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https://en.wikipedia.org/wiki/Singular%20trace | In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanishes
on operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operators on a finite-dimensional Hilbert space have only the zero functional as a singular trace since all operators have finite rank. For example, matrix algebras have no non-trivial singular traces and the matrix trace is the unique trace up to scaling.
American mathematician Gary Weiss and, later, British mathematician Nigel Kalton observed in the infinite-dimensional case that there are non-trivial singular traces on the ideal of trace class operators.
Therefore, in distinction to the finite-dimensional case, in infinite dimensions the canonical operator trace is not the unique trace up to scaling. The operator trace is the continuous extension of the matrix trace from finite rank operators to all trace class operators, and the term singular derives from the fact that a singular trace vanishes where the matrix trace is supported, analogous to a singular measure vanishing where Lebesgue measure is supported.
Singular traces measure the asymptotic spectral behaviour of operators and have found applications in the noncommutative geometry of French mathematician Alain Connes.
In heuristic terms, a singular trace corresponds to a way of summing
numbers a1, a2, a3, ... that is completely orthogonal or 'singular' with respect to the usual sum a1 + a2 + a3 + ... .
This allows mathematicians to sum sequences like the harmonic sequence (and operators with similar spectral behaviour) that are divergent for the usual sum. In similar terms a (noncommutative) measure theory or probability theory can be built for distributions like the Cauchy distribution (and operators with similar spectral behaviour) that do not have finite expectation in the usual sense.
Origin
By 1950 French mathematician Jacques Dixmier, a founder of the semifinite theory of von Neumann algebras,
thought that a trace on the bounded operators of a separable Hilbert space would automatically be normal up to some trivial counterexamples. Over the course of 15 years Dixmier, aided by a suggestion of Nachman Aronszajn and inequalities proved by Joseph Hersch, developed an example of a non-trivial yet non-normal trace on weak trace-class operators,
disproving his earlier view.
Singular traces based on Dixmier's construction are called Dixmier traces.
Independently and by different methods, German mathematician Albrecht Pietsch (de) investigated traces on ideals of operators on Banach spaces.
In 1987 Nigel Kalton answered a question of Pietsch by showing that the operator trace is not
the unique trace on quasi-normed proper subideals of the trace-class operators on a Hilbert space. József Varga independently studied a similar question.
To solve the question of uniqueness of th |
https://en.wikipedia.org/wiki/O%27Nan%E2%80%93Scott%20theorem | In mathematics, the O'Nan–Scott theorem is one of the most influential theorems of permutation group theory; the classification of finite simple groups is what makes it so useful. Originally the theorem was about maximal subgroups of the symmetric group. It appeared as an appendix to a paper by Leonard Scott written for The Santa Cruz Conference on Finite Groups in 1979, with a footnote that Michael O'Nan had independently proved the same result. Michael Aschbacher and Scott later gave a corrected version of the statement of the theorem.
The theorem states that a maximal subgroup of the symmetric group Sym(Ω), where |Ω| = n, is one of the following:
Sk × Sn−k the stabilizer of a k-set (that is, intransitive)
Sa wr Sb with n = ab, the stabilizer of a partition into b parts of size a (that is, imprimitive)
primitive (that is, preserves no nontrivial partition) and of one of the following types:
AGL(d,p)
Sl wr Sk, the stabilizer of the product structure Ω = Δk
a group of diagonal type
an almost simple group
In a survey paper written for the Bulletin of the London Mathematical Society,
Peter J. Cameron seems to have been the first to recognize that the real power in the O'Nan–Scott theorem is in the ability to split the finite primitive groups into various types.
A complete version of the theorem with a self-contained proof was given by M.W. Liebeck, Cheryl Praeger and Jan Saxl. The theorem is now a standard part of textbooks on permutation groups.
O'Nan–Scott types
The eight O'Nan–Scott types of finite primitive permutation groups are as follows:
HA (holomorph of an abelian group): These are the primitive groups which are subgroups of the affine general linear group AGL(d,p), for some prime p and positive integer d ≥ 1. For such a group G to be primitive, it must contain the subgroup of all translations, and the stabilizer G0 in G of the zero vector must be an irreducible subgroup of GL(d,p). Primitive groups of type HA are characterized by having a unique minimal normal subgroup which is elementary abelian and acts regularly.
HS (holomorph of a simple group): Let T be a finite nonabelian simple group. Then M = T×T acts on Ω = T by t(t1,t2) = t1−1tt2. Now M has two minimal normal subgroups N1, N2, each isomorphic to T and each acts regularly on Ω, one by right multiplication and one by left multiplication. The action of M is primitive and if we take α = 1T we have Mα = {(t,t)|t ∈ T}, which includes Inn(T) on Ω. In fact any automorphism of T will act on Ω. A primitive group of type HS is then any group G such that M ≅ T.Inn(T) ≤ G ≤ T.Aut(T). All such groups have N1 and N2 as minimal normal subgroups.
HC (holomorph of a compound group): Let T be a nonabelian simple group and let N1 ≅ N2 ≅ Tk for some integer k ≥ 2. Let Ω = Tk. Then M = N1 × N2 acts transitively on Ω via x(n1,n2) = n1−1xn2 for all x ∈ Ω, n1 ∈ N1, n2 ∈ N2. As in the HS case, we have M ≅ Tk.Inn(Tk) and any automorphism of Tk also acts on Ω. A primitive group of type HC is a |
https://en.wikipedia.org/wiki/Quadrisecant | In geometry, a quadrisecant or quadrisecant line of a space curve is a line that passes through four points of the curve. This is the largest possible number of intersections that a generic space curve can have with a line, and for such curves the quadrisecants form a discrete set of lines. Quadrisecants have been studied for curves of several types:
Knots and links in knot theory, when nontrivial, always have quadrisecants, and the existence and number of quadrisecants has been studied in connection with knot invariants including the minimum total curvature and the ropelength of a knot.
The number of quadrisecants of a non-singular algebraic curve in complex projective space can be computed by a formula derived by Arthur Cayley.
Quadrisecants of arrangements of skew lines touch subsets of four lines from the arrangement. They are associated with ruled surfaces and the Schläfli double six configuration.
Definition and motivation
A quadrisecant is a line that intersects a curve, surface, or other set in four distinct points. It is analogous to a secant line, a line that intersects a curve or surface in two points; and a trisecant, a line that intersects a curve or surface in three points.
Compared to secants and trisecants, quadrisecants are especially relevant for space curves, because they have the largest possible number of intersection points of a line with a generic curve. In the plane, a generic curve can be crossed arbitrarily many times by a line; for instance, small generic perturbations of the sine curve are crossed infinitely often by the horizontal axis. In contrast, if an arbitrary space curve is perturbed by a small distance to make it generic, there will be no lines through five or more points of the perturbed curve. Nevertheless, any quadrisecants of the original space curve will remain present nearby in its perturbation.
One explanation for this phenomenon is visual: looking at a space curve from far away, the space of such points of view can be described as a two-dimensional sphere, one point corresponding to each direction. Pairs of strands of the curve may appear to cross from all of these points of view, or from a two-dimensional subset of them. Three strands will form a triple crossing when the point of view lies on a trisecant, and four strands will form a quadruple crossing from a point of view on a quadrisecant. Each constraint that the crossing of a pair of strands lies on another strand reduces the number of degrees of freedom by one (for a generic curve), so the points of view on trisecants form a one-dimensional (continuously infinite) subset of the sphere, while the points of view on quadrisecants form a zero-dimensional (discrete) subset. C. T. C. Wall writes that the fact that generic space curves are crossed at most four times by lines is "one of the simplest theorems of the kind", a model case for analogous theorems on higher-dimensional transversals.
Additionally, for generic space curves, the quadrisecants f |
https://en.wikipedia.org/wiki/Descartes%20number | In number theory, a Descartes number is an odd number which would have been an odd perfect number if one of its composite factors were prime. They are named after René Descartes who observed that the number would be an odd perfect number if only were a prime number, since the sum-of-divisors function for would satisfy, if 22021 were prime,
where we ignore the fact that 22021 is composite ().
A Descartes number is defined as an odd number where and are coprime and , whence is taken as a 'spoof' prime. The example given is the only one currently known.
If is an odd almost perfect number, that is, and is taken as a 'spoof' prime, then is a Descartes number, since . If were prime, would be an odd perfect number.
Properties
Banks et al. showed in 2008 that if is a cube-free Descartes number not divisible by , then has over a million distinct prime divisors.
Tóth showed in 2021 that if denotes a Descartes number (other than Descartes’ example), with pseudo-prime factor , then .
Generalizations
John Voight generalized Descartes numbers to allow negative bases. He found the example . Subsequent work by a group at Brigham Young University found more examples similar to Voight's example, and also allowed a new class of spoofs where one is allowed to also not notice that a prime is the same as another prime in the factorization.
See also
Erdős–Nicolas number, another type of almost-perfect number
Notes
References
.
Divisor function
Integer sequences |
https://en.wikipedia.org/wiki/Csaba%20Vachtler | Csaba Vachtler (born 16 March 1993) is a Hungarian professional footballer who plays for Austrian club SVg Pitten.
Club statistics
Updated to games played as of 27 June 2020.
References
External links
MLSZ
HLSZ
1993 births
People from Mór
Living people
Hungarian men's footballers
Men's association football defenders
Fehérvár FC players
Puskás Akadémia FC players
Balmazújvárosi FC players
Kaposvári Rákóczi FC players
Tiszakécske FC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Nemzeti Bajnokság III players
Footballers from Fejér County
21st-century Hungarian people
Hungarian expatriate men's footballers
Expatriate men's footballers in Austria
Hungarian expatriate sportspeople in Austria |
https://en.wikipedia.org/wiki/Faculty%20of%20Science%2C%20University%20of%20Zagreb | Faculty of Science (, abbr: PMF) is a faculty of the University of Zagreb that comprises seven departments - biology, physics, chemistry, mathematics, geophysics, geography and geology. The Faculty has 288 full professors, associate and assistant professors, 180 junior researchers and about 6000 students.
The Faculty of Science was formally established in 1946, although the teaching of these subjects had existed in the university since 1876. The Faculty offers undergraduate, graduate, and postgraduate study programmes, and pursues research in the fields of natural sciences and mathematics. It also encompasses the seismological service, the mareographic and meteorological stations, and the Zagreb Botanical Garden.
The Faculty of Science is engaged in excellent cooperation with numerous universities and institutes abroad. Professors of the Faculty have been invited as visiting lecturers to European and American universities, and young staff members, as well as postgraduate students, are regularly sent to international universities and institutes for further research.
History
On 23 September 1669. Leopold I certified at the Jesuit Neoacademica Zagrebiensis, a three-year higher education institution, which gradually developed the studies of Philosophy, Law and Theology. At the Jesuit School philosophy was taught even earlier, and part of its first year studies were logic, physics, and metaphysics. Neither Jesuit School (until 1773), nor royal Regia Scientiarum Academica (until 1850) represented a real university. Croatian Parliament and Franz Joseph I of Austria, introduced the Law on founding the University of Zagreb. Soon after the establishing of the University of Zagreb, Faculties of Law, Theology and Philosophy started operating. The Chairs of the Faculty of Philosophy were appointed gradually. In the field of natural sciences the teaching started in 1876, with first lectures in mineralogy and geology, and then in botany, physics, mathematics, chemistry and zoology and geography. Dr. Fran Tućan (1878 - 1954), a popularizer of science in Croatia, who was also president of Matica hrvatska, was appointed as the first dean of the Faculty of Science.
A long endeavour of the Science Department of the Faculty of Philosophy to attain the status of Faculty, finally materialized in 1946, when the Faculty of Science was established.
Departments
The Faculty consists of following departments:
Department of Biology
Department of Physics
Department of Chemistry
Department of Mathematics
Department of Geophysics
Department of Geography
Department of Geology
Department of Geography
Department of Geography at the Faculty of Science in Zagreb is the oldest and the biggest geographic department in Croatia. The Department of Geography consists of three divisions: physical geography, human geography, and regional geography and teaching methods. The Cartographic-technical Centre with a rich Cartographic Collection and the Central Geographic Library are also |
https://en.wikipedia.org/wiki/Norbert%20Farkas%20%28footballer%2C%20born%201992%29 | Norbert Farkas (born 29 June 1992) is a Hungarian professional footballer who plays for Iváncsa.
Club statistics
Updated to games played as of 16 December 2018.
References
External links
HLSZ
1992 births
Living people
Footballers from Székesfehérvár
Hungarian men's footballers
Hungary men's youth international footballers
Hungary men's under-21 international footballers
Men's association football midfielders
Puskás Akadémia FC players
Zalaegerszegi TE players
BFC Siófok players
Balmazújvárosi FC players
MTK Budapest FC players
Monori SE players
Mosonmagyaróvári TE footballers
Tiszakécske FC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Nemzeti Bajnokság III players |
https://en.wikipedia.org/wiki/Gerg%C5%91%20Vaszicsku | Gergő Vaszicsku (born 30 June 1991) is a Hungarian professional footballer who plays for Budafoki MTE.
Club statistics
Updated to games played as of 15 May 2021.
References
MLSZ
HLSZ
1991 births
Living people
Footballers from Debrecen
Hungarian men's footballers
Men's association football defenders
Jászberényi SE footballers
FC Felcsút players
Fehérvár FC players
Puskás Akadémia FC players
Aqvital FC Csákvár players
Budafoki MTE footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/S%C3%A1ndor%20Moln%C3%A1r | Sándor Molnár (born 29 June 1994) is a Hungarian professional footballer who plays for Komárom VSE.
Club statistics
Updated to games played as of 4 August 2013.
References
External links
HLSZ
1994 births
Living people
Footballers from Budapest
Hungarian men's footballers
Men's association football defenders
Újpest FC players
BKV Előre SC footballers
FC Dabas footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/Diminished%20trapezohedron | In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular base face, triangle faces around the base, and kites meeting on top. The kites can also be replaced by rhombi with specific proportions.
Along with the set of pyramids and elongated pyramids, these figures are topologically self-dual.
It can also be seen as an augmented antiprism, with a pyramid augmented onto one of the faces, and whose height is adjusted so the upper antiprism triangle faces can be made coparallel to the pyramid faces and merged into kite-shaped faces.
They're also related to the gyroelongated pyramids, as augmented antiprisms and which are Johnson solids for . This sequence has sets of two triangles instead of kite faces.
Examples
Special cases
There are three special case geometries of the diminished trigonal trapezohedron. The simplest is a diminished cube. The Chestahedron, named after artist Frank Chester, is constructed with equilateral triangles around the base, and the geometry adjusted so the kite faces have the same area as the equilateral triangles. The last can be seen by augmenting a regular tetrahedron and an octahedron, leaving 10 equilateral triangle faces, and then merging 3 sets of coparallel equilateral triangular faces into 3 (60 degree) rhombic faces. It can also be seen as a tetrahedron with 3 of 4 of its vertices rectified. The three rhombic faces fold out flat to form half of a hexagram.
See also
Elongated pyramid
Gyroelongated bipyramid
Elongated bipyramid
Gyroelongated pyramid
Tetrahedrally diminished dodecahedron
References
Symmetries of Canonical Self-Dual Polyhedra 7F,C3v: 9,C4v: 11,C5v:, 13,C6v:, 15,C7v:.
Polyhedra |
https://en.wikipedia.org/wiki/Vilmos%20Szalai | Vilmos Szalai (born 11 August 1991) is a Hungarian football player who plays for III. Kerületi TVE in the Nemzeti Bajnokság II.
He played his first league match in 2010.
Club statistics
Honours
Mezőkövesd
NB II Kelet (1): 2012–13
References
External links
1991 births
Living people
Footballers from Budapest
Hungarian men's footballers
Men's association football defenders
Újpest FC players
Wormatia Worms players
Mezőkövesdi SE footballers
Diósgyőri VTK players
Soproni VSE players
Nyíregyháza Spartacus FC players
BFC Siófok players
Vecsési FC footballers
III. Kerületi TVE footballers
Nemzeti Bajnokság I players
Hungarian expatriate men's footballers
Expatriate men's footballers in Germany
Hungarian expatriate sportspeople in Germany |
https://en.wikipedia.org/wiki/Csaba%20Bogd%C3%A1ny | Csaba Bogdány (born 15 May 1981) is a Hungarian former football player.
Club statistics
Honours
Mezőkövesd
NB II Kelet (1): 2012–13
References
External links
MLSZ
1981 births
People from Balassagyarmat
Sportspeople from Nógrád County
Living people
Hungarian men's footballers
Men's association football midfielders
Mezőkövesdi SE footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Nemzeti Bajnokság III players |
https://en.wikipedia.org/wiki/Viktor%20V%C3%A1rosi | Viktor Városi (born 21 October 1993) is a Hungarian football player who currently plays for Kozármisleny SE.
Club statistics
Updated to games played as of 26 October 2014.
External links
HLSZ.hu
1993 births
Living people
Footballers from Pécs
Hungarian men's footballers
Men's association football midfielders
Pécsi MFC players
Kozármisleny SE footballers
Szombathelyi Haladás footballers
Nemzeti Bajnokság I players
Hungarian expatriate men's footballers
Expatriate men's footballers in Austria
Hungarian expatriate sportspeople in Austria |
https://en.wikipedia.org/wiki/Musa%20Al-Zubi | Musa Khaled Ismail Al-Zubi (; born 11 February 1993) is a Jordanian professional footballer who plays as a left-back for Jordanian club Al-Salt.
Career statistics
International
References
External links
Living people
Jordanian men's footballers
1993 births
Al-Ahli SC (Amman) players
Shabab Al-Ordon SC players
Al-Ramtha SC players
Men's association football defenders
Al-Salt SC players
Jordanian Pro League players |
https://en.wikipedia.org/wiki/Iliya%20Munin | Iliya Munin (; born 16 January 1993) is a Bulgarian footballer who plays as a right-back for PFC Bansko.
Career statistics
Club
References
External links
Living people
1993 births
Bulgarian men's footballers
Men's association football defenders
FC Lyubimets players
PFC Beroe Stara Zagora players
PFC Litex Lovech players
PFC Bansko players
FC Vereya players
FC Dunav Ruse players
FC Septemvri Simitli players
First Professional Football League (Bulgaria) players |
https://en.wikipedia.org/wiki/African%20Cup%20of%20Champions%20Clubs%20and%20CAF%20Champions%20League%20records%20and%20statistics | This page details statistics of the African Cup of Champions Clubs and CAF Champions League.
General performances
By club
By nation
By semi-final appearances
years from 1997 to 2000 the two winners of the two groups were qualifying to the final directly with no semi final stage.
Records and statistics of Champions League era
Participation and group stage qualification
The following table shows teams that took part in the Champions League since its inception in 1997 (up to 2019-20 season), number and years of their appearances and group stage qualification. Number in bracket next to country name denotes number of teams that represented that country in the competition, while countries in red did not have group stage representative.
Total of 406 clubs participated in the Champions League era (teams included are those that found themselves in the draw, regarding of whether they played or not in that specific season), 183 teams participated only once (25 editions, including 2020-21 edition). 80 teams from 27 countries qualified to group stage (including 2020-21 season) while 28 countries did not have group stage representative.
After 2018 edition CAF moved its club competitions to autumn-spring format, meaning that editions after 2018 were played through two years (e.g. 2018–19, 2019–20). In the table below 2019 stands for 2018–19 season, 2020 for 2019–20 season and so on.
Teams are sorted by number of appearances. If the number is same for two or more teams, team that appeared before in their first appearance are listed first.
W denotes team that was part of the draw, but withdrew (or was ejected by the Confederation) before playing any game.
WG denotes team that qualified to group stage but was disqualified with all of its results annulled.
Records
Last updated on June11, 2023
Most titles: 11
Al-Ahly in 1982, 1987, 2001, 2005, 2006, 2008, 2012, 2013, 2020, 2021, 2023
Most appearances: 25
Al-Ahly (1998 to 2002 and 2004 to 2022-23)
Most consecutive appearances: 20
Al-Ahly (2004 to 2022-23)
Al-Hilal (2004 to 2021-22)
Most consecutive matches without losing: 20
Espérance recorded best undefeated streak through three seasons: 2018 (1 match), 2019 (12 matches), 2020 (7 matches)
Undefeated through entire season:
Espérance in 1994 in 10 matches (7-3-0) and in 2018-19 in 12 matches (8-4-0)
Al-Ahly in 2005 in 14 matches (9-5-0 record)
Most goals scored in a season: 36
Al-Ahly in 2020, 19 goals in preliminary rounds, 7 in group stage, 9 in knockout stage
Most goals conceded in a season: 25
Young Africans SC in 1998, 6 goals in preliminary rounds, 19 in group stage
Biggest win: 10 goals margin
Mamelodi Sundowns - Cote d'Or FC 11-1 (27 September 2019, First round)
Difaâ El Jadidi - Sport Bissau e Benfica 10-0 (10 February 2018 , First round)
Biggest aggregate win: 15 goals margin
Mamelodi Sundowns - Cote d'Or FC 16-1 (5-0 away, 11-1 at home; 14 September, 27 September 2019, First round)
Most goals scored in a single match: 12
|
https://en.wikipedia.org/wiki/2013%E2%80%9314%20FK%20Vardar%20season | The 2013–14 season was FK Vardar's 22nd consecutive season in First League. This article shows player statistics and all official matches that the club was played during the 2013–14 season.
In the winter break of the season, Vardar was faced a major ownership changes. Russian businessman and an owner of ŽRK Vardar and RK Vardar Sergei Samsonenko takes over the football club, with an ambitious plans to enter a group stage of UEFA Champions League.
Squad
As of 10 February 2014
Left club during season
Competitions
Supercup
First League
League table
Results summary
Results by round
Matches
Macedonian Cup
First round
Second round
UEFA Champions League
Second qualifying round
Statistics
Top scorers
References
FK Vardar seasons
Vardar
Vardar |
https://en.wikipedia.org/wiki/Weak%20trace-class%20operator | In mathematics, a weak trace class operator is a compact operator on a separable Hilbert space H with singular values the same order as the harmonic sequence.
When the dimension of H is infinite, the ideal of weak trace-class operators is strictly larger than the ideal of trace class operators, and has fundamentally different properties. The usual operator trace on the trace-class operators does not extend to the weak trace class. Instead the ideal of weak trace-class operators admits an infinite number of linearly independent quasi-continuous traces, and it is the smallest two-sided ideal for which all traces on it are singular traces.
Weak trace-class operators feature in the noncommutative geometry of French mathematician Alain Connes.
Definition
A compact operator A on an infinite dimensional separable Hilbert space H is weak trace class if μ(n,A) O(n−1), where μ(A) is the sequence of singular values. In mathematical notation the two-sided ideal of all weak trace-class operators is denoted,
where are the compact operators. The term weak trace-class, or weak-L1, is used because the operator ideal corresponds, in J. W. Calkin's correspondence between two-sided ideals of bounded linear operators and rearrangement invariant sequence spaces, to the weak-l1 sequence space.
Properties
the weak trace-class operators admit a quasi-norm defined by
making L1,∞ a quasi-Banach operator ideal, that is an ideal that is also a quasi-Banach space.
See also
Lp space
Spectral triple
Singular trace
Dixmier trace
References
Operator algebras
Hilbert spaces
Von Neumann algebras |
https://en.wikipedia.org/wiki/Calkin%20correspondence | In mathematics, the Calkin correspondence, named after mathematician John Williams Calkin, is a bijective correspondence between two-sided ideals of bounded linear operators of a separable infinite-dimensional Hilbert space and Calkin sequence spaces (also called rearrangement invariant sequence spaces). The correspondence is implemented by mapping an operator to its singular value sequence.
It originated from John von Neumann's study of symmetric norms on matrix algebras. It provides a fundamental classification and tool for the study of two-sided ideals of compact operators and their traces, by reducing problems about operator spaces to (more resolvable) problems on sequence spaces.
Definitions
A two-sided ideal J of the bounded linear operators B(H) on a separable Hilbert space H is a linear subspace such that AB and BA belong to J for all operators A from J and B from B(H).
A sequence space j within l∞ can be embedded in B(H) using an arbitrary orthonormal basis {en }n=0∞. Associate to a sequence a from j the bounded operator
where bra–ket notation has been used for the one-dimensional projections onto the subspaces spanned by individual basis vectors. The sequence of absolute values of the entries of a in decreasing order is called the decreasing rearrangement of a. The decreasing rearrangement can be denoted μ(n,a), n = 0, 1, 2, ... Note that it is identical to the singular values of the operator diag(a). Another notation for the decreasing rearrangement is a*.
A Calkin (or rearrangement invariant) sequence space is a linear subspace j of the bounded sequences l∞ such that if a is a bounded sequence and μ(n,a) ≤ μ(n,b), n 0, 1, 2, ..., for some b in j, then a belongs to j.
Correspondence
Associate to a two-sided ideal J the sequence space j given by
Associate to a sequence space j the two-sided ideal J given by
Here μ(A) and μ(a) are the singular values of the operators A and diag(a), respectively.
Calkin's Theorem states that the two maps are inverse to each other. We obtain,
Calkin correspondence: The two-sided ideals of bounded operators on an infinite dimensional separable Hilbert space and the Calkin sequence spaces are in bijective correspondence.
It is sufficient to know the association only between positive operators and positive sequences, hence the map μ: J+ → j+ from a positive operator to its singular values implements the Calkin correspondence.
Another way of interpreting the Calkin correspondence, since the sequence space j is equivalent as a Banach space to the operators in the operator ideal J that are diagonal with respect to an arbitrary orthonormal basis, is that two-sided ideals are completely determined by their diagonal operators.
Examples
Suppose H is a separable infinite-dimensional Hilbert space.
Bounded operators. The improper two-sided ideal B(H) corresponds to l∞.
Compact operators. The proper and norm closed two-sided ideal K(H) corresponds to c0, the space of sequences converging to zer |
https://en.wikipedia.org/wiki/Commutator%20subspace | In mathematics, the commutator subspace of a two-sided ideal of bounded linear operators on a separable Hilbert space is the linear subspace spanned by commutators of operators in the ideal with bounded operators.
Modern characterisation of the commutator subspace is through the Calkin correspondence and it involves the invariance of the Calkin sequence space of an operator ideal to taking Cesàro means. This explicit spectral characterisation reduces problems and questions about commutators and traces on two-sided ideals to (more resolvable) problems and conditions on sequence spaces.
History
Commutators of linear operators on Hilbert spaces came to prominence in the 1930s as they featured in the matrix mechanics, or Heisenberg, formulation of quantum mechanics. Commutator subspaces, though, received sparse attention until the 1970s. American mathematician Paul Halmos in 1954 showed that every bounded operator on a separable infinite dimensional Hilbert space is the sum of two commutators of bounded operators.
In 1971 Carl Pearcy and David Topping revisited the topic and studied commutator subspaces for Schatten ideals. As a student American mathematician Gary Weiss began to investigate spectral conditions for commutators of Hilbert–Schmidt operators.
British mathematician Nigel Kalton, noticing the spectral condition of Weiss, characterised all trace class commutators.
Kalton's result forms the basis for the modern characterisation of the commutator subspace.
In 2004 Ken Dykema, Tadeusz Figiel, Gary Weiss and Mariusz Wodzicki published the spectral characterisation of normal operators in the commutator subspace for every two-sided ideal of compact operators.
Definition
The commutator subspace of a two-sided ideal J of the bounded linear operators B(H) on a separable Hilbert space H is the linear span of operators in J of the form [A,B] = AB − BA for all operators A from J and B from B(H).
The commutator subspace of J is a linear subspace of J denoted by Com(J) or [B(H),J].
Spectral characterisation
The Calkin correspondence states that a compact operator A belongs to a two-sided ideal J if and only if the singular values μ(A) of A belongs to the Calkin sequence space j associated to J. Normal operators that belong to the commutator subspace Com(J) can characterised as those A such that μ(A) belongs to j and the Cesàro mean of the sequence μ(A) belongs to j. The following theorem is a slight extension to differences of normal operators (setting B 0 in the following gives the statement of the previous sentence).
Theorem. Suppose A,B are compact normal operators that belong to a two-sided ideal J. Then A − B belongs to the commutator subspace Com(J) if and only if
where j is the Calkin sequence space corresponding to J and μ(A), μ(B) are the singular values of A and B, respectively.
Provided that the eigenvalue sequences of all operators in J belong to the Calkin sequence space j there is a spectral characterisation for arbitrar |
https://en.wikipedia.org/wiki/Separation%20axiom | In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff.
The separation axioms are not fundamental axioms like those of set theory, but rather defining properties which may be specified to distinguish certain types of topological spaces. The separation axioms are denoted with the letter "T" after the German Trennungsaxiom ("separation axiom"), and increasing numerical subscripts denote stronger and stronger properties.
The precise definitions of the separation axioms have varied over time. Especially in older literature, different authors might have different definitions of each condition.
Preliminary definitions
Before we define the separation axioms themselves, we give concrete meaning to the concept of separated sets (and points) in topological spaces. (Separated sets are not the same as separated spaces, defined in the next section.)
The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points. It's not enough for elements of a topological space to be distinct (that is, unequal); we may want them to be topologically distinguishable. Similarly, it's not enough for subsets of a topological space to be disjoint; we may want them to be separated (in any of various ways). The separation axioms all say, in one way or another, that points or sets that are distinguishable or separated in some weak sense must also be distinguishable or separated in some stronger sense.
Let X be a topological space. Then two points x and y in X are topologically distinguishable if they do not have exactly the same neighbourhoods (or equivalently the same open neighbourhoods); that is, at least one of them has a neighbourhood that is not a neighbourhood of the other (or equivalently there is an open set that one point belongs to but the other point does not). That is, at least one of the points does not belong to the other's closure.
Two points x and y are separated if each of them has a neighbourhood that is not a neighbourhood of the other; that is, neither belongs to the other's closure. More generally, two subsets A and B of X are separated if each is disjoint from the other's closure, though the closures themselves do not have to be disjoint. Equivalently, each subset is included in an open set disjoint from the other subset. All of the remaining conditions for separation of sets may also be applied to points (or to a point and a set) by using singleton sets. Points x and y will be considered separated, by neighbourhoods, by closed neighbourhoods, by a continuous function, precisely by a function, if and only if their singleton sets {x} and {y} are separated according to the corresponding criterion.
Subsets A and B are separated by neighbourhoods if they have disjo |
https://en.wikipedia.org/wiki/Stuart%20Geman | Stuart Alan Geman (born March 23, 1949) is an American mathematician, known for influential contributions to computer vision, statistics, probability theory, machine learning, and the neurosciences. He and his brother, Donald Geman, are well known for proposing the Gibbs sampler, and for the first proof of convergence of the simulated annealing algorithm.
Biography
Geman was born and raised in Chicago. He was educated at the University of Michigan (B.S., Physics, 1971), Dartmouth Medical College (MS, Neurophysiology, 1973), and the Massachusetts Institute of Technology (Ph.D, Applied Mathematics, 1977).
Since 1977, he has been a member of the faculty at Brown University, where he has worked in the Pattern Theory group, and is currently the James Manning Professor of Applied Mathematics. He has received many honors and awards, including selection as a Presidential Young Investigator and as an ISI Highly Cited researcher. He is an elected member of the International Statistical Institute, and a fellow of the Institute of Mathematical Statistics and of the American Mathematical Society. He was elected to the US National Academy of Sciences in 2011.
Work
Geman's scientific contributions span work in probabilistic and statistical approaches to artificial intelligence, Markov random fields, Markov chain Monte Carlo (MCMC) methods, nonparametric inference, random matrices, random dynamical systems, neural networks, neurophysiology, financial markets, and natural image statistics. Particularly notable works include: the development of the Gibbs sampler, proof of convergence of simulated annealing, foundational contributions to the Markov random field ("graphical model") approach to inference in vision and machine learning, and work on the compositional foundations of vision and cognition.
Notes
Members of the United States National Academy of Sciences
1949 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Probability theorists
American statisticians
Fellows of the American Mathematical Society
Brown University faculty
Geisel School of Medicine alumni
University of Michigan College of Literature, Science, and the Arts alumni
Massachusetts Institute of Technology School of Science alumni |
https://en.wikipedia.org/wiki/BIT%20Numerical%20Mathematics | BIT Numerical Mathematics is a quarterly peer-reviewed mathematics journal that covers research in numerical analysis. It was established in 1961 by Carl Erik Fröberg and is published by Springer Science+Business Media. The name "BIT" is a reverse acronym of Tidskrift för Informationsbehandling (Swedish: Journal of Information Processing).
Previous editors-in-chief have been Carl Erik Fröberg (1961-1992), Åke Björck (1993-2002), Axel Ruhe (2003-2015), and Lars Eldén (2016). , the editor-in-chief is Gunilla Kreiss.
Peter Naur served as a member of the editorial board between the years 1960 and 1993, and Germund Dahlquist between 1962 and 1991.
Abstracting and indexing
The journal is abstracted and indexed in:
According to the Journal Citation Reports, the journal has a 2020 impact factor of 1.663.
References
External links
Mathematics journals
Academic journals established in 1961
Springer Science+Business Media academic journals
English-language journals
Quarterly journals |
https://en.wikipedia.org/wiki/Terry%20Cavanagh%20%28developer%29 | Terry Cavanagh ( ; born 1984) is an Irish video game designer based in London, England. After studying mathematics at Trinity College in Dublin, Cavanagh worked briefly as a market risk analyst before focusing on game development full-time. Many of his titles share a primitive, minimalist aesthetic. He has created over two dozen games, most notably VVVVVV, Super Hexagon, and Dicey Dungeons. He is credited as a programmer for Alphaland, a platform game by Jonas Kyratzes.
Cavanagh has stated that he prefers the personal nature of independent game development, its smaller scale enabling the personality of the creator to shine through in the final product.
Influences
Cavanagh cites the 1997 Japanese RPG Final Fantasy VII as his favorite game, crediting it as his inspiration for becoming a video game developer. In 2009 Cavanagh named interactive fiction writer Adam Cadre as his favorite developer.
Awards
Cavanagh's game VVVVVV won the 2010 IndieCade Festival in the category of "Fun/Compelling".
In 2014, Cavanagh was named to Forbes' annual "30 Under 30" list in the Games category.
In 2019, Cavanagh's game Dicey Dungeons won the 2019 IndieCade Grand Jury award.
Games
References
External links
The Escapist interview with Terry Cavanagh
The Spelunky Showlike — Making Generous Games with Terry Cavanagh
1984 births
Alumni of Trinity College Dublin
Browser game developers
Indie game developers
Irish expatriates in England
Irish video game designers
Living people
Video game programmers |
https://en.wikipedia.org/wiki/Zhiliang%20Ying | Zhiliang Ying (; born April 1960) is a Professor of Statistics in the Department of Statistics, Columbia University. He served as co-chair of the department.
He received his PhD from Columbia University in 1987, with Tze Leung Lai as his doctoral advisor. He was the Director of the Institute of Statistics at Rutgers University from 1997 to 2001. His wide research interests cover Survival Analysis, Sequential Analysis, Longitudinal Data Analysis, Stochastic Processes, Semiparametric Inference, Biostatistics and Educational Statistics. He is a co-editor of Statistica Sinica and has been Associate Editor of JASA, Statistica Sinica, Annals of Statistics, Biometrics, and Lifetime Data Analysis.
Ying has supervised, collaborated with and encouraged many researchers. He has written or co-authored more than 100 research articles in professional journals.
Selected honours and awards
Fellow, Institute of Mathematical Statistics (1995 election)
Fellow, American Statistical Association (1999 election)
The Morningside Gold Medal of Applied Mathematics 2004
The Distinguished Achievement Award 2007, International Chinese Statistical Association
Selected papers
Lin, D. Y., Wei, L. J., & Ying, Z. (1993). Checking the Cox model with cumulative sums of martingale-based residuals. Biometrika, 80(3), 557–572.
Lin, D. Y., & Ying, Z. (1994). Semiparametric analysis of the additive risk model. Biometrika, 81(1), 61–71.
Chang, H. H., & Ying, Z. (1996). A global information approach to computerized adaptive testing. Applied Psychological Measurement, 20(3), 213–229.
Jin, Z., Lin, D. Y., Wei, L. J., & Ying, Z. (2003). Rank‐based inference for the accelerated failure time model. Biometrika, 90(2), 341–353.
Ying, Z. (1993), A large sample study of rank estimation for censored regression data. The Annals of Statistics, 76–99.
References
1960 births
Columbia University faculty
Chinese statisticians
American statisticians
Living people
Fellows of the American Statistical Association
Fellows of the Institute of Mathematical Statistics
Mathematicians from Shanghai
Educators from Shanghai
Chinese science writers
Writers from Shanghai |
https://en.wikipedia.org/wiki/J-2%20ring | In commutative algebra, a J-0 ring is a ring such that the set of regular points, that is, points of the spectrum at which the localization is a regular local ring, contains a non-empty open subset, a J-1 ring is a ring such that the set of regular points is an open subset, and a J-2 ring is a ring such that any finitely generated algebra over the ring is a J-1 ring.
Examples
Most rings that occur in algebraic geometry or number theory are J-2 rings, and in fact it is not trivial to construct any examples of rings that are not. In particular all excellent rings are J-2 rings; in fact this is part of the definition of an excellent ring.
All Dedekind domains of characteristic 0 and all local Noetherian rings of dimension at most 1 are J-2 rings. The family of J-2 rings is closed under taking localizations and finitely generated algebras.
For an example of a Noetherian domain that is not a J-0 ring, take R to be the subring of the polynomial ring k[x1,x2,...] in infinitely many generators generated by the squares and cubes of all generators, and form the ring S from R by adjoining inverses to all elements not in any of the ideals generated by some xn. Then S is a 1-dimensional Noetherian domain that is not a J-0 ring. More precisely S has a cusp singularity at every closed point, so the set of non-singular points consists of just the ideal (0) and contains no nonempty open sets.
See also
Excellent ring
References
H. Matsumura, Commutative algebra , chapter 12.
Commutative algebra |
https://en.wikipedia.org/wiki/List%20of%20Johnson%20solids | In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. In 1966, Norman Johnson published a list which included all 92 solids, and gave them their names and numbers. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller proved in 1969 that Johnson's list was complete.
Other polyhedra can be constructed that have only approximately regular planar polygon faces, and are informally called near-miss Johnson solids; there can be no definitive count of them.
The various sections that follow have tables listing all 92 Johnson solids, and values for some of their most important properties. Each table allows sorting by column so that numerical values, or the names of the solids, can be sorted in order.
Vertices, edges, faces, and symmetry
Legend:
Jn – Johnson solid number
Net – Flattened (unfolded) image
V – Number of vertices
E – Number of edges
F – Number of faces (total)
F3–F10 – Number of faces by side counts
The square pyramid has the fewest vertices (5), the fewest edges (8), and the fewest faces (5).
The triaugmented truncated dodecahedron has the most vertices (75) and the most edges (135). It also has the highest number of faces (62), along with the gyrate rhombicosidodecahedron , the parabigyrate rhombicosidodecahedron , the metabigyrate rhombicosidodecahedron , and the trigyrate rhombicosidodecahedron .
Surface area
Since all faces of Johnson solids are regular polygons with 3, 4, 5, 6, 8, or 10 sides, and since all these polygons have the same edge length , the surface area of a Johnson solid can be calculated as
where the are the polygonal face counts in the previous table and
is the area of a regular polygon with sides of length . In terms of radicals, one has
resulting in the following table of surface areas.
For a fixed edge length, the triangular dipyramid has the smallest surface area and the triaugmented truncated dodecahedron has the largest, more than 40 times larger.
Volume
The following table lists the volume of each Johnson solid. Here is the volume (not the number of vertices, as in the first table) and is the edge length.
The source for this table is the PolyhedronData[..., "Volume"] command in Wolfram Research's Mathematica.
These volumes can be calculated from a set of vertex coordinates; such coordinates are known for all 92 Johnson solids. A conceptually simple approach is to triangulate the surface of the solid (for example, by adding an extra point in the center of each non-triangular face) and choose some interior point as an "origin" so that the interior can be subdivided into irregular tetrahedra. Each tetrahedron has one vertex at the origin inside and three vertices on the surface. The volume of the solid is then the sum of the volumes of these tetrahedra. There is a simple formula for the volume |
https://en.wikipedia.org/wiki/OFF%20%28file%20format%29 | OFF (Object File Format) is a geometry definition file format containing the description of the composing polygons of a geometric object. It can store 2D or 3D objects, and simple extensions allow it to represent higher-dimensional objects as well. Though originally developed for Geomview, a geometry visualization software, other software has adapted the simple standard.
Composition
The composition of a standard OFF file is as follows:
First line (optional): the letters OFF to mark the file type.
Second line: the number of vertices, number of faces, and number of edges, in order (the latter can be ignored by writing 0 instead).
List of vertices: X, Y and Z coordinates.
List of faces: number of vertices, followed by the indexes of the composing vertices, in order (indexed from zero).
Optionally, the RGB values for the face color can follow the elements of the faces.
The four-dimensional OFF format, most notably used by Stella4D, which allows visualization of four-dimensional objects, has a few minor differences:
First line (optional): the letters 4OFF to mark the file type.
Second line: the number of vertices, number of faces, number of edges, and number of cells, in order (the number of edges can be ignored).
List of vertices: X, Y, Z and W coordinates.
List of faces: number of vertices, followed by the indexes of the composing vertices, in order (indexed from zero).
List of cells: number of faces, followed by the indexes of the composing faces, in order (indexed from zero).
Optionally, the RGB values for the cell color can follow the elements of the cells.
Comments are marked with a pound sign (#): these are not read by the software.
Example
OFF
# cube.off
# A cube
8 6 12
1.0 0.0 1.4142
0.0 1.0 1.4142
-1.0 0.0 1.4142
0.0 -1.0 1.4142
1.0 0.0 0.0
0.0 1.0 0.0
-1.0 0.0 0.0
0.0 -1.0 0.0
4 0 1 2 3 255 0 0 #red
4 7 4 0 3 0 255 0 #green
4 4 5 1 0 0 0 255 #blue
4 5 6 2 1 0 255 0
4 3 2 6 7 0 0 255
4 6 5 4 7 255 0 0
See also
Wavefront .obj file
STL (file format)
PLY (file format) is an alternative file format offering more flexibility than most stereolithography applications.
References
External links
Description of the OFF format in the Geomview manual
CAD file formats |
https://en.wikipedia.org/wiki/Devex%20algorithm | In applied mathematics, the devex algorithm is a pivot rule for the simplex method developed by Paula M. J. Harris. It identifies the steepest-edge approximately in its search for the optimal solution.
References
Algorithms |
https://en.wikipedia.org/wiki/Manuel%20Kauers | Manuel Kauers (born 20 February 1979 in Lahnstein, West Germany) is a German mathematician and computer scientist. He is
working on computer algebra and its applications to discrete mathematics. He is currently
professor for algebra at Johannes Kepler University (JKU) in Linz, Austria, and leader of the Institute for Algebra at JKU.
Before that, he was affiliated with that university's Research Institute for Symbolic Computation (RISC).
Kauers studied computer science at the University of Karlsruhe in Germany from 1998 to 2002 and then moved to RISC, where he completed his PhD in symbolic computation in 2005 under the supervision of Peter Paule. He earned his habilitation in mathematics from JKU in 2008.
Together with Doron Zeilberger and Christoph Koutschan, Kauers proved two famous open conjectures in combinatorics using large scale
computer algebra calculations. Both proofs appeared in the Proceedings of the National Academy of Sciences. The first concerned a conjecture
formulated by Ira Gessel on the number of certain lattice walks restricted to the quarter plane. This result was later generalized by Alin Bostan and Kauers when they showed, also using computer algebra, that the generating function for these walks is algebraic. The second conjecture proven by Kauers, Koutschan and Zeilberger was the so-called q-TSPP conjecture, a product formula for the orbit generating function of totally symmetric plane partitions, which was formulated by George Andrews and David Robbins in the early 1980s.
In 2009, Kauers received the Start-Preis, which is considered the most prestigious award for young scientists in Austria.
In 2016, with Christoph Koutschan and Doron Zeilberger he received the David P. Robbins prize of the American Mathematical Society.
References
1979 births
Living people
20th-century German mathematicians
21st-century German mathematicians
German expatriates in Austria
German computer scientists
Karlsruhe Institute of Technology alumni
Academic staff of Johannes Kepler University Linz |
https://en.wikipedia.org/wiki/Rich%C3%A1rd%20Kozma | Richárd Kozma (born 1 October 1994 in Nyíregyháza) is a Hungarian football player who currently plays for Budapest Honvéd FC.
Club statistics
Updated to games played as of 4 December 2013.
References
MLSZ
1994 births
Living people
Footballers from Nyíregyháza
Hungarian men's footballers
Men's association football midfielders
Budapest Honvéd FC players
Nemzeti Bajnokság I players |
https://en.wikipedia.org/wiki/Bundesliga%20records%20and%20statistics | The Bundesliga was founded as the top tier of German football at the start of the 1963–64 season. The following is a list of records attained in the Bundesliga since the league's inception.
Statistics are accurate as of the 2023–24 season.
Club records
Titles
Highest number of titles won: 32 by Bayern Munich (1968–69, 1971–72, 1972–73, 1973–74, 1979–80, 1980–81, 1984–85, 1985–86, 1986–87, 1988–89, 1989–90, 1993–94, 1996–97, 1998–99, 1999–2000 2000–01, 2002–03, 2004–05, 2005–06, 2007–08, 2009–10, 2012–13, 2013–14, 2014–15, 2015–16, 2016–17, 2017–18, 2018–19, 2019–20, 2020–21, 2021–22, 2022–23)
Champions
Highest number of games left when becoming champions: 7 by Bayern Munich (2013–14)
Earliest point of time in a year for a team to be crowned champions: 25 March by Bayern Munich (2013–14)
Latest point of time in a year for a team to be crowned champions: 28 June by Bayern Munich (1971–72)
Highest number of matchdays being league leaders: 858 by Bayern Munich
Highest number of matchdays being league leaders in a season: 34 by Bayern Munich (1968–69, 1972–73, 1984–85, 2007–08 and 2012–13)
Lowest number of matchdays being league leaders in a season for the champions: 1 by Bayern Munich (1985–86)
Lowest number of seasons before becoming champions after being promoted: 1 by 1. FC Kaiserslautern (Promotion: 1997; Champions: 1997–98)
Lowest number of seasons before getting relegated for the champions: 1 by 1. FC Nürnberg (Champions: 1967–68; Relegation: 1968–69)
Points
Highest number of points: 3,995 by Bayern Munich
Highest number of points in a season: 91 by Bayern Munich (2012–13)
Highest number of points in a season for the runners-up: 78 by Borussia Dortmund (2015–16)
Highest number of points in a season opening half: 47 by Bayern Munich (2013–14)
Highest number of points in a season closing half: 49 by Bayern Munich (2012–13 and 2019–20)
Highest number of points in a season away: 47 by Bayern Munich (2012–13)
Highest number of points in a season at home: 49 by Schalke 04 (33:1) (1971–72), Bayern Munich (33:1) (1972–73) and VfL Wolfsburg (2008–09)
Highest percentage of total possible points in a season: 89.22 by Bayern Munich (2012–13) (91 points out of a possible 102)
Highest percentage of total possible points in a season opening half: 92.16 by Bayern Munich (2013–14) (47 points out of a possible 51)
Highest percentage of total possible points in a season closing half: 96.08 by Bayern Munich (2012–13 and 2019–20) (49 points out of a possible 51)
Highest percentage of total possible points in a season at home: 96.08 by Schalke 04 (1971–72), Bayern Munich (1972–73) and VfL Wolfsburg (2008–09) (49 points out of a possible 51) (Based on 16 wins and a draw with 3 points per win)
Highest percentage of total possible points in a season away: 92.16 by Bayern Munich (2012–13) (47 points out of a possible 51)
Biggest lead in points after a season opening half: 11 by Bayern Munich (45) upon VfL Wolfsburg (34) (2014–15)
Biggest m |
https://en.wikipedia.org/wiki/Bi-twin%20chain | In number theory, a bi-twin chain of length k + 1 is a sequence of natural numbers
in which every number is prime.
The numbers form a Cunningham chain of the first kind of length , while forms a Cunningham chain of the second kind. Each of the pairs is a pair of twin primes. Each of the primes for is a Sophie Germain prime and each of the primes for is a safe prime.
Largest known bi-twin chains
q# denotes the primorial 2×3×5×7×...×q.
, the longest known bi-twin chain is of length 8.
Relation with other properties
Related chains
Cunningham chain
Related properties of primes/pairs of primes
Twin primes
Sophie Germain prime is a prime such that is also prime.
Safe prime is a prime such that is also prime.
Notes and references
Prime numbers |
https://en.wikipedia.org/wiki/Senad%20Husi%C4%87 | Senad Husić (born 12 April 1990) is a Bosnian footballer who plays for German amateur club SC Pfullendorf.
Career statistics
Honours
Diósgyőr
Hungarian League Cup (1): 2013–14
References
External links
MLSZ
Profile – Pfullendorf
1990 births
Living people
People from Kalesija
Sportspeople from Tuzla Canton
Men's association football fullbacks
Bosnia and Herzegovina men's footballers
Bosnia and Herzegovina men's under-21 international footballers
NK Zvijezda Gradačac players
Diósgyőri VTK players
FK Željezničar Sarajevo players
NK Čelik Zenica players
IFK Åmål players
KF Llapi players
SC Pfullendorf players
Premier League of Bosnia and Herzegovina players
Nemzeti Bajnokság I players
Football Superleague of Kosovo players
Bosnia and Herzegovina expatriate men's footballers
Expatriate men's footballers in Hungary
Expatriate men's footballers in Sweden
Expatriate men's footballers in Kosovo
Bosnia and Herzegovina expatriate sportspeople in Hungary
Bosnia and Herzegovina expatriate sportspeople in Sweden
Bosnia and Herzegovina expatriate sportspeople in Kosovo
Expatriate men's footballers in Germany
Bosnia and Herzegovina expatriate sportspeople in Germany |
https://en.wikipedia.org/wiki/Edge-contracted%20icosahedron | In geometry, an edge-contracted icosahedron is a polyhedron with 18 triangular faces, 27 edges, and 11 vertices.
Construction
It can be constructed from the regular icosahedron, with one edge contraction, removing one vertex, 3 edges, and 2 faces. This contraction distorts the circumscribed sphere original vertices. With all equilateral triangle faces, it has 2 sets of 3 coplanar equilateral triangles (each forming a half-hexagon), and thus is not a Johnson solid.
If the sets of three coplanar triangles are considered a single face (called a triamond), it has 10 vertices, 22 edges, and 14 faces, 12 triangles and 2 triamonds .
It may also be described as having a hybrid square-pentagonal antiprismatic core (an antiprismatic core with one square base and one pentagonal base); each base is then augmented with a pyramid.
Related polytopes
The dissected regular icosahedron is a variant topologically equivalent to the sphenocorona with the two sets of 3 coplanar faces as trapezoids. This is the vertex figure of a 4D polytope, grand antiprism. It has 10 vertices, 22 edges, and 12 equilateral triangular faces and 2 trapezoid faces.
In chemistry
In chemistry, this polyhedron is most commonly called the octadecahedron, for 18 triangular faces, and represents the closo-boranate .
Related polyhedra
The elongated octahedron is similar to the edge-contracted icosahedron, but instead of only one edge contracted, two opposite edges are contracted.
References
External links
The Convex Deltahedra, And the Allowance of Coplanar Faces
Polyhedra |
https://en.wikipedia.org/wiki/Elongated%20octahedron | In geometry, an elongated octahedron is a polyhedron with 8 faces (4 triangular, 4 isosceles trapezoidal), 14 edges, and 8 vertices.
As a deltahedral hexadecahedron
A related construction is a hexadecahedron, 16 triangular faces, 24 edges, and 10 vertices. Starting with the regular octahedron, it is elongated along one axes, adding 8 new triangles. It has 2 sets of 3 coplanar equilateral triangles (each forming a half-hexagon), and thus is not a Johnson solid.
If the sets of coplanar triangles are considered a single isosceles trapezoidal face (a triamond), it has 8 vertices, 14 edges, and 8 faces - 4 triangles and 4 triamonds . This construction has been called a triamond stretched octahedron.
As a folded hexahedron
Another interpretation can represent this solid as a hexahedron, by considering pairs of trapezoids as a folded regular hexagon. It will have 6 faces (4 triangles, and 2 hexagons), 12 edges, and 8 vertices.
It could also be seen as a folded tetrahedron also seeing pairs of end triangles as a folded rhombus. It would have 8 vertices, 10 edges, and 4 faces.
Cartesian coordinates
The Cartesian coordinates of the 8 vertices of an elongated octahedron, elongated in the x-axis, with edge length 2 are:
( ±1, 0, ±2 )
( ±2, ±1, 0 ).
The 2 extra vertices of the deltahedral variation are:
( 0, ±1, 0 ).
Related polyhedra and honeycombs
In the special case, where the trapezoid faces are squares or rectangles, the pairs of triangles becoming coplanar and the polyhedron's geometry is more specifically a right rhombic prism.
This polyhedron has a highest symmetry as D2h symmetry, order 8, representing 3 orthogonal mirrors. Removing one mirror between the pairs of triangles divides the polyhedron into two identical wedges, giving the names octahedral wedge, or double wedge. The half-model has 8 triangles and 2 squares.
It can also be seen as the augmentation of 2 octahedrons, sharing a common edge, with 2 tetrahedrons filling in the gaps. This represents a section of a tetrahedral-octahedral honeycomb. The elongated octahedron can thus be used with the tetrahedron as a space-filling honeycomb.
See also
Orthobifastigium
Edge-contracted icosahedron
Elongated dodecahedron
Elongated gyrobifastigium
References
p.172 tetrahedra-octahedral packing
H. Martyn Cundy Deltahedra. Math. Gaz. 36, 263-266, Dec 1952.
H. Martyn Cundy and A. Rollett. "Deltahedra". §3.11 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 142–144, 1989.
Charles W. Trigg An Infinite Class of Deltahedra, Mathematics Magazine, Vol. 51, No. 1 (Jan., 1978), pp. 55–57
Contains the original enumeration of the 92 solids and the conjecture that there are no others.
The first proof that there are only 92 Johnson solids: see also
External links
The Convex Deltahedra, And the Allowance of Coplanar Faces
Polyhedra |
https://en.wikipedia.org/wiki/Polykay | In statistics, a polykay, or generalised k-statistic, (denoted ) is a statistic defined as a linear combination of sample moments.
Etymology
The word polykay was coined by American mathematician John Tukey in 1956, from poly, "many" or "much", and kay, the phonetic spelling of the letter "k", as in k-statistic.
References
Symmetric functions
Statistical inference |
https://en.wikipedia.org/wiki/Burstiness | In statistics, burstiness is the intermittent increases and decreases in activity or frequency of an event.
One measure of burstiness is the Fano factor—a ratio between the variance and mean of counts.
Burstiness is observable in natural phenomena, such as natural disasters, or other phenomena, such as network/data/email network traffic or vehicular traffic. Burstiness is, in part, due to changes in the probability distribution of inter-event times. Distributions of bursty processes or events are characterised by heavy, or fat, tails.
Burstiness of inter-contact time between nodes in a time-varying network can decidedly slow spreading processes over the network. This is of great interest for studying the spread of information and disease.
Burstiness score
One relatively simple measure of burstiness is burstiness score. The burstiness score of a subset of time period relative to an event is a measure of how often appears in compared to its occurrences in . It is defined by
Where is the total number of occurrences of event in subset and is the total number of occurrences of in .
Burstiness score can be used to determine if is a "bursty period" relative to . A positive score says that occurs more often during subset than over total time , making a bursty period. A negative score implies otherwise.
See also
Burst transmission
Poisson clumping
Time-varying network
References
Markov processes
Applied statistics |
https://en.wikipedia.org/wiki/Fifth%20power%20%28algebra%29 | In arithmetic and algebra, the fifth power or sursolid of a number n is the result of multiplying five instances of n together:
.
Fifth powers are also formed by multiplying a number by its fourth power, or the square of a number by its cube.
The sequence of fifth powers of integers is:
0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, 3200000, 4084101, 5153632, 6436343, 7962624, 9765625, ...
Properties
For any integer n, the last decimal digit of n5 is the same as the last (decimal) digit of n, i.e.
By the Abel–Ruffini theorem, there is no general algebraic formula (formula expressed in terms of radical expressions) for the solution of polynomial equations containing a fifth power of the unknown as their highest power. This is the lowest power for which this is true. See quintic equation, sextic equation, and septic equation.
Along with the fourth power, the fifth power is one of two powers k that can be expressed as the sum of k − 1 other k-th powers, providing counterexamples to Euler's sum of powers conjecture. Specifically,
(Lander & Parkin, 1966)
See also
Eighth power
Seventh power
Sixth power
Fourth power
Cube (algebra)
Square (algebra)
Perfect power
Footnotes
References
Integers
Number theory
Elementary arithmetic
Integer sequences
Unary operations
Figurate numbers |
https://en.wikipedia.org/wiki/Poretsky%27s%20law%20of%20forms | In Boolean algebra, Poretsky's law of forms shows that the single Boolean equation is equivalent to if and only if , where represents exclusive or.
The law of forms was discovered by Platon Poretsky.
See also
Archie Blake (mathematician)
Blake–Poretsky law
References
(NB. This publication is also referred to as "On methods of solution of logical equalities and on inverse method of mathematical logic".)
[http://www2.fiit.stuba.sk/~kvasnicka/Free%20books/Brown_Boolean%20Reasoning.pdf]
External links
"Transhuman Reflections - Poretsky Form to Solve"
Boolean algebra |
https://en.wikipedia.org/wiki/Inclusion%20%28Boolean%20algebra%29 | In Boolean algebra, the inclusion relation is defined as and is the Boolean analogue to the subset relation in set theory. Inclusion is a partial order.
The inclusion relation can be expressed in many ways:
The inclusion relation has a natural interpretation in various Boolean algebras: in the subset algebra, the subset relation; in arithmetic Boolean algebra, divisibility; in the algebra of propositions, material implication; in the two-element algebra, the set { (0,0), (0,1), (1,1) }.
Some useful properties of the inclusion relation are:
The inclusion relation may be used to define Boolean intervals such that . A Boolean algebra whose carrier set is restricted to the elements in an interval is itself a Boolean algebra.
References
, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition, 2003, p. 34, 52
Boolean algebra |
https://en.wikipedia.org/wiki/Paul%20Poulet | Paul Poulet (1887–1946) was a self-taught Belgian mathematician who made several important contributions to number theory, including the discovery of sociable numbers in 1918. He is also remembered for calculating the pseudoprimes to base two, first up to 50 million in 1926, then up to 100 million in 1938. These are now often called Poulet numbers in his honour (they are also known as Fermatians or Sarrus numbers). In 1925, he published forty-three new multiperfect numbers, including the first two known octo-perfect numbers. His achievements are particularly remarkable given that he worked without the aid of modern computers and calculators.
Career
Poulet published at least two books about his mathematical work, Parfaits, amiables et extensions (1918) (Perfect and Amicable Numbers and Their Extensions) and La chasse aux nombres (1929) (The Hunt for Numbers). He wrote the latter in the French village of Lambres-lez-Aire in the Pas-de-Calais, a short distance across the border with Belgium. Both were published by éditions Stevens of Brussels.
Sociable chains
In a sociable chain, or aliquot cycle, a sequence of divisor-sums returns to the initial number. These are the two chains Poulet described in 1918:
12496 → 14288 → 15472 → 14536 → 14264 → 12496 (5 links)
14316 → 19116 → 31704 → 47616 → 83328 → 177792 → 295488 → 629072 → 589786 → 294896 → 358336 → 418904 → 366556 → 274924 → 275444 → 243760 → 376736 → 381028 → 285778 → 152990 → 122410 → 97946 → 48976 → 45946 → 22976 → 22744 → 19916 → 17716 → 14316 (28 links)
The second chain remains by far the longest known, despite the exhaustive computer searches begun by the French mathematician Henri Cohen in 1969. Poulet introduced sociable chains in a paper in the journal L'Intermédiaire des Mathématiciens #25 (1918). The paper ran like this:
If one considers a whole number a, the sum b of its proper divisors, the sum c of the proper divisors of b, the sum d of the proper divisors of c, and so on, one creates a sequence that, continued indefinitely, can develop in three ways:
The most frequent is to arrive at a prime number, then at unity [i.e., 1]. The sequence ends here.
One arrives at a previously calculated number. The sequence is indefinite and periodic. If the period is one, the number is perfect. If the period is two, the numbers are amicable. But the period can be longer than two, involving what I will call, to keep the same terminology, sociable numbers. For example, the number 12496 creates a period of four terms, the number 14316 a period of 28 terms.
Finally, in some cases a sequence creates very large numbers that become impossible to resolve into divisors. For example, the number 138.
This being so, I ask:
If this third case really exists or if, calculating long enough, one would not necessarily end in one of the two other cases, as I am driven to believe.
If sociable chains other than those above can be found, especially chains of three terms. (It will be pointless, I think, to |
https://en.wikipedia.org/wiki/Andrei%20Coroian | Andrei Ion Coroian (born 28 January 1991) is a Romanian footballer.
Club statistics
Updated to games played as of 30 November 2014.
External links
profile at the playersagent.com
MLSZ
1991 births
Living people
People from Câmpia Turzii
Romanian men's footballers
Romania men's youth international footballers
Men's association football midfielders
Brescia Calcio players
Liga II players
CF Liberty Oradea players
FC Bihor Oradea (1958) players
Nemzeti Bajnokság I players
Kaposvári Rákóczi FC players
Pápai FC footballers
CS Minaur Baia Mare (football) players
Romanian expatriate men's footballers
Romanian expatriate sportspeople in Italy
Expatriate men's footballers in Italy
Romanian expatriate sportspeople in Hungary
Expatriate men's footballers in Hungary
Footballers from Cluj County |
https://en.wikipedia.org/wiki/Adrian%20M%C4%83rku%C8%99 | Adrian Alexandru Mărkuș (born 4 October 1992, in Oțelu Roșu) is a Romanian footballer who plays as a forward.
Club statistics
Updated to games played as of 8 December 2013.
External links
1992 births
Living people
Footballers from Caraș-Severin County
Romanian men's footballers
Men's association football forwards
Romania men's youth international footballers
Romania men's under-21 international footballers
CF Liberty Oradea players
FC UTA Arad players
Liga I players
Liga II players
Nemzeti Bajnokság I players
FC Bihor Oradea (1958) players
Kaposvári Rákóczi FC players
CS Gaz Metan Mediaș players
FC Viitorul Constanța players
FC Olimpia Satu Mare players
Metropolitan Police F.C. players
Haringey Borough F.C. players
Romanian expatriate men's footballers
Romanian expatriate sportspeople in Hungary
Expatriate men's footballers in Hungary
Romanian expatriate sportspeople in England
Expatriate men's footballers in England |
https://en.wikipedia.org/wiki/Chebyshev%20integral | In mathematics, the Chebyshev integral, named after Pafnuty Chebyshev, is
where is an incomplete beta function.
References
Gamma and related functions |
https://en.wikipedia.org/wiki/K-statistic | In statistics, a k-statistic is a minimum-variance unbiased estimator of a cumulant.
References
External links
k-Statistic on Wolfram MathWorld
kStatistics, an R package for calculating k-statistics
Estimator |
https://en.wikipedia.org/wiki/Beto%20Almeida | Roberto de Almeida (born April 5, 1955), commonly known as Beto Almeida, is a Brazilian football manager.
Managerial statistics
References
External links
1955 births
Living people
Footballers from Porto Alegre
Brazilian football managers
Expatriate football managers in Japan
Expatriate football managers in Bahrain
Expatriate football managers in Paraguay
J2 League managers
Kawasaki Frontale managers
Esporte Clube São José managers
Clube Esportivo Bento Gonçalves managers
Esporte Clube Juventude managers
Grêmio Esportivo Brasil managers
Veranópolis Esporte Clube Recreativo e Cultural managers
Club Guaraní managers
Esporte Clube Pelotas managers
Esporte Clube São Luiz managers
Centro Sportivo Alagoano managers
Agremiação Sportiva Arapiraquense managers |
https://en.wikipedia.org/wiki/F%C3%A1bio%20Guar%C3%BA | Fábio Nascimento de Oliveira (born 3 September 1987 in Guarulhos), known as Fábio Guarú, is a Brazilian professional footballer who plays for Monori SE in Hungary.
Club statistics
Updated to games played as of 28 September 2014.
References
MLSZ
External links
1987 births
Living people
Footballers from Guarulhos
Brazilian men's footballers
Men's association football defenders
Campeonato Brasileiro Série B players
Nemzeti Bajnokság I players
Clube Náutico Capibaribe players
Associação Ferroviária de Esportes players
Clube Atlético Bragantino players
Szigetszentmiklósi TK footballers
Puskás Akadémia FC players
Békéscsaba 1912 Előre footballers
FK Csíkszereda Miercurea Ciuc players
Tiszakécske FC footballers
Monori SE players
Brazilian expatriate men's footballers
Brazilian expatriate sportspeople in Hungary
Expatriate men's footballers in Hungary
Brazilian expatriate sportspeople in Romania
Expatriate men's footballers in Romania |
https://en.wikipedia.org/wiki/Zero-truncated%20Poisson%20distribution | In probability theory, the zero-truncated Poisson (ZTP) distribution is a certain discrete probability distribution whose support is the set of positive integers. This distribution is also known as the conditional Poisson distribution or the positive Poisson distribution. It is the conditional probability distribution of a Poisson-distributed random variable, given that the value of the random variable is not zero. Thus it is impossible for a ZTP random variable to be zero. Consider for example the random variable of the number of items in a shopper's basket at a supermarket checkout line. Presumably a shopper does not stand in line with nothing to buy (i.e., the minimum purchase is 1 item), so this phenomenon may follow a ZTP distribution.
Since the ZTP is a truncated distribution with the truncation stipulated as , one can derive the probability mass function from a standard Poisson distribution ) as follows:
The mean is
and the variance is
Parameter estimation
The method of moments estimator for the parameter is obtained by solving
where is the sample mean.
This equation does not have a closed-form solution. In practice, a solution may be found using numerical methods.
Generating zero-truncated Poisson-distributed random variables
Random variables sampled from the Zero-truncated Poisson distribution may be achieved using algorithms derived from Poisson distributing sampling algorithms.
init:
Let k ← 1, t ← e−λ / (1 - e−λ) * λ, s ← t.
Generate uniform random number u in [0,1].
while s < u do:
k ← k + 1.
t ← t * λ / k.
s ← s + t.
return k.
The cost of the procedure above is linear in k, which may be large for large values of . Given access to an efficient sampler for non-truncated Poisson random variates, a non-iterative approach involves sampling from a truncated exponential distribution representing the time of the first event in a Poisson point process, conditional on such an event existing. A simple NumPy implementation is:
def sample_zero_truncated_poisson(rate):
u = np.random.uniform(np.exp(-rate), 1)
t = -np.log(u)
return 1 + np.random.poisson(rate - t)
References
Discrete distributions
Poisson distribution |
https://en.wikipedia.org/wiki/Devavrat%20Shah | Devavrat Shah is a professor in the Electrical Engineering and Computer Science department at MIT. He is director of the Statistics and Data Science Center at MIT. He received a B.Tech. degree in computer science from IIT Bombay in 1999 and a Ph.D. in computer science from Stanford University in 2004, where his thesis was completed under the supervision of Balaji Prabhakar.
Research
Shah's research focuses on the theory of large complex networks which includes network algorithms, stochastic networks, network information theory and large scale statistical inference. His work has had significant impact both in the development of theoretical tools and in its practical application. This is highlighted by the "Best Paper" awards he has received from top publication venues such as ACM SIGMETRICS, IEEE INFOCOM and NIPS. Additionally, his work has been recognized by the INFORMS Applied Probability Society via the Erlang Prize, given for outstanding contributions to applied probability by a researcher not more than 9 years from their PhD and the ACM SIGMETRICS Rising Star award, given for outstanding contributions to computer/communication performance evaluation by a research not more than 7 years from their PhD. He is a young distinguished alumnus of his alma mater IIT Bombay.
Awards
Shah has received many awards, including
Erlang Prize from Applied Probability Society of INFORMS 2010
ACM SIGMETRICS/Performance best student paper award 2009 (supervised)
ACM SIGMETRICS Rising Star Award 2008
Neural Information Processing System (NIPS) outstanding paper award 2008 (supervised)
ACM SIGMETRICS/Performance best paper award 2006
NSF CAREER Award 2006
George B. Dantzig best dissertation award from INFORMS 2005
IEEE INFOCOM best paper award 2004
President of India Gold Medal at Indian Institute of Technology-Bombay 1999
Industry
Shah co-founded Celect, Inc. in 2013.
References
Stanford University alumni
MIT School of Engineering faculty
Indian computer scientists
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/List%20of%20algebras | This is a list of possibly nonassociative algebras. An algebra is a module, wherein you can also multiply two module elements. (The multiplication in the module is compatible with multiplication-by-scalars from the base ring).
*-algebra
Akivis algebra
Algebra for a monad
Albert algebra
Alternative algebra
Azumaya algebra
Banach algebra
Birman–Wenzl algebra
Boolean algebra
Borcherds algebra
Brauer algebra
C*-algebra
Central simple algebra
Clifford algebra
Cluster algebra
Dendriform algebra
Differential graded algebra
Differential graded Lie algebra
Exterior algebra
F-algebra
Filtered algebra
Flexible algebra
Freudenthal algebra
Genetic algebra
Geometric algebra
Gerstenhaber algebra
Graded algebra
Griess algebra
Group algebra
Group algebra of a locally compact group
Hall algebra
Hecke algebra of a locally compact group
Heyting algebra
Hopf algebra
Hurwitz algebra
Hypercomplex algebra
Incidence algebra
Iwahori–Hecke algebra
Jordan algebra
Kac–Moody algebra
Kleene algebra
Leibniz algebra
Lie algebra
Lie superalgebra
Malcev algebra
Matrix algebra
Non-associative algebra
Octonion algebra
Pre-Lie algebra
Poisson algebra
Process algebra
Quadratic algebra
Quaternion algebra
Rees algebra
Relation algebra
Relational algebra
Schur algebra
Semisimple algebra
Separable algebra
Shuffle algebra
Sigma-algebra
Simple algebra
Structurable algebra
Supercommutative algebra
Symmetric algebra
Tensor algebra
Universal enveloping algebra
Vertex operator algebra
von Neumann algebra
Weyl algebra
Zinbiel algebra
This is a list of fields of algebra.
Linear algebra
Homological algebra
Universal algebra
Algebras |
https://en.wikipedia.org/wiki/Science%20Expo | Science Expo was a Canadian national student run non-profit organization that connected high-achieving youth to innovators and STEM (Science, Technology, Engineering and Mathematics) opportunities. In 2017, they merged with another Canadian non-profit, the Foundation for Student Science and Technology (FSST) which ran a similar platform, RISE (Research in Science and Engineering). They exist to bridge the gap between the two parties and provide opportunities to excel which features youth-to-youth interactions to help them reach their potential. It is the place where ambitious youth can be at the forefront of innovation, and where young people have the chance to enrich their skills, be empowered to learn and grow, and be inspired to explore their passions and dreams.
Science Expo is active from Vancouver to Toronto, with a network of 150 leaders reaching 80 high schools, representing a student body of over 120,000. They are best known for their annual conference, which has attracted thousands of high school students across Ontario to convene and connect with fellow like-minded students as well as leading innovators and scientists.
History
Five individuals who met at the 2009 Canada Wide Science Fair and wanted to encourage more students to get involved in STEM opportunities founded Science Expo in 2010. Two months later, the first Science Expo, a 2-hour symposium, was held in a Guelph, Ontario high school auditorium. It attracted over 200 students, parents, and teachers, and featured Ph.D candidate and science fair guru Mubdi Rahman as the keynote speaker.
The next year, the Science Expo team held its first full day conference at the University of Waterloo, and marked a milestone by successfully registering as a non-for-profit organization. In addition, a group of 5 interns joined the team that summer, 4 of whom stayed on as executives in the following year. Science Expo experienced a rapid surge in growth in 2012, expanding both its team size to a group of 13 high school and university students, and its reach to include the Greater Toronto Area. Science Expo 2012 was held at the Ontario Science Centre, and featured Nobel Prize winner Dr. Brad Bass as one of its keynote speakers.
Today, Science Expo has grown to include several other outreach programs. These include a high school ambassador program, a teacher outreach program, and EXPOtential, the Science Expo alumni exclusive mentorship program. Through these outreach programs, the organization hopes to build a network of passionate students who are involved in STEM.
Programs
Science Expo features a variety of programs. High school students are invited to gather annually in February to participate in various workshops and are given opportunities to network with fellow peers and members of the STEM community. Keynote speakers have also been an exciting component to these conferences, with past speakers such as Dr. Steve Mann (Father of Wearable Computing) in 2013 and Dr. Brad Bass in 201 |
https://en.wikipedia.org/wiki/Eugen%20Jahnke | Paul Rudolf Eugen Jahnke (November 30, 1861, in Berlin – October 18, 1921, in Berlin) was a German mathematician.
Jahnke studied mathematics and physics at the Humboldt University of Berlin, where he graduated in 1886. In 1889 he received his doctorate from Martin-Luther-Universität Halle-Wittenberg under Albert Wangerin on the integration of first-order ordinary differential equations. After that, he was a teacher at secondary schools in Berlin, where he simultaneously in 1901 taught at the Technische Hochschule Berlin-Charlottenburg and in 1905 he became a professor at the Mining Academy in Berlin, which merged in 1916 with the Berlin Institute of Technology. In 1919 he was rector of the Berlin Institute of Technology.
In 1900 Jahnke read a paper at the International Congress of Mathematicians in Paris. He was editor of the Archives of Mathematics and Physics and contributor to the Yearbook for the Progress of Mathematics. He wrote an early book on vector calculus but is now known primarily for his function tables, which first appeared in 1909. This was also translated into English and was in print into the 1960s. (Professor of Electrical Engineering at the Technical University of Stuttgart) contributed to later editions, as did others.
Selected works
Jahnke: Zur Integration von Differentialgleichungen erster Ordnung, in welchen die unabhängige Veränderliche explicite nicht vorkommt, durch eindeutige doppeltperiodische Funktionen (dissertation), 1889
Jahnke: Vorlesungen über die Vektorenrechnung – mit Anwendungen auf Geometrie, Mechanik und mathematische Physik (lectures on vector analysis, with applications to geometry, mechanics, and mathematical physics), Teubner 1905
Jahnke (jointly with ): Funktionentafeln mit Formeln und Kurven (tables of functions with formulas and graphs), Teubner. 1909, 1933, 1945, 7. Auflage 1966, edited by Fritz Emde until his death in 1951 and later by Friedrich Lösch as "Tafeln höherer Funktionen". In America, published as Tables of Functions With Formulas and Curves by Eugene Jahnke and Fritz Emde, Dover June 1945
References
Humboldt University of Berlin alumni
Academic staff of the Technical University of Berlin
19th-century German mathematicians
1861 births
1921 deaths
20th-century German mathematicians |
https://en.wikipedia.org/wiki/Badri%20Nath%20Prasad | Badri Nath Prasad (1899-1966) was an Indian parliamentarian. He wrote many books on mathematics and was awarded the Padma Bhushan in 1963. He was a nominated member of the Rajya Sabha from 1964 till his death in 1966.
References
Sources
Brief Biodata
Nominated members of the Rajya Sabha
1899 births
1966 deaths
Recipients of the Padma Bhushan in literature & education |
https://en.wikipedia.org/wiki/W.%20C.%20Robinson%20%28educator%29 | William Claiborne Robinson, known as W. C. Robinson (April 25, 1861 – April 1, 1914), was a mathematics professor paid $800 per year who was elevated for one year, 1889 to 1900, as the second president of Louisiana Tech University in Ruston, Louisiana.
Robinson began teaching at Louisiana Tech when the institution consisted of one building with eight classrooms, a science laboratory, an auditorium, two offices, and a small frame building which served as a workshop. One of the offices was used as a reading room, the modest forerunner of Prescott Memorial Library, named for the first college president, Arthur T. Prescott, who provided the original reading materials at his own expense.
In 1900, Robinson resigned as president to return to the classroom. He was succeeded by the English professor James B. Aswell, later a state superintendent of education, president of Northwestern State University in Natchitoches, Louisiana, and a member of the United States House of Representatives.
Little is known of Robinson himself, but much attention has been placed on the campus building, Robinson Hall, named in his honor. Constructed in 1939, twenty-five years after Robinson's death, the building was originally a men's dormitory which had fallen into disrepair by the middle 1960s. With a refurbished interior but with the exterior still unaltered and easily identified by former alumni, Robinson Hall is the home of the Louisiana Tech speech and hearing center. The three-story red brick structure is built in the Colonial Revival style of architecture. The contractor was the T. L. James Company of Ruston.
Robinson and his wife, Etta Asenah Moore Robinson (1869-1917), were both originally from Virginia. They had two children, Hunt Vaughn Robinson and Ealvise Conner Robinson, who died before their first birthdays and six children who survived into adulthood, Herbert Lynn, Fred B., Esther, Virginia, Robert Hicks, and Luther William Robinson. The Robinsons are interred at Mt. Lebanon Cemetery near Gibsland in Bienville Parish, Louisiana. His gravestone has a Woodmen of the World emblem at the base.
A later Louisiana Tech president also has a tie to Mount Lebanon. Claybrook Cottingham was the last president of Southern Baptist-affiliated Mount Lebanon College from 1905 to 1906, when he became from 1906 to 1910 a founding professor of the new Louisiana College in Pineville. His tenure as LC president stretched from 1910 to 1941, when he accepted the highest position at Louisiana Tech, then known as Louisiana Polytechnic Institute. And sixty-two years after Robinson left the Louisiana Tech presidency, F. Jay Taylor, a Gibsland native, became president and held the position for 25 years.
References
1861 births
1914 deaths
Educators from Louisiana
People from Bienville Parish, Louisiana
People from Ruston, Louisiana
Presidents of Louisiana Tech University |
https://en.wikipedia.org/wiki/Vincenzo%20Cecere | Vincenzo Cecere (1897–1955) was an Italian painter, active in a Realist style.
Biography
He was born in Aversa and initially a pupil of Luigi Pastore, but later enrolled at the Institute of Geometry of Caserta. He served as a soldier in the Austrian front, followed by an exile in Marseille due to his political leanings. He returned to earn a diploma in Naples only by the 1930s. He painted Dopo il bagno, a subject previously treated by the Scapigliatura painter Girolamo Induno. He painted a Portrait of this cousin Amelia, Un bue al pascolo, Head of a Dog, and Portrait of a Girl, likely his cousin Ersilia. He also wrote poetry in Italian and dialect.
References
1897 births
1955 deaths
20th-century Italian painters
Italian male painters
Painters from Naples
Italian genre painters
20th-century Italian male artists |
https://en.wikipedia.org/wiki/Jean-Loup%20Gervais | Jean-Loup Gervais (born 10 September 1936 in Paris) is a French theoretical physicist.
Gervais studied physics and mathematics in Paris, where he graduated in 1961 and got his Ph.D. in 1965 as a student of Claude Bouchiat and Philippe Meyer in Orsay. From 1966 to 1968 he was a post-doctoral researcher at New York University. Already since 1960 he was employed at the CNRS, from 1970 on as Maître de conférences. During 1973–1985 he was Maître de conférences at École polytechnique.
From 1979 to 1983 and from 1995 to 1998 he was director of the Laboratory of Theoretical Physics of the École Normale Supérieure. He had been a guest professor at the City College of New York and also partly at University of California, Berkeley, at the Isaac Newton Institute in Cambridge (1997), at University of California, Los Angeles (UCLA) and at University of California, Santa Barbara.
Gervais worked on quantum field theory, supersymmetry and string theory. In 1969, he investigated (together with Benjamin W. Lee) renormalisability of theories of spontaneous symmetry breaking. In 1971, he presented with Bunji Sakita a supersymmetric invariant Lagrangian in the framework of a precursor of string theory, called the dual resonance models.
In 1969, he calculated one-loop diagrams in the early string theory, with Daniele Amati and Bouchiat. In the beginning of the 1970s, he also studied, with Sakita, string theories as conformal field theories in two dimensions and then soliton theories as field theories of collective excitations, e.g., in the context of WKB wave functions.
In the 1980s he studied soliton (Skyrmion) models of quarks in the limit of many color degrees of freedom (large-N limit). He then also considered conformal field theories such as the Liouville field theory, string theories and two-dimensional quantum gravity from the point of view of exactly integrable systems. With André Neveu, he investigated in the 1980s also non-critical string theories.
In 1997 he was awarded the highly reputed Prix Créé par l'État from the French Académie des sciences.
Among his Ph.D. students are particle physicists Antal Jevicki (now professor at Brown University) and Adel Bilal.
External links
Homepage with details on work of J-L Gervais
References
J-L Gervais, M Jacob (Eds): Non-linear and collective phenomena in quantum physics. A reprint volume from Physics reports. World Scientific 1983
J-L Gervais, A Jevicki and B Sakita: A collective coordinate method for the quantization of extended systems. In: Physics Reports 23 (1976), p. 237
French physicists
1936 births
Living people
Scientists from Paris
Academic staff of the École Normale Supérieure
City College of New York faculty
University of California, Berkeley faculty
University of California, Los Angeles faculty
University of California, Santa Barbara faculty |
https://en.wikipedia.org/wiki/Maier%27s%20matrix%20method | Maier's matrix method is a technique in analytic number theory due to Helmut Maier that is used to demonstrate the existence of intervals of natural numbers within which the prime numbers are distributed with a certain property. In particular it has been used to prove Maier's theorem and also the existence of chains of large gaps between consecutive primes . The method uses estimates for the distribution of prime numbers in arithmetic progressions to prove the existence of a large set of intervals where the number of primes in the set is well understood and hence that at least one of the intervals contains primes in the required distribution.
The method
The method first selects a primorial and then constructs an interval in which the distribution of integers coprime to the primorial is well understood. By looking at copies of the interval translated by multiples of the primorial an array (or matrix) of integers is formed where the rows are the translated intervals and the columns are arithmetic progressions where the difference is the primorial. By Dirichlet's theorem on arithmetic progressions the columns will contain many primes if and only if the integer in the original interval was coprime to the primorial. Good estimates for the number of small primes in these progressions due to allows the estimation of the primes in the matrix which guarantees the existence of at least one row or interval with at least a certain number of primes.
References
Analytic number theory |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Paul%20Dirac | Below is a list of things, primarily in the fields of mathematics and physics, named in honour of Paul Adrien Maurice Dirac.
Physics
Dirac large numbers hypothesis
Dirac monopole
Dirac string
Dirac's string trick
Quantum physics
Notations
Dirac notation
Dirac bracket
Equations and related objects
Dirac adjoint
Dirac cone
Dirac points
Dirac constant, see reduced Planck constant
Dirac–Coulomb–Breit Hamiltonian
Dirac equation
Dirac equation in curved spacetime
Dirac equation in the algebra of physical space
Nonlinear Dirac equation
Two-body Dirac equations
Dirac fermion
Dirac field
Dirac gauge
Dirac hole theory
Dirac Lagrangian
Dirac matrices
Dirac matter
Dirac membrane
Dirac picture
Dirac sea
Dirac spectrum
Dirac spinor
Formalisms
Fermi–Dirac statistics
Dirac–von Neumann axioms
Effects
Abraham–Lorentz–Dirac force
Kapitsa–Dirac effect
Pure and applied mathematics
Complete Fermi–Dirac integral
Incomplete Fermi–Dirac integral
Dirac delta function
Dirac comb
Dirac measure
Dirac operator
Dirac algebra
Other uses
5997 Dirac, an asteroid
The various Dirac Medals
Dirac (software)
DiRAC supercomputing research facility of the Science and Technology Facilities Council
Dirac Science Library, at Florida State University
Dirac road, Bristol, ().
References
Dirac
Paul Dirac |
https://en.wikipedia.org/wiki/Sven%20Kreyer | Sven Kreyer (born 14 May 1991) is a German footballer who plays as a striker for Rot-Weiß Oberhausen.
Career
Statistics
References
External links
1991 births
Living people
Bayer 04 Leverkusen II players
VfL Bochum II players
VfL Bochum players
Rot-Weiss Essen players
Rot-Weiß Oberhausen players
German men's footballers
2. Bundesliga players
Regionalliga players
Footballers from Düsseldorf
Men's association football forwards
3. Liga players |
https://en.wikipedia.org/wiki/Pfister%27s%20sixteen-square%20identity | In algebra, Pfister's sixteen-square identity is a non-bilinear identity of form
It was first proven to exist by H. Zassenhaus and W. Eichhorn in the 1960s, and independently by Albrecht Pfister around the same time. There are several versions, a concise one of which is
If all and with are set equal to zero, then it reduces to Degen's eight-square identity (in blue). The are
and,
The identity shows that, in general, the product of two sums of sixteen squares is the sum of sixteen rational squares. Incidentally, the also obey,
No sixteen-square identity exists involving only bilinear functions since Hurwitz's theorem states an identity of the form
with the bilinear functions of the and is possible only for n ∈ {1, 2, 4, 8} . However, the more general Pfister's theorem (1965) shows that if the are rational functions of one set of variables, hence has a denominator, then it is possible for all . There are also non-bilinear versions of Euler's four-square and Degen's eight-square identities.
See also
Brahmagupta–Fibonacci identity
Euler's four-square identity
Degen's eight-square identity
Sedenions
References
External links
Pfister's 16-Square Identity
Analytic number theory
Mathematical identities |
https://en.wikipedia.org/wiki/Giovanni%20Bordiga | Giovanni Bordiga (2 April 1854 in Novara – 16 June 1933 in Venice) was an Italian mathematician who worked on algebraic and projective geometry at the university of Padua.
He introduced the Bordiga surface.
Giovanni as the son of Carlo and Amalia Adami. He matriculated at a young age to Turin University, graduating in 1874 in civil engineering.
From 22 December 1929 until his death, he was president of the Ateneo Veneto of Science, Letters and Arts.
He was the paternal uncle of Italian Left Communist theorist Amadeo Bordiga.
References
19th-century Italian mathematicians
1854 births
1933 deaths
20th-century Italian mathematicians
University of Padua alumni
University of Turin alumni
Italian civil engineers
19th-century Italian engineers |
https://en.wikipedia.org/wiki/2013%E2%80%9314%20FK%20Horizont%20Turnovo%20season | The 2013–14 season is FK Turnovo's 6th consecutive season in First League. This article shows player statistics and all official matches that the club will play during the 2013–14 season.
Squad
As of 10 February 2014
Left club during season
Competitions
First League
Results summary
Results by round
Results
Table
Macedonian Cup
First round
Second round
Europa League
Qualifying rounds
First qualifying round
Second qualifying round
Statistics
Top scorers
References
FK Horizont Turnovo seasons
Turnovo
Turnovo |
https://en.wikipedia.org/wiki/Tibor%20Moln%C3%A1r%20%28footballer%2C%20born%201993%29 | Tibor Molnár (born 12 May 1993) is a Hungarian professional footballer who plays for Budaörsi SC.
Club statistics
Updated to games played as of 13 November 2013.
References
External links
Tibor Molnár at HLSZ
1993 births
Footballers from Székesfehérvár
Living people
Hungarian men's footballers
Men's association football forwards
FC Felcsút players
Fehérvár FC players
Puskás Akadémia FC players
Aqvital FC Csákvár players
FC Ajka players
Budaörsi SC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Hungarian expatriate men's footballers
Expatriate men's footballers in Italy
Hungarian expatriate sportspeople in Italy |
https://en.wikipedia.org/wiki/List%20of%20Celtic%20F.C.%20records%20and%20statistics | Celtic Football Club are a Scottish professional association football club based in Glasgow. They have played at their home ground, Celtic Park, since 1892. Celtic were founding members of the Scottish Football League in 1890, and the Scottish Premier League in 1998 as well as the Scottish Professional Football League in 2013.
The list encompasses the major honours won by Celtic, records set by the club, their managers and their players. The player records section includes details of the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by Celtic players on the international stage, and the highest transfer fees paid and received by the club. Attendance records at Celtic Park, and also at Hampden Park which has on occasion been used for home games, are also included.
Celtic have won 53 top-flight titles, and hold the record for most Scottish Cup wins with 41. The club's record appearance maker is Billy McNeill, who made 822 appearances between 1957 and 1975. Jimmy McGrory is the club's record goalscorer, scoring 522 goals during his Celtic career.
All figures are correct as of 3 June 2023
Honours
Celtic's first ever silverware was won in 1889 when they defeated Cowlairs 6–1 in the final of the North-Eastern Cup. A year later they won the Glasgow Cup, before winning their first major national honour in 1892 by defeating Queen's Park 5–1 in the final of the Scottish Cup. Celtic won their first league title in 1892–93. In 1906–07 Celtic became the first club to win the league and cup double in Scotland, a feat they have now accomplished on 12 occasions. They won their first domestic treble in 1966–67, the same season they became the first British club to win the European Cup with their 2–1 victory over Inter Milan in the final. Celtic's most recent success was their win in the 2022-23 Scottish Cup. Celtic have won a total of 116 trophies.
In all, Celtic have won the Scottish League Championship 53 times, the Scottish Cup a record 41 times, the Scottish League Cup 21 times and the European Cup once. They have completed a World Record, eight domestic trebles, including an unprecedented quadruple treble between the 16/17 and 19/20 seasons.
Domestic
League
Scottish League Championship:
Winners (53): 1893, 1894, 1896, 1898,1905, 1906, 1907, 1908, 1909, 1910, 1914, 1915, 1916, 1917, 1919, 1922, 1926, 1936, 1938, 1954, 1966, 1967, 1968, 1969, 1970, 1971, 1972, 1973, 1974, 1977, 1979, 1981, 1982,1986, 1988, 1998, 2001, 2002, 2004, 2006, 2007, 2008, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2022, 2023
Runners-up (32): 1892, 1895, 1900, 1901, 1902, 1912, 1913, 1918, 1920, 1921, 1928, 1929, 1930, 1935, 1939, 1955, 1976, 1980, 1983, 1984, 1985, 1987, 1996, 1997, 1999, 2000, 2003, 2005, 2009, 2010, 2011, 2021
Cups
Scottish Cup:
Winners (41): 1892, 1899, 1900, 1904, 1907, 1908, 1911, 1912, 1914, 1923, 1925, 1927, 1931, 1933, 1937, 1951, 1954, 1965, 1967, 1969, 197 |
https://en.wikipedia.org/wiki/Bence%20Somodi | Bence Somodi (born 25 November 1988) is a Hungarian professional footballer who plays for Csákvár.
Club statistics
Updated to games played as of 15 May 2021.
References
HLSZ
1988 births
Living people
Footballers from Eger
Hungarian men's footballers
Men's association football goalkeepers
Ferencvárosi TC footballers
Vecsési FC footballers
Diósgyőri VTK players
Kazincbarcikai SC footballers
Gyirmót FC Győr players
Fehérvár FC players
Puskás Akadémia FC players
Kaposvári Rákóczi FC players
MTK Budapest FC players
Aqvital FC Csákvár players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Hungarian expatriate men's footballers
Expatriate men's footballers in England
Hungarian expatriate sportspeople in England |
https://en.wikipedia.org/wiki/Gary%20Hicks | Gary E. Hicks (born November 30, 1953) is the Senior Associate Justice of the New Hampshire Supreme Court. He was sworn in January 31, 2006.
Hicks obtained his degree in mathematics from Bucknell University in 1975. He is a 1978 graduate of Boston University School of Law
Before being sworn in to the New Hampshire Supreme Court, Hicks worked for Wiggin & Nourie for 23 years. He is the former chairman of the American Inns of Court Leadership Council, and the American Inns Nomination Committee.
Hicks was presented with the 2021 Civil Justice Award by the New Hampshire Access to Justice Commission.
References
1953 births
20th-century American lawyers
21st-century American judges
Boston University School of Law alumni
Bucknell University alumni
Living people
New Hampshire lawyers
New Hampshire state court judges
Justices of the New Hampshire Supreme Court
Place of birth missing (living people)
21st-century American politicians |
https://en.wikipedia.org/wiki/Variational%20analysis | In mathematics, the term variational analysis usually denotes the combination and extension of methods from convex optimization and the classical calculus of variations to a more general theory. This includes the more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized derivatives.
In the Mathematics Subject Classification scheme (MSC2010), the field of "Set-valued and variational analysis" is coded by "49J53".
History
While this area of mathematics has a long history, the first use of the term "Variational analysis" in this sense was in an eponymous book by R. Tyrrell Rockafellar and Roger J-B Wets.
Existence of Minima
A classical result is that a lower semicontinuous function on a compact set attains its minimum. Results from variational analysis such as Ekeland's variational principle allow us to extend this result of lower semicontinuous functions on non-compact sets provided that the function has a lower bound and at the cost of adding a small perturbation to the function.
Generalized derivatives
The classical Fermat's theorem says that if a differentiable function attains its minimum at a point, and that point is an interior point of its domain, then its derivative must be zero at that point. For problems where a smooth function must be minimized subject to constraints which can be expressed in the form of other smooth functions being equal to zero, the method of Lagrange multipliers, another classical result, gives necessary conditions in terms of the derivatives of the function.
The ideas of these classical results can be extended to nondifferentiable convex functions by generalizing the notion of derivative to that of subderivative. Further generalization of the notion of the derivative such as the Clarke generalized gradient allow the results to be extended to nonsmooth locally Lipschitz functions.
See also
Citations
References
External links |
https://en.wikipedia.org/wiki/List%20of%20Primera%20Divisi%C3%B3n%20de%20Nicaragua%20records | The following is a compilation of notable records and statistics for teams and players in and seasons of Primera División de Nicaragua.
All time League Records
Titles
Most top-flight League titles: 26, Diriangén FC
Most consecutive League titles: 7,
Real Esteli F.C.
Top flight appearances
Most Appearances: 80, (1933-)
Diriangén FC
Goals
Individual
All-time leading goalscorer: Manuel "Catarrito" Cuadra (742 goals)
Team
Most league goals scored in a season: 88,
Fewest league goals scored in a season:
Most league goals conceded in a season:
Fewest league goals conceded in a season:, ()
Biggest Win:
Records 1933-1994
Most league goals scored in a season (excluding playoffs): goals, TBD (TBD)
Fewest league goals scored in a season:TBD (TBD-TBD)
Most league goals conceded in a season:TBD (TBD-TBD)
Fewest league goals conceded in a season: 6, TBD (TBD)
Most goals scored in one season: 44 goals, Oscar "Chiqui" Calvo playing for Flor de Caña in 1967.
Most goals scored in one game by a player: 6, Manuel “Catarrito” Cuadra vs Corinto
Biggest Win: UCA 15-1 Esteli, August 6, 1972, and UCA 15-1 Corinto, November 9, 1980
Record away win:
Highest scoring game: UCA 15-1 Esteli, August 6, 1972, and UCA 15-1 Corinto, November 9, 1980, and UCA 13-3 ISA, November 23, 1980
Most wins in a row:
Most championships won by a player:
Most Championship by a coach:
Longest Period of time by a coach (in the first division):
Most consecutive championship: 5, Diriangén FC (1940, 1941, 1942, 1943, 1944, 1945,)
Most seasons appearance: 80, Diriangén FC (1933–present)
Most participants from one place:
Most points in a season: points, TBD (TBD)
Fewest points in a season: points, TBD (TBD)
Most goals scored in a finals game:
Most goals scored in a final game:
Highest scoring game in a finals game:
Most appearances (team) in the finals:
Most defeats in a final series:
Most defeats in a final:
Most appearances in a final series without winning a championship:
Lowest ranked winners:
Lowest ranked finalists:
Biggest win (aggregate):
Most final series goals by an individual:
Most goals by a losing side in a final games:
Lowest finish by the previous season's champions:
TBD that have won a championship in the season following their promotion to the Primera. They did so in
Records Short fomat/Clausura and Apertura 1995-
Most league goals scored in a season (excluding playoffs):
Fewest league goals scored in a season:
Most league goals conceded in a season:
Fewest league goals conceded in a season:
Biggest Win: Diriangén FC 14-0 Pinares, February 2, 1997
Record away win:
Highest scoring game: Real Estelí F.C. 13-2 Chinandega FC, December 23, 2001
Most wins in a row:
Best undefeated streak:
Most consecutive minutes without conceding a goal : 741 minutes, Denis Espinoza (, Deportivo Walter Ferretti , 2014 Apertura).
Most championships won by a player:
Most Championship by a coach: 6 by Ramón Otoniel Olivas with Real Esteli F. |
https://en.wikipedia.org/wiki/Rajmond%20Toricska | Rajmond Toricska (born 11 May 1993 in Cegléd) is a Hungarian professional footballer who plays for Újpest FC.
Club statistics
Updated to games played as of 12 April 2014.
References
MLSZ
1993 births
Living people
People from Cegléd
Hungarian men's footballers
Men's association football midfielders
Újpest FC players
Kozármisleny SE footballers
Nemzeti Bajnokság I players
Footballers from Pest County |
https://en.wikipedia.org/wiki/Milan%20Vojnovic | Milan Vojnovic is a professor of data science with the Department of Statistics at the London School of Economics, where he is also director of the MSc Data Science Programme. Prior to this, he worked as a researcher with Microsoft Research from 2004 to 2016.
He received his Ph.D. degree in Technical Sciences from École Polytechnique Fédérale de Lausanne in 2003, and both M.Sc. and B.Sc. degrees in Electrical Engineering from the University of Split, Croatia, in 1995 and 1998, respectively. He undertook an internship with the Mathematical Research Centre at Bell Labs in 2001. From 2005 to 2014, he was a visiting professor at the University of Split, Croatia. From 2014 to 2016, he was an affiliated lecturer at the Statistical Laboratory, University of Cambridge.
Research
His research interests include data science, machine learning, artificial intelligence, game theory, multi-agent systems and information networks. He has made contributions to the theory and the design of computation platforms for processing large-scale data.
He received several prizes for his work. In 2010, he was awarded the ACM SIGMETRICS Rising Star Researcher Award, and in 2005, the ERCIM Cor Baayen Award. He received the IEEE IWQoS 2007 Best Student Paper Award (with Shao Liu and Dinan Gunawardena), the IEEE INFOCOM 2005 Best Paper Award (with Jean-Yves Le Boudec), the ACM SIGMETRICS 2005 Best Paper Award (with Laurent Massoulie) and the ITC 2001 Best Student Paper Award (with Jean-Yves Le Boudec).
Vojnovic authored the book Contest Theory: Incentive Mechanisms and Ranking Methods.
References
Computer scientists
Croatian engineers
Living people
Academics of the London School of Economics
1971 births |
https://en.wikipedia.org/wiki/Kim%20Plofker | Kim Leslie Plofker (born November 25, 1964) is an American historian of mathematics, specializing in Indian mathematics.
Education and career
Born in Chennai, India, Plofker received her bachelor's degree in mathematics from Haverford College. She received her Ph.D. in 1995 while studying with adviser David Pingree (Mathematical Approximation by Transformation of Sine Functions in Medieval Sanskrit Astronomical Texts) from Brown University, where she conducted research and later joined as a guest professor.
In the late 1990s, she was Technical Director of the American Committee for South Asian Manuscripts of the American Oriental Society, where she was also concerned with the development of programs for the text comparison. From 2000 to 2004, she was at the Dibner Institute for the History of Science and Technology at the Massachusetts Institute of Technology. During 2004 and 2005, she was a visiting professor in Utrecht and at the same time Fellow of the International Institute for Asian Studies in Leiden. She is currently an associate professor at Union College in Schenectady.
Contributions
Plofker deals with the history of Indian mathematics, the topic of her 2009 book Mathematics in India. She is particularly interested in the exchange of mathematics and astronomy between India and Islam in the Middle Ages and generally in the exact sciences between Europe and Asia from antiquity to the 20th Century.
With Clemency Montelle, she is the coauthor of Sanskrit Astronomical Tables (Springer, 2019).
Recognition
In 2010 she gave a plenary lecture at the International Congress of Mathematicians, Hyderabad (Indian rules, Yavana rules: foreign identity and the transmission of mathematics). In 2011, she was awarded the Brouwer Medal of the Royal Dutch Mathematical Society.
References
1964 births
Living people
American historians of mathematics
20th-century American mathematicians
21st-century American mathematicians
Haverford College alumni
Brown University alumni
Brouwer Medalists
Massachusetts Institute of Technology faculty
American women mathematicians
American Indologists
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/List%20of%20most%20valuable%20crops%20and%20livestock%20products | The following list, derived from the statistics of the United Nations' Food and Agriculture Organization (FAO), lists the most valuable agricultural products produced by the countries of the world. The data in this article, unless otherwise noted, was reported for 2016. The value and production of individual crops varies substantially from year to year as prices fluctuate on the world and country markets and weather and other factors influence production.
This list includes the top 50 most valuable crops and livestock products but does not necessarily include the top 50 most heavily produced crops and livestock products. Indigenous meat values have been omitted from this table.
References
Agricultural production
Most Valuable Crops
Agricultural production by commodity
Economy-related lists of superlatives |
https://en.wikipedia.org/wiki/Marc%20A.%20Suchard | Marc Adam Suchard (born December 23, 1972 in California) is an American statistician. He is Professor in the Departments of Biomathematics and of Human Genetics in the David Geffen School of Medicine at UCLA and in the Department of Biostatistics in the UCLA Fielding School of Public Health at the University of California, Los Angeles. He was elected as a Fellow of the American Statistical Association in 2012, and he received the COPSS Presidents' Award in 2013.
Selected publications
Suchard, M. A., Weiss, R. E., & Sinsheimer, J. S. (2001). Bayesian selection of continuous-time Markov chain evolutionary models. Molecular Biology and Evolution, 18(6), 1001-1013.
Awards
COPSS Presidents' Award (2013)
Mitchell Prize (2011)
John Simon Guggenheim Fellowship (2008)
Alfred P. Sloan Fellowship (2007)
Mitchell Prize (2006)
British Marshall Scholarship (1995)
References
External links
Publications and Presentations
1972 births
Living people
American statisticians
David Geffen School of Medicine at UCLA faculty
Fellows of the American Statistical Association |
https://en.wikipedia.org/wiki/Georg%20Bohlmann | Georg Bohlmann (23 April 1869 – 25 April 1928) was a German mathematician who specialized in probability theory and actuarial mathematics.
Life and career
Georg Bohlmann went to school in Berlin and Leipzig and took his Abitur at the Wilhelms-Gymnasium in Berlin in 1888. After that, he began studying mathematics at the University of Berlin under Leopold Kronecker, Lazarus Fuchs, and Wilhelm Dilthey. As he advanced in his studies, Lie groups became the focus of his interest. Since this area was poorly represented at Berlin, he moved to the University of Halle, where he obtained his doctorate in 1892 under Albert Wangerin with a dissertation on the topic Ueber eine gewisse Klasse continuierlicher Gruppen und ihren Zusammenhang mit den Additionstheoremen ("On a certain class of continuous groups and their relation to addition theorems"). After that, he worked at the Meteorological Institute of Berlin, where presumably his interest in applied mathematics developed. At the invitation of Felix Klein, he moved to the University of Göttingen, where he habilitated in 1894. In 1895, he was involved in starting a seminar on actuarial science at Göttingen. However, since he held no permanent position there, he went to Berlin in 1903 to work as the Chief Actuary for the German subsidiary of the New York Mutual Life Insurance Company.
In 1901, he wrote the entry on life insurance mathematics in the Enzyklopädie der mathematischen Wissenschaften ("Encyclopaedia of Mathematical Sciences") in which he gave axioms for probability theory long before Andrey Kolmogorov did so in 1933. In particular, he was the first to give the modern definition of statistical independence. Compared to the current structure of probability theory, his work only lacked the technical condition of sigma additivity. However, in contrast to Kolmogorov, Bohlmann failed to prove significant theorems within his axiomatic framework. As a result, his fundamental contributions to probability theory gained very little attention. In particular, though Kolmogorov had visited Göttingen several times in the late 1920s, he had no knowledge of Bohlmann's work.
Bohlmann was an invited speaker in the International Congress of Mathematicians in 1908 at Rome.
Publications
Lebensversicherungsmathematik (Life Insurance Mathematics), Enzyklopädie der Mathematischen Wissenschaften, 1901
Continuierliche Gruppen von quadratischen Transformationen der Ebene (Continuous groups of quadratic transformations of the plane), Göttinger Nachrichten, 1896, pp. 44–54
Ein Ausgleichungsproblem (A stabilization problem), Göttinger Nachrichten, 1899, pp. 260–271
Die Grundbegriffe der Wahrscheinlichkeitsrechnung in ihrer Anwendung auf die Lebensversicherung (The basic concepts of probability theory and its applications to life insurance), Atti del IV Congresso internazionale dei Matematici III, Rome 1909, pp. 244–278
Anthropometrie und Lebensversicherung (Anthropometry and life insurance), Zeitschrift für die ges |
https://en.wikipedia.org/wiki/Gabriel%20%C3%81valos | Gabriel Ávalos Stumpfs (born 12 October 1990) is a Paraguayan footballer currently playing for Argentine Primera División club Argentinos Juniors and the Paraguay national team.
Career statistics
International career
Ávalos represented the Paraguay national team in a 0–0 2022 FIFA World Cup qualification tie with Uruguay on 3 June 2021.
International goals
Notes
References
External links
1990 births
Living people
Paraguayan men's footballers
Paraguay men's international footballers
Paraguayan expatriate men's footballers
Men's association football forwards
Club Libertad footballers
Club General Díaz (Luque) footballers
Club Atlético Tembetary footballers
Independiente F.B.C. footballers
Defensores de Cambaceres footballers
Club de Gimnasia y Esgrima La Plata footballers
Club Atlético Tigre footballers
Deportes Concepción (Chile) footballers
Crucero del Norte footballers
Peñarol players
Club Atlético Nueva Chicago footballers
Godoy Cruz Antonio Tomba footballers
Club Atlético Patronato footballers
Argentinos Juniors footballers
Primera B de Chile players
Argentine Primera División players
Uruguayan Primera División players
Expatriate men's footballers in Chile
Expatriate men's footballers in Argentina
Expatriate men's footballers in Uruguay
Paraguayan expatriate sportspeople in Argentina
Paraguayan expatriate sportspeople in Chile
Paraguayan expatriate sportspeople in Uruguay
People from Itapúa Department |
https://en.wikipedia.org/wiki/S.I.N.%20Theory | S.I.N. Theory (abbreviation for social insurance number theory) is a 2012 Canadian science fiction drama film about a mathematics professor creating an algorithm capable of predicting an individual's future. The film was written and directed by Richie Mitchell, and stars Jeremy Larter and Allison Dawn Doiron. S.I.N. Theory was produced on a shoestring budget and makes notable use of existing mathematical theories to affirm the concept's plausibility. The film has been screened at science fiction film festivals in Canada and the US, whereas it has been picked up for distribution by MCTV with an anticipated release Fall 2013.
Plot
Michael, a mathematics professor having dedicated his career to creating the ultimate game theory, is finally let go for being a long time black mark of the faculty. Despite becoming a laughing stock, Michael continues his work from home and is now free from legal parameters the faculty has enforced. Determined to prove all naysayers wrong, and to reclaim if not heighten his name, Michael gives the go-ahead to his anonymous colleague, a hacker, to obtain access to the populous’ full credit and health report information. This illegal database proves to be the last piece of the puzzle, and with it, Michael is able to accurately calculate and therefore predict the outcome to nearly any situation. As it was initially thought to be of use for stock markets and political predicaments, Michael however curiously formulates the life span of his favorite student, a young woman he loves, and finds out she only has days to live. The further he unravels the possibilities and dangers the algorithm beholds, his morality, its proper use, and reporting of it comes into question. Each time he uses the algorithm, he taps into the credit report database and his illegal presence becomes vulnerable, and ultimately discovered by a competing-dirty corporation, hell bent on the same goal for monetary purposes. Michael must choose what to do with this illegal and very powerful equation; either to publish it for his own glorious demise, or save the woman he loves and risk annihilation by the threatening competitor.
Cast
Jeremy Larter as Dr. Michael Liemann
Allison Dawn Doiron as Evelyn Palmer
Farid Yazdani as David
Richard Guppy as Sean
Stephen Jacob Hogan as Thug 1
Ed Lewis as Thug 2
Kevin Stonefield as The Dean
Production
In writing the script, a contained sci-fi written explicitly for the accommodation of the budget, Mitchell wished to incorporate existing mathematical theories to showcase the plausibility of such an equation, and how to a certain extent, already exists today. In a viral video interview posted on the film's website, Mitchell talks of drawing comparisons to observing human behavior to the study of fluid behavior, as depicted by the Navier–Stokes equations, an attribution to Mitchell's engineering education.
Having undergone development hell on a couple other of his projects, Mitchell decided to finance the film alo |
https://en.wikipedia.org/wiki/Thiago%20Ribeiro%20%28footballer%2C%20born%201985%29 | Thiago Vasconcelos Ribeiro Da Silva (born 23 January 1985 in Rio de Janeiro), known as just Thiago Ribeiro, is a Brazilian professional footballer who plays for Royal Pari FC.
Club statistics
Updated to games played as of 9 December 2014.
References
External links
MLSZ
HLSZ
1985 births
Living people
Footballers from Rio de Janeiro (city)
Brazilian men's footballers
Men's association football midfielders
Clube Atlético Juventus players
BFC Siófok players
Barcsi SC footballers
FC Veszprém footballers
Dunaújváros PASE players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Bolivian Primera División players
Brazilian expatriate men's footballers
Expatriate men's footballers in Hungary
Brazilian expatriate sportspeople in Hungary
Expatriate men's footballers in Bolivia
Brazilian expatriate sportspeople in Bolivia |
https://en.wikipedia.org/wiki/Scattered%20space | In mathematics, a scattered space is a topological space X that contains no nonempty dense-in-itself subset. Equivalently, every nonempty subset A of X contains a point isolated in A.
A subset of a topological space is called a scattered set if it is a scattered space with the subspace topology.
Examples
Every discrete space is scattered.
Every ordinal number with the order topology is scattered. Indeed, every nonempty subset A contains a minimum element, and that element is isolated in A.
A space X with the particular point topology, in particular the Sierpinski space, is scattered. This is an example of a scattered space that is not a T1 space.
The closure of a scattered set is not necessarily scattered. For example, in the Euclidean plane take a countably infinite discrete set A in the unit disk, with the points getting denser and denser as one approaches the boundary. For example, take the union of the vertices of a series of n-gons centered at the origin, with radius getting closer and closer to 1. Then the closure of A will contain the whole circle of radius 1, which is dense-in-itself.
Properties
In a topological space X the closure of a dense-in-itself subset is a perfect set. So X is scattered if and only if it does not contain any nonempty perfect set.
Every subset of a scattered space is scattered. Being scattered is a hereditary property.
Every scattered space X is a T0 space. (Proof: Given two distinct points x, y in X, at least one of them, say x, will be isolated in . That means there is neighborhood of x in X that does not contain y.)
In a T0 space the union of two scattered sets is scattered. Note that the T0 assumption is necessary here. For example, if with the indiscrete topology, and are both scattered, but their union, , is not scattered as it has no isolated point.
Every T1 scattered space is totally disconnected. (Proof: If C is a nonempty connected subset of X, it contains a point x isolated in C. So the singleton is both open in C (because x is isolated) and closed in C (because of the T1 property). Because C is connected, it must be equal to . This shows that every connected component of X has a single point.)
Every second countable scattered space is countable.
Every topological space X can be written in a unique way as the disjoint union of a perfect set and a scattered set.
Every second countable space X can be written in a unique way as the disjoint union of a perfect set and a countable scattered open set. (Proof: Use the perfect + scattered decomposition and the fact above about second countable scattered spaces, together with the fact that a subset of a second countable space is second countable.) Furthermore, every closed subset of a second countable X can be written uniquely as the disjoint union of a perfect subset of X and a countable scattered subset of X. This holds in particular in any Polish space, which is the contents of the Cantor–Bendixson theorem.
Notes
Refe |
https://en.wikipedia.org/wiki/D%C3%A1rius%20Csillag | Dárius Csillag (born 29 January 1995) is a Hungarian professional footballer who plays for Budaörs.
Club statistics
Updated to games played as of 15 October 2014.
References
External links
1995 births
Living people
People from Gyöngyös
Hungarian men's footballers
Men's association football forwards
Kecskeméti TE players
Budaörsi SC footballers
Soroksár SC players
Vác FC players
Dorogi FC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Footballers from Heves County |
https://en.wikipedia.org/wiki/Sir%20Isaac%20Newton%20Sixth%20Form | Sir Isaac Newton Sixth Form is a specialist maths and science sixth form with free school status located in Norwich, owned by the Inspiration Trust. It has the capacity for 480 students aged 16–19. It specialises in mathematics and science.
History
Prior to becoming a Sixth Form College the building functioned as a fire station serving the central Norwich area until August 2011 when it closed down. Two years later the Sixth Form was created within the empty building with various additions being made to the existing structure. The sixth form was ranked the 7th best state sixth form in England by the Times in 2022.
Curriculum
At Sir Isaac Newton Sixth Form, students can study a choice of either Maths, Further Maths, Core Maths, Biology, Chemistry, Physics, Computer Science, Environmental Science or Psychology. Additionally, students can also study any of the subjects on offer at the partner free school Jane Austen College, also located in Norwich and specialising in humanities, Arts and English.
References
External links
Ofsted reports
Free schools in England
Schools in Norwich
Education in Norfolk
2013 establishments in England
Inspiration Trust
Educational institutions established in 2013
Mathematics education in the United Kingdom |
https://en.wikipedia.org/wiki/Affinity%20propagation | In statistics and data mining, affinity propagation (AP) is a clustering algorithm based on the concept of "message passing" between data points.
Unlike clustering algorithms such as -means or -medoids, affinity propagation does not require the number of clusters to be determined or estimated before running the algorithm. Similar to -medoids, affinity propagation finds "exemplars," members of the input set that are representative of clusters.
Algorithm
Let through be a set of data points, with no assumptions made about their internal structure, and let be a function that quantifies the similarity between any two points, such that iff is more similar to than to . For this example, the negative squared distance of two data points was used i.e. for points and ,
The diagonal of (i.e. ) is particularly important, as it represents the instance preference, meaning how likely a particular instance is to become an exemplar. When it is set to the same value for all inputs, it controls how many classes the algorithm produces. A value close to the minimum possible similarity produces fewer classes, while a value close to or larger than the maximum possible similarity produces many classes. It is typically initialized to the median similarity of all pairs of inputs.
The algorithm proceeds by alternating between two message-passing steps, which update two matrices:
The "responsibility" matrix has values that quantify how well-suited is to serve as the exemplar for , relative to other candidate exemplars for .
The "availability" matrix contains values that represent how "appropriate" it would be for to pick as its exemplar, taking into account other points' preference for as an exemplar.
Both matrices are initialized to all zeroes, and can be viewed as log-probability tables. The algorithm then performs the following updates iteratively:
First, responsibility updates are sent around:
Then, availability is updated per
for and
.
Iterations are performed until either the cluster boundaries remain unchanged over a number of iterations, or some predetermined number (of iterations) is reached. The exemplars are extracted from the final matrices as those whose 'responsibility + availability' for themselves is positive (i.e. ).
Applications
The inventors of affinity propagation showed it is better for certain computer vision and computational biology tasks, e.g. clustering of pictures of human faces and identifying regulated transcripts, than -means, even when -means was allowed many random restarts and initialized using PCA.
A study comparing affinity propagation and Markov clustering on protein interaction graph partitioning found Markov clustering to work better for that problem. A semi-supervised variant has been proposed for text mining applications. Another recent application was in economics, when the affinity propagation was used to find some temporal patterns in the output multipliers of the US economy between 1997 and 2017.
S |
https://en.wikipedia.org/wiki/Tushar%20Raheja | Tushar Raheja (born 1984) is an Indian storyteller and mathematics researcher based in Delhi. His first book Anything for you, Ma'am, a comedy, was published in 2006 while he was an undergraduate student in Indian Institute of Technology Delhi. His first feature film The Bizarre Murder of Mr Tusker , a sci-fi, psychological noir, starring BAFTA nominee Victor Banerjee, is due for release. His writing has been compared to that of P. G. Wodehouse by The Hindu and The Times of India and his books have gone on to achieve massive success, consistently remaining on the national best-selling charts. Raheja chose not to climb on the bandwagon of formulaic books but instead devoted himself to mathematical research and the study of narration.
Romi and Gang (published July 2013 by Pirates), previously titled Run Romi Run is only his second book in the market. The book about the unalloyed dreams of the young in the Indian hinterland revolves around cricket. It has been praised by The Hindu, Hindustan Times, The Daily Telegraph among other publications. Raheja is one of the few authors in India to combine widespread popularity with critical acclaim. In 2015, he obtained his PhD from IIT Delhi in the field of applied probability.
Personal life
Raheja was born and brought up at Faridabad. His parents are doctors, his father a graduate of Armed Forces Medical College, Pune (AFMC). Raheja did his schooling from Apeejay School and DPS Faridabad. He obtained his B.Tech in industrial engineering from IIT Delhi in 2006. Anything for you, Ma'am, his first novel was also published in the same year. He followed it up with research in applied mathematics and completed Masters of Science in Operations research in 2010. In 2015, Raheja was awarded a PhD by IIT Delhi in the field of applied probability. Kiran Seth of SPIC MACAY and Sandeep Juneja were his thesis advisors.
Books
Anything for you, Ma'am
Anything for you, Ma'am shot to national fame after its review in The Hindu headlined Outsourcing Wodehouse. The Times of India compared the plot to a classic Jeeves Wooster saga. The main protagonist Tejas has a propensity to land himself into comical troubles like Wooster and has an array of Jeeveses around him in the form of his friends and family. The book was especially praised for 'cleverly localising the Wooster persona. So English aristocracy, the idle rich, the lad sent down from Oxford, the young man with great expectations and little ability, the chappie whose only survival tool is a smart gentleman's gentleman called Jeeves – all this is turned into rich material for humour of a local kind.' There has been criticism of the book's ending which is compared to a Bollywood movie.
Romi and Gang
Romi and Gang, while it has been likened to Enid Blyton's stories for its innocence and the sense of nostalgia it evokes, and has been considered by Hindustan Times to be 'the equivalent of watching Lagaan', it is closer in spirit to Swami and Friends. It is the story o |
https://en.wikipedia.org/wiki/Stefan%20Kutschke | Stefan Kutschke (born 3 November 1988) is a German professional footballer who plays as a forward for Dynamo Dresden.
Career statistics
References
External links
1988 births
Living people
Footballers from Dresden
German men's footballers
Men's association football forwards
Bundesliga players
2. Bundesliga players
3. Liga players
SV Babelsberg 03 players
RB Leipzig players
VfL Wolfsburg players
VfL Wolfsburg II players
SC Paderborn 07 players
Dynamo Dresden players |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Pierre-Simon%20Laplace | This is a list of things named after Pierre-Simon Laplace
Probability theory
de Moivre-Laplace theorem that approximates binomial distribution with a normal distribution
Laplace–Bayes estimator
Laplace distribution
Laplace–Gauss distribution
Asymmetric Laplace distribution
Log-Laplace distribution
Multivariate Laplace distribution
Wrapped asymmetric Laplace distribution
Laplace functional
Laplace motion
Laplace's rule of succession
Laplace smoothing
Mathematical analysis
Laplace principle (large deviations theory)
Laplace series
Laplace transform
Two-sided Laplace transform
Laplace–Carson transform
Laplace–Stieltjes transform
Inverse Laplace transform
Laplace's method for approximating integrals
Laplace limit, concerning series solutions to Kepler's equation
Laplacian vector field
Differential equations
Laplace's equation
Laplace operator
Discrete Laplace operator
Laplace–Beltrami operator
Laplacian, see Laplace operator
Infinity Laplacian
p-Laplacian
Laplace operators in differential geometry
Young–Laplace equation
Laplace invariant
Spherical harmonics
Laplace series (Fourier–Laplace series)
Laplace expansion (potential)
Laplace coefficient: see Laplace expansion (potential)
Algebra
Laplace expansion of determinants of matrices
Discrete mathematics
Laplace matrices in graph theory
Physics
Laplace's demon
Laplace equation for irrotational flow
Laplace force
Laplace number
Laplace plane
Laplace's invariable plane
Laplace pressure
Laplace-Runge-Lenz vector
Laplace resonance
Laplace's tidal equations
Computer science
Harris–Laplace detector
Laplace mechanism
Laplacian smoothing
Others
The asteroid 4628 Laplace is named for Laplace.
A spur of the Montes Jura on the moon is known as Promontorium Laplace.
The tentative working name of the European Space Agency Europa Jupiter System Mission is the "Laplace" space probe.
French submarine Laplace
LaplacesDemon is Bayesian software
Institut Pierre Simon Laplace
In popular culture
Laplace no Ma, a video game about Laplace's demon
In Kamen Rider Fourze the Libra Horoscopes develops an ability called "The eye of Laplace"
In Mega Man Star Force 3 Solo gains a wizard named Laplace.
In Pokémon, the Japanese name of Lapras is Laplace (ラプラス).
The idea of the Laplace Demon has been cited several times in Japanese pop culture:
In the Super Robot Wars serial, Elemental Lord of the Wind Cybuster is said to be equipped with the Laplace Demon which can alter the Laws of Probabilities.
In Gundam UC, the titular machine, the Gundam Unicorn, has the La+ (Laplus; Laplace) operative system, which is the key to obtain the Box of Laplace—a repository of secret information whose possession could change the course of the world.
In Mushoku Tensei, the Demon God who tried to rule the world was called Laplace.
See also
References
Laplace
L
Pierre-Simon Laplace |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Hendrik%20Antoon%20Lorentz | List of things named after Dutch physicist Hendrik Antoon Lorentz:
Mathematics and science
Abraham–Lorentz force
Abraham–Lorentz–Dirac Force
Cauchy–Lorentz distribution
Lorentzian
Drude–Lorentz model
Fock–Lorentz symmetry
Lorentz–Berthelot rules
Lorentz covariance
Lorentz symmetry
Lorentz–FitzGerald contraction
Heaviside–Lorentz units
Lorentz–Lorenz equation
Lorentz aether theory
Lorentz factor
Lorentz force
Lorentz force velocimetry
Lorentz group
Lorentz manifold
Lorentz metric
Lorentz pendulum
Lorentz oscillator model
Lorentz scalar
Lorentz surface
Lorentz transformation
Lorentz-violating electrodynamics
Tauc–Lorentz model
Others
Lorentz Centre
Lorentz (crater)
Lorentz Institute
Lorentz Medal
Lorentz locks, lock in the Afsluitdijk in the Netherlands
Lorentz Casimir Lyceum (nl)
Lorentz Driver (Exotic-tier Linear Fusion Rifle found in Destiny 2)
L
Hendrik Lorentz |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Ludwig%20Boltzmann | This is a list of things named after the Austrian physicist and philosopher Ludwig Eduard Boltzmann (20 February 1844 – 5 September 1906).
Science and mathematics
Boltzmann codes
Boltzmann's entropy formula
Boltzmann's principle
Boltzmann's H-theorem
Boltzmann brain
Boltzmann constant
Boltzmann distribution
Boltzmann equation
Quantum Boltzmann equation
Boltzmann factor
Boltzmann machine
Deep Boltzmann machine
Restricted Boltzmann machine
Boltzmann–Matano analysis
Boltzmann relation
Boltzmann sampler
Boltzmann selection
Lattice Boltzmann methods
Maxwell–Boltzmann distribution
Maxwell–Boltzmann statistics
Poisson–Boltzmann equation
Stefan–Boltzmann law
Stefan–Boltzmann constant
Williams–Boltzmann equation
Other
24712 Boltzmann, a main-belt asteroid
Boltzmann (crater), an old lunar crater
Boltzmann Medal
Ludwig Boltzmann Gesellschaft
Ludwig Boltzmann Institute for Neo-Latin Studies
Ludwig Boltzmann Institut für Menschenrechte
Ludwig Boltzmann Prize
Streets, Houses
7 streets in Austria and 5 in Germany are named after him:
Boltzmanngasse in Austria: Vienna (since 27 February 1913)
Boltzmannstraße in Austria: Linz, Klagenfurt; in Germany: Berlin-Dahlem, Garching near Munich, Rhede.
Boltzmann-Straße in Austria: Baden
Ludwig-Boltzmann-Gasse in Graz
Ludwig-Boltzmann-Straße in Neusiedl am See; Germany: Potsdam, Berlin-Adlershof.
Ludwig Boltzmann-Straße in Wiener Neustadt
One house in Austria:
Hotel Boltzmann, Vienna, is named after the address: Boltzmanngasse.
See also
Boltzmann (disambiguation)
References
boltzman |
https://en.wikipedia.org/wiki/Hexagonal%20tiling%20honeycomb | In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity.
The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex.
Images
Viewed in perspective outside of a Poincaré disk model, the image above shows one hexagonal tiling cell within the honeycomb, and its mid-radius horosphere (the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary, asymptoting towards a single ideal point. It can be seen as similar to the order-3 apeirogonal tiling, {∞,3} of H2, with horocycles circumscribing vertices of apeirogonal faces.
Symmetry constructions
It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular: [6,3,3], [3,6,3], [6,3,6], [6,3[3]] and [3[3,3]] , having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)*] (remove 3 mirrors, index 24 subgroup); [3,6,3*] or [3*,6,3] (remove 2 mirrors, index 6 subgroup); [1+,6,3,6,1+] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3[3,3]]. The ringed Coxeter diagrams are , , , and , representing different types (colors) of hexagonal tilings in the Wythoff construction.
Related polytopes and honeycombs
The hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb.
It is part of a sequence of regular polychora, which include the 5-cell {3,3,3}, tesseract {4,3,3}, and 120-cell {5,3,3} of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral vertex figures.
It is also part of a sequence of regular honeycombs of the form {6,3,p}, which are each composed of hexagonal tiling cells:
Rectified hexagonal tiling honeycomb
The rectified hexagonal tiling honeycomb, t1{6,3,3}, has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure. The half-symmetry construction alternates two types of tetrahedra.
Truncated hexagonal tiling honeycomb
The truncated hexagonal tiling honeycomb, t0,1{6,3,3}, has tetrahedral and truncated hexagonal |
https://en.wikipedia.org/wiki/Order-4%20hexagonal%20tiling%20honeycomb | In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.
The Schläfli symbol of the order-4 hexagonal tiling honeycomb is {6,3,4}. Since that of the hexagonal tiling is {6,3}, this honeycomb has four such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the octahedron is {3,4}, the vertex figure of this honeycomb is an octahedron. Thus, eight hexagonal tilings meet at each vertex of this honeycomb, and the six edges meeting at each vertex lie along three orthogonal axes.
Images
Symmetry
The order-4 hexagonal tiling honeycomb has three reflective simplex symmetry constructions.
The half-symmetry uniform construction {6,31,1} has two types (colors) of hexagonal tilings, with Coxeter diagram ↔ . A quarter-symmetry construction also exists, with four colors of hexagonal tilings: .
An additional two reflective symmetries exist with non-simplectic fundamental domains: [6,3*,4], which is index 6, with Coxeter diagram ; and [6,(3,4)*], which is index 48. The latter has a cubic fundamental domain, and an octahedral Coxeter diagram with three axial infinite branches: . It can be seen as using eight colors to color the hexagonal tilings of the honeycomb.
The order-4 hexagonal tiling honeycomb contains , which tile 2-hypercycle surfaces and are similar to the truncated infinite-order triangular tiling, :
Related polytopes and honeycombs
The order-4 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including this regular form, and its dual, the order-6 cubic honeycomb.
The order-4 hexagonal tiling honeycomb has a related alternated honeycomb, ↔ , with triangular tiling and octahedron cells.
It is a part of sequence of regular honeycombs of the form {6,3,p}, all of which are composed of hexagonal tiling cells:
This honeycomb is also related to the 16-cell, cubic honeycomb and order-4 dodecahedral honeycomb, all of which have octahedral vertex figures.
The aforementioned honeycombs are also quasiregular:
Rectified order-4 hexagonal tiling honeycomb
The rectified order-4 hexagonal tiling honeycomb, t1{6,3,4}, has octahedral and trihexagonal tiling facets, with a square prism vertex figure.
It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{∞,4}, which alternates apeirogonal and square faces:
Truncated order-4 hexagonal tiling honeycomb
The truncated order-4 hexagonal tiling honeycomb, t0,1{6,3,4}, has octahedron and truncated hexagonal tiling facets, with a square pyramid vertex figure.
It is similar to the 2D hyperbolic truncated order-4 |
https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3%20T%C3%B3th%20%28footballer%29 | László Tóth (born 9 July 1995) is a Hungarian professional footballer who plays for Szolnok.
Club statistics
Updated to games played as of 9 September 2014.
References
Player profile at HLSZ
1995 births
People from Jászberény
Living people
Hungarian men's footballers
Men's association football midfielders
Hungary men's youth international footballers
Fehérvár FC players
Puskás Akadémia FC players
Balmazújvárosi FC players
Mezőkövesdi SE footballers
Szolnoki MÁV FC footballers
Kazincbarcikai SC footballers
Vasas SC players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Footballers from Jász-Nagykun-Szolnok County |
https://en.wikipedia.org/wiki/Order-5%20hexagonal%20tiling%20honeycomb | In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere, a flat plane in hyperbolic space that approaches a single ideal point at infinity.
The Schläfli symbol of the order-5 hexagonal tiling honeycomb is {6,3,5}. Since that of the hexagonal tiling is {6,3}, this honeycomb has five such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the icosahedron is {3,5}, the vertex figure of this honeycomb is an icosahedron. Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.
Symmetry
A lower-symmetry construction of index 120, [6,(3,5)*], exists with regular dodecahedral fundamental domains, and an icosahedral Coxeter-Dynkin diagram with 6 axial infinite-order (ultraparallel) branches.
Images
The order-5 hexagonal tiling honeycomb is similar to the 2D hyperbolic regular paracompact order-5 apeirogonal tiling, {∞,5}, with five apeirogonal faces meeting around every vertex.
Related polytopes and honeycombs
The order-5 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
There are 15 uniform honeycombs in the [6,3,5] Coxeter group family, including this regular form, and its regular dual, the order-6 dodecahedral honeycomb.
The order-5 hexagonal tiling honeycomb has a related alternation honeycomb, represented by ↔ , with icosahedron and triangular tiling cells.
It is a part of sequence of regular hyperbolic honeycombs of the form {6,3,p}, with hexagonal tiling facets:
It is also part of a sequence of regular polychora and honeycombs with icosahedral vertex figures:
Rectified order-5 hexagonal tiling honeycomb
The rectified order-5 hexagonal tiling honeycomb, t1{6,3,5}, has icosahedron and trihexagonal tiling facets, with a pentagonal prism vertex figure.
It is similar to the 2D hyperbolic infinite-order square tiling, r{∞,5} with pentagon and apeirogonal faces. All vertices are on the ideal surface.
Truncated order-5 hexagonal tiling honeycomb
The truncated order-5 hexagonal tiling honeycomb, t0,1{6,3,5}, has icosahedron and truncated hexagonal tiling facets, with a pentagonal pyramid vertex figure.
Bitruncated order-5 hexagonal tiling honeycomb
The bitruncated order-5 hexagonal tiling honeycomb, t1,2{6,3,5}, has hexagonal tiling and truncated icosahedron facets, with a digonal disphenoid vertex figure.
Cantellated order-5 hexagonal tiling honeycomb
The cantellated order-5 hexagonal tiling honeycomb, t0,2{6,3,5}, has icosidodecahedron, rhombitrihexagonal tiling, and pentagonal prism facets, with a wedge vertex figure.
Cantitruncated order-5 hexagonal tiling honeycomb
The cantitruncated order-5 hexagonal tiling honeycomb, t0,1,2{6,3,5}, has truncated icosahedron, truncated trih |
https://en.wikipedia.org/wiki/Order-6%20hexagonal%20tiling%20honeycomb | In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells with an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.
The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,6}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has six such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the triangular tiling is {3,6}, the vertex figure of this honeycomb is a triangular tiling. Thus, infinitely many hexagonal tilings meet at each vertex of this honeycomb.
Related tilings
The order-6 hexagonal tiling honeycomb is analogous to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface.
It contains and that tile 2-hypercycle surfaces, which are similar to the paracompact tilings and (the truncated infinite-order triangular tiling and order-3 apeirogonal tiling, respectively):
Symmetry
The order-6 hexagonal tiling honeycomb has a half-symmetry construction: .
It also has an index-6 subgroup, [6,3*,6], with a non-simplex fundamental domain. This subgroup corresponds to a Coxeter diagram with six order-3 branches and three infinite-order branches in the shape of a triangular prism: .
Related polytopes and honeycombs
The order-6 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs in 3-space.
There are nine uniform honeycombs in the [6,3,6] Coxeter group family, including this regular form.
This honeycomb has a related alternated honeycomb, the triangular tiling honeycomb, but with a lower symmetry: ↔ .
The order-6 hexagonal tiling honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures:
It is also part of a sequence of regular polychora and honeycombs with hexagonal tiling cells:
It is also part of a sequence of regular polychora and honeycombs with regular deltahedral vertex figures:
Rectified order-6 hexagonal tiling honeycomb
The rectified order-6 hexagonal tiling honeycomb, t1{6,3,6}, has triangular tiling and trihexagonal tiling facets, with a hexagonal prism vertex figure.
it can also be seen as a quarter order-6 hexagonal tiling honeycomb, q{6,3,6}, ↔ .
It is analogous to 2D hyperbolic order-4 apeirogonal tiling, r{∞,∞} with infinite apeirogonal faces, and with all vertices on the ideal surface.
Related honeycombs
The order-6 hexagonal tiling honeycomb is part of a series of honeycombs with hexagonal prism vertex figures:
It is also part of a matrix of 3-dimensional quarter honeycombs: q{2p,4,2q}
Truncated order-6 hexagonal tiling honeycomb
The truncated order-6 hexagonal tiling honeycomb, t0,1{6,3,6}, has triangul |
https://en.wikipedia.org/wiki/Vincenzo%20Brunacci | Vincenzo Brunacci (3 March 1768 – 16 June 1818) was an Italian mathematician born in Florence. He was professor of Matematica sublime (infinitesimal calculus) in Pavia. He transmitted Lagrange's ideas to his pupils, including Ottaviano Fabrizio Mossotti, Antonio Bordoni and Gabrio Piola.
Biography
He studied medicine, astronomy and mathematics at the University of Pisa. In 1788 he earned his laurea and the same year he started teaching mathematics at the Naval Institute of Livorno. In 1796, when Napoleon entered Italy, he endorsed the new order. Upon the reinstatement of the Austrian rule, he moved to France between 1799 and 1800. On returning he attained a chair at the University of Pisa. In 1801 he moved to the University of Pavia with the office of professor of infinitesimal calculus and become its dean.
Brunacci believed that Lagrange's approach, developed in the "Théorie des fonctions analytiques", was the correct one and that the infinitesimal concept was to be banned from analysis and mechanics. In Brunacci's university teaching infinitesimal calculus differently from Lagrange's principles was even prohibited as a rule. Brunacci passed his idea of analysis on to his students, among which Fabrizio Ottaviano Mossotti, Gabrio Piola and Antonio Bordoni.
He cooperated with the public administration, in 1805 he was in the Committee for the Naviglio Pavese (Pavia Canal) project and the following year as inspector of Waters and Roads.
In 1809 he joined the Committee for the new measurements and weights system and from 1811 he was inspector general of Public Education for the entire Italian Kingdom.
He died in Pavia in 1818.
Writings
Opuscolo analitico, (1792).
Calcolo integrale delle equazioni lineari, (1798).
Corso di matematica sublime, in four volumes, Firenze, (1804–1807).
Elementi di algebra e di geometria, in two volumes, Firenze, (1809).
Trattato dell'ariete idraulico, (1810).
Notes
External links
An Italian short biography Vincenzo Brunacci in Edizione Nazionale Mathematica Italiana online.
18th-century Italian mathematicians
19th-century Italian mathematicians
University of Pisa alumni
University of Pavia alumni
1768 births
1818 deaths |
https://en.wikipedia.org/wiki/Tom%C3%A1%C5%A1%20Berdych%20career%20statistics | This is a list of the main career statistics of Czech professional tennis player Tomáš Berdych.
Significant finals
Grand Slam finals
Singles: 1 (1 runner-up)
Masters 1000 finals
Singles: 4 (1 title, 3 runners-up)
ATP career finals
Singles: 32 (13 titles, 19 runner-ups)
Doubles: 3 (2 titles, 1 runner-up)
Team competition finals: 6 (3 titles, 3 runners-up)
Performance timelines
Davis Cup matches are included in the statistics. Walkovers or qualifying matches are neither official wins nor losses.
Singles
Notes:
1Berdych withdrew before the semifinals of the 2014 Miami Masters.
2Berdych received a second round walkover at the 2016 Miami Masters.
Doubles
Best Grand Slam results details
Record against top 10 players
* Statistics correct as of August 2019.
Top 10 wins
He has a 53–124 record against players who were, at the time the match was played, ranked in the top 10.
Career Grand Slam tournament seedings
Notes
Tennis career statistics |
https://en.wikipedia.org/wiki/List%20of%20Cultural%20Properties%20of%20Yuzawa%2C%20Niigata | This list is of the Cultural Properties of Japan located within the town of Yuzawa in Niigata Prefecture.
Statistics
12 Properties have been designated and a further 1 Property registered.
Designated Cultural Properties
Registered Cultural Properties
See also
Cultural Properties of Japan
Snow Country
References
External links
Outline of the Cultural Administration of Japan
Cultural Properties of Niigata Prefecture
Cultural Properties of Yuzawa
Yuzawa, Niigata
Lists of Cultural Properties of Japan |
https://en.wikipedia.org/wiki/Cyclotruncated%207-simplex%20honeycomb | In seven-dimensional Euclidean geometry, the cyclotruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, truncated 7-simplex, bitruncated 7-simplex, and tritruncated 7-simplex facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb.
Structure
It can be constructed by eight sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 6-simplex honeycomb divisions on each hyperplane.
Related polytopes and honeycombs
See also
Regular and uniform honeycombs in 7-space:
7-cubic honeycomb
7-demicubic honeycomb
7-simplex honeycomb
Omnitruncated 7-simplex honeycomb
331 honeycomb
Notes
References
Norman Johnson Uniform Polytopes, Manuscript (1991)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Honeycombs (geometry)
8-polytopes |
https://en.wikipedia.org/wiki/Cyclotruncated%208-simplex%20honeycomb | In eight-dimensional Euclidean geometry, the cyclotruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, truncated 8-simplex, bitruncated 8-simplex, tritruncated 8-simplex, and quadritruncated 8-simplex facets. These facet types occur in proportions of 2:2:2:2:1 respectively in the whole honeycomb.
Structure
It can be constructed by nine sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 7-simplex honeycomb divisions on each hyperplane.
Related polytopes and honeycombs
See also
Regular and uniform honeycombs in 8-space:
8-cubic honeycomb
8-demicubic honeycomb
8-simplex honeycomb
Omnitruncated 8-simplex honeycomb
521 honeycomb
251 honeycomb
152 honeycomb
Notes
References
Norman Johnson Uniform Polytopes, Manuscript (1991)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Honeycombs (geometry)
9-polytopes |
https://en.wikipedia.org/wiki/Square%20tiling%20honeycomb | In the geometry of hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.
Rectified order-4 square tiling
It is also seen as a rectified order-4 square tiling honeycomb, r{4,4,4}:
Symmetry
The square tiling honeycomb has three reflective symmetry constructions: as a regular honeycomb, a half symmetry construction ↔ , and lastly a construction with three types (colors) of checkered square tilings ↔ .
It also contains an index 6 subgroup [4,4,3*] ↔ [41,1,1], and a radial subgroup [4,(4,3)*] of index 48, with a right dihedral-angled octahedral fundamental domain, and four pairs of ultraparallel mirrors: .
This honeycomb contains that tile 2-hypercycle surfaces, which are similar to the paracompact order-3 apeirogonal tiling :
Related polytopes and honeycombs
The square tiling honeycomb is a regular hyperbolic honeycomb in 3-space. It is one of eleven regular paracompact honeycombs.
There are fifteen uniform honeycombs in the [4,4,3] Coxeter group family, including this regular form, and its dual, the order-4 octahedral honeycomb, {3,4,4}.
The square tiling honeycomb is part of the order-4 square tiling honeycomb family, as it can be seen as a rectified order-4 square tiling honeycomb.
It is related to the 24-cell, {3,4,3}, which also has a cubic vertex figure.
It is also part of a sequence of honeycombs with square tiling cells:
Rectified square tiling honeycomb
The rectified square tiling honeycomb, t1{4,4,3}, has cube and square tiling facets, with a triangular prism vertex figure.
It is similar to the 2D hyperbolic uniform triapeirogonal tiling, r{∞,3}, with triangle and apeirogonal faces.
Truncated square tiling honeycomb
The truncated square tiling honeycomb, t{4,4,3}, has cube and truncated square tiling facets, with a triangular pyramid vertex figure. It is the same as the cantitruncated order-4 square tiling honeycomb, tr{4,4,4}, .
Bitruncated square tiling honeycomb
The bitruncated square tiling honeycomb, 2t{4,4,3}, has truncated cube and truncated square tiling facets, with a digonal disphenoid vertex figure.
Cantellated square tiling honeycomb
The cantellated square tiling honeycomb, rr{4,4,3}, has cuboctahedron, square tiling, and triangular prism facets, with an isosceles triangular prism vertex figure.
Cantitruncated square tiling honeycomb
The cantitruncated square tiling honeycomb, tr{4,4,3}, has truncated cube, truncated square tiling, and triangular prism facets, with an isosceles triangular pyramid vertex figure.
Runcinated square tiling honeycomb
The runcinated square tiling honeycomb, t0,3{4,4,3}, has octahedron, triangular prism, cube, and square tilin |
https://en.wikipedia.org/wiki/Order-4%20square%20tiling%20honeycomb | In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb is one of 11 paracompact regular honeycombs. It is paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {4,4,4}, it has four square tilings around each edge, and infinite square tilings around each vertex in a square tiling vertex figure.
Symmetry
The order-4 square tiling honeycomb has many reflective symmetry constructions: as a regular honeycomb, ↔ with alternating types (colors) of square tilings, and with 3 types (colors) of square tilings in a ratio of 2:1:1.
Two more half symmetry constructions with pyramidal domains have [4,4,1+,4] symmetry: ↔ , and ↔ .
There are two high-index subgroups, both index 8: [4,4,4*] ↔ [(4,4,4,4,1+)], with a pyramidal fundamental domain: [((4,∞,4)),((4,∞,4))] or ; and [4,4*,4], with 4 orthogonal sets of ultra-parallel mirrors in an octahedral fundamental domain: .
Images
The order-4 square tiling honeycomb is analogous to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface.
It contains and that tile 2-hypercycle surfaces, which are similar to these paracompact order-4 apeirogonal tilings :
Related polytopes and honeycombs
The order-4 square tiling honeycomb is a regular hyperbolic honeycomb in 3-space. It is one of eleven regular paracompact honeycombs.
There are nine uniform honeycombs in the [4,4,4] Coxeter group family, including this regular form.
It is part of a sequence of honeycombs with a square tiling vertex figure:
It is part of a sequence of honeycombs with square tiling cells:
It is part of a sequence of quasiregular polychora and honeycombs:
Rectified order-4 square tiling honeycomb
The rectified order-4 hexagonal tiling honeycomb, t1{4,4,4}, has square tiling facets, with a cubic vertex figure. It is the same as the regular square tiling honeycomb, {4,4,3}, .
Truncated order-4 square tiling honeycomb
The truncated order-4 square tiling honeycomb, t0,1{4,4,4}, has square tiling and truncated square tiling facets, with a square pyramid vertex figure.
Bitruncated order-4 square tiling honeycomb
The bitruncated order-4 square tiling honeycomb, t1,2{4,4,4}, has truncated square tiling facets, with a tetragonal disphenoid vertex figure.
Cantellated order-4 square tiling honeycomb
The cantellated order-4 square tiling honeycomb, is the same thing as the rectified square tiling honeycomb, . It has cube and square tiling facets, with a triangular prism vertex figure.
Cantitruncated order-4 square tiling honeycomb
The cantitruncated order-4 square tiling honeycomb, is the same as the truncated square tiling honeycomb, . It contains cube and truncated square tiling facets, with a mirrored sphenoid vertex figure.
It is the same as the truncated square tiling honeycomb, .
Runcinated order-4 square tiling honeycomb
The runcinate |
https://en.wikipedia.org/wiki/Lambda%20g%20conjecture | In algebraic geometry, the -conjecture gives a particularly simple formula for certain integrals on the Deligne–Mumford compactification of the moduli space of curves with marked points. It was first found as a consequence of the Virasoro conjecture by . Later, it was proven by using virtual localization in Gromov–Witten theory. It is named after the factor of , the gth Chern class of the Hodge bundle, appearing in its integrand. The other factor is a monomial in the , the first Chern classes of the n cotangent line bundles, as in Witten's conjecture.
Let be positive integers such that:
Then the -formula can be stated as follows:
The -formula in combination withge
where the B2g are Bernoulli numbers, gives a way to calculate all integrals on involving products in -classes and a factor of .
References
Algebraic curves
Moduli theory |
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