U4(3).22
M23 – fixes a 2-3-4 triangle.
35:(2 × M11) - fixes or reflects a 3-3-3 triangle.
2.Sp6(2) – centralizer of involution class 2A (trace 8), which moves 240 of the 276 type 2-2-3 triangles
U3(5):S3
31+4:4S6
24.A8
PSL(3,4):(2 × S3)
2 × M12 – centralizer of involution class 2B (trace 0), which moves 264 of the 276 type 2-2-3 triangles
[210.33]
S3 × PSL(2,8):3 - normalizer of 3-subgroup generated by class 3C (trace 0) element
A4 × S5
Conjugacy classes
Traces of matrices in a standard 24-dimensional representation of Co3 are shown. The names of conjugacy classes are taken from the Atlas of Finite Group Representations.
The cycle structures listed act on the 276 2-2-3 triangles that share the fixed type 3 side.
Generalized Monstrous Moonshine
In analogy to monstrous moonshine for the monster M, for Co3, the relevant McKay-Thompson series is where one can set the constant term a(0) = 24 (),
and η(τ) is the Dedekind eta function.
References
Reprinted in
External links
MathWorld: Conway Groups
Atlas of Finite Group Representations: Co3 version 2
Atlas of Finite Group Representations: Co3 version 3
Sporadic groups
John Horton Conway |
https://en.wikipedia.org/wiki/Conway%20group%20Co1 | {{DISPLAYTITLE:Conway group Co1}}
In the area of modern algebra known as group theory, the Conway group Co1 is a sporadic simple group of order
221395472111323
= 4157776806543360000
≈ 4.
History and properties
Co1 is one of the 26 sporadic groups and was discovered by John Horton Conway in 1968. It is the largest of the three sporadic Conway groups and can be obtained as the quotient of Co0 (group of automorphisms of the Leech lattice Λ that fix the origin) by its center, which consists of the scalar matrices ±1. It also appears at the top of the automorphism group of the even 26-dimensional unimodular lattice II25,1. Some rather cryptic comments in Witt's collected works suggest that he found the Leech lattice and possibly the order of its automorphism group in unpublished work in 1940.
The outer automorphism group is trivial and the Schur multiplier has order 2.
Involutions
Co0 has 4 conjugacy classes of involutions; these collapse to 2 in Co1, but there are 4-elements in Co0 that correspond to a third class of involutions in Co1.
An image of a dodecad has a centralizer of type 211:M12:2, which is contained in a maximal subgroup of type 211:M24.
An image of an octad or 16-set has a centralizer of the form 21+8.O8+(2), a maximal subgroup.
Representations
The smallest faithful permutation representation of Co1 is on the 98280 pairs {v,–v} of norm 4 vectors.
There is a matrix representation of dimension 24 over the field .
The centralizer of an involution of type 2B in the monster group is of the form 21+24Co1.
The Dynkin diagram of the even Lorentzian unimodular lattice II1,25 is isometric to the (affine) Leech lattice Λ, so the group of diagram automorphisms is split extension Λ,Co0 of affine isometries of the Leech lattice.
Maximal subgroups
found the 22 conjugacy classes of maximal subgroups of Co1, though there were some errors in this list, corrected by .
Co2
3.Suz:2 The lift to Aut(Λ) = Co0 fixes a complex structure or changes it to the complex conjugate structure. Also, top of Suzuki chain.
211:M24 Image of monomial subgroup from Aut(Λ), that subgroup stabilizing the standard frame of 48 vectors of form (±8,023) .
Co3
21+8.O8+(2) centralizer of involution class 2A (image of octad from Aut(Λ))
Fi21:S3 ≈ U6(2):S3 The lift to Aut(Λ) is the symmetry group of a coplanar hexagon of 6 type 2 points.
(A4 × G2(4)):2 in Suzuki chain.
22+12:(A8 × S3)
24+12.(S3 × 3.S6)
32.U4(3).D8
36:2.M12 (holomorph of ternary Golay code)
(A5 × J2):2 in Suzuki chain
31+4:2.PSp4(3).2
(A6 × U3(3)).2 in Suzuki chain
33+4:2.(S4 × S4)
A9 × S3 in Suzuki chain
(A7 × L2(7)):2 in Suzuki chain
(D10 × (A5 × A5).2).2
51+2:GL2(5)
53:(4 × A5).2
72:(3 × 2.S4)
52:2A5
References
Reprinted in
External links
MathWorld: Conway Groups
Atlas of Finite Group Representations: Co1 version 2
Atlas of Finite Group Representations: Co1 version 3
Sporadic groups
John Horton Conway |
https://en.wikipedia.org/wiki/1955%E2%80%9356%20Rochester%20Royals%20season | The 1955–56 NBA season was the Royals eighth season in the NBA.
Regular season
Season standings
x – clinched playoff spot
Record vs. opponents
Game log
Player statistics
Awards and records
Maurice Stokes, NBA Rookie of the Year Award
Maurice Stokes, All-NBA Second Team
References
Sacramento Kings seasons
Rochester
Rochester Royals
Rochester Royals |
https://en.wikipedia.org/wiki/Minimal-entropy%20martingale%20measure | In probability theory, the minimal-entropy martingale measure (MEMM) is the risk-neutral probability measure that minimises the entropy difference between the objective probability measure, , and the risk-neutral measure, . In incomplete markets, this is one way of choosing a risk-neutral measure (from the infinite number available) so as to still maintain the no-arbitrage conditions.
The MEMM has the advantage that the measure will always be equivalent to the measure by construction. Another common choice of equivalent martingale measure is the minimal martingale measure, which minimises the variance of the equivalent martingale. For certain situations, the resultant measure will not be equivalent to .
In a finite probability model, for objective probabilities and risk-neutral probabilities then one must minimise the Kullback–Leibler divergence subject to the requirement that the expected return is , where is the risk-free rate.
References
M. Frittelli, Minimal Entropy Criterion for Pricing in One Period Incomplete Markets, Working Paper. University of Brescia, Italy (1995).
Martingale theory
Game theory |
https://en.wikipedia.org/wiki/Eddie%20Denis | Eddie Denis (born 14 October 1970) is an Australian water polo player who competed in the 2000 Summer Olympics.
See also
Australia men's Olympic water polo team records and statistics
List of men's Olympic water polo tournament goalkeepers
References
External links
1970 births
Living people
Australian male water polo players
Water polo goalkeepers
Olympic water polo players for Australia
Water polo players at the 2000 Summer Olympics
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Guy%20Newman%20%28water%20polo%29 | Guy Newman (born 25 March 1969) is an Australian former water polo player who competed in the 1992 Summer Olympics.
See also
Australia men's Olympic water polo team records and statistics
List of men's Olympic water polo tournament goalkeepers
References
External links
1969 births
Living people
Australian male water polo players
Water polo goalkeepers
Olympic water polo players for Australia
Water polo players at the 1992 Summer Olympics
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Andrew%20Steward | Andrew Steward (born 10 December 1954) is an Australian former water polo player who competed in the 1980 Summer Olympics.
See also
Australia men's Olympic water polo team records and statistics
List of men's Olympic water polo tournament goalkeepers
References
External links
1954 births
Living people
Australian male water polo players
Water polo goalkeepers
Olympic water polo players for Australia
Water polo players at the 1980 Summer Olympics
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Connective%20constant | In mathematics, the connective constant is a numerical quantity associated with self-avoiding walks on a lattice. It is studied in connection with the notion of universality in two-dimensional statistical physics models. While the connective constant depends on the choice of lattice so itself is not universal (similarly to other lattice-dependent quantities such as the critical probability threshold for percolation), it is nonetheless an important quantity that appears in conjectures for universal laws. Furthermore, the mathematical techniques used to understand the connective constant, for example in the recent rigorous proof by Duminil-Copin and Smirnov that the connective constant of the hexagonal lattice has the precise value , may provide clues to a possible approach for attacking other important open problems in the study of self-avoiding walks, notably the conjecture that self-avoiding walks converge in the scaling limit to the Schramm–Loewner evolution.
Definition
The connective constant is defined as follows. Let denote the number of n-step self-avoiding walks starting from a fixed origin point in the lattice. Since every n + m step self avoiding walk can be decomposed into an n-step self-avoiding walk and an m-step self-avoiding walk, it follows that . Then by applying Fekete's lemma to the logarithm of the above relation, the limit can be shown to exist. This number is called the connective constant, and clearly depends on the particular lattice chosen for the walk since does. The value of is precisely known only for two lattices, see below. For other lattices, has only been approximated numerically. It is conjectured that as n goes to infinity, where and , the critical amplitude, depend on the lattice, and the exponent , which is believed to be universal and dependent on the dimension of the lattice, is conjectured to be .
Known values
These values are taken from the 1998 Jensen–Guttmann paper and a more recent paper by Jacobsen, Scullard and Guttmann.
The connective constant of the lattice, since each step on the hexagonal lattice corresponds to either two or three steps in it, can be expressed exactly as the largest real root of the polynomial
given the exact expression for the hexagonal lattice connective constant. More information about these lattices can be found in the percolation threshold article.
Duminil-Copin–Smirnov proof
In 2010, Hugo Duminil-Copin and Stanislav Smirnov published the first rigorous proof of the fact that for the hexagonal lattice. This had been conjectured by Nienhuis in 1982 as part of a larger study of O(n) models using renormalization techniques. The rigorous proof of this fact came from a program of applying tools from complex analysis to discrete probabilistic models that has also produced impressive results about the Ising model among others. The argument relies on the existence of a parafermionic observable that satisfies half of the discrete Cauchy–Riemann equations for the hex |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20NK%20Osijek%20season | This article shows statistics of individual players for the Osijek football club. It also lists all matches that Osijek played in the 2012–13 season.
First-team squad
Competitions
Overall
Prva HNL
Results summary
Results by round
Matches
Prva HNL
Europa League
Croatian Cup
Sources: Prva-HNL.hr
Player seasonal records
Competitive matches only. Updated to games played 20 April 2013.
Top scorers
Source: Competitive matches
Appearances and goals
Sources: Prva-HNL.hr, UEFA.com
References
2012-13
Croatian football clubs 2012–13 season |
https://en.wikipedia.org/wiki/Mathieu%20groupoid | In mathematics, the Mathieu groupoid M13 is a groupoid acting on 13 points such that the stabilizer of each point is the Mathieu group M12. It was introduced by and studied in detail by .
Construction
The projective plane of order 3 has 13 points and 13 lines, each containing 4 points. The Mathieu groupoid can be visualized as a sliding block puzzle by placing 12 counters on 12 of the 13 points of the projective plane. A move consists of moving a counter from any point x to the empty point y, then exchanging the 2 other counters on the line containing x and y. The Mathieu groupoid consists of the permutations that can be obtained by composing several moves.
This is not a group because two operations A and B can only be composed if the empty point after carrying out A is the empty point at the beginning of B. It is in fact a groupoid (a category such that every morphism is invertible) whose 13 objects are the 13 points, and whose morphisms from x to y are the operations taking the empty point from x to y. The morphisms fixing the empty point form a group isomorphic to the Mathieu group M12 with 12×11×10×9×8 elements.
References
External links
The Mathieu groupoid
Sporadic groups
John Horton Conway |
https://en.wikipedia.org/wiki/List%20of%20unitary%20authorities%20of%20England | This is a list of unitary authorities of England ordered by population.
Figures are mid-year estimates for from the Office for National Statistics.
Areas from UK Standard Area Measurements
The list does not include North Northamptonshire and West Northamptonshire unitary authorities, created in 2021, for which statistics are not yet available.
In July 2021 plans were announced to create unitary authorities in Cumbria, North Yorkshire and Somerset in 2023. The North Yorkshire authority will be the largest in England by population (618,054 in 2019) and by far the largest by area (8,053 km²).
References
Unitary authorities
Unitary authorities
Unitary authorities of England, List of
Population |
https://en.wikipedia.org/wiki/Bauerian%20extension | In mathematics, in the field of algebraic number theory, a Bauerian extension is a field extension of an algebraic number field which is characterized by the prime ideals with inertial degree one in the extension.
For a finite degree extension L/K of an algebraic number field K we define P(L/K) to be the set of primes p of K which have a factor P with inertial degree one (that is, the residue field of P has the same order as the residue field of p).
Bauer's theorem states that if M/K is a finite degree Galois extension, then P(M/K) ⊇ P(L/K) if and only if M ⊆ L. In particular, finite degree Galois extensions N of K are characterised by set of prime ideals which split completely in N.
An extension F/K is Bauerian if it obeys Bauer's theorem: that is, for every finite extension L of K, we have P(F/K) ⊇ P(L/K) if and only if L contains a subfield K-isomorphic to F.
All field extensions of degree at most 4 over Q are Bauerian.
An example of a non-Bauerian extension is the Galois extension of Q by the roots of 2x5 − 32x + 1, which has Galois group S5.
See also
Splitting of prime ideals in Galois extensions
References
Theorems in algebraic number theory |
https://en.wikipedia.org/wiki/Mathspy | Mathspy is a 1988 BBC Maths Educational programme.
Episodes
Needle and Thread.
More Waste, Less Speed.
Play Your Cards.
Solid Clues.
To Make the Pattern Fit.
1 Across, 1 Down.
F2 to B4.
Locks and Box.
Seven Times Able.
The Fourth Term.
Final challenge.
Cast
Notes
English-language television shows |
https://en.wikipedia.org/wiki/Geological%20compass | There are a number of different specialized magnetic compasses used by geologists to measure orientation of geological structures, as they map in the field, to analyze and document the geometry of bedding planes, joints, and/or metamorphic foliations and lineations. In this aspect the most common device used to date is the analogue compass.
Classic geological compasses
Classic geological compasses that are of practical use combine two functions, direction finding and navigation (especially in remote areas), and the ability to measure strike and dip of bedding surfaces and/or metamorphic foliation planes. Structural geologists (i.e. those concerned with geometry and the pattern of relative movement) also have a need to measure the plunge and plunge direction of lineations.
Compasses in common use include the Brunton compass and the Silva compass.
Modern geological compasses
The concept of modern geological compass was developed by Eberhard Clar of the University of Vienna during his work as structural geologist, which he published it in 1954. An advantage of his concept is that strike and dip is measured in one step, using the vertical circle for dip angle and the compass for the strike direction.
The first implementation was by the VEB Freiberger Präzisionsmechanik in Freiberg, Germany. The details of the design were made in a close cooperation with the Freiberg University of Mining and Technology. In 2016 Brunton Inc. introduced the Axis Pocket Transit which, for the first time, offered simultaneous measurements of both strike and dip and trend and plunge in a variety of configurations. It featured an unconventional lid design that swung a full 360 degrees in both directions and two axes that allow precise measurement of vertical and horizontal angles on all configurations of bedding surfaces.
Usage
Geological compasses are distinctive because of the anti-clockwise direction of the numbers on the compass dial. This is because the compass is used to determine dip and dip-direction of surfaces (foliations), and plunge and plunge-direction of lines (lineations). To use the compass one aligns the lid of the compass with the orientation of the surface to be measured (to obtain dip and dip direction), or the edge of the lid of the compass with the orientation of the line (to obtain plunge and plunge direction). The compass must be twisted so that the base of the compass becomes horizontal, as accomplished using the spirit level incorporated in it. The needle of the compass is then freed by using the side button, and allowed to spin until the damping action slows its movement, and then stabilises. The side button is released and the needle is then firmly held in place, allowing the user thereafter to conveniently read the orientation measured. One first reads the scale that shows the angle subtended by the lid of the compass, and then depending on the colour shown (red or black) the end of the compass needle with the corresponding colour. D |
https://en.wikipedia.org/wiki/Wythoff%20array | In mathematics, the Wythoff array is an infinite matrix of integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff. Every positive integer occurs exactly once in the array, and every integer sequence defined by the Fibonacci recurrence can be derived by shifting a row of the array.
The Wythoff array was first defined by using Wythoff pairs, the coordinates of winning positions in Wythoff's game. It can also be defined using Fibonacci numbers and Zeckendorf's theorem, or directly from the golden ratio and the recurrence relation defining the Fibonacci numbers.
Values
The Wythoff array has the values
.
Equivalent definitions
Inspired by a similar Stolarsky array previously defined by , defined the Wythoff array as follows. Let denote the golden ratio; then the th winning position in Wythoff's game is given by the pair of positive integers , where the numbers on the left and right sides of the pair define two complementary Beatty sequences that together include each positive integer exactly once. Morrison defines the first two numbers in row of the array to be the Wythoff pair given by the equation , and where the remaining numbers in each row are determined by the Fibonacci recurrence relation. That is, if denotes the entry in row and column of the array, then
,
, and
for .
The Zeckendorf representation of any positive integer is a representation as a sum of distinct Fibonacci numbers, no two of which are consecutive in the Fibonacci sequence. As describes, the numbers within each row of the array have Zeckendorf representation that differ by a shift operation from each other, and the numbers within each column have Zeckendorf representations that all use the same smallest Fibonacci number. In particular the entry of the array is the th smallest number whose Zeckendorf representation begins with the th Fibonacci number.
Properties
Each Wythoff pair occurs exactly once in the Wythoff array, as a consecutive pair of numbers in the same row, with an odd index for the first number and an even index for the second. Because each positive integer occurs in exactly one Wythoff pair, each positive integer occurs exactly once in the array .
Every sequence of positive integers satisfying the Fibonacci recurrence occurs, shifted by at most finitely many positions, in the Wythoff array. In particular, the Fibonacci sequence itself is the first row, and the sequence of Lucas numbers appears in shifted form in the second row .
References
.
.
.
External links
Triangles of numbers
Fibonacci numbers |
https://en.wikipedia.org/wiki/The%20Universal%20Book%20of%20Mathematics | The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes (2004) is a bestselling book by British author David Darling.
Summary
The book is presented in a dictionary format. The book is divided into headwords, which, as the title suggests, run from Abracadabra to Zeno's paradoxes.
The book also provides relevant diagrams and illustrations.
Errors
The first edition of the book had several errors which were fixed in later editions. Several famous scientists have sent in corrections to the author of the book. These include Warren Johnson and Freeman Dyson.
Reception
The book has been praised by BoingBoing and British newspaper The Independent.
Problems and Puzzles mentioned in the book have been discussed and debated several times by several major mathematicians.
See also
David Darling (astronomer)
Mathematics
References
External links
boingboing.com
math.com
google.com
2004 non-fiction books
Books about mathematics |
https://en.wikipedia.org/wiki/Preradical | In mathematics, a preradical is a subfunctor of the identity functor in the category of left modules over a ring with identity. The class of all preradicals over R-mod is denoted by R-pr. There is a natural order in R-pr given by, for any two preradicals and , , if for any left R-module M, . With this order R-pr becomes a big lattice.
References
Stenstrom, Bo Rings of Quotients: An Introduction To Methods Of Ring Theory – Chapter 6, Springer,
Bican, L., Kepka, T. and Nemec, P. Rings, Modules, and Preradicals, Lecture Notes in Pure and Applied Mathematics, M. Dekker, 1982,
Functors |
https://en.wikipedia.org/wiki/Fundamental%20increment%20lemma | In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative of a function at a point :
The lemma asserts that the existence of this derivative implies the existence of a function such that
for sufficiently small but non-zero . For a proof, it suffices to define
and verify this meets the requirements.
The lemma says, at least when is sufficiently close to zero, that the difference quotient
can be written as the derivative f plus an error term that vanishes at .
I.e. one has,
Differentiability in higher dimensions
In that the existence of uniquely characterises the number , the fundamental increment lemma can be said to characterise the differentiability of single-variable functions. For this reason, a generalisation of the lemma can be used in the definition of differentiability in multivariable calculus. In particular, suppose f maps some subset of to . Then f is said to be differentiable at a if there is a linear function
and a function
such that
for non-zero h sufficiently close to 0. In this case, M is the unique derivative (or total derivative, to distinguish from the directional and partial derivatives) of f at a. Notably, M is given by the Jacobian matrix of f evaluated at a'.
We can write the above equation in terms of the partial derivatives as
See also
Generalizations of the derivative
References
Differential calculus |
https://en.wikipedia.org/wiki/Central%20line%20%28geometry%29 | In geometry, central lines are certain special straight lines that lie in the plane of a triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This special property is related to the concept of triangle center also. The concept of a central line was introduced by Clark Kimberling in a paper published in 1994.
Definition
Let be a plane triangle and let be the trilinear coordinates of an arbitrary point in the plane of triangle .
A straight line in the plane of whose equation in trilinear coordinates has the form
where the point with trilinear coordinates
is a triangle center, is a central line in the plane of relative to .
Central lines as trilinear polars
The geometric relation between a central line and its associated triangle center can be expressed using the concepts of trilinear polars and isogonal conjugates.
Let be a triangle center. The line whose equation is
is the trilinear polar of the triangle center . Also the point
is the isogonal conjugate of the triangle center .
Thus the central line given by the equation
is the trilinear polar of the isogonal conjugate of the triangle center
Construction of central lines
Let be any triangle center of .
Draw the lines and their reflections in the internal bisectors of the angles at the vertices respectively.
The reflected lines are concurrent and the point of concurrence is the isogonal conjugate of .
Let the cevians meet the opposite sidelines of at respectively. The triangle is the cevian triangle of .
The and the cevian triangle are in perspective and let be the axis of perspectivity of the two triangles. The line is the trilinear polar of the point . is the central line associated with the triangle center .
Some named central lines
Let be the th triangle center in Clark Kimberling's Encyclopedia of Triangle Centers. The central line associated with is denoted by . Some of the named central lines are given below.
Central line associated with X1, the incenter: Antiorthic axis
The central line associated with the incenter (also denoted by ) is
This line is the antiorthic axis of .
The isogonal conjugate of the incenter of is the incenter itself. So the antiorthic axis, which is the central line associated with the incenter, is the axis of perspectivity of and its incentral triangle (the cevian triangle of the incenter of ).
The antiorthic axis of is the axis of perspectivity of and the excentral triangle of .
The triangle whose sidelines are externally tangent to the excircles of is the extangents triangle of . and its extangents triangle are in perspective and the axis of perspectivity is the antiorthic axis of .
Central line associated with X2, the centroid: Lemoine axis
The trilinear coordinates of the centroid (also denoted by ) of are:
So the central line associated with the centroid is the line whose trilinear equation is
This line |
https://en.wikipedia.org/wiki/Singular%20integral%20operators%20of%20convolution%20type | In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rn and Tn through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on L2 is evident because the Fourier transform converts them into multiplication operators. Continuity on Lp spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed by a number of authors to give general criteria for continuity on Lp spaces. This article explains the theory for the classical operators and sketches the subsequent general theory.
L2 theory
Hilbert transform on the circle
The theory for L2 functions is particularly simple on the circle. If f ∈ L2(T), then it has a Fourier series expansion
Hardy space H2(T) consists of the functions for which the negative coefficients vanish, an = 0 for n < 0. These are precisely the square-integrable functions that arise as boundary values of holomorphic functions in the open unit disk. Indeed, f is the boundary value of the function
in the sense that the functions
defined by the restriction of F to the concentric circles |z| = r, satisfy
The orthogonal projection P of L2(T) onto H2(T) is called the Szegő projection. It is a bounded operator on L2(T) with operator norm 1. By Cauchy's theorem
Thus
When r = 1, the integrand on the right-hand side has a singularity at θ = 0. The truncated Hilbert transform is defined by
where δ = |1 – eiε|. Since it is defined as convolution with a bounded function, it is a bounded operator on L2(T). Now
If f is a polynomial in z then
By Cauchy's theorem the right-hand side tends to 0 uniformly as ε, and hence δ, tends to 0. So
uniformly for polynomials. On the other hand, if u(z) = z it is immediate that
Thus if f is a polynomial in z−1 without constant term
uniformly.
Define the Hilbert transform on the circle by
Thus if f is a trigonometric polynomial
uniformly.
It follows that if f is any L2 function
in the L2 norm.
This is an immediate consequence of the result for trigonometric polynomials once it is established that the operators Hε are uniformly bounded in operator norm. But on [–π,π]
The first term is bounded on the whole of [–π,π], so it suffices to show that the convolution operators Sε defined by
are uniformly bounded. With respect to the orthonormal basis einθ convolution operators are diagonal and their operator norms are given by taking the supremum of the moduli of the Fo |
https://en.wikipedia.org/wiki/CRC%20Concise%20Encyclopedia%20of%20Mathematics | CRC Concise Encyclopedia of Mathematics is a bestselling book by American author Eric W. Weisstein.
Summary
The book is presented in a dictionary format. The book is divided into headwords. The book also provides relevant diagrams and illustrations.
Lawsuits
The book became the subject of a lawsuit between CRC Press and Eric W. Weisstein. The CRC Press claimed Weisstein's website MathWorld violated the copyright on the CRC Concise Encyclopedia of Mathematics. During the dispute, a court order shut down MathWorld for over a year starting October 23, 2000. According to Eric Weisstein's personal site, he restarted MathWorld on November 6, 2001. Wolfram Research, Stephen Wolfram, and Eric Weisstein settled with the CRC Press for an undisclosed financial award and several benefits. Among these benefits are the legal rights to reproduce MathWorld in book format again.
Reception
The book has consistently received good reviews.
Editions
1st edition, CRC Press, 1999,
2nd edition, CRC Press, 2002,
3rd edition, CRC Press, 2005,
References
External links
wolfram.com
crcpress.com
crcpress.com
crcnetbase.com
google.com
2002 non-fiction books
Books about mathematics
Encyclopedias of mathematics
CRC Press books |
https://en.wikipedia.org/wiki/Umbral%20moonshine | In mathematics, umbral moonshine is a mysterious connection between Niemeier lattices and Ramanujan's mock theta functions. It is a generalization of the Mathieu moonshine phenomenon connecting representations of the Mathieu group M24 with K3 surfaces.
Mathieu moonshine
The prehistory of Mathieu moonshine starts with a theorem of Mukai, asserting that any group of symplectic automorphisms of a K3 surface embeds in the Mathieu group M23. The moonshine observation arose from physical considerations: any K3 sigma-model conformal field theory has an action of the N=(4,4) superconformal algebra, arising from a hyperkähler structure. When computed the first few terms of the decomposition of the elliptic genus of a K3 CFT into characters of the N=(4,4) superconformal algebra, they found that the multiplicities matched well with simple combinations of representations of M24. However, by the Mukai–Kondo classification, there is no faithful action of this group on any K3 surface by symplectic automorphisms, and by work of Gaberdiel–Hohenegger–Volpato, there is no faithful action on any K3 CFT, so the appearance of an action on the underlying Hilbert space is still a mystery.
Eguchi and Hikami showed that the N=(4,4) multiplicities are mock modular forms, and Miranda Cheng suggested that characters of elements of M24 should also be mock modular forms. This suggestion became the Mathieu Moonshine conjecture, asserting that the virtual representation of N=(4,4) given by the K3 elliptic genus is an infinite dimensional graded representation of M24 with non-negative multiplicities in the massive sector, and that the characters are mock modular forms. In 2012, Terry Gannon proved that the representation of M24 exists.
Umbral moonshine
In 2012, amassed numerical evidence of an extension of Mathieu moonshine, where families of mock modular forms were attached to divisors of 24. After some group-theoretic discussion with Glauberman, found that this earlier extension was a special case (the A-series) of a more natural encoding by Niemeier lattices. For each Niemeier root system X, with corresponding lattice LX, they defined an umbral group GX, given by the quotient of the automorphism group of LX by the subgroup of reflections- these are also known as the stabilizers of deep holes in the Leech lattice. They conjectured that for each X, there is an infinite dimensional graded representation KX of GX, such that the characters of elements are given by a list of vector-valued mock modular forms that they computed. The candidate forms satisfy minimality properties quite similar to the genus-zero condition for Monstrous moonshine. These minimality properties imply the mock modular forms are uniquely determined by their shadows, which are vector-valued theta series constructed from the root system. The special case where X is the A124 root system yields precisely Mathieu Moonshine. The umbral moonshine conjecture has been proved in .
The name of umb |
https://en.wikipedia.org/wiki/Transgression%20map | In algebraic topology, a transgression map is a way to transfer cohomology classes.
It occurs, for example in the inflation-restriction exact sequence in group cohomology, and in integration in fibers. It also naturally arises in many spectral sequences; see spectral sequence#Edge maps and transgressions.
Inflation-restriction exact sequence
The transgression map appears in the inflation-restriction exact sequence, an exact sequence occurring in group cohomology. Let G be a group, N a normal subgroup, and A an abelian group which is equipped with an action of G, i.e., a homomorphism from G to the automorphism group of A. The quotient group acts on
Then the inflation-restriction exact sequence is:
The transgression map is the map .
Transgression is defined for general ,
,
only if for .
References
External links
Homological algebra
Algebraic topology |
https://en.wikipedia.org/wiki/John%20Williamson%20%28mathematician%29 | John Williamson (23 May 1901 – 1949) was a Scottish mathematician who worked in the fields of algebra, invariant theory, and linear algebra. Among other contributions, he is known for the Williamson construction of Hadamard matrices. Williamson graduated from the University of Edinburgh with first-class honours in 1922. Awarded a Commonwealth Fellowship in 1925, he studied at the University of Chicago under the direction of L. E. Dickson and E. H. Moore, receiving the Ph.D. in 1927. He held a Lectureship in Mathematics at the University of St Andrews and an Associate Professorship in Mathematics at Johns Hopkins University.
See also
Williamson conjecture
References
External links
1901 births
1949 deaths
Alumni of the University of Edinburgh
Johns Hopkins University faculty
University of Chicago fellows
Academics of the University of St Andrews
Matrices
20th-century Scottish mathematicians |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20F.C.%20Copenhagen%20season | This article shows statistics of individual players for the football club F.C. Copenhagen. It also lists all matches that F.C. Copenhagen played in the 2012–13 season.
Players
Squad information
This section show the squad as currently, considering all players who are confirmedly moved in and out (see section Players in / out).
Squad stats
Players in / out
In
Out
Club
Coaching staff
Kit
|
|
|
|
|
Other information
Competitions
Overall
Danish Superliga
Classification
Results summary
Results by round
UEFA Champions League
Third qualifying round
Play-off round
Results summary
UEFA Europa League
Group E
Classification
Results by round
Results summary
Matches
Competitive
References
External links
F.C. Copenhagen official website
2012–13
Danish football clubs 2012–13 season
2012–13 UEFA Europa League participants seasons
2012–13 UEFA Champions League participants seasons |
https://en.wikipedia.org/wiki/1935%E2%80%9336%20Hong%20Kong%20Second%20Division%20League | Statistics of Hong Kong Second Division League in the 1935/1936 season.
Overview
Royal Navy won the championship.
League table
References
RSSSF
2
Hong Kong Second Division League seasons |
https://en.wikipedia.org/wiki/1935%E2%80%9336%20Hong%20Kong%20Third%20Division%20League | Statistics of Hong Kong Third Division League in the 1935/1936 season.
Overview
Eastern Lancashire Regiment won the championship.
League table
References
RSSSF
3
Hong Kong Third Division League seasons
1935–36 in Asian association football leagues |
https://en.wikipedia.org/wiki/Al-Bayuk | Al-Bayuk (, also spelled al-Buyuk) is a Palestinian village in the Rafah Governorate located south of Rafah in the southern Gaza Strip. According to the Palestinian Central Bureau of Statistics (PCBS), it had a population of 5,648 in 2006.
References
Villages in the Gaza Strip
Municipalities of the State of Palestine |
https://en.wikipedia.org/wiki/Forum%20of%20Mathematics | Forum of Mathematics, Pi and Forum of Mathematics, Sigma are open-access peer-reviewed journals for mathematics published under a creative commons license by Cambridge University Press.
The founding managing editor was Rob Kirby. He was succeeded by Robert Guralnick, who is currently the managing editor of both journals.
Forum of Mathematics, Pi publishes articles of interest to a wide audience of mathematicians, while Forum of Mathematics, Sigma is intended for more specialized articles, with clusters of editors in different areas of mathematics.
Abstracting and indexing
Both journals are abstracted and indexed in Science Citation Index Expanded, MathSciNet, and Scopus.
References
External links
A new open-access venture from Cambridge University Press, Tim Gowers, 2 July 2012
Forum of Mathematics, Pi and Forum of Mathematics, Sigma, Terry Tao, 2 July 2012
The Forum of Mathematics, blessing or curse?, Peter Krautzberger, 11 November 2012
Cambridge University Press academic journals
Creative Commons Attribution-licensed journals
English-language journals
Mathematics education in the United Kingdom
Mathematics journals |
https://en.wikipedia.org/wiki/Cyrus%20Derman | Cyrus Derman (July 16, 1925 – April 27, 2011) was an American mathematician and amateur musician who did research in Markov decision process, stochastic processes, operations research, statistics and a variety of other fields.
Early life
Derman grew up in Collingdale Pennsylvania. He was the son of a grocery store owner who came to the US from Lithuania. As a young boy he was often invited to at a Philadelphia radio show for talented children. Although his initial dream was to become a concert violinist, in the end he chose to study mathematics. Indeed, after he finished his undergraduate degree at the University of Pennsylvania in music and mathematics, he went on to Columbia University for his graduate work in mathematical statistics. At Columbia he was privileged to work with many of the important US statisticians and probabilists of that time.
Career
After taking his Ph.D., Derman joined the Department of Industrial Engineering at Columbia University in 1954 as an instructor in Operations Research. He rose to the rank of Professor of Operations Research in 1965 and retired in 1992. He was a key figure in operations research at Columbia during his 38 years there. He was instrumental in the formation of the Columbia Industrial Engineering and Operations Research Department in 1977, which rose to be one of the top departments in that field. In addition, professor Derman held visiting appointments and taught at Syracuse University, Stanford University, University of California Berkeley, University of California Davis, Imperial College (London) and The Technion (Israel). Dr. Derman was an excellent teacher at all levels who managed to make difficult ideas easy for students to learn. He was also a dedicated and helpful advisor to 17 Ph.D. students and he has 260 descendants listed at the Mathematics Genealogy Project.
Dr. Derman did fundamental research in Markov decision processes, i.e., sequential decisions under uncertainty, including an important book on the subject. He also did significant work in optimal maintenance, stochastic assignment, surveillance, quality control, clinical trials, queueing and inventory depletion management among others. For his sustained fundamental contributions to theory in operations research and the management sciences, he was a co-recipient of the 2002 John von Neumann Theory Prize of the Institute for Operations Research and the Management Science. His significant contributions to probability and statistics which were recognized by his election as a Fellow of the Institute of Mathematical Statistics and the American Statistical Association.
Selected writings
Books by Cyrus Derman:
"Statistical aspects of quality control", with Sheldon M. Ross, 1997.
"Probability models and applications", with , 1994.
"A guide to probability theory and application", with L. Gleser and I. Olkin, 1973,
"Finite state Markovian decision processes", 1970.
"Probability and statistical inference for engineers: a first |
https://en.wikipedia.org/wiki/Joel%20Hass | Joel Hass is an American mathematician and a professor of mathematics and at the University of California, Davis. His work focuses on geometric and topological problems in dimension 3.
Biography
Hass received his Ph.D. from the University of California, Berkeley in 1981 under the supervision of Robion Kirby. He joined the Davis faculty in 1988.
In 2012 he became a fellow of the American Mathematical Society. From 2010 to 2014 he served as the chair of the UC Davis mathematics department.
Research contributions
Hass is known for proving the equal-volume special case of the double bubble conjecture, for proving that the unknotting problem is in NP, and for giving an exponential bound on the number of Reidemeister moves needed to reduce the unknot to a circle.
Selected publications
Research papers
.
.
.
.
Books
.
.
2004: Student Solutions Manual, Maurice D. Weir, Joel Hass, George B. Thomas, Frank R Giordano
References
External links
Home page at UC Davis
Google scholar profile
Year of birth missing (living people)
Living people
20th-century American mathematicians
University of California, Berkeley alumni
University of California, Davis faculty
Fellows of the American Mathematical Society
Topologists
21st-century American mathematicians |
https://en.wikipedia.org/wiki/Hilary%20Shuard | Hilary Bertha Shuard CBE (14 November 192824 December 1992) was "an internationally known expert on mathematics in primary schools".
She was deputy principal of Homerton College, Cambridge, England and was president of the Mathematical Association for 1985–1986.
References
1928 births
1992 deaths
Commanders of the Order of the British Empire
Fellows of Homerton College, Cambridge
Mathematics educators
Schoolteachers from Cheshire
British women mathematicians
20th-century English mathematicians
20th-century women mathematicians
People from Cheshire |
https://en.wikipedia.org/wiki/James%20Milne%20%28mathematician%29 | James S. Milne (born 10 October 1942 in Invercargill, New Zealand) is a New Zealand mathematician working in arithmetic geometry.
Life
Milne attended the High School in Invercargill in New Zealand until 1959, and then studied at the University of Otago in Dunedin (B.A. 1964) and Harvard University (Masters 1966, Ph.D. 1967 under John Tate). From then to 1969 he was a lecturer at University College London. After that he was at the University of Michigan, as Assistant Professor (1969–1972), Associate Professor (1972–1977), Professor (1977–2000), and Professor Emeritus (since 2000). He has also been a visiting professor at King's College London, at the Institut des hautes études scientifiques in Paris (1975, 1978), at the Mathematical Sciences Research Institute in Berkeley, California (1986–87), and the Institute for Advanced Study in Princeton, New Jersey (1976–77, 1982, 1988).
In his dissertation, entitled "The conjectures of Birch and Swinnerton-Dyer for constant abelian varieties over function fields," he proved the conjecture of Birch and Swinnerton–Dyer for constant abelian varieties over function fields in one variable over a finite field. He also gave the first examples of nonzero abelian varieties with finite Tate–Shafarevich group. He went on to study Shimura varieties (certain hermitian symmetric spaces, low-dimensional examples being modular curves) and motives.
His students include Piotr Blass, Michael Bester, Matthew DeLong, Pierre Giguere, William Hawkins Jr, Matthias Pfau, Victor Scharaschkin, Stefan Treatman, Anthony Vazzana, and Wafa Wei.
Milne is also an avid mountain climber.
Writings
Abelian Varieties, Jacobian Varieties, in Arithmetic Geometry Proc. Conference Storrs 1984, Springer 1986
With Pierre Deligne, Arthur Ogus, Kuang-yen Shih, Hodge Cycles, Motives and Shimura Varieties, Springer Verlag, Lecture Notes in Mathematics vol. 900, 1982 (therein by Deligne: Tannakian Categories)
Arithmetic Duality Theorems, Academic Press, Perspectives in Mathematics, 1986
Editor with Laurent Clozel, Automorphic Forms, Shimura Varieties and L-Functions, 2 volumes, Elsevier 1988 (Conference University of Michigan, 1988)
Elliptic Curves, BookSurge Publishing 2006
Shimura Varieties and Motives in Jannsen, Kleiman, Serre (editor) motif, Proc. Symp. Pure vol. 55 Math, AMS, 1994
References
The original article was a Google translation of the corresponding article in German Wikipedia.
External links
Personal website
New Zealand mathematicians
Living people
1942 births
University of Michigan faculty
Harvard University alumni
University of Otago alumni
People from Invercargill
Arithmetic geometers |
https://en.wikipedia.org/wiki/Ofer%20Gabber | Ofer Gabber (עופר גאבר; born May 16, 1958) is a mathematician working in algebraic geometry.
Life
In 1978 Gabber received a Ph.D. from Harvard University for the thesis Some theorems on Azumaya algebras, written under the supervision of Barry Mazur. Gabber has been at the Institut des Hautes Études Scientifiques in Bures-sur-Yvette in Paris since 1984 as a
CNRS senior researcher. He won the Erdős Prize in 1981 and the Prix Thérèse Gautier from the French Academy of Sciences in 2011. In 1981 Gabber with Victor Kac published a proof of a conjecture stated by Kac in 1968.
Books
With Lorenzo Ramero: Almost Ring Theory, Springer, Lecture Notes in Computer Science, vol 1800, 2003.
With Brian Conrad, Gopal Prasad: Pseudo-reductive Groups, Cambridge University Press, 2010; 2015, 2nd edition
See also
Gabber rigidity
Almost ring theory
t-structure
Theorem of absolute purity
References
Israeli mathematicians
Algebraic geometers
Living people
1958 births
Harvard University alumni
Erdős Prize recipients |
https://en.wikipedia.org/wiki/Adi%20Adilovi%C4%87 | Adi Adilović (born 20 February 1983) is a Bosnian retired goalkeeper and current goalkeeping coach at Ludogorets Razgrad.
Career statistics
Club
Honours
Player
Željezničar
Bosnian Cup: 2002–03
Interblock
Slovenian Cup: 2007–08
External links
1983 births
Living people
Footballers from Zenica
Men's association football goalkeepers
Bosnia and Herzegovina men's footballers
FK Željezničar Sarajevo players
NK Koprivnica players
NK Hrvatski Dragovoljac players
NK Travnik players
FK Sloboda Tuzla players
NK IB 1975 Ljubljana players
NK Ivančna Gorica players
FK Olimpik players
FK Sarajevo players
Panthrakikos F.C. players
NK Čelik Zenica players
Premier League of Bosnia and Herzegovina players
First Football League (Croatia) players
Slovenian PrvaLiga players
Super League Greece players
Bosnia and Herzegovina expatriate men's footballers
Expatriate men's footballers in Croatia
Bosnia and Herzegovina expatriate sportspeople in Croatia
Expatriate men's footballers in Slovenia
Bosnia and Herzegovina expatriate sportspeople in Slovenia
Expatriate men's footballers in Greece
Bosnia and Herzegovina expatriate sportspeople in Greece |
https://en.wikipedia.org/wiki/1999%E2%80%932000%20Galatasaray%20S.K.%20season | The 1999–2000 season was Galatasaray's 96th in existence and the 42nd consecutive season in the 1. Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Galatasaray completed a treble of the 1.Lig, Turkish Cup and UEFA Cup.
Club
Board of directors
Elected as of 25 March 2000
Facilities
Squad statistics
Players in / out
In
Out
1. Lig
Standings
Matches
Türkiye Kupası
Kick-off listed in local time (EET)
Third round
Fourth round
Quarter-final
Semi-final
Final
UEFA Champions League
Third qualifying round
Group stage
UEFA Cup
Third round
Fourth round
Quarter-final
Semi-final
Final
Friendlies
TSYD Kupası
Attendance
References
Galatasaray S.K. (football) seasons
Galatasaray S.K.
UEFA Europa League-winning seasons
Turkish football championship-winning seasons
1990s in Istanbul
2000s in Istanbul
Galatasaray Sports Club 1999–2000 season |
https://en.wikipedia.org/wiki/Strongly%20regular | In mathematics, strongly regular might refer to:
Strongly regular graph
Strongly regular ring, or "strongly von Neumann regular" ring |
https://en.wikipedia.org/wiki/Roberta%20Vinci%20career%20statistics | This is a list of the main career statistics of Italian professional tennis player Roberta Vinci.
Performance timelines
Only main-draw results in WTA Tour, Grand Slam tournaments, Fed Cup and Olympic Games are included in win–loss records.
Singles
Doubles
Significant finals
Grand Slam finals
Singles: 1 (runner-up)
Doubles: 8 (5 titles, 3 runner-ups)
WTA Premier Mandatory & 5 finals
Doubles: 12 (5 titles, 7 runner-ups)
WTA career finals
Singles: 15 (10 titles, 5 runner–ups)
Doubles: 43 (25 titles, 18 runner-ups)
ITF Circuit finals
Singles: 11 (9 titles, 2 runner–ups)
Doubles: 14 (9 titles, 5 runner–ups)
Junior Grand Slam finals
Doubles: 1 (title)
Record against other players
Record against top 10 players
Vinci's match record against players who have been ranked in the top 10, with those who have been ranked No. 1 in boldface
Dominika Cibulková 6–3
Daniela Hantuchová 4–3
Lucie Šafářová 4–3
Sara Errani 4–6
Flavia Pennetta 4–6
Ana Ivanovic 4–6
Petra Kvitová 3–4
Nadia Petrova 3–3
Svetlana Kuznetsova 3–4
Ekaterina Makarova 3–3
Jelena Janković 3–5
Ai Sugiyama 2–0
Karolína Plíšková 2–1
Angelique Kerber 2–3
Maria Kirilenko 2–3
Caroline Wozniacki 2–4
Simona Halep 2–5
Agnieszka Radwańska 2–8
Barbora Krejčíková 1–0
Eugenie Bouchard 1–0
Jelena Dokic 1–0
Jelena Dokic 1–0
Timea Bacsinszky 1–1
Marion Bartoli 1–1
Alicia Molik 1–1
Andrea Petkovic 1–1
Nicole Vaidišová 1–1
Samantha Stosur 1–2
Anastasia Myskina 1–2
Patty Schnyder 1–3
Francesca Schiavone 1–4
Carla Suárez Navarro 1–4
Serena Williams 1–4
Vera Zvonareva 1–5
Justine Henin 0–1
Garbiñe Muguruza 0-1
Elena Dementieva 0–2
Amélie Mauresmo 0–2
Maria Sharapova 0–3
Kim Clijsters 0–3
Victoria Azarenka 0–4
Li Na 0–4
Venus Williams 0–5
Dinara Safina 0–5
No. 1 wins
Top 10 wins
Longest winning streaks
9–match singles winning streak (2012)
21–match doubles winning streak (2012)
Team competition
2006, 2009, 2010 and 2013 Fed Cup
See also
Italy Fed Cup team
Italy at the 2004 Summer Olympics
Italy at the 2008 Summer Olympics
Sara Errani career statistics
References
External links
Tennis career statistics |
https://en.wikipedia.org/wiki/Linear%20arboricity | In graph theory, a branch of mathematics, the linear arboricity of an undirected graph is the smallest number of linear forests its edges can be partitioned into. Here, a linear forest is an acyclic graph with maximum degree two; that is, it is a disjoint union of path graphs. Linear arboricity is a variant of arboricity, the minimum number of forests into which the edges can be partitioned.
The linear arboricity of any graph of maximum degree is known to be at least and is conjectured to be at most . This conjecture would determine the linear arboricity exactly for graphs of odd degree, as in that case both expressions are equal. For graphs of even degree it would imply that the linear arboricity must be one of only two possible values, but determining the exact value among these two choices is NP-complete.
Relation to degree
The linear arboricity of a graph with maximum degree is always at least , because each linear forest can use only two of the edges at a maximum-degree vertex. The linear arboricity conjecture of is that this lower bound is also tight: according to their conjecture, every graph has linear arboricity at most . However, this remains unproven, with the best proven upper bound on the linear arboricity being somewhat larger,
for some constant due to Ferber, Fox and Jain.
In order for the linear arboricity of a graph to equal , must be even and each linear forest must have two edges incident to each vertex of degree . But at a vertex that is at the end of a path, the forest containing that path has only one incident edge, so the degree at that vertex cannot equal . Thus, a graph whose linear arboricity equals must have some vertices whose degree is less than maximum. In a regular graph, there are no such vertices, and the linear arboricity cannot equal . Therefore, for regular graphs, the linear arboricity conjecture implies that the linear arboricity is exactly .
Related problems
Linear arboricity is a variation of arboricity, the minimum number of forests that the edges of a graph can be partitioned into. Researchers have also studied linear -arboricity, a variant of linear arboricity in which each path in the linear forest can have at most edges.
Another related problem is Hamiltonian decomposition, the problem of decomposing a regular graph of even degree into exactly Hamiltonian cycles. A given graph has a Hamiltonian decomposition if and only if the subgraph formed by removing an arbitrary vertex from the graph has linear arboricity .
Computational complexity
Unlike arboricity, which can be determined in polynomial time, linear arboricity is NP-hard. Even recognizing the graphs of linear arboricity two is NP-complete. However, for cubic graphs and other graphs of maximum degree three, the linear arboricity is always two, and a decomposition into two linear forests can be found in linear time using an algorithm based on depth-first search.
References
Graph invariants |
https://en.wikipedia.org/wiki/Linear%20forest | In graph theory, a branch of mathematics, a linear forest is a kind of forest formed from the disjoint union of path graphs. It is an undirected graph with no cycles in which every vertex has degree at most two. Linear forests are the same thing as claw-free forests. They are the graphs whose Colin de Verdière graph invariant is at most 1.
The linear arboricity of a graph is the minimum number of linear forests into which the graph can be partitioned. For a graph of maximum degree , the linear arboricity is always at least , and it is conjectured that it is always at most .
A linear coloring of a graph is a proper graph coloring in which the induced subgraph formed by each two colors is a linear forest. The linear chromatic number of a graph is the smallest number of colors used by any linear coloring. The linear chromatic number is at most proportional to , and there exist graphs for which it is at least proportional to this quantity.
References
Trees (graph theory)
Graph families |
https://en.wikipedia.org/wiki/Lukas%20Th%C3%BCrauer | Lukas Thürauer (born 21 December 1987) is an Austrian footballer who plays for Kremser SC.
Career statistics
References
External links
Austrian men's footballers
Austrian Football Bundesliga players
FC Admira Wacker Mödling players
1987 births
Living people
SKN St. Pölten players
Kremser SC players
Men's association football midfielders
People from Krems an der Donau
Footballers from Lower Austria |
https://en.wikipedia.org/wiki/Sophie%20Morel | Sophie Morel (born 1979) is a French mathematician, specializing in number theory. She is a CNRS directrice de recherches in mathematics at École normale supérieure de Lyon. In 2012 she received one of the ten prizes of the European Mathematical Society.
Biography
In a 2011 interview, Morel credited a math magazine bought while in 9th grade as well as summer camps for developing her interest in mathematics and in a 2012 interview she mentioned being a keen distance runner.
She studied in Paris at the École Normale Supérieure, graduating in 1999. In 2005 she finished her Ph.D. at the University of Paris-Sud, under the supervision of Gérard Laumon. Her thesis made progress on the Langlands program.
After her Ph.D., she was a Clay Research Fellow between 2005 and 2011. In December 2009 she was appointed as a professor of mathematics at Harvard University, becoming the first woman in mathematics to be tenured there. From 2012 to 2020, she was a professor of mathematics in Princeton University, where she was also the Henry Burchard Fine Professor in 2015. Morel moved to École Normale supérieure de Lyon as an CNRS directrice de recherches in mathematics in 2020.
Recognition
She gave an invited talk at the International Congress of Mathematicians in 2010, in the "Number Theory" section.
In 2012 she received one of the prestigious European Mathematical Society Prize for young researchers, and in May 2013 she was announced as the winner of the inaugural 2014 AWM-Microsoft Research Prize in Algebra and Number Theory.
Selected publications
References
External links
20th-century French mathematicians
21st-century French mathematicians
Harvard University Department of Mathematics faculty
Harvard University faculty
Princeton University faculty
1979 births
Living people
Lycée Louis-le-Grand alumni
École Normale Supérieure alumni
Number theorists
French women mathematicians
20th-century women mathematicians
21st-century women mathematicians
20th-century French women
21st-century French women |
https://en.wikipedia.org/wiki/Nagao%27s%20theorem | In mathematics, Nagao's theorem, named after Hirosi Nagao, is a result about the structure of the group of 2-by-2 invertible matrices over the ring of polynomials over a field. It has been extended by Serre to give a description of the structure of the corresponding matrix group over the coordinate ring of a projective curve.
Nagao's theorem
For a general ring R we let GL2(R) denote the group of invertible 2-by-2 matrices with entries in R, and let R* denote the group of units of R, and let
Then B(R) is a subgroup of GL2(R).
Nagao's theorem states that in the case that R is the ring K[t] of polynomials in one variable over a field K, the group GL2(R) is the amalgamated product of GL2(K) and B(K[t]) over their intersection B(K).
Serre's extension
In this setting, C is a smooth projective curve C over a field K. For a closed point P of C let R be the corresponding coordinate ring of C with P removed. There exists a graph of groups (G,T) where T is a tree with at most one non-terminal vertex, such that GL2(R) is isomorphic to the fundamental group π1(G,T).
References
Theorems in group theory |
https://en.wikipedia.org/wiki/Square%20class | In mathematics, specifically abstract algebra, a square class of a field is an element of the square class group, the quotient group of the multiplicative group of nonzero elements in the field modulo the square elements of the field. Each square class is a subset of the nonzero elements (a coset of the multiplicative group) consisting of the elements of the form xy2 where x is some particular fixed element and y ranges over all nonzero field elements.
For instance, if , the field of real numbers, then is just the group of all nonzero real numbers (with the multiplication operation) and is the subgroup of positive numbers (as every positive number has a real square root). The quotient of these two groups is a group with two elements, corresponding to two cosets: the set of positive numbers and the set of negative numbers. Thus, the real numbers have two square classes, the positive numbers and the negative numbers.
Square classes are frequently studied in relation to the theory of quadratic forms. The reason is that if is an -vector space and is a quadratic form and is an element of such that , then for all , and thus it is sometimes more convenient to talk about the square classes which the quadratic form represents.
Every element of the square class group is an involution. It follows that, if the number of square classes of a field is finite, it must be a power of two.
References
Field (mathematics) |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 2012–13.
Scottish First Division
Scottish Second Division
Scottish Third Division
See also
2012–13 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/Topological%20complexity | In mathematics, topological complexity of a topological space X (also denoted by TC(X)) is a topological invariant closely connected to the motion planning problem, introduced by Michael Farber in 2003.
Definition
Let X be a topological space and be the space of all continuous paths in X. Define the projection
by . The topological complexity is the minimal number k such that
there exists an open cover of ,
for each , there exists a local section
Examples
The topological complexity: TC(X) = 1 if and only if X is contractible.
The topological complexity of the sphere is 2 for n odd and 3 for n even. For example, in the case of the circle , we may define a path between two points to be the geodesic between the points, if it is unique. Any pair of antipodal points can be connected by a counter-clockwise path.
If is the configuration space of n distinct points in the Euclidean m-space, then
The topological complexity of the Klein bottle is 5.
References
Armindo Costa: Topological Complexity of Configuration Spaces, Ph.D. Thesis, Durham University (2010), online
Topology
ko:단면 범주#위상 복잡도 |
https://en.wikipedia.org/wiki/Tom%20Brooks%20%28writer%29 | Tom Brooks (writer and theorist), born in London, England, is British author, draftsman and a proponent of Prehistoric geometry theories.
Brooks was born in London and attended East Sheen Grammar School before returning to his family home in Devon where he attended Colyton Grammar School. His career included time spent in the Royal British Navy along with being a Marketing director and a draftsman.
Brooks has concentrated on theorizing upon the layout and geometry of ancient sites in Britain and has published three books on the subject: The Hand of Man; Prehistoric Geometry in Britain; and 'Seeing Around Corners' – Geometry in Stone Age Britain – The Proof. In a survey of over 1500 ancient sites in Britain, Brooks claims that many were constructed by prehistoric man on a connecting grid of isosceles triangles spiraling outwards from Silbury Hill (pictured) with each triangle pointing to the next site. Monuments that comprised the grid included hillforts, standing stones, churches and stone circles such as Stonehenge. Archaeologists have made the criticism that many such patterns can be easily found as Britain is so rich in ancient sites of different types from different periods.
Publications
The Hand of Man. Now out of print, published Edward Gaskell.
Prehistoric Geometry in Britain. Self-published. Out of print.
Seeing Around Corners. Geometry in Stone Age Britain- the Proof. Currently available from http://www.prehistoric-geometry.co.uk/.
See also
Alexander Thom
Anne Macaulay
References
External links
"Did aliens help to line up Woolworths stores?"
www.prehistoric-geometry.co.uk Prehistoric Geometry – The discoveries of Tom Brooks
Prehistoric Geometry in Britain part 1 on YouTube
Prehistoric Geometry in Britain part 2 on YouTube
Prehistoric Geometry in Britain part 3 on YouTube
Prehistoric Geometry in Britain part 4 on YouTube
Prehistoric Geometry in Britain part 5 on YouTube
British writers
British historians
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Tsen%20rank | In mathematics, the Tsen rank of a field describes conditions under which a system of polynomial equations must have a solution in the field. The concept is named for C. C. Tsen, who introduced their study in 1936.
We consider a system of m polynomial equations in n variables over a field F. Assume that the equations all have constant term zero, so that (0, 0, ... ,0) is a common solution. We say that F is a Ti-field if every such system, of degrees d1, ..., dm has a common non-zero solution whenever
The Tsen rank of F is the smallest i such that F is a Ti-field. We say that the Tsen rank of F is infinite if it is not a Ti-field for any i (for example, if it is formally real).
Properties
A field has Tsen rank zero if and only if it is algebraically closed.
A finite field has Tsen rank 1: this is the Chevalley–Warning theorem.
If F is algebraically closed then rational function field F(X) has Tsen rank 1.
If F has Tsen rank i, then the rational function field F(X) has Tsen rank at most i + 1.
If F has Tsen rank i, then an algebraic extension of F has Tsen rank at most i.
If F has Tsen rank i, then an extension of F of transcendence degree k has Tsen rank at most i + k.
There exist fields of Tsen rank i for every integer i ≥ 0.
Norm form
We define a norm form of level i on a field F to be a homogeneous polynomial of degree d in n=di variables with only the trivial zero over F (we exclude the case n=d=1). The existence of a norm form on level i on F implies that F is of Tsen rank at least i − 1. If E is an extension of F of finite degree n > 1, then the field norm form for E/F is a norm form of level 1. If F admits a norm form of level i then the rational function field F(X) admits a norm form of level i + 1. This allows us to demonstrate the existence of fields of any given Tsen rank.
Diophantine dimension
The Diophantine dimension of a field is the smallest natural number k, if it exists, such that the field of is class Ck: that is, such that any homogeneous polynomial of degree d in N variables has a non-trivial zero whenever N > dk. Algebraically closed fields are of Diophantine dimension 0; quasi-algebraically closed fields of dimension 1.
Clearly if a field is Ti then it is Ci, and T0 and C0 are equivalent, each being equivalent to being algebraically closed. It is not known whether Tsen rank and Diophantine dimension are equal in general.
See also
Tsen's theorem
References
Field (mathematics)
Diophantine geometry |
https://en.wikipedia.org/wiki/Martin%20Bridson | Martin Robert Bridson (born 22 October 1964) is a Manx mathematician. He is Whitehead Professor of Pure Mathematics at the University of Oxford, and the president of the Clay Mathematics Institute. He was previously the head of Oxford's Mathematical Institute. He is a fellow of Magdalen College, Oxford and an honorary fellow of Hertford College, Oxford. Specializing in geometry, topology and group theory, Bridson is best known for his work in geometric group theory.
Education and early life
Bridson is a native of the Isle of Man. He was educated at St Ninian's High School, Douglas, then Hertford College, Oxford, and Cornell University, receiving a Master of Arts degree from Oxford in 1986, and a Master of Science degree in 1988 followed by a PhD in 1991 from Cornell. His PhD thesis was supervised by Karen Vogtmann, and was entitled Geodesics and Curvature in Metric Simplicial Complexes.
Career and research
He was an assistant professor at Princeton University until 1996, was twice a visiting professor at the University of Geneva (1992 and 2006), and was Professor of Mathematics at Imperial College London from 2002 to 2007. From 1993 to 2002 he was a Tutorial Fellow of Pembroke College, Oxford, and Reader (1996) then Professor of Topology (2000) in the University of Oxford. He remains a Supernumerary Fellow of Pembroke College. In 2016, Bridson became only the second Manxman to ever be elected to the Royal Society, after Edward Forbes. In 2020, he was elected to Academia Europaea. With André Haefliger, he won
the 2020 Steele Prize for Mathematical Exposition for the highly influential book Metric Spaces of Non-positive Curvature, published by Springer-Verlag in 1999.
Honours and awards
Bridson was an invited lecturer at the International Congress of Mathematicians in 2006.
2016 Elected a Fellow of the Royal Society.
2014 Elected a Fellow of the American Mathematical Society.
1999 Whitehead Prize
2012 Royal Society Wolfson Research Merit Award
2020 Elected Member of Academia Europaea
2020 Steele Prize of the American Mathematical Society
References
1964 births
Living people
People educated at St Ninian's High School, Douglas
Alumni of Hertford College, Oxford
20th-century British mathematicians
21st-century British mathematicians
Fellows of Magdalen College, Oxford
Fellows of Pembroke College, Oxford
Statutory Professors of the University of Oxford
Whitehead Prize winners
Fellows of the American Mathematical Society
Fellows of the Royal Society
People from Douglas, Isle of Man
Members of Academia Europaea |
https://en.wikipedia.org/wiki/Irreligion%20in%20Ghana | Irreligion in Ghana is difficult to measure in the country, as regular demographic polling is not widespread and available statistics are often many years old. Most Ghanaian nationals claim the Christian (71%) or Muslim (18%) faiths. Many atheists in Ghana are not willing to openly express their beliefs due to the fear of persecution. Most secondary educational institutions also have some form of religious affiliation. This is evident in the names of schools like Presbyterian Boys School, Holy Child School and many others. Atheists form a very small minority in Ghana.
In the Ghana census taken in 2010, Christians make up 71.2% of the population, Islam 17.6%, Irreligion 5.3%, Traditional religion 5.2%. Other faiths include Hinduism, Buddhism and Nichiren Buddhism, Taoism, Sōka Gakkai, Shinto and Judaism.
Contrary to the generally accepted view that all Ghanaians profess one religion or the other, there is a small group of outspoken atheists, freethinkers and skeptics who form the Humanist Association of Ghana. The group organized a Humanist conference in November, 2012 which brought together Humanists from around the world to discuss issues relevant to the advancement of Humanism in Ghana.
A second International Humanist conference was hosted by the same organization in December 2014. It featured discussions on additional topics relevant to Humanism such as feminism, witchcraft accusations in West Africa and Humanist ceremonies. The organization currently has about fifty members and attracts limited media coverage. Work is still in progress to officially register the association and make it more broadly known in civic society.
Humanism is not well known in Ghana and this, coupled with high levels of religious belief in Ghana makes it difficult for nonbelievers to share their opinions freely without fear of stigma. There have been a few debates conducted by humanists in the country regarding what should be considered core humanist principles and what should be shifted to the broad spectrum of secularity.
Openly professing atheism or antireligious beliefs can lead to public outrage, such as when the popular hiplife artist Mzbel stated that Jesus was a fake.
See also
Religion in Ghana
Christianity in Ghana
Islam in Ghana
Demographics of Ghana
References
Religion in Ghana
Ghana
Ghana |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20PFC%20Botev%20Plovdiv%20season | The 2012–13 season is Botev Plovdiv's 1st season in A Group after their return to the top division of the Bulgarian football league system. This article shows player statistics and all matches (official and friendly) that the club will play during the 2012–13 season.
Players
Squad stats
Appearances for competitive matches only
|-
|colspan="14"|Players sold or loaned out after the start of the season:
|}
As of 25 May 2013
Players in/out
Summer transfers
In:
Out:
Winter transfers
In:
Out:
Matches
A Group
League table
Results summary
Results by round
Results
Bulgarian Cup
Botev won 9−1 on aggregate and qualified for the Third Round.
Slavia won 4−0 on aggregate. Botev is eliminated.
Friendlies
See also
PFC Botev Plovdiv
References
External links
Botev Official Site
Botev Fan Page with up-to-date information
Bulgarian A Professional Football Group
UEFA Profile
Botev Plovdiv seasons
Botev Plovdiv |
https://en.wikipedia.org/wiki/North%20Wales%20Crusaders%20statistics%20and%20records | This page details statistics and records regarding the North Wales Crusaders Rugby League club. This includes competitive matches following their inception in 2012.
Team Records
Seasons
As of 02/09/12. Round is the round reached in the competition.
Wins & Losses
As of 2/9/12
Opposition
As of 02/9/12
Attendance
Highest Attendance: 1,513 V Barrow Raiders (11 March 2012)
As of 12/8/12
Player Records
For a list of North Wales Crusaders players,
Player summary
As of 2/9/12
Appearances
As of 10/4/18
Points
As of 10/4/18
Tries
As of 2/9/12
Goals
As of 2/9/12
Firsts
Games
First game: Friendly V Leigh East at Leigh Sports Village on 20/1/12. Won 12-34
First competitive game: Championship 1 game V Barrow Raiders at Racecourse Ground on 11/3/12. Lost 24-26
First Challenge Cup game: V Toulouse Olympique at Racecourse Ground on 24/3/12. Won 28-10
First win: Challenge Cup game V Toulouse Olympique at Racecourse Ground on 34/3/12. Won 28-10
Players
First Try: Lee Hudson v Leigh East on 20/1/12.
First competitive Try: Andy Moulsdale v Barrow Raiders on 11/3/12.
Most Try's in a Game: 3 Leon Brennan V Oldham, Andy Moulsdale V Gateshead Thunder & Billy Sheen V Gateshead Thunder
Oldest player: Christiaan Roets 31 years, 11 months, 28 days - V Whitehaven 2/9/2012
Youngest player: Lewys Weaver 18 years, 10 months, 29 days - V London Skolars 12/8/2012
Awards, honours and nominations
End-of-season awards
Awards are presented at the end of season dinner.
Nominations for 'Try of the year' were on the Official website after the final game.
Nominations for 'Smash of the year' were on the Official website after the final game.
Voted for by coaching staff.
Voted for on the official website
Voted for by the playing staff
Voted for by members of the Crusaders Rugby League Supporters Club.
Internationals
Players who have won international caps whilst playing for the Crusaders. Stats also include total caps during their career.
Updated 22 October 2012.
References
Statistics and records |
https://en.wikipedia.org/wiki/Manin%20matrix | In mathematics, Manin matrices, named after Yuri Manin who introduced them around 1987–88, are a class of matrices with elements in a not-necessarily commutative ring, which in a certain sense behave like matrices whose elements commute. In particular there is natural definition of the determinant for them and most linear algebra theorems like Cramer's rule, Cayley–Hamilton theorem, etc. hold true for them. Any matrix with commuting elements is a Manin matrix. These matrices have applications in representation theory in particular to Capelli's identity, Yangian and quantum integrable systems.
Manin matrices are particular examples of Manin's general construction of "non-commutative symmetries" which can be applied to any algebra.
From this point of view they are "non-commutative endomorphisms" of polynomial algebra C[x1, ...xn].
Taking (q)-(super)-commuting variables one will get (q)-(super)-analogs of Manin matrices, which are closely related to quantum groups. Manin works were influenced by the quantum group theory.
He discovered that quantized algebra of functions Funq(GL) can be defined by the requirement that T and Tt are simultaneously q-Manin matrices.
In that sense it should be stressed that (q)-Manin matrices are defined only by half of the relations of related quantum group Funq(GL), and these relations are enough for many linear algebra theorems.
Definition
Context
Matrices with generic noncommutative elements do not admit a natural construction of the determinant with values in a ground ring and basic theorems of the linear algebra fail to hold true. There are several modifications of the determinant theory: Dieudonné determinant which takes values in the abelianization K*/[K*, K*] of the multiplicative group K* of ground ring K; and theory of quasideterminants. But the analogy between these determinants and commutative determinants is not complete. On the other hand, if one considers certain specific classes of matrices with non-commutative elements, then there are examples where one can define the determinant and prove linear algebra theorems which are very similar to their commutative analogs. Examples include: quantum groups and q-determinant; Capelli matrix and Capelli determinant; super-matrices and Berezinian.
Manin matrices is a general and natural class of matrices with not-necessarily commutative elements which admit natural definition of the determinant and generalizations of the linear algebra theorems.
Formal definition
An n by m matrix M with entries Mij over a ring R (not necessarily commutative) is a Manin matrix if all elements in a given column commute and if for all i,j,k,l it holds that [Mij,Mkl] = [Mkj,Mil]. Here [a,b] denotes (ab − ba) the commutator of a and b.
The definition can be better seen from the following formulas.
A rectangular matrix M is called a Manin matrix if for any 2×2 submatrix, consisting of rows i and k, and columns j and l:
the following commutation relations hold
Ubiquity of 2 × 2 M |
https://en.wikipedia.org/wiki/Let%20expression | In computer science, a "let" expression associates a function definition with a restricted scope.
The "let" expression may also be defined in mathematics, where it associates a Boolean condition with a restricted scope.
The "let" expression may be considered as a lambda abstraction applied to a value. Within mathematics, a let expression may also be considered as a conjunction of expressions, within an existential quantifier which restricts the scope of the variable.
The let expression is present in many functional languages to allow the local definition of expression, for use in defining another expression. The let-expression is present in some functional languages in two forms; let or "let rec". Let rec is an extension of the simple let expression which uses the fixed-point combinator to implement recursion.
History
Dana Scott's LCF language was a stage in the evolution of lambda calculus into modern functional languages. This language introduced the let expression, which has appeared in most functional languages since that time.
The languages Scheme, ML, and more recently Haskell have inherited let expressions from LCF.
Stateful imperative languages such as ALGOL and Pascal essentially implement a let expression, to implement restricted scope of functions, in block structures.
A closely related "where" clause, together with its recursive variant "where rec", appeared already in Peter Landin's The mechanical evaluation of expressions.
Description
A "let" expression defines a function or value for use in another expression. As well as being a construct used in many functional programming languages, it is a natural language construct often used in mathematical texts. It is an alternate syntactical construct for a where clause.
In both cases the whole construct is an expression whose value is 5. Like the if-then-else the type returned by the expression is not necessarily Boolean.
A let expression comes in 4 main forms,
In functional languages the let expression defines functions which may be called in the expression. The scope of the function name is limited to the let expression structure.
In mathematics, the let expression defines a condition, which is a constraint on the expression. The syntax may also support the declaration of existentially quantified variables local to the let expression.
The terminology, syntax and semantics vary from language to language. In Scheme, let is used for the simple form and let rec for the recursive form. In ML let marks only the start of a block of declarations with fun marking the start of the function definition. In Haskell, let may be mutually recursive, with the compiler figuring out what is needed.
Definition
A lambda abstraction represents a function without a name. This is a source of the inconsistency in the definition of a lambda abstraction. However lambda abstractions may be composed to represent a function with a name. In this form the inconsistency is removed. The |
https://en.wikipedia.org/wiki/2007%E2%80%9308%20Parma%20FC%20season |
Squad
Competitions
Serie A
Results
Notes
De Vezze was an unused sub.
Domenico Morfeo was an unused sub.
League table
Coppa Italia
Squad statistics
Appearances and goals
|-
|colspan="14"|Players who appeared for Parma that left during the season:
|}
Top scorers
Disciplinary record
References
Sources
RSSSF - Italy 2007/08
Parma Calcio 1913 seasons
Parma |
https://en.wikipedia.org/wiki/Statistical%20manifold | In mathematics, a statistical manifold is a Riemannian manifold, each of whose points is a probability distribution. Statistical manifolds provide a setting for the field of information geometry. The Fisher information metric provides a metric on these manifolds. Following this definition, the log-likelihood function is a differentiable map and the score is an inclusion.
Examples
The family of all normal distributions can be thought of as a 2-dimensional parametric space parametrized by the expected value μ and the variance σ2 ≥ 0. Equipped with the Riemannian metric given by the Fisher information matrix, it is a statistical manifold with a geometry modeled on hyperbolic space. A way of picturing the manifold is done by inferring the parametric equations via the Fisher Information rather than starting from the likelihood-function.
A simple example of a statistical manifold, taken from physics, would be the canonical ensemble: it is a one-dimensional manifold, with the temperature T serving as the coordinate on the manifold. For any fixed temperature T, one has a probability space: so, for a gas of atoms, it would be the probability distribution of the velocities of the atoms. As one varies the temperature T, the probability distribution varies.
Another simple example, taken from medicine, would be the probability distribution of patient outcomes, in response to the quantity of medicine administered. That is, for a fixed dose, some patients improve, and some do not: this is the base probability space. If the dosage is varied, then the probability of outcomes changes. Thus, the dosage is the coordinate on the manifold. To be a smooth manifold, one would have to measure outcomes in response to arbitrarily small changes in dosage; this is not a practically realizable example, unless one has a pre-existing mathematical model of dose-response where the dose can be arbitrarily varied.
Definition
Let X be an orientable manifold, and let be a measure on X. Equivalently, let be a probability space on , with sigma algebra and probability .
The statistical manifold S(X) of X is defined as the space of all measures on X (with the sigma-algebra held fixed). Note that this space is infinite-dimensional; it is commonly taken to be a Fréchet space. The points of S(X) are measures.
Rather than dealing with an infinite-dimensional space S(X), it is common to work with a finite-dimensional submanifold, defined by considering a set of probability distributions parameterized by some smooth, continuously-varying parameter . That is, one considers only those measures that are selected by the parameter. If the parameter is n-dimensional, then, in general, the submanifold will be as well. All finite-dimensional statistical manifolds can be understood in this way.
See also
Chentsov's theorem
References
Manifolds
Information theory |
https://en.wikipedia.org/wiki/George%20P%C3%B3lya%20Award | The George Pólya Award is presented annually by the Mathematical Association of America (MAA) for articles of expository excellence that have been published in The College Mathematics Journal. The award was established in 1976 and up to two awards of $1,000 each are given in each year. The award is named after Hungarian mathematician George Pólya.
Recipients
Recipients of the George Pólya Award have included:
See also
List of mathematics awards
References
Awards of the Mathematical Association of America |
https://en.wikipedia.org/wiki/Dian%20Agus | Dian Agus Prasetyo (born 3 August 1985) is an Indonesian professional footballer who plays as a goalkeeper.
Club career
On November 30, 2014, he was signed by Sriwijaya.
Career statistics
International appearances
References
External links
Dian Agus Prasetyo at Liga Indonesia
1985 births
Living people
Footballers from East Java
People from Ponorogo Regency
Men's association football goalkeepers
Indonesian men's footballers
Indonesian Premier Division players
Liga 1 (Indonesia) players
Liga 2 (Indonesia) players
Gresik United F.C. players
Madura United F.C. players
Arema F.C. players
PS Barito Putera players
PS Mitra Kukar players
Sriwijaya F.C. players
Borneo F.C. Samarinda players
Persiba Balikpapan players
Persela Lamongan players
Persik Kediri players
PSPS Riau players
Indonesia men's youth international footballers
Indonesia men's international footballers
Footballers at the 2006 Asian Games
Asian Games competitors for Indonesia |
https://en.wikipedia.org/wiki/Anneli%20Cahn%20Lax | Anneli Cahn Lax (23 February 1922, Katowice – 24 September 1999, New York City) was an American mathematician, who was known for being an editor of the Mathematics Association of America's New Mathematical Library Series, and for her work in reforming mathematics education with the inclusion of language skills. Anneli Lax received a bachelor's degree in 1942 from Adelphi University and her doctorate in 1956. She was a professor of mathematics at New York University's Courant Institute. She was married to the mathematician Peter Lax.
Life
Career
In 1942, she earned a bachelor's degree in mathematics at Adelphi University in Long Island. In 1956, she earned a PhD from New York University with the dissertation Cauchy's Problem for a Partial Differential Equation with Real Multiple Characteristics with thesis adviser Richard Courant.
She became a mathematics professor at NYU and was the editor of the Mathematics Association of America's New Mathematical Library Series.
In 1961, the series started, with 36 published volumes by 1995. It was planned, by Professor Lax and others, to make mathematics accessible to the average reader while still being technically accurate. The first two books were, Numbers: Rational and Irrational by Ivan Niven and What Is Calculus About? by W. W. Sawyer.
In 1977, she won the George Pólya Award for her article: Linear Algebra, A Potent Tool, Vol. 7 (1976), 3–15.
In 1980, the mathematics department of New York University assigned Lax to design a remedial course in mathematics for freshmen. The course she devised and called "Mathematical Thinking" presented mathematics not as a body of facts but as a set of problems to be analyzed and resolved.
"There is a misconception among people and schoolchildren about the nature of mathematics," she said, in a 1979 interview. "They consider it a matter of rules and regulations instead of thinking." The pressure, she said, was for pupils to come up with the right answer quickly, without time to analyze.
She teamed up with John Devine, working with teachers in inner-city New York schools. Together they got funding from the Ford Foundation to train teachers from these schools in the methods Lax had pioneered at New York University.
Though she was involved with reforming mathematics education for high school and college students in New York, she didn't like panel discussions at conferences. Especially when she was meant to reply immediately to preceding remarks by fellow panelists. Anneli said she was a slow listener and reader. She believed her responses were “not ready for public consumption when my turn comes.”
An insert from her writing at the Mathematics as a Humanistic Discipline Session stated:
“I am convinced that the use of language- reading, writing, listening and speaking is essential part of learning anything, and especially mathematics.”
Anneli knew from her experience as teacher, that students learn new material easily when they are able to connect to their past |
https://en.wikipedia.org/wiki/Irreligion%20in%20Finland | Irreligion in Finland: according to Statistics Finland in 2020, 29.4% of the population in Finland were non-religious, or about 1,628,000 people. The Union of Freethinkers of Finland and other organisations have acted as interest organisations, legal protection organisations and cultural organisations for non-religious people. In a 2018 international ISSP survey, 40% of the Finnish population said they did not believe in God, 34% said they believed in God and 26% did not know. Nearly one out of every five people in the country is not a member of a religious organisation, and the number of people with no religious affiliation has doubled in two decades.
History
In Finland, the Enlightenment mainly influenced the educated classes. In the late 19th century, Darwinism came to Finland. The secularist philosophy of materialism was represented solely by Wilhelm Bolin, a friend of Ludwig Feuerbach, who worked as a librarian at the University of Helsinki, and who had a doctorate in Spinoza's philosophy. Bolin was one of the pioneers of cremation in Finland. Naturalistic evolutionary theory inspired, among others, Hjalmar Neiglick and Edvard Westermarck in Finland as early as the 1880s.
In 1887, Viktor Heikel and 50 other citizens attempted to found the Finnish Association for Religious Freedom and Tolerance (, ). The Senate sent the association's statutes to the Lutheran Church's chapter for approval, which rejected the proposal. In 1889, the newspaper , edited by Eero Erkko, published a series of articles on evolution. It led to a heated debate, involving Juhani Aho and Minna Canth, among others. The same year, Canth and A.B. Mäkelä began publication of the magazine ('Free Ideas') in Kuopio, which presented development theory. The magazine was discontinued because censors removed a number of the articles in the magazine.
At the turn of the 20th century, the labor movement was headed by several critics of religion. The labour movement in Finland adopted the program of Marxist atheism as a challenge to the State Church. The 1903 Forssa Program of the Finnish Social Democratic Party stated:
The magazine Euterpe, published between 1902 and 1905, was written by Rolf Lagerborg, Gunnar Castrén and Georg Schauman, among others. From 1909 to 1917, S.E. Kristiansson published . A.B. Sarlin wrote a series of books critical of religion under the pseudonym Asa Jalas.
In 1905, the Prometheus Society, a student society, was founded at the University of Helsinki with the aim of promoting freedom of thought. The association (1905–1914) included Edvard Westermarck, Rafael Karsten, Rolf Lagerborg, Knut Tallqvist, Wilhelm Bolin, Yrjö Hirn, Georg Schauman, Hjalmar Magnus Eklund, Harry Federley, Alma Söderhjelm, Gunnar Castrén, K.H. Wiik, Viktor Heikel and Ernst Lampén. The association demanded the abolition of confessional religious education in schools and the acceptance of civil marriage. Westermarck was also a member of the English Freethinkers' Association. The ten |
https://en.wikipedia.org/wiki/List%20of%20definite%20integrals | In mathematics, the definite integral
is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.
The fundamental theorem of calculus establishes the relationship between indefinite and definite integrals and introduces a technique for evaluating definite integrals.
If the interval is infinite the definite integral is called an improper integral and defined by using appropriate limiting procedures. for example:
A constant, such pi, that may be defined by the integral of an algebraic function over an algebraic domain is known as a period.
The following is a list of some of the most common or interesting definite integrals. For a list of indefinite integrals see List of indefinite integrals.
Definite integrals involving rational or irrational expressions
Definite integrals involving trigonometric functions
(see Dirichlet integral)
Definite integrals involving exponential functions
(see also Gamma function)
(the Gaussian integral)
(where !! is the double factorial)
(where is Euler–Mascheroni constant)
Definite integrals involving logarithmic functions
Definite integrals involving hyperbolic functions
Frullani integrals
holds if the integral exists and is continuous.
See also
List of integrals
Indefinite sum
Gamma function
List of limits
References
Integrals
Integrals |
https://en.wikipedia.org/wiki/Hasse%20invariant%20of%20an%20algebra | In mathematics, the Hasse invariant of an algebra is an invariant attached to a Brauer class of algebras over a field. The concept is named after Helmut Hasse. The invariant plays a role in local class field theory.
Local fields
Let K be a local field with valuation v and D a K-algebra. We may assume D is a division algebra with centre K of degree n. The valuation v can be extended to D, for example by extending it compatibly to each commutative subfield of D: the value group of this valuation is (1/n)Z.
There is a commutative subfield L of D which is unramified over K, and D splits over L. The field L is not unique but all such extensions are conjugate by the Skolem–Noether theorem, which further shows that any automorphism of L is induced by a conjugation in D. Take γ in D such that conjugation by γ induces the Frobenius automorphism of L/K and let v(γ) = k/n. Then k/n modulo 1 is the Hasse invariant of D. It depends only on the Brauer class of D.
The Hasse invariant is thus a map defined on the Brauer group of a local field K to the divisible group Q/Z. Every class in the Brauer group is represented by a class in the Brauer group of an unramified extension of L/K of degree n, which by the Grunwald–Wang theorem and the Albert–Brauer–Hasse–Noether theorem we may take to be a cyclic algebra (L,φ,πk) for some k mod n, where φ is the Frobenius map and π is a uniformiser. The invariant map attaches the element k/n mod 1 to the class. This exhibits the invariant map as a homomorphism
The invariant map extends to Br(K) by representing each class by some element of Br(L/K) as above.
For a non-Archimedean local field, the invariant map is a group isomorphism.
In the case of the field R of real numbers, there are two Brauer classes, represented by the algebra R itself and the quaternion algebra H. It is convenient to assign invariant zero to the class of R and invariant 1/2 modulo 1 to the quaternion class.
In the case of the field C of complex numbers, the only Brauer class is the trivial one, with invariant zero.
Global fields
For a global field K, given a central simple algebra D over K then for each valuation v of K we can consider the extension of scalars Dv = D ⊗ Kv The extension Dv splits for all but finitely many v, so that the local invariant of Dv is almost always zero. The Brauer group Br(K) fits into an exact sequence
where S is the set of all valuations of K and the right arrow is the sum of the local invariants. The injectivity of the left arrow is the content of the Albert–Brauer–Hasse–Noether theorem. Exactness in the middle term is a deep fact from global class field theory.
References
Further reading
Field (mathematics)
Algebraic number theory |
https://en.wikipedia.org/wiki/Adrian%20Ioana | Adrian Ioana (born 18 January 1981, Târgu Jiu) is a Romanian mathematician. He is currently a professor at the University of California, San Diego.
Ioana earned a BS in Mathematics from the University of Bucharest in 2003, and completed his PhD at the University of California, Los Angeles in 2007, under the supervision of Sorin Popa. He then was a postdoc at the California Institute of Technology and a Research Fellow supported by the Clay Mathematics Institute, after which he joined UC San Diego in 2011.
For his contributions to von Neumann algebras and representation theory of groups, he was awarded a 2012 EMS Prize. In 2018 he was an invited speaker at the International Congress of Mathematicians (ICM) in Rio de Janeiro (on "Rigidity for von Neumann algebras").
References
External links
Website at UCSD
1981 births
Living people
21st-century Romanian mathematicians
University of California, Los Angeles alumni
University of California, San Diego faculty
Place of birth missing (living people)
International Mathematical Olympiad participants
People from Târgu Jiu
University of Bucharest alumni |
https://en.wikipedia.org/wiki/Web%20Coverage%20Processing%20Service | The Web Coverage Processing Service (WCPS) defines a language for filtering and processing of multi-dimensional raster coverages, such as sensor, simulation, image, and statistics data. The Web Coverage Processing Service is maintained by the Open Geospatial Consortium (OGC). This raster query language allows clients to obtain original coverage data, or derived information, in a platform-neutral manner over the Web.
Overview
WCPS allows to generate pictures suitable for displaying to humans and information concise enough for further consumption by programs. In particular, the formally defined syntax and semantics make WCPS amenable to program-generated queries and automatic service chaining.
As the WCPS language is not tied to any particular transmission protocol, the query paradigm can be embedded into any service framework, such as OGC Web Coverage Service (WCS) and OGC Web Processing Service (WPS).
The current WCPS version is 1.0. The standards document, available from the OGC WCPS standards page, presents a condensed definition of syntax and semantics. In addition, there is an introduction to the concepts along with design rationales.
Currently, WCPS is constrained to multi-dimensional raster data, but an activity is under work in OGC to extend it to all coverage types, i.e., digital geospatial information representing space-varying phenomena as defined in OGC Abstract Specification Topic 6: Schema for Coverage Geometry and Functions (which is identical to ISO 19123) and refined to a concrete, interoperable model in the OGC GML 3.2.1 Application Schema - Coverages (GMLCOV) Standard.
WCPS language in a nutshell
WCPS establishes a protocol to send a query string to a server and obtain, as a result of the server's processing, a set of coverages.
The query string can be expressed in either Abstract Syntax or XML. In the following examples, Abstract Syntax will be used as it is more apt for human consumption.
The WCPS syntax tentatively has been crafted close to the XQuery language – as metadata more and more are established in XML, and OGC heavily relies on XML (such as Geography Markup Language), it is anticipated that eventually a combination of XQuery and WCPS will be established. This will unify data and metadata retrieval.
The following example may serve to illustrate these principles. Task is to inspect three coverages M1, M2, and M3; for each one, deliver the pixelwise difference of red and near-infrared (nir) channel; return the result encoded in HDF5:
for $c in ( M1, M2, M3 )
return
encode( abs( $c.red - $c.nir ), "hdf5" )
This will return three coverages, that is: three HDF5 files.
Next, we are interested only in those coverages where nir exceeds 127 somewhere:
for $c in ( M1, M2, M3 )
where
some( $c.nir > 127 )
return
encode( abs( $c.red - $c.nir ), "hdf5" )
The result might be only two coverages that pass the filter.
Finally, we want to constrain the filter predicate through a pixel mask acting as filter:
for $ |
https://en.wikipedia.org/wiki/Anatoli%20Gospodinov | Anatoli Gospodinov (; born 21 March 1994) is a Bulgarian football goalkeeper who plays for Arda Kardzhali.
Career statistics
Club
Honours
Club
CSKA Sofia
Bulgarian Cup: 2015–16
References
External links
1994 births
Living people
Bulgarian men's footballers
Bulgarian expatriate men's footballers
Expatriate men's footballers in Poland
First Professional Football League (Bulgaria) players
I liga players
OFC Sliven 2000 players
PFC CSKA Sofia players
FC Vitosha Bistritsa players
Chrobry Głogów players
SFC Etar Veliko Tarnovo players
Men's association football goalkeepers
Sportspeople from Sliven |
https://en.wikipedia.org/wiki/Factor%20system | In mathematics, a factor system (sometimes called factor set) is a fundamental tool of Otto Schreier’s classical theory for group extension problem. It consists of a set of automorphisms and a binary function on a group satisfying certain condition (so-called cocycle condition). In fact, a factor system constitutes a realisation of the cocycles in the second cohomology group in group cohomology.
Introduction
Suppose is a group and is an abelian group. For a group extension
there exists a factor system which consists of a function and homomorphism such that it makes the cartesian product a group as
So must be a "group 2-cocycle" (and thus define an element in H(G, A), as studied in group cohomology). In fact, does not have to be abelian, but the situation is more complicated for non-abelian groups
If is trivial, then splits over , so that is the semidirect product of with .
If a group algebra is given, then a factor system f modifies that algebra to a skew-group algebra by modifying the group operation to .
Application: for Abelian field extensions
Let G be a group and L a field on which G acts as automorphisms. A cocycle or (Noether) factor system is a map c: G × G → L* satisfying
Cocycles are equivalent if there exists some system of elements a : G → L* with
Cocycles of the form
are called split. Cocycles under multiplication modulo split cocycles form a group, the second cohomology group H2(G,L*).
Crossed product algebras
Let us take the case that G is the Galois group of a field extension L/K. A factor system c in H2(G,L*) gives rise to a crossed product algebra A, which is a K-algebra containing L as a subfield, generated by the elements λ in L and ug with multiplication
Equivalent factor systems correspond to a change of basis in A over K. We may write
The crossed product algebra A is a central simple algebra (CSA) of degree equal to [L : K]. The converse holds: every central simple algebra over K that splits over L and such that deg A = [L : K] arises in this way. The tensor product of algebras corresponds to multiplication of the corresponding elements in H2. We thus obtain an identification of the Brauer group, where the elements are classes of CSAs over K, with H2.
Cyclic algebra
Let us further restrict to the case that L/K is cyclic with Galois group G of order n generated by t. Let A be a crossed product (L,G,c) with factor set c. Let u = ut be the generator in A corresponding to t. We can define the other generators
and then we have un = a in K. This element a specifies a cocycle c by
It thus makes sense to denote A simply by (L,t,a). However a is not uniquely specified by A since we can multiply u by any element λ of L* and then a is multiplied by the product of the conjugates of λ. Hence A corresponds to an element of the norm residue group K*/NL/KL*. We obtain the isomorphisms
References
Cohomology theories
Group theory |
https://en.wikipedia.org/wiki/Emmanuel%20Breuillard | Emmanuel Breuillard (born 25 June 1977) is a French mathematician. He was the Sadleirian Professor of Pure Mathematics in the
Department of Pure Mathematics and Mathematical Statistics (DPMMS) at the University of Cambridge, and is now Professor of Pure Mathematics at the Mathematical Institute, University of Oxford as of January 1, 2022. He had previously been professor at Paris-Sud 11 University.
In 2012, he won an EMS Prize for his contributions to combinatorics and other fields. His area of research has been in group theoretic aspects of geometry, number theory and combinatorics. In 2014 he was an invited speaker at the International Congress of Mathematicians in Seoul.
References
External links
Website at the University of Cambridge
Website at Paris-Sud 11 University
1977 births
Living people
20th-century French mathematicians
21st-century French mathematicians
École Normale Supérieure alumni
Yale University alumni
Academic staff of Paris-Sud University
International Mathematical Olympiad participants
Fellows of Worcester College, Oxford
Sadleirian Professors of Pure Mathematics |
https://en.wikipedia.org/wiki/David%20Ferrer%20career%20statistics | This is a list of the main career statistics of professional tennis player David Ferrer.
Performance timelines
Singles
Doubles
Significant finals
Grand Slam finals
Singles: 1 (1 runner-up)
Year-end championship finals
Singles: 1 (1 runner-up)
Olympics finals
Men's doubles: 1 Bronze Medal match (0–1)
Masters 1000 finals
Singles: 7 (1 title, 6 runners-up)
ATP career finals
Singles: 52 (27 titles, 25 runner-ups)
Doubles: 3 (2 titles, 1 runner-up)
Team competition: 4 (3 titles, 1 runner-up)
Record against other players
Ferrer's record against those who have been ranked in the top 2, with active players in boldface
* Statistics include Davis Cup matches.
Record against top 20 players
Ferrer's match record against players who were ranked world No. 20 or higher at the time is as follows, with those who have been No. 1 in boldface:
Nicolás Almagro 15–1
Fernando Verdasco 14–7
Fabio Fognini 11–0
Philipp Kohlschreiber 11–3
Feliciano López 11–8
Richard Gasquet 10–3
Alexandr Dolgopolov 10–4
Andreas Seppi 9–1
David Nalbandian 9–5
Gilles Simon 8–2
Tommy Robredo 8–2
Radek Štěpánek 8–3
Tomáš Berdych 8–8
Juan Carlos Ferrero 7–2
John Isner 7–2
Jürgen Melzer 7–2
Andy Roddick 7–4
Stanislas Wawrinka 7–7
Ivan Ljubičić 6–1
Janko Tipsarević 6–1
Albert Ramos Viñolas 6–2
Juan Martín del Potro 6–7
Andy Murray 6–14
Rafael Nadal 6–26
Grigor Dimitrov 5–1
Marcel Granollers 5–1
Florian Mayer 5–3
Dominik Hrbatý 5–3
Igor Andreev 5–4
Juan Mónaco 5–4
Fernando González 5–5
Novak Djokovic 5–16
Pablo Cuevas 4–0
Tommy Haas 4–0
Milos Raonic 4–0
Ernests Gulbis 4–0
Ivo Karlović 4–1
Viktor Troicki 4–1
Marcos Baghdatis 4–2
Marin Čilić 4–2
Bernard Tomic 4–2
Mardy Fish 4–4
Mikhail Youzhny 4–5
Kei Nishikori 4–10
Robin Söderling 4–10
Roberto Bautista Agut 3–1
Lleyton Hewitt 3–1
Jo-Wilfried Tsonga 3–1
Jonas Björkman 3–1
Sébastien Grosjean 3–1
Benoît Paire 3–1
Sam Querrey 3–1
Agustín Calleri 3–2
Kevin Anderson 3–3
Gaël Monfils 3–3
Gastón Gaudio 3–4
José Acasuso 3–5
Alexander Zverev 3–5
Albert Costa 2–0
David Goffin 2–0
Jerzy Janowicz 2–0
Vincent Spadea 2–0
Mariano Puerta 2–0
Mario Ančić 2–1
Thomas Johansson 2–1
Nicolas Kiefer 2–1
Jarkko Nieminen 2–1
Fabrice Santoro 2–1
Albert Portas 2–1
Nicolás Massú 2–1
Nicolás Lapentti 2–1
Jack Sock 2–2
Juan Ignacio Chela 2–2
Jiří Novák 2–2
Pablo Carreño Busta 2–2
Kyle Edmund 2–2
Xavier Malisse 2–3
Nikolay Davydenko 2–4
Robby Ginepri 2–4
Carlos Moyá 2–6
Andre Agassi 1–0
Nikoloz Basilashvili 1–0
Guillermo Cañas 1–0
Hyeon Chung 1–0
Karol Kučera 1–0
Arnaud Clément 1–1
Gustavo Kuerten 1–1
Nick Kyrgios 1–1
Max Mirnyi 1–1
Marat Safin 1–1
Dominic Thiem 1–1
Andrea Gaudenzi 1–1
Paradorn Srichaphan 1–1
James Blake 1–2
Lucas Pouille 1–2
Guillermo Coria 1–4
Tim Henman 0–1
Todd Martin 0–1
Jan Michael Gambill 0–1
Wayne Ferreira 0–1
Stefan Koubek 0–1
Casper Ruud 0–1
Sjeng Schalken 0–1
Diego Schwartzman 0–2
Karen Khachanov 0–2
Félix Mantilla 0–2
Paul |
https://en.wikipedia.org/wiki/1991%20FC%20Dinamo%20Tbilisi%20season | Dinamo Tbilisi's second season in the Umaglesi Liga.
Season report
Dinamo Tbilisi played by the name FC Iberia Tbilisi.
Current squad
Statistics
Appearances, goals and disciplinary record
Umaglesi Liga
League table
Matches
External links
Archive of FC Dinamo Tbilisi matches by seasons
FC Dinamo Tbilisi seasons
Din |
https://en.wikipedia.org/wiki/Witt%20equivalence | In mathematics, Witt equivalence is either of two concepts in the theory of quadratic spaces:
For fields: having isomorphic Witt rings
For quadratic forms: having isomorphic core forms in a Witt decomposition |
https://en.wikipedia.org/wiki/Hwang%20Song-su | Hwang Song-Su (黄誠秀, 10 July 1987) is a Zainichi Korean football player, who has represented North Korea in international competition. He currently features for Criacao Shinjuku.
Club statistics
Updated to 23 February 2019.
References
External links
Profile at Oita Trinita
1987 births
Living people
Association football people from Tokyo
North Korean men's footballers
J1 League players
J2 League players
J3 League players
Júbilo Iwata players
Thespakusatsu Gunma players
Oita Trinita players
Men's association football midfielders
Zainichi Korean men's footballers |
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