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https://en.wikipedia.org/wiki/Wilhelm%20Josef%20Oomens | Wilhelm Josef Oomens SJ (14 September 1918 – 27 June 2008) was a Dutch Jesuit and painter.
Wilhelm Josef Oomens was born in The Hague, Netherlands. He studied mathematics, physics and psychology. In 1949, he entered the congregation of the Society of Jesus (Jesuits) and begun a study of philosophy and Catholic theology.
In 1957 he was ordained as a priest. From then on he acted as teacher and director of a grammar school and was longstanding director of the centre of orientation and cultural studies in Limburg. Following heart surgery (1978), he was compelled to leave teaching, but continued as a priest in pastoral care in Sittard, Netherlands, near the border with Germany.
He enjoyed the work, so he asked the Bishop of Aachen, Klaus Hemmerle, to assign him a rectorate. Thus, the Bishop of Aachen appointed him as pastor at St. Antony, Eschweiler-Roehe, where he served until 2005. Here, he worked closely with the composer and church musician Franz Surges. Meanwhile, continued as a painter. He died 27 June 2008 in Nijmegen.
Works
The emphasis of his art was the rendering and interpretation of biblical motifs. He painted mythological scenes (for example, "Phaeton", 1978; "The three Graces", 1983), but this was of less importance that his historic and scriptural motifs.
He painted the Christmas story ("In dulci jubilo", 1991, "The Adoration of the Magi") and the history of Jesus' suffering (for example "Man of Sorrows", "Stations of the Cross", 1994, "The body of Jesus taken off the cross", 1981 and 1992 and of particular importance "Ecce Homo") and his resurrection and the coronation of the Eucharist ("The doubting Thomas", 1993; "The path to the Light", 1998)
The manner of painting shows the clear influence of such classical Dutch masters as Lucas van Leiden and Hieronymus Bosch.
Philosophy
A focus of Wilhelm Joseph Oomens was the idea of wondering (in the tradition of Aristotle and Fathers of the church). The importance of this he accentuated in a multiplicity of homilies and unpublished writings). For him, he function of painting derived from his confidence in the importance of wondering and reverence.
Literature
Wilhelm Josef Oomens: Worte in Bildern erleben. Verlag Dohr, 2000 (an anthology of selected works of W.J. Oomens, with meditative texts in English and German)
150 Jahre Pfarre St. Antonius Eschweiler-Röhe (published by parish, St. Antony, Eschweiler-Roehe, 1995)
External links
Example of the artist's work
1918 births
2008 deaths
Painters from The Hague
21st-century Dutch Jesuits
20th-century Dutch painters
Dutch male painters
20th-century Dutch Jesuits
20th-century Dutch male artists |
https://en.wikipedia.org/wiki/Hualaihu%C3%A9 | Hualaihué () is a Chilean commune located in Palena Province, Los Lagos Region. The communal capital is the town of Hornopirén.
Demographics
According to the 2002 census of the National Statistics Institute, Hualaihué spans an area of and has 8,273 inhabitants (4,457 men and 3,816 women). Of these, 2,406 (29.1%) lived in urban areas and 5,867 (70.9%) in rural areas. The population grew by 2.1% (169 persons) between the 1992 and 2002 censuses.
Administration
As a commune, Hualaihué is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2008-2012 alcalde is Freddy Ibacache Muñoz (PPD).
Within the electoral divisions of Chile, Hualaihué is represented in the Chamber of Deputies by Gabriel Ascencio (PDC) and Alejandro Santana (RN) as part of the 58th electoral district, together with Castro, Ancud, Quemchi, Dalcahue, Curaco de Vélez, Quinchao, Puqueldón, Chonchi, Queilén, Quellón, Chaitén, Futaleufú and Palena. The commune is represented in the Senate by Camilo Escalona Medina (PS) and Carlos Kuschel Silva (RN) as part of the 17th senatorial constituency (Los Lagos Region).
References
External links
Municipality of Hualaihué
Communes of Chile |
https://en.wikipedia.org/wiki/1977%201.%20deild | Statistics of 1. deild in the 1977 season.
Overview
It was contested by 7 teams, and TB Tvøroyri won the championship.
League standings
Results
The schedule consisted of a total of 12 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Regular home games
References
1. deild seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1978%201.%20deild | Statistics of 1. deild in the 1978 season.
Overview
It was contested by 7 teams, and Havnar Bóltfelag won the championship.
League standings
Results
The schedule consisted of a total of 12 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
1. deild seasons
Faroe
Faroe |
https://en.wikipedia.org/wiki/1979%201.%20deild | Statistics of 1. deild in the 1979 season.
Overview
It was contested by 8 teams, and ÍF Fuglafjørður won the championship.
League standings
Results
The schedule consisted of a total of 14 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
1. deild seasons
Faroe
Faroe
Football |
https://en.wikipedia.org/wiki/1980%201.%20deild | Statistics of 1. deild in the 1980 season.
Overview
It was contested by 8 teams, and TB Tvøroyri won the championship.
League standings
Results
The schedule consisted of a total of 14 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
1. deild seasons
Faroe
Faroe
Football |
https://en.wikipedia.org/wiki/1981%201.%20deild | Statistics of 1. deild in the 1981 season.
Overview
It was contested by 8 teams, and Havnar Bóltfelag won the championship.
League standings
Results
The schedule consisted of a total of 14 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
1. deild seasons
Faroe
Faroe
Football |
https://en.wikipedia.org/wiki/1982%201.%20deild | Statistics of 1. deild in the 1982 season.
Overview
It was contested by 8 teams, and Havnar Bóltfelag won the championship.
League standings
Results
The schedule consisted of a total of 14 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
1. deild seasons
Faroe
Faroe
Football |
https://en.wikipedia.org/wiki/1985%201.%20deild | Statistics of 1. deild in the 1985 season.
Overview
It was contested by 8 teams, and B68 Toftir won the championship.
League standings
Results
The schedule consisted of a total of 14 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Top goalscorers
1. deild seasons
Faroe
Faroe
Football |
https://en.wikipedia.org/wiki/Tucker%27s%20lemma | In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker.
Let T be a triangulation of the closed n-dimensional ball . Assume T is antipodally symmetric on the boundary sphere . That means that the subset of simplices of T which are in provides a triangulation of where if σ is a simplex then so is −σ.
Let be a labeling of the vertices of T which is an odd function on , i.e, for every vertex .
Then Tucker's lemma states that T contains a complementary edge - an edge (a 1-simplex) whose vertices are labelled by the same number but with opposite signs.
Proofs
The first proofs were non-constructive, by way of contradiction.
Later, constructive proofs were found, which also supplied algorithms for finding the complementary edge. Basically, the algorithms are path-based: they start at a certain point or edge of the triangulation, then go from simplex to simplex according to prescribed rules, until it is not possible to proceed any more. It can be proved that the path must end in a simplex which contains a complementary edge.
An easier proof of Tucker's lemma uses the more general Ky Fan lemma, which has a simple algorithmic proof.
The following description illustrates the algorithm for . Note that in this case is a disc and there are 4 possible labels: , like the figure at the top-right.
Start outside the ball and consider the labels of the boundary vertices. Because the labeling is an odd function on the boundary, the boundary must have both positive and negative labels:
If the boundary contains only or only , there must be a complementary edge on the boundary. Done.
Otherwise, the boundary must contain edges. Moreover, the number of edges on the boundary must be odd.
Select an edge and go through it. There are three cases:
You are now in a simplex. Done.
You are now in a simplex. Done.
You are in a simplex with another edge. Go through it and continue.
The last case can take you outside the ball. However, since the number of edges on the boundary must be odd, there must be a new, unvisited edge on the boundary. Go through it and continue.
This walk must end inside the ball, either in a or in a simplex. Done.
Run-time
The run-time of the algorithm described above is polynomial in the triangulation size. This is considered bad, since the triangulations might be very large. It would be desirable to find an algorithm which is logarithmic in the triangulation size. However, the problem of finding a complementary edge is PPA-complete even for dimensions. This implies that there is not too much hope for finding a fast algorithm.
Equivalent results
See also
Topological combinatorics
References
Combinatorics
Topology
Lemmas |
https://en.wikipedia.org/wiki/1986%201.%20deild | Statistics of 1. deild in the 1986 season.
Overview
It was contested by 8 teams, and GÍ Gøta won the championship.
League standings
Results
The schedule consisted of a total of 14 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Top goalscorers
1. deild seasons
Faroe
Faroe
Football |
https://en.wikipedia.org/wiki/1987%201.%20deild | Statistics of 1. deild in the 1987 season.
Overview
It was contested by 8 teams, and TB Tvøroyri won the championship.
League standings
Results
The schedule consisted of a total of 14 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
1. deild seasons
Faroe
Faroe
Football |
https://en.wikipedia.org/wiki/1989%201.%20deild | In 1989, 1. deild was the top tier league in Faroe Islands football (since 2005, the top tier has been the Faroe Islands Premier League, with 1. deild becoming the second tier).
Statistics of 1. deild in the 1989 season.
Overview
It was contested by 10 teams, and B71 Sandoy won the championship.
League standings
Results
Top goalscorers
References
1. deild seasons
Faroe
Faroe
1 |
https://en.wikipedia.org/wiki/1990%201.%20deild | Statistics of 1. deild in the 1990 season.
Overview
It was contested by 10 teams, and Havnar Bóltfelag won the championship.
League standings
Results
The schedule consisted of a total of 14 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Top goalscorers
1. deild seasons
Faroe
Faroe
1 |
https://en.wikipedia.org/wiki/1991%201.%20deild | Statistics of 1. deild in the 1991 season.
Overview
It was contested by 10 teams, and KÍ Klaksvík won the championship.
League standings
Results
The schedule consisted of a total of 14 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Top goalscorers
Source: faroesoccer.com
Top goalscorers
1. deild seasons
Faroe
Faroe
1 |
https://en.wikipedia.org/wiki/1992%201.%20deild | Statistics of 1. deild in the 1992 season.
Overview
It was contested by 10 teams, and B68 Toftir won the championship.
League standings
Results
The schedule consisted of a total of 18 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Top goalscorers
Source: faroesoccer.com
14 goals
Símun Petur Justinussen (GÍ)
11 goals
Uni Arge (HB)
10 goals
Øssur Hansen (B68)
Olgar Danielsen (KÍ)
8 goals
Jákup Símun Simonsen (B36)
Aksel Højgaard (B68)
Bogi Johannesen (TB)
7 goals
Jens Kristian Hansen (B36)
Gunnar Mohr (HB)
1. deild seasons
Faroe
Faroe
1 |
https://en.wikipedia.org/wiki/1993%201.%20deild | Statistics of 1. deild in the 1993 season.
Overview
It was contested by 10 teams, and GÍ Gøta won the championship.
League standings
Results
The schedule consisted of a total of 18 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Top goalscorers
Source: faroesoccer.com
11 goals
Uni Arge (HB)
8 goals
Torbjørn Jensen (B71)
7 goals
Súni Fríði Barbá (B68)
Øssur Hansen (B68)
Henning Jarnskor (GÍ)
6 goals
Kári Gullfoss (B36)
Gunnar Mohr (HB)
Sámal Joensen (GÍ)
Pól Thorsteinsson (B36)
1. deild seasons
Faroe
Faroe
1 |
https://en.wikipedia.org/wiki/1994%201.%20deild | Statistics of 1. deild in the 1994 season.
Overview
It was contested by 10 teams, and GÍ Gøta won the championship.
League standings
Results
The schedule consisted of a total of 18 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Top goalscorers
Source: faroesoccer.com
21 goals
John Petersen (GÍ)
14 goals
Gunnar Mohr (HB)
12 goals
Eyðun Klakstein (KÍ)
10 goals
Allan Mørkøre (KÍ)
9 goals
Bogi Johannesen (TB)
Kurt Mørkøre (KÍ)
8 goals
Djóni Joensen (NSÍ)
Sámal Joensen (GÍ)
1. deild seasons
Faroe
Faroe
1 |
https://en.wikipedia.org/wiki/1995%201.%20deild | Statistics of 1. deild in the 1995 season.
Overview
It was contested by 10 teams, and GÍ Gøta won the championship.
League standings
Results
The schedule consisted of a total of 18 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Top goalscorers
Source: faroesoccer.com
24 goals
Súni Fríði Barbá (B68)
11 goals
Jan Allan Müller (Sumba/VB)
10 goals
Kurt Mørkøre (B68)
Eli Hentze (B71)
Magni Jarnskor (GÍ)
9 goals
Torbjørn Jensen (B71)
8 goals
Sigfríður Clementsen (HB)
Uni Arge (HB)
Olgar Danielsen (KÍ)
1. deild seasons
Faroe
Faroe
1 |
https://en.wikipedia.org/wiki/1996%201.%20deild | Statistics of 1. deild in the 1996 season.
Overview
It was contested by 10 teams, and GÍ Gøta won the championship.
League standings
Results
The schedule consisted of a total of 18 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Top goalscorers
Source: faroesoccer.com
Top goalscorers
References
1. deild seasons
Faroe
Faroe
1 |
https://en.wikipedia.org/wiki/1997%201.%20deild | Statistics of 1. deild in the 1997 season.
Overview
It was contested by 10 teams, and B36 Tórshavn won the championship.
League standings
Results
The schedule consisted of a total of 18 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Top goalscorers
Source: faroesoccer.com
24 goals
Uni Arge (HB)
16 goals
Henning Jarnskor (GÍ)
John Petersen (B36)
12 goals
Kurt Mørkøre (KÍ)
10 goals
Pól Thorsteinsson (VB)
9 goals
Heðin á Lakjuni (KÍ)
8 goals
Gunnar Mohr (HB)
Jens Kristian Hansen (B36)
Julian Johnsson (B36)
Óli Johannesen (B36)
Óli Hansen (NSÍ)
References
1. deild seasons
Faroe
Faroe
1 |
https://en.wikipedia.org/wiki/2001%201.%20deild | In 2001, 1. deild was the top-tier league in Faroe Islands football (since 2005, the top tier has been the Faroe Islands Premier League, with 1. deild becoming the second tier).
Statistics of 1. deild in the 2001 season.
Overview
It was contested by 10 teams, and B36 Tórshavn won the championship.
League standings
Results
The schedule consisted of a total of 18 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Top goalscorers
Source: faroesoccer.com
1. deild seasons
Faroe
Faroe
1 |
https://en.wikipedia.org/wiki/2002%201.%20deild | Statistics of 1. deild in the 2002 season.
Overview
It was contested by 10 teams, and Havnar Bóltfelag won the championship.
League standings
Results
The schedule consisted of a total of 18 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Top goalscorers
Source: faroesoccer.com
18 goals
Andrew av Fløtum (HB)
15 goals
John Petersen (B36)
12 goals
Jón Rói Jacobsen (HB)
11 goals
Hjalgrím Elttør (KÍ)
Øssur Hansen (B68)
9 goals
Sámal Joensen (GÍ)
8 goals
Heðin á Lakjuni (KÍ)
Kurt Mørkøre (KÍ)
1. deild seasons
Faroe
Faroe
1 |
https://en.wikipedia.org/wiki/2004%201.%20deild | Statistics of 1. deild in the 2004 season.
Overview
It was contested by 10 teams, and Havnar Bóltfelag won the championship.
League standings
Results
The schedule consisted of a total of 18 games. Each team played two games against every opponent in no particular order. One of the games was at home and one was away.
Top goalscorers
Source: faroesoccer.com
13 goals
Sonni L. Petersen (EB/Streymur)
12 goals
Súni Olsen (GÍ)
Rógvi Jacobsen (HB)
9 goals
Jacob Bymar (KÍ)
Jónhard Frederiksberg (Skála)
8 goals
John Petersen (B36)
Heine Fernandez (HB)
Bogi Gregersen (Skála)
7 goals
Egil á Bø (B36)
Anderson Cardena (B68)
Sorin Anghel (EB/Streymur)
Heðin á Lakjuni (HB)
Høgni Zachariassen (ÍF)
Erling Fles (KÍ)
References
1. deild seasons
Faroe
Faroe
1 |
https://en.wikipedia.org/wiki/2008%E2%80%9309%20F.C.%20Copenhagen%20season | This article shows statistics of individual players for the football club F.C. Copenhagen. It also lists all matches that F.C. Copenhagen played in the 2008–09 season.
Events
2 April: Forward Marcus Allbäck agrees a 1½ year contract with Swedish club Örgryte IS, meaning he will leave FCK on a free transfer on 1 July.
26 May: Midfielder Michael Silberbauer joins FC Utrecht on a free transfer.
26 May: Assistant coach Peter Nielsen stops.
4 June: Bård Wiggen signs as new assistant coach.
6 July: Captain Michael Gravgaard is sold to FC Nantes for DKK 29,000,000
8 July: Midfielder Morten Bertolt is sold to SønderjyskE for DKK 1,000,000.
14 July: It was published that midfielder Thomas Kristensen will join FCK on 1 January on a free transfer.
15 July: F.C. Copenhagen and FC Nordsjælland agrees on a transfer fee for Thomas Kristensen, so he can transfer immediately.
18 July: Forward César Santin is bought from Kalmar FF for DKK 15,000,000. He will join on 24 July.
25 July: Defender Peter Larsson is bought from Halmstads BK for DKK 15,000,000.
28 July: Goalkeeper Johan Wiland is bought from IF Elfsborg. He will join the club 1 January 2009.
22 December: Midfielder Martin Vingaard is bought from Esbjerg fB for DKK 7,000,000.
12 January: Forward Dame N'Doye is bought from OFI Crete for DKK 15,000,000.
14 January: Midfielder Mads Laudrup is sold to Herfølge Boldklub.
Players
Squad information
This section show the squad as currently, considering all players who are confirmedly moved in and out (see section Players in / out).
Squad stats
Players in / out
In
Out
Club
Coaching staff
Kit
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Other information
*During rebuilt
Competitions
Overall
Danish Superliga
Classification
Results summary
Results by round
Matches
Competitive
Friendlies
References
External links
F.C. Copenhagen official website
F.C. Copenhagen seasons
Copenhagen |
https://en.wikipedia.org/wiki/Discrete%20Morse%20theory | Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology computation, denoising, mesh compression, and topological data analysis.
Notation regarding CW complexes
Let be a CW complex and denote by its set of cells. Define the incidence function in the following way: given two cells and in , let be the degree of the attaching map from the boundary of to . The boundary operator is the endomorphism of the free abelian group generated by defined by
It is a defining property of boundary operators that . In more axiomatic definitions one can find the requirement that
which is a consequence of the above definition of the boundary operator and the requirement that .
Discrete Morse functions
A real-valued function is a discrete Morse function if it satisfies the following two properties:
For any cell , the number of cells in the boundary of which satisfy is at most one.
For any cell , the number of cells containing in their boundary which satisfy is at most one.
It can be shown that the cardinalities in the two conditions cannot both be one simultaneously for a fixed cell , provided that is a regular CW complex. In this case, each cell can be paired with at most one exceptional cell : either a boundary cell with larger value, or a co-boundary cell with smaller value. The cells which have no pairs, i.e., whose function values are strictly higher than their boundary cells and strictly lower than their co-boundary cells are called critical cells. Thus, a discrete Morse function partitions the CW complex into three distinct cell collections: , where:
denotes the critical cells which are unpaired,
denotes cells which are paired with boundary cells, and
denotes cells which are paired with co-boundary cells.
By construction, there is a bijection of sets between -dimensional cells in and the -dimensional cells in , which can be denoted by for each natural number . It is an additional technical requirement that for each , the degree of the attaching map from the boundary of to its paired cell is a unit in the underlying ring of . For instance, over the integers , the only allowed values are . This technical requirement is guaranteed, for instance, when one assumes that is a regular CW complex over .
The fundamental result of discrete Morse theory establishes that the CW complex is isomorphic on the level of homology to a new complex consisting of only the critical cells. The paired cells in and describe gradient paths between adjacent critical cells which can be used to obtain the boundary operator on . Some details of this construction are provided in the next section.
The Morse complex
A gradient path is a sequence of paired cells
satisfying and . The index of this gradient path is defined to be the integer
The division here m |
https://en.wikipedia.org/wiki/Bertram%20Huppert | Bertram Huppert (born 22 October 1927 in Worms, Germany) is a German mathematician specializing in group theory and the representation theory of finite groups. His Endliche Gruppen (finite groups) is an influential textbook in group theory, and he has over 50 doctoral descendants.
Life
Education
Bertram Huppert went to school in Bonn from 1934 until 1945. In 1950, he wrote his diploma thesis in mathematics at the University of Mainz. The thesis discussed "nicht fortsetzbare Potenzreihen" (discontinuous power series), and was written under the direction of Helmut Wielandt.
When Wielandt moved to the University of Tübingen in April 1951, Huppert followed him later in the year, and gained his doctorate (as Wielandt's first doctoral student) with the work "Produkte von paarweise vertauschbaren zyklischen Gruppen" (products of pairwise permutable cyclic groups), in which he showed, among other things, that such groups were supersoluble. This was the first of more than forty further scientific works, not including his books and monographs. The focus of the dissertation was very close to Wielandt's interests at the time, whose 1951 work shows that the product of pairwise permutable nilpotent groups is solvable.
Academic career
Huppert spent the years 1963/64 as a visiting professor at the University of Illinois at Urbana-Champaign and at the California Institute of Technology (Caltech) in Pasadena. In January 1965, he became a professor of pure mathematics at the University of Mainz, where he later became a professor emeritus in 1994. He put a lot of effort into building up the Mainz group theory and abstract algebra research groups.
Following an assignment, he wrote a monumental standard text in the theory of finite groups, Endliche Gruppen I. The group around Wolfgang Gaschütz in Kiel provided important contributions in discussions to that volume. Volumes II and III appeared 14 years later in English with co-author Norman Blackburn.
In 1984, Huppert founded, together with Gerhard Michler, the first Deutsche Forschungsgemeinschaft priority programme in Mathematics at the German universities of Aachen, Bielefeld, Essen and Mainz.
From 1964 to 1985, Huppert was a member of the editorial board of the Journal of Algebra. Together with Wolfgang Gaschütz and Karl W. Gruenberg, he organised Oberwolfach workshops on group theory over many years, and with Michler the Oberwolfach workshop on representation theory.
He was a founding member of the Institute of Experimental Mathematics of the University of Essen and is a member of the Akademie gemeinnütziger Wissenschaften zu Erfurt.
Selected bibliography
Endliche Gruppen (Springer, 1967)
Finite Groups II, III (with N. Blackburn, Springer, 1981/82) und
Stochastische Matrizen (with F.J. Fritz, W.Willems, Springer, 1979)
Angewandte Lineare Algebra (de Gruyter, 1990),
Character Theory of Finite Groups (de Gruyter, 1998),
Lineare Algebra (with Wolfgang Willems) (Teubner, 2006),
References
Wol |
https://en.wikipedia.org/wiki/Coulomb%20wave%20function | In mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument.
Coulomb wave equation
The Coulomb wave equation for a single charged particle of mass is the Schrödinger equation with Coulomb potential
where is the product of the charges of the particle and of the field source (in units of the elementary charge, for the hydrogen atom), is the fine-structure constant, and is the energy of the particle. The solution, which is the Coulomb wave function, can be found by solving this equation in parabolic coordinates
Depending on the boundary conditions chosen, the solution has different forms. Two of the solutions are
where is the confluent hypergeometric function, and is the gamma function. The two boundary conditions used here are
which correspond to -oriented plane-wave asymptotic states before or after its approach of the field source at the origin, respectively. The functions are related to each other by the formula
Partial wave expansion
The wave function can be expanded into partial waves (i.e. with respect to the angular basis) to obtain angle-independent radial functions . Here .
A single term of the expansion can be isolated by the scalar product with a specific spherical harmonic
The equation for single partial wave can be obtained by rewriting the laplacian in the Coulomb wave equation in spherical coordinates and projecting the equation on a specific spherical harmonic
The solutions are also called Coulomb (partial) wave functions or spherical Coulomb functions. Putting changes the Coulomb wave equation into the Whittaker equation, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments and . The latter can be expressed in terms of the confluent hypergeometric functions and . For , one defines the special solutions
where
is called the Coulomb phase shift. One also defines the real functions
In particular one has
The asymptotic behavior of the spherical Coulomb functions , , and at large is
where
The solutions correspond to incoming and outgoing spherical waves. The solutions and are real and are called the regular and irregular Coulomb wave functions.
In particular one has the following partial wave expansion for the wave function
Properties of the Coulomb function
The radial parts for a given angular momentum are orthonormal. When normalized on the wave number scale (k-scale), the continuum radial wave functions satisfy
Other common normalizations of continuum wave functions are on the reduced wave number scale (-scale),
and on the energy scale
The radial wave functions defined in the previous section are normalized to
as a consequence of the normalization
The continuum (or scattering) Coulomb wave |
https://en.wikipedia.org/wiki/SPASS | SPASS is an automated theorem prover for first-order logic with equality developed at the Max Planck Institute for Computer Science and using the superposition calculus. The name originally stood for Synergetic Prover Augmenting Superposition with Sorts. The theorem-proving system is released under the FreeBSD license.
An extension of SPASS called SPASS-XDB added support for on-the-fly retrieval of positive unit axioms from external sources. SPASS-XDB can thus incorporate facts coming from relational databases, web services, or linked data servers. Support for arithmetic using Mathematica was also added.
References
Sources
.
External links
Free theorem provers
Unix programming tools
Max Planck Institute for Informatics |
https://en.wikipedia.org/wiki/Anabta | Anabta () is a Palestinian town in the Tulkarm Governorate in the northern West Bank, located 9 kilometers east of Tulkarm. According to the Palestinian Central Bureau of Statistics, Anabta had a population of 8,077 inhabitants in 2017. Anabta is administered by a municipal council and is one of the oldest municipalities in the Tulkarm Governorate.
Etymology
The name is derived from the word Enabta, which meant "grape" or "berry" in Aramaic and Syriac. Many grape presses have been found in the hills around the village.
History
Roman and Byzantine era
Sherds from the Hellenistic, early and late Roman and the Byzantine eras have been found here.
During Roman and Byzantine rule, Anabta was a Samaritan village. A tradition connects the village with Dositheos, a Samaritan religious leader possibly active during the 1st-century CE. The Samaritan chronicler Abu l-Fath (14th century) mentions that Dositheos died of starvation after going to 'Anbata where he hid in a cave, fasting in an effort to gain wisdom. Some olive trees still existing in Anabta are said to date back to Roman times.
According to the PEF's Survey of Western Palestine, the site appeared "ancient", and rock-cut tombs and a tank of good masonry had been found.
Pottery sherds from the early Muslim and Medieval eras have also been found here.
Mamluk and Ottoman eras
During the reign of Mamluk Sultan Baibars al-Bunduqdari in the 13th century, Anabta served as a central staging point from which to supply the Muslim armies fighting Crusader and Mongol incursions. The location was chosen because it was considered relatively easy to protect as the area is nestled between two large hills.
During Ottoman rule, Anabta was listed in the 1596 Ottoman tax register as being in the Nahiya of Jabal Sami of the Liwa of Nablus. It had a population of 55 Muslim households who paid a fixed tax rate of 33,3% on wheat, barley, summer crops, olives, goats or beehives, and presses for grapes or olives; a total of 13,757 akçe.
In 1852, the American scholar Edward Robinson visited the village. He described it as "large and well built", with two watermills by the stream. There were many camels there, as the village was on the main route for camels from Nablus to Ramleh.
In 1870/1871 (1288 AH), an Ottoman census listed the village in the nahiya (sub-district) of Wadi al-Sha'ir.
In 1882, the PEF's Survey of Western Palestine described it as a village of moderate size, in the valley, with olives around it. It also had a mill. A portion of the Hejaz Railway used to run through the centre of the town, parallel to the main street.
British Mandate era
The first local council in Anabta was established in 1922 during the mandate period. In the 1936 Anabta shooting, on the night of April 15, 1936, a prelude to the 1936–39 Arab revolt in Palestine, about 20 vehicles traveling on the road outside Anabta were stopped at a road block constructed for the purpose by armed villagers, and forced to hand over weapons a |
https://en.wikipedia.org/wiki/Fourier%E2%80%93Bros%E2%80%93Iagolnitzer%20transform | In mathematics, the FBI transform or Fourier–Bros–Iagolnitzer transform is a generalization of the Fourier transform developed by the French mathematical physicists Jacques Bros and Daniel Iagolnitzer in order to characterise the local analyticity of functions (or distributions) on Rn. The transform provides an alternative approach to analytic wave front sets of distributions, developed independently by the Japanese mathematicians Mikio Sato, Masaki Kashiwara and Takahiro Kawai in their approach to microlocal analysis. It can also be used to prove the analyticity of solutions of analytic elliptic partial differential equations as well as a version of the classical uniqueness theorem, strengthening the Cauchy–Kowalevski theorem, due to the Swedish mathematician Erik Albert Holmgren (1872–1943).
Definitions
The Fourier transform of a Schwartz function f in S(Rn) is defined by
The FBI transform of f is defined for a ≥ 0 by
Thus, when a = 0, it essentially coincides with the Fourier transform.
The same formulas can be used to define the Fourier and FBI transforms of tempered distributions in
S(Rn).
Inversion formula
The Fourier inversion formula
allows a function f to be recovered from its Fourier transform.
In particular
Similarly, at a positive value of a, f(0) can be recovered from the FBI transform of f(x) by the inversion formula
Criterion for local analyticity
Bros and Iagolnitzer showed that a distribution f is locally equal to a real analytic function at y, in the direction ξ
if and only if its FBI transform satisfies an inequality of the form
for |ξ| sufficiently large.
Holmgren's uniqueness theorem
A simple consequence of the Bros and Iagolnitzer characterisation of local analyticity is the following regularity result of Lars Hörmander and Mikio Sato ().Theorem. Let P be an elliptic partial differential operator with analytic coefficients defined on an open subset
X of Rn. If Pf is analytic in X, then so too is f.
When "analytic" is replaced by "smooth" in this theorem, the result is just Hermann Weyl's classical lemma on elliptic regularity, usually proved using Sobolev spaces (Warner 1983). It is a special case of more general results involving the analytic wave front set (see below), which imply Holmgren's classical strengthening of the Cauchy–Kowalevski theorem on linear partial differential equations with real analytic coefficients. In modern language, Holmgren's uniquess theorem states that any distributional solution of such a system of equations must be analytic and therefore unique, by the Cauchy–Kowalevski theorem.
The analytic wave front set
The analytic wave front set or singular spectrum WFA(f) of a distribution f (or more generally of a hyperfunction) can be defined in terms of the FBI transform () as the complement of the conical set of points (x, λ ξ) (λ > 0) such that the FBI transform satisfies the Bros–Iagolnitzer inequality
for y the point at which one would like to test for analyticity, and |ξ| suff |
https://en.wikipedia.org/wiki/Alice%20T.%20Schafer | Alice Turner Schafer (June 18, 1915 – September 27, 2009) was an American mathematician. She was one of the founding members of the Association for Women in Mathematics in 1971.
Early life
Alice Elizabeth Turner was born on June 18, 1915, in Richmond, Virginia. She received a full scholarship to study at the University of Richmond. She was the only female mathematics major. At the time, women were not allowed in the campus library. She was a brilliant student and won the department's James D. Crump Prize in mathematics in her junior year. She completed her B.A. degree in mathematics in 1936.
For three years Alice was a secondary school teacher, accruing savings to pay for graduate school.
At University of Chicago, Alice was a student of Ernest Preston Lane, author of Metric Differential Geometry of Curves and Surfaces (1940) and A Treatise on Projective Differential Geometry (1942).
Alice studied differential geometry of curves and implications of the singular point of a curve. When a curve has null binormal, it is planar at that point. Duke Mathematical Journal published her work in 1944. Alice continued her investigations into curves near an undulation point, publishing in American Journal of Mathematics in 1948.
When she was completing her studies at Chicago, she met Richard Schafer, who was also completing his Ph.D. in mathematics at Chicago. In 1942 Turner married Richard Schafer, after both had completed their doctorates. They had two sons.
Academic career
After completing her Ph.D., Alice Schafer taught at Connecticut College, Swarthmore College, the University of Michigan and several other institutions. In 1962 she joined the faculty of Wellesley College as a full professor. Her husband Richard was working at the Massachusetts Institute of Technology, researching non-associative algebras. In 1966 he published a book on them which he dedicated "To Alice".
As a teacher, Alice especially reached out to students who had difficulties with or were afraid of mathematics, by designing special classes for them. She took a special interest in helping high-school students, women in particular, achieve in mathematics.
In 1971, Schafer was one of the founding members of the Association for Women in Mathematics. She was elected as the second President of the Association. "Under the leadership of its second president Alice T. Schafer, [AWM] was legally incorporated in 1973 and received tax-exempt status in 1974."
Schafer was named Helen Day Gould Professor of Mathematics at Wellesley in 1980. She retired from Wellesley in 1980. However, she remained there for two more years during which she was chairman of Wellesley's Affirmative Action Program. After retiring from Wellesley, she taught at Simmons College and was also involved in the management program in the Radcliffe College Seminars. Her husband retired from MIT in 1988 and the couple moved to Arlington, Virginia. However, she still wanted to teach. She became professor of mathematics at Mar |
https://en.wikipedia.org/wiki/1959%20Mestaruussarja | Statistics of Mestaruussarja in the 1959 season.
Overview
It was contested by 10 teams, and HIFK Helsinki won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1960%20Mestaruussarja | Statistics of Mestaruussarja in the 1960 season.
Overview
It was contested by 12 teams, and Haka Valkeakoski won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1961%20Mestaruussarja | Statistics of Mestaruussarja in the 1961 season.
Overview
It was contested by 12 teams, and HIFK Helsinki won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1962%20Mestaruussarja | Statistics of Mestaruussarja in the 1962 season.
Overview
It was contested by 12 teams, and Haka Valkeakoski won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1963%20Mestaruussarja | Statistics of Mestaruussarja in the 1963 season.
Overview
It was contested by 12 teams, and Reipas Lahti won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1964%20Mestaruussarja | Statistics of Mestaruussarja in the 1964 season.
Overview
It was contested by 12 teams, and HJK Helsinki won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1965%20Mestaruussarja | Statistics of Mestaruussarja in the 1965 season.
Overview
It was contested by 12 teams, and Haka Valkeakoski won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1966%20Mestaruussarja | Statistics of Mestaruussarja in the 1966 season.
Overview
It was contested by 12 teams, and KuPS Kuopio won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1967%20Mestaruussarja | Statistics of Mestaruussarja in the 1967 season.
Overview
It was contested by 12 teams, and Reipas Lahti won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1968%20Mestaruussarja | Statistics of Mestaruussarja in the 1968 season.
Overview
It was contested by 12 teams, and TPS Turku won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1969%20Mestaruussarja | Statistics of Mestaruussarja in the 1969 season.
Overview
It was contested by 12 teams, and KPV Kokkola won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1970%20Mestaruussarja | Statistics of Mestaruussarja in the 1970 season.
Overview
It was contested by 12 teams, and Reipas Lahti won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1971%20Mestaruussarja | Statistics of Mestaruussarja in the 1971 season.
Overview
It was contested by 14 teams, and TPS Turku won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1972%20Mestaruussarja | Statistics of Mestaruussarja in the 1972 season.
Overview
It was contested by 12 teams, and TPS Turku won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1975%20Mestaruussarja | Statistics of Mestaruussarja in the 1975 season.
Overview
It was contested by 12 teams, and TPS Turku won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1976%20Mestaruussarja | Statistics of Mestaruussarja in the 1976 season.
Overview
It was contested by 12 teams, and KuPS Kuopio won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1977%20Mestaruussarja | This page provides statistics of the Mestaruussarja, the premier division of Finnish football, for the 1977 season.
Overview
It was contested by 12 teams, and Haka Valkeakoski won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1978%20Mestaruussarja | Statistics of Mestaruussarja in the 1978 season.
Overview
It was contested by 12 teams, and HJK Helsinki won the championship.
League standings
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1979%20Mestaruussarja | Statistics of Mestaruussarja in the 1979 season.
Overview
Preliminary Stage was contested by 12 teams, and higher 8 teams go into Championship Group. Lower 4 teams fought in promotion/relegation group with higher 4 teams of Ykkönen.
OPS Oulu won the championship.
Preliminary stage
Table
Results
Championship group
Table
Results
Promotion/relegation group
Table
The teams obtained bonus points on the basis of their preliminary stage position.
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1980%20Mestaruussarja | Statistics of Mestaruussarja in the 1980 season.
Overview
Preliminary Stage is performed in 12 teams, and higher 8 teams go into Championship Group. Lower 4 teams fought in promotion/relegation group with higher 4 teams of Ykkönen.
OPS Oulu won the championship.
Preliminary stage
Table
Results
Championship group
Table
The points were halved (rounded upwards in uneven cases) after the preliminary stage.
Results
Promotion and relegation group
Table
The teams obtained bonus points on the basis of their preliminary stage position.
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1981%20Mestaruussarja | Statistics of Mestaruussarja in the 1981 season.
Overview
Preliminary Stage is performed in 12 teams, and higher 8 teams go into Championship Group. Lower 4 teams fought in promotion/relegation group with higher 4 teams of Ykkönen.
HJK Helsinki won the championship.
Preliminary stage
Table
Results
Championship group
Table
The points were halved (rounded upwards in uneven cases) after the preliminary stage.
Results
Promotion and relegation group
Table
The teams obtained bonus points on the basis of their preliminary stage position.
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1982%20Mestaruussarja | Statistics of (Finnish football) Mestaruussarja in the 1982 season.
Overview
Preliminary Stage was contested by 12 teams, and higher 8 teams go into Championship Group. Lower 4 teams fought in promotion/relegation group with higher 4 teams of Ykkönen.
Kuusysi Lahti won the championship.
Preliminary stage
Table
Results
Championship group
Table
The points were halved (rounded upwards in uneven cases) after the preliminary stage.
Results
Promotion and relegation group
Table
The teams obtained bonus points on the basis of their preliminary stage position.
Results
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1983%20Mestaruussarja | Statistics of Mestaruussarja in the 1983 season.
Overview
Preliminary Stage is performed in 12 teams, and higher 8 teams go into Championship Group. Lower 4 teams fought in promotion/relegation group with higher 4 teams of Ykkönen.
Ilves Tampere won the championship.
Preliminary stage
Table
Results
Championship group
Table
The points were halved (rounded upwards in uneven cases) after the preliminary stage.
Results
Promotion/relegation group
Table
The teams obtained bonus points on the basis of their preliminary stage position.
Results
See also
Ykkönen (Tier 2)
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1984%20Mestaruussarja | Statistics of Mestaruussarja in the 1984 season.
Overview
It was contested by 12 teams, and Kuusysi Lahti won the championship.
Preliminary stage
Table
Results
Championship Playoffs
Semifinals
|}
For Third Place
|}
Finals
|}
See also
Ykkönen (Tier 2)
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1985%20Mestaruussarja | Statistics of Mestaruussarja in the 1985 season.
Overview
It was contested by 12 teams, and HJK Helsinki won the championship.
Preliminary stage
Table
Results
Championship Playoffs
Semifinals
|}
For Third Place
|}
Finals
|}
See also
Ykkönen (Tier 2)
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1986%20Mestaruussarja | Statistics of Mestaruussarja in the 1986 season.
Overview
It was contested by 12 teams, and Kuusysi Lahti won the championship.
League standings
Results
See also
Ykkönen (Tier 2)
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1987%20Mestaruussarja | Statistics of the Mestaruussarja, the premier division of Finnish football, in the 1987 season.
Overview
12 teams performed in the league, and HJK Helsinki won the championship.
League standings
Results
See also
Ykkönen (Tier 2)
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1988%20Mestaruussarja | Statistics of Mestaruussarja in the 1988 season.
Overview
It was contested by 12 teams, and HJK Helsinki won the championship.
Preliminary stage
Table
Results
Championship group
Table
Results
Relegation group
Table
Results
See also
Ykkönen (Tier 2)
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1989%20Mestaruussarja | Statistics of Mestaruussarja in the 1989 season. Mestaruussarja changed the name to Veikkausliiga from
season 1990.
Overview
It was contested by 12 teams, and Kuusysi Lahti won the championship.
Preliminary stage
Table
Results
Championship group
Table
Results
Relegation group
Table
Results
See also
Ykkönen (Tier 2)
References
Finland - List of final tables (RSSSF)
Mestaruussarja seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1990%20Futisliiga | The 1990 Futisliiga was won by Kuusysi Lahti. However during the playoff Helsingin JK won the Finland national championship.
Statistics of Futisliiga in the 1990 season.
Overview
It was contested by 12 teams, and HJK Helsinki won the championship.
Preliminary stage
Table
Results
Championship playoffs
Quarterfinals
|}
Semifinals
|}
For third place
|}
MP Mikkeli were qualified for the first round of the 1991–92 UEFA Cup.
Finals
|}
The champions HJK Helsinki were qualified for the first round of the 1991–92 European Cup, while the Kuusysi Lahti were qualified for the first round of the 1991–92 UEFA Cup.
See also
Ykkönen (Tier 2)
References
External links
Finland - List of final tables (RSSSF)
Veikkausliiga seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1991%20Futisliiga | Statistics of Futisliiga in the 1991 season.
Overview
It was contested by 12 teams, and Kuusysi Lahti won the championship.
League standings
Results
Matches 1–22
Matches 23–33
See also
Ykkönen (Tier 2)
References
Finland - List of final tables (RSSSF)
Veikkausliiga seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1992%20Veikkausliiga | Statistics of Veikkausliiga in the 1992 season. This was the first season under the Veikkausliiga brand.
Overview
It was contested by 12 teams, and HJK Helsinki won the championship.
League standings
Results
Matches 1–22
Matches 23–33
See also
Ykkönen (Tier 2)
References
Finland - List of final tables (RSSSF)
Veikkausliiga seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1993%20Veikkausliiga | This article contains statistics of Veikkausliiga in the 1993 season.
Overview
Preliminary Stage is performed in 12 teams, and higher 8 teams go into Championship Group. Lower 4 teams fought in promotion/relegation group with higher 4 teams of Ykkönen.
FC Jazz Pori won the championship.
Preliminary stage
Table
Results
Championship group
Table
Results
Promotion/relegation group
Table
NB: Top six to Premier Division 1994, the rest to Division One 1994.
See also
Ykkönen (Tier 2)
Footnotes
References
Finland - List of final tables (RSSSF)
Veikkausliiga seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1994%20Veikkausliiga | Statistics of Veikkausliiga in the 1994 season.
Overview
It was contested by 14 teams, and TPV Tampere won the championship.
League standings
Results
See also
Ykkönen (Tier 2)
References
Finland - List of final tables (RSSSF)
Veikkausliiga seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1996%20Veikkausliiga | Statistics of Veikkausliiga in the 1996 season.
Overview
It was contested by 12 teams, and FC Jazz Pori won the championship.
Preliminary stage
Table
Results
Championship group
Table
Results
Relegation group
Table
Results
See also
Ykkönen (Tier 2)
Suomen Cup 1996
References
Finland - List of final tables (RSSSF)
Veikkausliiga seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1997%20Veikkausliiga | Statistics of Veikkausliiga in the 1997 season.
Overview
It was contested by 10 teams, and HJK Helsinki won the championship.
League standings
Haka Valkeakoski as winners of the 1997 Finnish Cup from the lower division qualified for the qualifying round of the 1998–99 Cup Winners' Cup.
Results
Matches 1–18
Matches 19–27
See also
Suomen Cup 1997
References
Finland - List of final tables (RSSSF)
Veikkausliiga seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1998%20Veikkausliiga | Statistics of Veikkausliiga in the 1998 season.
Overview
It was contested by 10 teams, and Haka Valkeakoski won the championship.
League standings
Results
Matches 1–18
Matches 19–27
See also
Suomen Cup 1998
References
Finland - List of final tables (RSSSF)
Veikkausliiga seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/1999%20Veikkausliiga | Statistics of Veikkausliiga in the 1999 season.
Overview
It was contested by 12 teams, and Haka Valkeakoski won the championship.
Preliminary stage
Table
Results
Final stage
Championship group
Table
Results
Relegation group
Table
Results
See also
Suomen Cup 1999
References
Finland - List of final tables (RSSSF)
Veikkausliiga seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/2000%20Veikkausliiga | Statistics of Veikkausliiga in the 2000 season.
Overview
It was contested by 12 teams, and Haka Valkeakoski won the championship.
League standings
Results
Each team plays three times against every other team, either twice at home and once away or once at home and twice away, for a total of 33 matches played each.
Matches 1–22
Matches 23–33
References
Finland - List of final tables (RSSSF)
Veikkausliiga seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/2004%20Veikkausliiga | Statistics of Veikkausliiga in the 2004 season.
Overview
It was contested by 14 teams, and Haka Valkeakoski won the championship.
League table
Premier Division/Division One 2004, promotion/relegation playoff
MIFK Maarianhamina - Jazz Pori 1-0
Jazz Pori - MIFK Maarianhamina 2-2
MIFK Maarianhamina promoted, Jazz Pori relegated.
Results
References
Finland - List of final tables (RSSSF)
Veikkausliiga seasons
Fin
Fin
1 |
https://en.wikipedia.org/wiki/RTPA | RTPA or rtPA may refer to:
Radiotelevisión del Principado de Asturias, a public broadcasting company in Asturias, Spain
Real-time process algebra, a set of new mathematical notations for formally describing system architectures, and static and dynamic behaviors
Recombinant tissue plasminogen activator, a protein involved in the breakdown of blood clots
Release to Production Acceptance, (in IT systems management) a methodology used to consistently and successfully deploy application systems into a production environment regardless of platform. |
https://en.wikipedia.org/wiki/Newcastle%E2%80%93Ottawa%20scale | In statistics, the Newcastle–Ottawa scale is a tool used for assessing the quality of non-randomized studies included in a systematic review and/or meta-analyses. Using the tool, each study is judged on eight items, categorized into three groups: the selection of the study groups; the comparability of the groups; and the ascertainment of either the exposure or outcome of interest for case-control or cohort studies respectively. Stars awarded for each quality item serve as a quick visual assessment. Stars are awarded such that the highest quality studies are awarded up to nine stars. The method was developed as a collaboration between the University of Newcastle, Australia, and the University of Ottawa, Canada, using a Delphi process to define variables for data extraction. The scale was then tested on systematic reviews and further refined. Separate tools were developed for cohort and case–control studies. It has also been adapted for prevalence studies.
References
External links
http://www.ohri.ca/programs/clinical_epidemiology/oxford.htm
Meta-analysis |
https://en.wikipedia.org/wiki/Ohio%20Resource%20Center | The Ohio Resource Center (ORC) for mathematics, science and reading, a project of the State University Education Deans, has been funded by the Ohio General Assembly and established by the Ohio Board of Regents to:
Identify effective instructional and professional development resources and best practices and disseminate them to schools, school districts and higher education institutions.
Support sustained professional development for teachers and administrators in the effective adoption of best practices and teaching resources.
Foster an integrated educational research and development capacity for Ohio through collaboration with colleges and universities involved in teacher preparation.
The role of the ORC is to identify best practices, to disseminate this information to a wide variety of user audiences, and then to assist with implementing, institutionalizing and sustaining these pre-K to 12 practices across the Ohio educational system. ORC uses proposal processes and subcontracts with two- and four-year higher education institutions and other pre- K to 12 education providers for aggregating best practices information, developing interactive data and knowledge bases, disseminating best practices information, and conducting applied research on needed areas critical to the state educational system. Best practices determined through ORC represent a variety of practices, suitable for different purposes and for different audiences, but certified through a peer-review process that assures the validity of each practice for its intended purpose. ORC also makes important contributions to communication with public and business stakeholders of education and to the expansion of the knowledge and databases for informing educational policy.
The ORC provides links to peer-reviewed instructional resources that have been identified by a panel of Ohio educators as exemplifying best or promising practice. Available resources also include content and professional resources as well as assessment and general education resources that will support the work of pre-K to 12 classroom teachers and higher education faculty members. The resources are correlated with Ohio's academic content standards and with applicable national content standards.
The administrative site for the ORC is located in and administered through the College of Education and Human Ecology of the Ohio State University. Many two- and four-year public and private higher education institutions and several other agencies are involved in the design of ORC's structure, the development of its products, and the delivery of its services. ORC operates primarily as a virtual best practice center, with working groups and research teams drawn from faculty at Ohio colleges and universities in cooperation with schools and school districts across the state. ORC's resources are available primarily via the web and are coordinated with other state and regional efforts to improve student achievement and teacher effectiv |
https://en.wikipedia.org/wiki/Automatic%20semigroup | In mathematics, an automatic semigroup is a finitely generated semigroup equipped with several regular languages over an alphabet representing a generating set. One of these languages determines "canonical forms" for the elements of the semigroup, the other languages determine if two canonical forms represent elements that differ by multiplication by a generator.
Formally, let be a semigroup and be a finite set of generators. Then an automatic structure for with respect to consists of a regular language over such that every element of has at least one representative in and such that for each , the relation consisting of pairs with is regular, viewed as a subset of (A# × A#)*. Here A# is A augmented with a padding symbol.
The concept of an automatic semigroup was generalized from automatic groups by Campbell et al. (2001)
Unlike automatic groups (see Epstein et al. 1992), a semigroup may have an automatic structure with respect to one generating set, but not with respect to another. However, if an automatic semigroup has an identity, then it has an automatic structure with respect to any generating set (Duncan et al. 1999).
Decision problems
Like automatic groups, automatic semigroups have word problem solvable in quadratic time. Kambites & Otto (2006) showed that it is undecidable whether an element of an automatic monoid possesses a right inverse.
Cain (2006) proved that both cancellativity and left-cancellativity are undecidable for automatic semigroups. On the other hand, right-cancellativity is decidable for automatic semigroups (Silva & Steinberg 2004).
Geometric characterization
Automatic structures for groups have an elegant geometric characterization called the fellow traveller property (Epstein et al. 1992, ch. 2). Automatic structures for semigroups possess the fellow traveller property but are not in general characterized by it (Campbell et al. 2001). However, the characterization can be generalized to certain 'group-like' classes of semigroups, notably completely simple semigroups (Campbell et al. 2002) and group-embeddable semigroups (Cain et al. 2006).
Examples of automatic semigroups
Bicyclic monoid
Finitely generated subsemigroups of a free semigroup
References
.
.
.
.
Further reading
Semigroup theory
Computability theory |
https://en.wikipedia.org/wiki/Normal-inverse-gamma%20distribution | In probability theory and statistics, the normal-inverse-gamma distribution (or Gaussian-inverse-gamma distribution) is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.
Definition
Suppose
has a normal distribution with mean and variance , where
has an inverse-gamma distribution. Then
has a normal-inverse-gamma distribution, denoted as
( is also used instead of )
The normal-inverse-Wishart distribution is a generalization of the normal-inverse-gamma distribution that is defined over multivariate random variables.
Characterization
Probability density function
For the multivariate form where is a random vector,
where is the determinant of the matrix . Note how this last equation reduces to the first form if so that are scalars.
Alternative parameterization
It is also possible to let in which case the pdf becomes
In the multivariate form, the corresponding change would be to regard the covariance matrix instead of its inverse as a parameter.
Cumulative distribution function
Properties
Marginal distributions
Given as above, by itself follows an inverse gamma distribution:
while follows a t distribution with degrees of freedom.
In the multivariate case, the marginal distribution of is a multivariate t distribution:
Summation
Scaling
Suppose
Then for ,
Proof: To prove this let and fix . Defining , observe that the PDF of the random variable evaluated at is given by times the PDF of a random variable evaluated at . Hence the PDF of evaluated at is given by :
The right hand expression is the PDF for a random variable evaluated at , which completes the proof.
Exponential family
Normal-inverse-gamma distributions form an exponential family with natural parameters , , , and and sufficient statistics , , , and .
Information entropy
Kullback–Leibler divergence
Measures difference between two distributions.
Maximum likelihood estimation
Posterior distribution of the parameters
See the articles on normal-gamma distribution and conjugate prior.
Interpretation of the parameters
See the articles on normal-gamma distribution and conjugate prior.
Generating normal-inverse-gamma random variates
Generation of random variates is straightforward:
Sample from an inverse gamma distribution with parameters and
Sample from a normal distribution with mean and variance
Related distributions
The normal-gamma distribution is the same distribution parameterized by precision rather than variance
A generalization of this distribution which allows for a multivariate mean and a completely unknown positive-definite covariance matrix (whereas in the multivariate inverse-gamma distribution the covariance matrix is regarded as known up to the scale factor ) is the normal-inverse-Wishart distribution
See also
Compound probability distribution
References
Denison, David G. T. ; Holme |
https://en.wikipedia.org/wiki/Olimp%C3%ADada%20Brasileira%20de%20Matem%C3%A1tica | The Brazilian Mathematical Olympiad (, also known as OBM) is a mathematics competition held every year for students of Brazil. The participants are awarded gold, silver and bronze medals in accordance with their performance. The main purpose of this competition is to help in selecting students to represent Brazil at the International Mathematical Olympiad.
History
The Brazilian Mathematical Olympiad has been held since 1979. On that occasion, 11 students were awarded in 1st, 2nd, and 3rd places, and 15 were awarded in the other categories.
In 1991, the competition started to have two levels: Junior, for students with a maximum of 15 years; and Senior, for high school students. Over the years, there have been changes, such as the creation, in 2001, of the University level, with two phases.
In 2017, OBM was integrated into OBMEP. It is then carried out in a single phase, for levels 1, 2, and 3, only for invited students, considering, among other criteria, the score obtained in the second phase of the OBMEP. The University level is maintained, in two phases, but it counts on the individual registration of the undergraduate student.
Goals
Decisively interfere in the improvement of Mathematics teaching in Brazil, improving students and teachers through participation in the Olympics.
Discover young people with exceptional mathematical talent and put them in contact with professional mathematicians and high-level research institutions, providing favorable conditions for the formation and development of a research career.
Select and train students who will represent Brazil in international Mathematics Olympiads, based on their performance at OBM.
Support regional mathematics competitions throughout Brazil.
Organize international mathematics competitions based in Brazil.
Participation levels
The exams are subdivided into four levels, according to the student's education level.
Level 1: Middle school student from 6th or 7th year (grade level)
Level 2: Middle school student from 8th or 9th year or any student who finished Middle school less than one year and has not yet been admitted into the High school
Level 3: High school student from any year (1st, 2nd, or 3rd)
University: Higher education student from any course or any student who finished High school less than one year and has not yet gotten into the university
External links
Official site
See also
Olimpíada Brasileira de Matemática das Escolas Públicas
Mathematics competitions |
https://en.wikipedia.org/wiki/Introduction%20to%20systolic%20geometry | Systolic geometry is a branch of differential geometry, a field within mathematics, studying problems such as the relationship between the area inside a closed curve C, and the length or perimeter of C. Since the area A may be small while the length l is large, when C looks elongated, the relationship can only take the form of an inequality. What is more, such an inequality would be an upper bound for A: there is no interesting lower bound just in terms of the length.
Mikhail Gromov once voiced the opinion that the isoperimetric inequality was known already to the Ancient Greeks. The mythological tale of Dido, Queen of Carthage shows that problems about making a maximum area for a given perimeter were posed in a natural way, in past eras.
The relation between length and area is closely related to the physical phenomenon known as surface tension, which gives a visible form to the comparable relation between surface area and volume. The familiar shapes of drops of water express minima of surface area.
The purpose of this article is to explain another such relation between length and area. A space is called simply connected if every loop in the space can be contracted to a point in a continuous fashion. For example, a room with a pillar in the middle, connecting floor to ceiling, is not simply connected. In geometry, a systole is a distance which is characteristic of a compact metric space which is not simply connected. It is the length of a shortest loop in the space that cannot be contracted to a point in the space. In the room example, absent other features, the systole would be the circumference of the pillar. Systolic geometry gives lower bounds for various attributes of the space in terms of its systole.
It is known that the Fubini–Study metric is the natural metric for the geometrisation of quantum mechanics. In an intriguing connection to global geometric phenomena, it turns out that the Fubini–Study metric can be characterized as the boundary case of equality in Gromov's inequality for complex projective space, involving an area quantity called the 2-systole, pointing to a possible connection to quantum mechanical phenomena.
In the following, these systolic inequalities will be compared to the classical isoperimetric inequalities, which can in turn be motivated by physical phenomena observed in the behavior of a water drop.
Surface tension and shape of a water drop
Perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water. Namely, a drop will typically assume a symmetric round shape. Since the amount of water in a drop is fixed, surface tension forces the drop into a shape which minimizes the surface area of the drop, namely a round sphere. Thus the round shape of the drop is a consequence of the phenomenon of surface tension. Mathematically, this phenomenon is expressed by the isoperimetric inequality.
Isoperimetric inequality in the plane
The solution to |
https://en.wikipedia.org/wiki/List%20of%20Thor%20and%20Delta%20launches%20%281957%E2%80%931959%29 | Between 1957 and 1959, there were 79 Thor missiles launched, of which 51 were successful, giving a 64.5% success rate.
Launch statistics
Rocket configurations
Launch sites
Launch outcomes
1957
There were 9 Thor missiles launched in 1957. 4 of the 9 launches were successful, giving a 44.4% success rate.
1958
There were 20 Thor missiles launched in 1958. 8 of the 20 launches were successful, giving a 40% success rate.
1959
There were 50 Thor missiles launched in 1959. 39 of the 50 launches were successful, giving a 78% success rate.
Images
See also
Lists of Thor and Delta launches
Lists of Thor launches |
https://en.wikipedia.org/wiki/Az-Zawiya%2C%20Salfit | Az-Zawiya () is a Palestinian town in the Salfit Governorate in the northern West Bank, located west of Salfit and south of Qalqilya. According to the Palestinian Central Bureau of Statistics, az-Zawiya had a population of 6,033 in 2017. The town's population is made up of primarily three families: Shuqeir (45%), Muqadi (30%) and Raddad (20%), while the remaining 5% consists of Palestinian refugee families such as Shamlawi, Rabi and Yusif.
Location
Az Zawiya is located (horizontally) west of Salfit. It is bordered by Biddya to the east, Rafat to the south, Kafr Qasem to the west, and 'Azzun 'Atma and Mas-ha to the north.
History
Sherds from IA II, Roman and Byzantine eras have been found.
Ottoman era
Zawiya appeared in the 1596 Ottoman tax registers as being in the Nahiya of Jabal Qubal, part of the Sanjak of Nablus. It had a population of 4 households, all Muslim. The villagers paid a fixed tax-rate of 33.3% on agricultural products, including wheat, barley, fruit trees, goats and beehives, in addition to occasional revenues; a total of 800 akçe,
In the 18th and 19th centuries, al-Zawiya formed part of the highland region known as Jūrat ‘Amra or Bilād Jammā‘īn. Situated between Dayr Ghassāna in the south and the present Route 5 in the north, and between Majdal Yābā in the west and Jammā‘īn, Mardā and Kifl Ḥāris in the east, this area served, according to historian Roy Marom, "as a buffer zone between the political-economic-social units of the Jerusalem and the Nablus regions. On the political level, it suffered from instability due to the migration of the Bedouin tribes and the constant competition among local clans for the right to collect taxes on behalf of the Ottoman authorities.”
In 1838, Edward Robinson noted it as a village, es-Zawieh, in the Jurat Merda district, south of Nablus.
Victor Guérin visited the village in 1870, and described it as having about 200 inhabitants and a small mosque.
In 1870/1871 (1288 AH), an Ottoman census listed the village in the nahiya (sub-district) of Jamma'in al-Thani, subordinate to Nablus.
In 1882, the PEF's Survey of Western Palestine (SWP) described the village as being of moderate size, "probably an ancient place, having rock-cut tombs to the south."
British Mandate era
In the 1922 census of Palestine, conducted by the British Mandate authorities, Zawiya (called: Zawiyeh) had a population of 398, 396 Muslims and 2 Christians, both Orthodox, while in the 1931 census it had 122 occupied houses and a population of 513, all Muslim.
In the 1945 statistics the population was 720, all Muslims, while the total land area was 11,516 dunams, according to an official land and population survey. Of this, 964 were allocated for plantations and irrigable land, 2,055 for cereals, while 41 dunams were classified as built-up (urban) areas.
Jordanian era
In the wake of the 1948 Arab–Israeli War, and after the 1949 Armistice Agreements, Az-Zawiya came under Jordanian rule.
The Jordanian census of 1 |
https://en.wikipedia.org/wiki/Honduran%20Liga%20Nacional%20records%20and%20statistics | This page details Honduran football league records.
All-time table
From 1965–66 to 2018–19 Clausura
Only regular season computed
2 points per win
Highlighted in green currently active (2022-23 season)
Winning percentage
As of 2018–19
Method of calculation: 1 point per win, 0.5 points per draw; divided by games played
Records
Most League titles: 32
Olimpia: 1966–67, 1967–68, 1969–70, 1971–72, 1977–78, 1982–83, 1984–85, 1986–87, 1987–88, 1989–90, 1992–93, 1995–96, 1996–97, 1998–99, 2000–01 A, 2002–03 A, 2003–04 C, 2004–05 C, 2005–06 A, 2005–06 C, 2007–08 C, 2008–09 C, 2009–10 C, 2011–12 A, 2011–12 C, 2012–13 A, 2012–13 C, 2013–14 C, 2014–15 C, 2015–16 C, 2019–20 A, 2020–21 A
Most consecutive league titles: 4
Olimpia: 2011–12 A, 2011–12 C, 2012–13 A, 2012–13 C
Largest attendance: 38,256
17 December 2006, Olimpia 1–3 Motagua at San Pedro Sula
Most appearances: 55
Marathón, Motagua, Olimpia, Real España and Vida
Fewest defeats in season: 0
Olimpia: 1969–70
Goals scored so far: 21,003
As of 2019–20 C
Players
Top scorers
Updated 17 January 2021
By team
Updated 17 January 2021
Most appearances
Updated 17 January 2021
Most goals in one season
As of 2019–20 Apertura
Most goals in one game
As of 2019–20 Apertura
Most Hat-tricks
Most top scorer titles
As of 2018–19
Most appearances in a team
Mauricio Sabillón Marathon 486
Most consecutive matches scoring
Rubén Rodríguez Platense 8 (28 Jul to 15 September 1974)
Fastest 50 goals
Luciano Emilio Real C.D. España , Olimpia 70 (games)
Youngest goalscorer 50 goals
Roger Rojas Club Deportivo Olimpia 22 years 299 days vs Platense F.C.
Fastest goalscorer 100 goals
Rubilio Castillo | Deportes Savio , C.D.S. Vida , F.C. Motagua 188 games
Youngest goalscorer 100 goals
Rubilio Castillo '26 years 259 days vs Real C.D. España
Head to Head
List of Liga Nacional clubs head-to-head comparison (incomplete).
Qualifications by team
Updated 10 January 2021Titles
DL = Domestic leaguesDC = Domestic cupsSC = Domestic SupercupsCA = Central American championships (includes Copa Fraternidad, Torneo Grandes de Centroamérica and/or UNCAF Interclub Cup)CC = CONCACAF championships (includes CONCACAF Champions League, CONCACAF Cup Winners Cup, CONCACAF Giants Cup and/or CONCACAF League).
Regular season performance
From 1965–66 to 2020–21 Clausura
Top 10 attendances
Most active coaches
Updated 17 January 2021''
References
External links
RSSSF.com – Honduras – List of Champions
RSSSF.com – Honduras – Final Tables 1965/66-1994/95
Futhn – Campeones y Subcampeones de la Liga Nacional Desde 1965-66
Stats
Honduras
Association football league records and statistics |
https://en.wikipedia.org/wiki/Stuck%20unknot | In mathematics, a stuck unknot is a closed polygonal chain in three-dimensional space (a skew polygon) that is topologically equal to the unknot but cannot be deformed to a simple polygon when interpreted as a mechanical linkage, by rigid length-preserving and non-self-intersecting motions of its segments.
Similarly a stuck open chain is an open polygonal chain such that the segments may not be aligned by moving rigidly its segments. Topologically such a chain can be unknotted, but the limitation of using only rigid motions of the segments can create nontrivial knots in such a chain.
Consideration of such "stuck" configurations arises in the study of molecular chains in biochemistry.
References
Knots (knot theory) |
https://en.wikipedia.org/wiki/Palena%2C%20Chile | Palena is a Chilean commune located in Palena Province, Los Lagos Region. The commune is named after Palena Lake.
Demographics
According to the 2002 census of the National Statistics Institute, Palena spans an area of and has 1,690 inhabitants (904 men and 786 women), making the commune an entirely rural area. The population grew by 2.2% (37 persons) between the 1992 and 2002 censuses.
Administration
As a commune, Palena is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2008-2012 alcalde is Aladin Delgado Casanova (RN).
Within the electoral divisions of Chile, Palena is represented in the Chamber of Deputies by Gabriel Ascencio (PDC) and Alejandro Santana (RN) as part of the 58th electoral district, together with Castro, Ancud, Quemchi, Dalcahue, Curaco de Vélez, Quinchao, Puqueldón, Chonchi, Queilén, Quellón, Chaitén, Hualaihué and Futaleufú. The commune is represented in the Senate by Camilo Escalona Medina (PS) and Carlos Kuschel Silva (RN) as part of the 17th senatorial constituency (Los Lagos Region).
References
Communes of Chile
Populated places in Palena Province |
https://en.wikipedia.org/wiki/Newton%E2%80%93Pepys%20problem | The Newton–Pepys problem is a probability problem concerning the probability of throwing sixes from a certain number of dice.
In 1693 Samuel Pepys and Isaac Newton corresponded over a problem posed to Pepys by a school teacher named John Smith. The problem was:
Pepys initially thought that outcome C had the highest probability, but Newton correctly concluded that outcome A actually has the highest probability.
Solution
The probabilities of outcomes A, B and C are:
These results may be obtained by applying the binomial distribution (although Newton obtained them from first principles). In general, if P(n) is the probability of throwing at least n sixes with 6n dice, then:
As n grows, P(n) decreases monotonically towards an asymptotic limit of 1/2.
Example in R
The solution outlined above can be implemented in R as follows:
for (s in 1:3) { # looking for s = 1, 2 or 3 sixes
n = 6*s # ... in n = 6, 12 or 18 dice
q = pbinom(s-1, n, 1/6) # q = Prob( <s sixes in n dice )
cat("Probability of at least", s, "six in", n, "fair dice:", 1-q, "\n")
}
Newton's explanation
Although Newton correctly calculated the odds of each bet, he provided a separate intuitive explanation to Pepys. He imagined that B and C toss their dice in groups of six, and said that A was most favorable because it required a 6 in only one toss, while B and C required a 6 in each of their tosses. This explanation assumes that a group does not produce more than one 6, so it does not actually correspond to the original problem.
Generalizations
A natural generalization of the problem is to consider n non-necessarily fair dice, with p the probability that each die will select the 6 face when thrown (notice that actually the number of faces of the dice and which face should be selected are irrelevant). If r is the total number of dice selecting the 6 face, then is the probability of having at least k correct selections when throwing exactly n dice. Then the original Newton–Pepys problem can be generalized as follows:
Let be natural positive numbers s.t. . Is then not smaller than for all n, p, k?
Notice that, with this notation, the original Newton–Pepys problem reads as: is ?
As noticed in Rubin and Evans (1961), there are no uniform answers to the generalized Newton–Pepys problem since answers depend on k, n and p. There are nonetheless some variations of the previous questions that admit uniform answers:
(from Chaundy and Bullard (1960)):
If are positive natural numbers, and , then .
If are positive natural numbers, and , then .
(from Varagnolo, Pillonetto and Schenato (2013)):
If are positive natural numbers, and then .
References
Factorial and binomial topics
Probability problems
Isaac Newton
Mathematical problems |
https://en.wikipedia.org/wiki/Exact%20trigonometric%20values | In mathematics, the values of the trigonometric functions can be expressed approximately, as in , or exactly, as in . While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots.
Common angles
The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 90°. For angles outside of this range, trigonometric values can be found by applying the reflection and shift identities. In the table below, stands for the ratio 1:0. These values can also be considered to be undefined (see division by zero).
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Expressibility with square roots
Some exact trigonometric values, such as , can be expressed in terms of a combination of arithmetic operations and square roots. Such numbers are called constructible, because one length can be constructed by compass and straightedge from another if and only if the ratio between the two lengths is such a number. However, some trigonometric values, such as , have been proven to not be constructible.
The sine and cosine of an angle are constructible if and only if the angle is constructible. If an angle is a rational multiple of radians, whether or not it is constructible can be determined as follows. Let the angle be radians, where a and b are relatively prime integers. Then it is constructible if and only if the prime factorization of the denominator, b, consists of any number of Fermat primes, each with an exponent of 1, times any power of two. For example, and are constructible because they are equivalent to and radians, respectively, and 12 is the product of 3 and 4, which are a Fermat prime and a power of two, and 15 is the product of Fermat primes 3 and 5. On the other hand, is not constructible because it corresponds to a denominator of 9 = 32, and the Fermat prime cannot be raised to a power greater than one. As another example, is not constructible, because the denominator of 7 is not a Fermat prime.
Derivations of constructible values
The values of trigonometric numbers can be derived through a combination of methods. The values of sine and cosine of 30, 45, and 60 degrees are derived by analysis of the 30-60-90 and 90-45-45 triangles. If the angle is expressed in radians as , this takes care of the case where a is 1 and b is 2, 3, 4, or 6.
Half-angle formula
If the denominator, b, is multiplied by additional factors of 2, the sine and cosine can be derived with the half-angl |
https://en.wikipedia.org/wiki/Discrete%20differential%20geometry | Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics, geometry processing and topological combinatorics.
See also
Discrete Laplace operator
Discrete exterior calculus
Discrete Morse theory
Topological combinatorics
Spectral shape analysis
Abstract differential geometry
Analysis on fractals
Discrete calculus
References
Discrete differential geometry Forum
Alexander I. Bobenko, Yuri B. Suris (2008), "Discrete Differential Geometry", American Mathematical Society,
Differential geometry
Simplicial sets |
https://en.wikipedia.org/wiki/Chris%20de%20Ronde | Chris (Christiaan) de Ronde (1912 in Schiedam – 1996 in Buenos Aires) was a Dutch–Argentinian chess master.
He was a champion of Rotterdam. He had studied mathematics in Leyden and Paris.
De Ronde played for the Netherlands in the 8th Chess Olympiad at Buenos Aires 1939, scoring 8½ in his 14 games.
After the tournament, during which World War II broke out in Europe (September 1939), De Ronde, along with many other participants of the Olympiad (Miguel Najdorf, Gideon Ståhlberg, et al.) decided to stay permanently in Argentina.
He played in Buenos Aires in 1940, and tied for 12-13th at Buenos Aires (Circulo) 1945 (Miguel Najdorf won).
External links
Chris de Ronde: a Dutch immortal unearthed
References
1912 births
1996 deaths
Dutch chess players
Argentine chess players
Dutch emigrants
Immigrants to Argentina
People from Schiedam
20th-century chess players |
https://en.wikipedia.org/wiki/Hiroki%20Bandai | is a Japanese football player.
Club statistics
Updated to 1 January 2020.
References
External links
Profile at Mito HollyHock
1986 births
Living people
Association football people from Miyagi Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Vegalta Sendai players
Júbilo Iwata players
Sagan Tosu players
Thespakusatsu Gunma players
Montedio Yamagata players
Mito HollyHock players
AC Nagano Parceiro players
ReinMeer Aomori players
Footballers at the 2006 Asian Games
Asian Games competitors for Japan
Men's association football forwards |
https://en.wikipedia.org/wiki/Shingo%20Hoshino | is a former Japanese football player. Hoshino previously played for Ehime FC in the J2 League.
Club statistics
References
External links
1978 births
Living people
Aichi Gakuin University alumni
Association football people from Fukuoka Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Ehime FC players
Men's association football defenders |
https://en.wikipedia.org/wiki/Hideto%20Inoue | is a former Japanese football player.
Inoue spent most of his career playing for Ehime FC in the J2 League.
Club statistics
References
External links
1982 births
Living people
Chukyo University alumni
Association football people from Ehime Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Ehime FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Zonal%20spherical%20function | In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the spherical principal series, the irreducible representations appearing in the decomposition of the unitary representation of G on L2(G/K). In this case the commutant of G is generated by the algebra of biinvariant functions on G with respect to K acting by right convolution. It is commutative if in addition G/K is a symmetric space, for example when G is a connected semisimple Lie group with finite centre and K is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding
C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant L1 functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.
Zonal spherical functions have been explicitly determined for real semisimple groups by Harish-Chandra. For special linear groups, they were independently discovered by Israel Gelfand and Mark Naimark. For complex groups, the theory simplifies significantly, because G is the complexification of K, and the formulas are related to analytic continuations of the Weyl character formula on K. The abstract functional analytic theory of zonal spherical functions was first developed by Roger Godement. Apart from their group theoretic interpretation, the zonal spherical functions for a semisimple Lie group G also provide a set of simultaneous eigenfunctions for the natural action of the centre of the universal enveloping algebra of G on L2(G/K), as differential operators on the symmetric space G/K. For semisimple p-adic Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and Ian G. Macdonald. The analogues of the Plancherel theorem and Fourier inversion formula in this setting generalise the eigenfunction expansions of Mehler, Weyl and Fock for singular ordinary differential equations: they were obtained in full generality in the 1960s in terms of Harish-Chandra's c-function.
The name "zonal spherical function" comes from the case when G is SO(3,R) acting on a 2-sphere and K is the subgroup fixing a point: in this case the zonal spherical functions can be regarded as certain functions on the sphere invariant under rotation about a fixed axis.
Definitions
Let G be a locally compact unimodular topological group and K a compact subgroup and let H1 = L2(G/K). Thus, H1 admits a unitary representation π of G by left translation. This is a subr |
https://en.wikipedia.org/wiki/K.%20R.%20Sreenivasan | Katepalli Raju Sreenivasan is an aerospace scientist, fluid dynamicist, and applied physicist whose research includes physics and applied mathematics. He studies turbulence, nonlinear and statistical physics, astrophysical fluid mechanics, and cryogenic helium. He was the dean of engineering and executive vice provost for science and technology of New York University. Sreenivasan is also the Eugene Kleiner Professor for Innovation in Mechanical Engineering at New York University Tandon School of Engineering and a professor of physics and mathematics professor at the New York University Graduate School of Arts and Science and Courant Institute of Mathematical Sciences.
Education
Sreenivasan earned his bachelor's degree in mechanical engineering from University Visvesvaraya College of Engineering (UVCE), Bangalore University, in 1968. He attended the Indian Institute of Science, Bangalore, where he was awarded a master's degree in 1970 and a doctorate in aerospace engineering in 1975. His post-doctoral research was at the University of Sydney, the University of Newcastle, and Johns Hopkins University. Sreenivasan was awarded a Honoris Causa master's degree from Yale University in 1985. In 2006, he was awarded an Honoris Causa doctorate from University of Lucknow. He received an Honoris Causa doctorate from the University of Hyderabad in 2007 and the Romanian Academy in 2008.
Career
In 1979, he joined the faculty at Yale University, New Haven, Connecticut as an assistant professor. In 1985, he became a full professor. Sreenivasan became chairman of Mechanical Engineering in 1987. He became the Harold W. Cheel professor of mechanical engineering in 1988. In 1989, Sreenivasan was named acting chairman of the council of engineering. He became the Andrew W. Mellon Professor in 1991. He also served as a professor of physics, applied physics, and mathematics. In 1991, Sreenivasan was appointed to the Society of Scholars for Johns Hopkins University. At the American Physical Society (APS), he served as the chair of the Division of Fluid Dynamics, and the founding chairman of the Topical Group in Statistical and Nonlinear Physics. In 1995, he was awarded the APS Otto Laporte Memorial Award. In 1997, Sreenivasan became an American citizen.
In 2002, he joined the University of Maryland, College Park and became director of the Institute for Physical Sciences and Technology, which is a part of the University of Maryland College of Computer, Mathematical, and Natural Sciences. That same year, Sreenivasan was named director of the Abdus Salam International Centre for Theoretical Physics (ICTP) in Trieste, Italy where he held the Abdus Salam Honorary Professorship. He started the position in March 2003. While working at ICTP, he continued to hold his appointment at the University of Maryland as Glenn L. Martin Professor of Engineering and professor of physics.
In 2002, he received the Medal in Engineering Sciences from the Academy of Sciences for the Develo |
https://en.wikipedia.org/wiki/Superincreasing%20sequence | In mathematics, a sequence of positive real numbers is called superincreasing if every element of the sequence is greater than the sum of all previous elements in the sequence.
Formally, this condition can be written as
for all n ≥ 1.
Example
For example, (1, 3, 6, 13, 27, 52) is a superincreasing sequence, but (1, 3, 4, 9, 15, 25) is not. The following Python source code tests a sequence of numbers to determine if it is superincreasing:
sequence = [1, 3, 6, 13, 27, 52]
total = 0
test = True
for n in sequence:
print("Sum: ", total, "Element: ", n)
if n <= total:
test = False
break
total += n
print("Superincreasing sequence? ", test)
This produces the following output:
Sum: 0 Element: 1
Sum: 1 Element: 3
Sum: 4 Element: 6
Sum: 10 Element: 13
Sum: 23 Element: 27
Sum: 50 Element: 52
Superincreasing sequence? True
See also
Merkle-Hellman knapsack cryptosystem
References
Cryptography |
https://en.wikipedia.org/wiki/M%C3%B6bius%20energy | In mathematics, the Möbius energy of a knot is a particular knot energy, i.e., a functional on the space of knots. It was discovered by Jun O'Hara, who demonstrated that the energy blows up as the knot's strands get close to one another. This is a useful property because it prevents self-intersection and ensures the result under gradient descent is of the same knot type.
Invariance of Möbius energy under Möbius transformations was demonstrated by Michael Freedman, Zheng-Xu He, and Zhenghan Wang (1994) who used it to show the existence of a energy minimizer in each isotopy class of a prime knot. They also showed the minimum energy of any knot conformation is achieved by a round circle.
Conjecturally, there is no energy minimizer for composite knots. Robert B. Kusner and John M. Sullivan have done computer experiments with a discretized version of the Möbius energy and concluded that there should be no energy minimizer for the knot sum of two trefoils (although this is not a proof).
Recall that the Möbius transformations of the 3-sphere
are the ten-dimensional group of angle-preserving diffeomorphisms generated by inversion in 2-spheres. For example, the inversion in the sphere is defined by
Consider a rectifiable simple curve in the Euclidean
3-space , where belongs to or . Define its energy by
where is the shortest arc
distance between
and on the curve. The second term of the
integrand is called a
regularization. It is easy to see that is
independent of parametrization and is unchanged if is changed by a similarity of . Moreover, the energy of any line is 0, the energy of any circle is . In fact, let us use the arc-length parameterization. Denote by the length of the curve . Then
Let denote a unit circle. We have
and consequently,
since .
Knot invariant
A knot is created by beginning with a one-dimensional line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop. Mathematically, we can say a knot is an injective and continuous function with . Topologists consider knots and other entanglements such as links and braids to be equivalent if the knot can be pushed about smoothly, without intersecting itself, to coincide with another knot. The idea of knot equivalence is to give a precise definition of when two knots should be considered the same even when positioned quite differently in space. A mathematical definition is that two knots are equivalent if there is an orientation-preserving homeomorphism with , and this is known to be equivalent to existence of ambient isotopy.
The basic problem of knot theory, the recognition problem, is determining the equivalence of two knots. Algorithms exist to solve this problem, with the first given by Wolfgang Haken in the late 1960s. Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is. The special case of recognizing th |
https://en.wikipedia.org/wiki/1902%20Nemzeti%20Bajnoks%C3%A1g%20I | Statistics of Nemzeti Bajnokság I for the 1902 season.
Overview
It was contested by 5 teams, and Budapesti TC won the championship.
League standings
Results
References
Hungary - List of final tables (RSSSF)
Nemzeti Bajnokság I seasons
1902 in Hungarian football
Hun
Hun |
https://en.wikipedia.org/wiki/1903%20Nemzeti%20Bajnoks%C3%A1g%20I | Statistics of Nemzeti Bajnokság I for the 1903 season.
Overview
It was contested by 8 teams, and Ferencvárosi TC won the championship.
League standings
Results
References
Hungary - List of final tables (RSSSF)
Nemzeti Bajnokság I seasons
1903 in Hungarian football
Hun
Hun |
https://en.wikipedia.org/wiki/1904%20Nemzeti%20Bajnoks%C3%A1g%20I | Statistics of Nemzeti Bajnokság I for the 1904 season.
Overview
It was contested by 9 teams, and MTK Hungária FC won the championship.
League standings
Results
References
Hungary - List of final tables (RSSSF)
Nemzeti Bajnokság I seasons
1904 in Hungarian football
Hun
Hun |
https://en.wikipedia.org/wiki/1907%E2%80%9308%20Nemzeti%20Bajnoks%C3%A1g%20I | Statistics of Nemzeti Bajnokság I for the 1907–08 season.
Overview
It was contested by 9 teams, and MTK Hungária FC won the championship.
League standings
Results
References
Hungary - List of final tables (RSSSF)
1907-08
1907–08 in Hungarian football
1907–08 in European association football leagues |
https://en.wikipedia.org/wiki/1908%E2%80%9309%20Nemzeti%20Bajnoks%C3%A1g%20I | Statistics of Nemzeti Bajnokság I for the 1908–09 season.
Overview
It was contested by 9 teams, and Ferencvárosi TC won the championship.
League standings
Results
References
Hungary - List of final tables (RSSSF)
1908-09
1908–09 in Hungarian football
1908–09 in European association football leagues |
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