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https://en.wikipedia.org/wiki/Nikolai%20Nikolayevich%20Vorobyov%20%28mathematician%29 | Nikolai Nikolayevich Vorobyov (also Vorobiev) (, 18 September 1925, Leningrad — July 14, 1995) was a Soviet and Russian mathematician, an expert in the field of abstract algebra, mathematical logic and probability theory, the founder of the Soviet school of game theory. He is an author of two textbooks, three monographs, a large number of mathematical articles and a number of popular science books. He supervised over 30 kandidat and D.Sc (habilitation) dissertations.
References
1925 births
1995 deaths
Soviet mathematicians
Game theorists
20th-century Russian mathematicians |
https://en.wikipedia.org/wiki/Hinged%20dissection | In geometry, a hinged dissection, also known as a swing-hinged dissection or Dudeney dissection, is a kind of geometric dissection in which all of the pieces are connected into a chain by "hinged" points, such that the rearrangement from one figure to another can be carried out by swinging the chain continuously, without severing any of the connections. Typically, it is assumed that the pieces are allowed to overlap in the folding and unfolding process; this is sometimes called the "wobbly-hinged" model of hinged dissection.
History
The concept of hinged dissections was popularised by the author of mathematical puzzles, Henry Dudeney. He introduced the famous hinged dissection of a square into a triangle (pictured) in his 1907 book The Canterbury Puzzles. The Wallace–Bolyai–Gerwien theorem, first proven in 1807, states that any two equal-area polygons must have a common dissection. However, the question of whether two such polygons must also share a hinged dissection remained open until 2007, when Erik Demaine et al. proved that there must always exist such a hinged dissection, and provided a constructive algorithm to produce them. This proof holds even under the assumption that the pieces may not overlap while swinging, and can be generalised to any pair of three-dimensional figures which have a common dissection (see Hilbert's third problem). In three dimensions, however, the pieces are not guaranteed to swing without overlap.
Other hinges
Other types of "hinges" have been considered in the context of dissections. A twist-hinge dissection is one which use a three-dimensional "hinge" which is placed on the edges of pieces rather than their vertices, allowing them to be "flipped" three-dimensionally. As of 2002, the question of whether any two polygons must have a common twist-hinged dissection remains unsolved.
References
Bibliography
External links
An applet demonstrating Dudeney's hinged square-triangle dissection
A gallery of hinged dissections
Geometric dissection
Recreational mathematics
Discrete geometry
Euclidean plane geometry |
https://en.wikipedia.org/wiki/Equivariant%20differential%20form | In differential geometry, an equivariant differential form on a manifold M acted upon by a Lie group G is a polynomial map
from the Lie algebra to the space of differential forms on M that are equivariant; i.e.,
In other words, an equivariant differential form is an invariant element of
For an equivariant differential form , the equivariant exterior derivative of is defined by
where d is the usual exterior derivative and is the interior product by the fundamental vector field generated by X.
It is easy to see (use the fact the Lie derivative of along is zero) and one then puts
which is called the equivariant cohomology of M (which coincides with the ordinary equivariant cohomology defined in terms of Borel construction.) The definition is due to H. Cartan. The notion has an application to the equivariant index theory.
-closed or -exact forms are called equivariantly closed or equivariantly exact.
The integral of an equivariantly closed form may be evaluated from its restriction to the fixed point by means of the localization formula.
References
Differential geometry |
https://en.wikipedia.org/wiki/Cobordism%20hypothesis | In mathematics, the cobordism hypothesis, due to John C. Baez and James Dolan, concerns the classification of extended topological quantum field theories (TQFTs). In 2008, Jacob Lurie outlined a proof of the cobordism hypothesis, though the details of his approach have yet to appear in the literature as of 2022. In 2021, Daniel Grady and Dmitri Pavlov claimed a complete proof of the cobordism hypothesis, as well as a generalization to bordisms with arbitrary geometric structures.
Formulation
For a symmetric monoidal -category which is fully dualizable and every -morphism of which is adjointable, for , there is a bijection between the -valued symmetric monoidal functors of the cobordism category and the objects of .
Motivation
Symmetric monoidal functors from the cobordism category correspond to topological quantum field theories. The cobordism hypothesis for topological quantum field theories is the analogue of the Eilenberg–Steenrod axioms for homology theories. The Eilenberg–Steenrod axioms state that a homology theory is uniquely determined by its value for the point, so analogously what the cobordism hypothesis states is that a topological quantum field theory is uniquely determined by its value for the point. In other words, the bijection between -valued symmetric monoidal functors and the objects of is uniquely defined by its value for the point.
See also
Cobordism
References
Further reading
Seminar on the Cobordism Hypothesis and (Infinity,n)-Categories, 2013-04-22
Jacob Lurie (4 May 2009). On the Classification of Topological Field Theories
External links
Quantum field theory |
https://en.wikipedia.org/wiki/Simplicial%20space | In mathematics, a simplicial space is a simplicial object in the category of topological spaces. In other words, it is a contravariant functor from the simplex category Δ to the category of topological spaces.
References
Homotopy theory
Topological spaces |
https://en.wikipedia.org/wiki/Zhujianglu%20station | Zhujianglu station () is a station on Line 1 of the Nanjing Metro. It started operations on 3 September 2005 as part of the line's Phase I from to .
Statistics
The station covers an area of . It is long and wide. The roof of the station consists of thick soil which is deep on average; the bottom of the station is about deep.
Around the station
Fu Baoshi Memorial
Jizhaoying Mosque
John Rabe House
Nanjing University Library
References
Railway stations in China opened in 2005
Nanjing Metro stations |
https://en.wikipedia.org/wiki/Kang%20Yoon-goo | Kang Yoon-koo (Hangul: 강윤구, Hanja: 姜倫求) (born July 10, 1990 in Seoul) is a South Korean a starting pitcher for the NC Dinos of the KBO League.
External links
Career statistics from KBO Official Website
Kiwoom Heroes players
KBO League pitchers
South Korean baseball players
1990 births
Living people |
https://en.wikipedia.org/wiki/1993%20North%20Korean%20census | The 1993 North Korean census () was a census conducted by the Central Bureau of Statistics on 31 December 1993.
The population of the country, according to this census, was 21,213,478. The life expectancy at birth was of 70.7 years (67.8 for males and 73.9 for females).
The census was inconsistent internally and in comparison to previous censuses. According to Nicholas Eberstadt: "Quotation marks should attend the '1993' census because that enumeration was not actually conducted in 1993, but rather in early 1994, with respondents replying to questions about their circumstances as of year-end 1993; needless to say, such a procedure is highly unorthodox."
See also
Demographics of North Korea
2008 North Korea Census
References
1993
Census
December 1993 events in Asia
North Korea |
https://en.wikipedia.org/wiki/Fuss%E2%80%93Catalan%20number | In combinatorial mathematics and statistics, the Fuss–Catalan numbers are numbers of the form
They are named after N. I. Fuss and Eugène Charles Catalan.
In some publications this equation is sometimes referred to as Two-parameter Fuss–Catalan numbers or Raney numbers. The implication is the single-parameter Fuss-Catalan numbers are when and .
Uses
The Fuss-Catalan represents the number of legal permutations or allowed ways of arranging a number of articles, that is restricted in some way. This means that they are related to the Binomial Coefficient. The key difference between Fuss-Catalan and the Binomial Coefficient is that there are no "illegal" arrangement permutations within Binomial Coefficient, but there are within Fuss-Catalan. An example of legal and illegal permutations can be better demonstrated by a specific problem such as balanced brackets (see Dyck language).
A general problem is to count the number of balanced brackets (or legal permutations) that a string of m open and m closed brackets forms (total of 2m brackets). By legally arranged, the following rules apply:
For the sequence as a whole, the number of open brackets must equal the number of closed brackets
Working along the sequence, the number of open brackets must be greater than the number of closed brackets
As an numeric example how many combinations can 3 pairs of brackets be legally arranged? From the Binomial interpretation there are or numerically = 20 ways of arranging 3 open and 3 closed brackets. However, there are fewer legal combinations than these when all of the above restrictions apply. Evaluating these by hand, there are 5 legal combinations, namely: ()()(); (())(); ()(()); (()()); ((())). This corresponds to the Fuss-Catalan formula when p=2, r=1 which is the Catalan number formula or =5. By simple subtraction, there are or =15 illegal combinations. To further illustrate the subtlety of the problem, if one were to persist with solving the problem just using the Binomial formula, it would be realised that the 2 rules imply that the sequence must start with an open bracket and finish with a closed bracket. This implies that there are or =6 combinations. This is inconsistent with the above answer of 5, and the missing combination is: ())((), which is illegal and would complete the binomial interpretation.
Whilst the above is a concrete example Catalan numbers, similar problems can be evaluated using Fuss-Catalan formula:
Computer Stack: ways of arranging and completing a computer stack of instructions, each time step 1 instruction is processed and p new instructions arrive randomly. If at the beginning of the sequence there are r instructions outstanding.
Betting: ways of losing all money when betting. A player has a total stake pot that allows them to make r bets, and plays a game of chance that pays p times the bet stake.
Tries: Calculating the number of order m tries on n nodes.
Special Cases
Below is listed a few formulae, al |
https://en.wikipedia.org/wiki/1964%20S%C3%A3o%20Paulo%20FC%20season | The 1964 football season was São Paulo's 35th season since club's existence.
Statistics
Overall
{|class="wikitable"
|-
|Games played || 63 (9 Torneio Rio-São Paulo, 30 Campeonato Paulista, 24 Friendly match)
|-
|Games won ||29 (1 Torneio Rio-São Paulo, 12 Campeonato Paulista, 16 Friendly match)
|-
|Games drawn ||16 (2 Torneio Rio-São Paulo, 9 Campeonato Paulista, 5 Friendly match)
|-
|Games lost ||18 (6 Torneio Rio-São Paulo, 9 Campeonato Paulista, 3 Friendly match)
|-
|Goals scored || 115
|-
|Goals conceded || 86
|-
|Goal difference || +29
|-
|Best result || 6–0 (H) v XV de Piracicaba - Campeonato Paulista - 1964.12.02
|-
|Worst result || 1–5 (A) v Santos - Campeonato Paulista - 1964.07.19
|-
|Most appearances ||
|-
|Top scorer ||
|-
Friendlies
I Torneo Internacional de El Salvador
Torneo Internacional Ciudad de Mexico
II Torneo Internacional de El Salvador
Coppa Città di Firenze
Official competitions
Torneio Rio-São Paulo
Record
Campeonato Paulista
Record
External links
Official website
Sao Paulo
1964 |
https://en.wikipedia.org/wiki/Sly%20%28surname%29 | Sly or Slye is the surname of:
People:
Allan Sly (born 1951), sculptor
Allan Sly (mathematician), probability theorist and MacArthur Fellow
Darryl Sly (1939–2007), Canadian National Hockey League player
Damon Slye (born 1962), computer game designer, director and programmer
Harold Sly (1904–1996), English professional association football player
James Calvin Sly (1807–1864), Mormon pioneer, scout, settler and missionary
Leonard Slye, birth name of Roy Rogers (1911–1998), America singer and actor
Maud Slye (1879–1954), American pathologist
Philippe Sly, Canadian singer
Richard Meares Sly (1849–1929), Australian judge
Tony Sly (1970–2012), American singer, songwriter and guitarist, best known as the frontman of the punk rock band No Use for a Name
William Sly (died 1608), Elizabethan actor and colleague of William Shakespeare
Fictional characters:
Christopher Sly, in Shakespeare's play The Taming of the Shrew |
https://en.wikipedia.org/wiki/Fundamental%20theorem%20of%20algebraic%20K-theory | In algebra, the fundamental theorem of algebraic K-theory describes the effects of changing the ring of K-groups from a ring R to or . The theorem was first proved by Hyman Bass for and was later extended to higher K-groups by Daniel Quillen.
Description
Let be the algebraic K-theory of the category of finitely generated modules over a noetherian ring R; explicitly, we can take , where is given by Quillen's Q-construction. If R is a regular ring (i.e., has finite global dimension), then the i-th K-group of R. This is an immediate consequence of the resolution theorem, which compares the K-theories of two different categories (with inclusion relation.)
For a noetherian ring R, the fundamental theorem states:
(i) .
(ii) .
The proof of the theorem uses the Q-construction. There is also a version of the theorem for the singular case (for ); this is the version proved in Grayson's paper.
See also
basic theorems in algebraic K-theory
Notes
References
Daniel Grayson, Higher algebraic K-theory II [after Daniel Quillen], 1976
Algebraic K-theory
Theorems in algebraic topology |
https://en.wikipedia.org/wiki/1963%20S%C3%A3o%20Paulo%20FC%20season | The 1963 football season was São Paulo's 34th season since club's existence.
Statistics
Overall
{|class="wikitable"
|-
|Games played || 57 (9 Torneio Rio-São Paulo, 30 Campeonato Paulista, 18 Friendly match)
|-
|Games won || 37 (3 Torneio Rio-São Paulo, 18 Campeonato Paulista, 16 Friendly match)
|-
|Games drawn || 11 (2 Torneio Rio-São Paulo, 8 Campeonato Paulista, 1 Friendly match)
|-
|Games lost || 9 (4 Torneio Rio-São Paulo, 4 Campeonato Paulista, 1 Friendly match)
|-
|Goals scored || 118
|-
|Goals conceded || 63
|-
|Goal difference || +55
|-
|Best result || 7–2 (A) v Nacional - Friendly match - 1963.04.28
|-
|Worst result || 2–6 (A) v Santos - Torneio Rio-São Paulo - 1963.03.07
|-
|Most appearances ||
|-
|Top scorer ||
|-|}
Friendlies
Pequeña Copa del Mundo de Clubes 1963
Official competitions
Torneio-Rio São Paulo
Record
Campeonato Paulista
Record
External links
official website
Association football clubs 1963 season
1963
1963 in Brazilian football |
https://en.wikipedia.org/wiki/Homotopy%20group%20with%20coefficients | In topology, a branch of mathematics, for , the i-th homotopy group with coefficients in an abelian group G of a based space X is the pointed set of homotopy classes of based maps from the Moore space of type to X, and is denoted by . For , is a group. The groups are the usual homotopy groups of X.
References
Algebraic topology
Homotopy theory
ko:호모토피 군#계수가 있는 호모토피 |
https://en.wikipedia.org/wiki/Quillen%27s%20theorems%20A%20and%20B | In topology, a branch of mathematics, Quillen's Theorem A gives a sufficient condition for the classifying spaces of two categories to be homotopy equivalent. Quillen's Theorem B gives a sufficient condition for a square consisting of classifying spaces of categories to be homotopy Cartesian. The two theorems play central roles in Quillen's Q-construction in algebraic K-theory and are named after Daniel Quillen.
The precise statements of the theorems are as follows.
In general, the homotopy fiber of is not naturally the classifying space of a category: there is no natural category such that . Theorem B constructs in a case when is especially nice.
References
Theorems in topology |
https://en.wikipedia.org/wiki/Marty%20Golubitsky | Martin Aaron Golubitsky is an American Distinguished professor of mathematics at Ohio State University and the former director of the Mathematical Biosciences Institute.
Biography
Education
Marty Golubitsky was born on April 5, 1945, in Philadelphia, Pennsylvania. He graduated with bachelor's degree in 1966 from the University of Pennsylvania and the same year got his master's there as well. He obtained his Ph.D. from Massachusetts Institute of Technology in 1970 where his advisor was Victor Guillemin.
Career
Full-time
From September 1974 to December 1976 he was an assistant professor at the Queens College and from January of next year to August 1979 served as an associate professor there. Starting from the same month of 1979 he relocated himself to the Arizona State University where he became a professor and served there till August 1983. In September of the same year he held the same position at the University of Houston where he remained till November 2008. From then until 2016 he served as the director of the Mathematical Biosciences Institute at Ohio State University where he retains a distinguished professorship in mathematics. He affiliates himself with such organizations as the American Association for the Advancement of Science, American Mathematical Society, Association for Women in Mathematics and Society for Industrial and Applied Mathematics. He served as the President of the Society for Industrial and Applied Mathematics (SIAM) 2005–2006. In 2012 he became an inaugural fellow of the American Mathematical Society. and in 2009 a SIAM Fellow.
Visiting
From January 1980 to June of the same year he worked at University of Nice Sophia Antipolis as a visiting professor and then from September to December of the next year worked at the Duke University. Following that he worked at the University of California, Berkeley, with the same position which lasted him for two months in summer of 1982 and then from January to June 1989 he worked at the Institute for Mathematics and Applications, a division of the University of Minnesota. He continued to hold that position even four years later when from January to June 1993 he was working at the division of University of Waterloo called Fields Institute. From August to November 2005 he worked at both Newton Institute and Trinity College in Cambridge and then from January to June 2006 worked at the University of Toronto as a distinguished professor. As of July 2005 he works as an adjunct professor at the Computational and Applied Mathematics division of Rice University.
Publications
In 1992, he and Ian Stewart wrote a book called Fearful Symmetry: Is God a Geometer? which was published by Blackwell Publishers in Oxford. In 1994 it was translated into Dutch by Hans van Cuijlenborg where it came out under a title of Turings tijger by Epsilon Uitgaven in Utrecht. In 1995 the same work was translated into Italian by Libero Sosio as Terribili simmetrie: Dio è un geometra? and was published in Turin, I |
https://en.wikipedia.org/wiki/Albrecht%20Schrauf | Albrecht Schrauf (14 December 1837, Vienna – 29 November 1897, Vienna) was an Austrian mineralogist and crystallographer.
Biography
Schrauf studied mathematics, physics and mineralogy at the University of Vienna, where one of his instructors was Wilhelm Josef Grailich. Several years later, he became "custos-adjunct" at the "Imperial Hofmineralien Cabinet" in Vienna. In 1867 he was named first curator of the mineral cabinet, and in 1874 was appointed professor and director of the mineralogical museum at the University of Vienna.
Known for his investigations in the field of crystallography, he was a proponent of the crystallographic index developed by William Hallowes Miller. In the mid-1860s, he published his best works, "Atlas der Krystallformen des Mineralreiches" and an award-winning textbook titled "Lehrbuch der physikalischen Mineralogie". In Vienna, he collaborated with Gustav Tschermak in publication of the journal "Mineralogische Mitteilungen". A rare mineral known as albrechtschraufite is named in his honor.
In 1896 Schrauf lost sight in his left eye due to sudden exposure of sunlight in the course of performing crystallographic measurements.
Principal works
Atlas der krystall-formen des mineralreiches, 1865 - Atlas of crystal forms.
Lehrbuch der physikalischen Mineralogie, 1866 - Textbook of physical mineralogy.
Physikalische Studien. Die gesetzmässigen Beziehungen von Materie und licht, mit specieller Berucksichtigung der Molecular-constitution organischer Reihen und Krystallisirter Körper, 1867 - Physics studies. the lawful relationships of matter and light, etc.
Handbuch der Edelsteinkunde, 1869 - Handbook of gemstone types.
References
1837 births
1897 deaths
Scientists from Vienna
Crystallographers
Austrian mineralogists
Academic staff of the University of Vienna |
https://en.wikipedia.org/wiki/Approximate%20tangent%20space | In geometric measure theory an approximate tangent space is a measure theoretic generalization of the concept of a tangent space for a differentiable manifold.
Definition
In differential geometry the defining characteristic of a tangent space is that it approximates the smooth manifold to first order near the point of tangency. Equivalently, if we zoom in more and more at the point of tangency the manifold appears to become more and more straight, asymptotically tending to approach the tangent space. This turns out to be the correct point of view in geometric measure theory.
Definition for sets
Definition. Let be a set that is measurable with respect to m-dimensional Hausdorff measure , and such that the restriction measure is a Radon measure. We say that an m-dimensional subspace is the approximate tangent space to at a certain point , denoted , if
as
in the sense of Radon measures. Here for any measure we denote by the rescaled and translated measure:
Certainly any classical tangent space to a smooth submanifold is an approximate tangent space, but the converse is not necessarily true.
Multiplicities
The parabola
is a smooth 1-dimensional submanifold. Its tangent space at the origin is the horizontal line . On the other hand, if we incorporate the reflection along the x-axis:
then is no longer a smooth 1-dimensional submanifold, and there is no classical tangent space at the origin. On the other hand, by zooming in at the origin the set is approximately equal to two straight lines that overlap in the limit. It would be reasonable to say it has an approximate tangent space with multiplicity two.
Definition for measures
One can generalize the previous definition and proceed to define approximate tangent spaces for certain Radon measures, allowing for multiplicities as explained in the section above.
Definition. Let be a Radon measure on . We say that an m-dimensional subspace is the approximate tangent space to at a point with multiplicity , denoted with multiplicity , if
as
in the sense of Radon measures. The right-hand side is a constant multiple of m-dimensional Hausdorff measure restricted to .
This definition generalizes the one for sets as one can see by taking for any as in that section. It also accounts for the reflected paraboloid example above because for we have with multiplicity two.
Relation to rectifiable sets
The notion of approximate tangent spaces is very closely related to that of rectifiable sets. Loosely speaking, rectifiable sets are precisely those for which approximate tangent spaces exist almost everywhere. The following lemma encapsulates this relationship:
Lemma. Let be measurable with respect to m-dimensional Hausdorff measure. Then is m-rectifiable if and only if there exists a positive locally -integrable function such that the Radon measure
has approximate tangent spaces for -almost every .
References
, particularly Chapter 3, Section 11 "'Basic Notions, Tangent P |
https://en.wikipedia.org/wiki/1962%20S%C3%A3o%20Paulo%20FC%20season | The 1962 football season was São Paulo's 33rd season since club's existence.
Statistics
Overall
{|class="wikitable"
|-
|Games played || 71 (7 Torneio Rio-São Paulo, 30 Campeonato Paulista, 34 Friendly match)
|-
|Games won || 41 (2 Torneio Rio-São Paulo, 19 Campeonato Paulista, 20 Friendly match)
|-
|Games drawn || 16 (4 Torneio Rio-São Paulo, 5 Campeonato Paulista, 7 Friendly match)
|-
|Games lost || 14 (1 Torneio Rio-São Paulo, 6 Campeonato Paulista, 7 Friendly match)
|-
|Goals scored || 158
|-
|Goals conceded || 98
|-
|Goal difference || +60
|-
|Best result || 5–0 (A) v Once Caldas - Friendly match - 1962.10.205–0 (H) v Prudentina - Campeonato Paulista - 1962.11.08
|-
|Worst result || 1–5 (A) v Corinthians - Friendly match - 1962.06.03
|-
|Most appearances ||
|-
|Top scorer ||
|-
Friendlies
Troféu Lourenço Fló Junior
Taça São Paulo
Official competitions
Torneio Rio-São Paulo
Record
Campeonato Paulista
Record
External links
official website
Association football clubs 1962 season
1962
1962 in Brazilian football |
https://en.wikipedia.org/wiki/United%20States%20national%20rugby%20union%20team%20player%20statistics | The following is a list of selected United States national rugby union team player statistics. For additional statistics, see the United States national rugby union team main page.
Most caps
Last updated: August 19, 2023. Statistics include officially capped matches only.
Most tries
Last updated: August 19, 2023. Statistics include officially capped matches only.
Most points
Last updated: August 19, 2023. Statistics include officially capped matches only.
Most matches as captain
Last updated: USA vs Portugal, 13 August 2023. Statistics include officially capped matches only.
Youngest players
Last updated: USA vs Portugal, 18 November 2022. Statistics include officially capped matches only.
Source: ESPN Scrum
Oldest players
Last updated: USA vs Portugal, 18 November 2022. Statistics include officially capped matches only.
Source: ESPN Scrum
Most points in a match
Last updated: USA vs Portugal, 18 November 2022. Statistics include officially capped matches only.
Source: ESPN Scrum
Most tries in a match
Last updated: USA vs Portugal, 18 November 2022. Statistics include officially capped matches only.
External links
Source: ESPN Scrum
Player |
https://en.wikipedia.org/wiki/Omar%20Khalil%20Al-Hasani | Omar Khalil Ismaeel Al-Hasani (; born 4 February 1992) is a Jordanian former professional footballer who played as an attacking midfielder.
Career statistics
References
Khalil Al-Hasani: "My Presence in Jordan U-22 Means A Lot to Me"
External links
Living people
Jordanian men's footballers
Jordan men's youth international footballers
Men's association football midfielders
1994 births
Al-Salt SC players
Al-Jazeera SC (Amman) players
Jordanian Pro League players
Al-Tai FC players
Saudi First Division League players
Jordanian expatriate men's footballers
Jordanian expatriate sportspeople in Saudi Arabia
Expatriate men's footballers in Saudi Arabia |
https://en.wikipedia.org/wiki/Filip%20Starzy%C5%84ski | Filip Starzyński (born 27 May 1991) is a Polish professional footballer who plays as an attacking midfielder for Ekstraklasa club Ruch Chorzów.
Career statistics
Club
International
Scores and results list Poland's goal tally first, score column indicates score after each Starzyński goal.
References
External links
Living people
1991 births
Footballers from Szczecin
Polish men's footballers
Men's association football midfielders
Poland men's international footballers
Poland men's under-21 international footballers
UEFA Euro 2016 players
Ekstraklasa players
Belgian Pro League players
Ruch Chorzów players
Zagłębie Lubin players
K.S.C. Lokeren Oost-Vlaanderen players
Polish expatriate men's footballers
Polish expatriate sportspeople in Belgium
Expatriate men's footballers in Belgium |
https://en.wikipedia.org/wiki/Universal%20set%20%28disambiguation%29 | Universal set may refer to:
Mathematics
Universal set, the set of all objects, an object whose existence conflicts with the axioms of standard set theory but may exist in other variants
Universe (mathematics), the proper class of all objects in a domain of discourse
Universal point set, in graph drawing, a set that can be used for the vertices of drawings of all n-vertex planar graphs
Sample space, in probability theory and statistics, the set of all possible outcomes of an observation or experiment
Other
Universal Character Set, a set of nearly 100,000 characters on which many character encodings are based |
https://en.wikipedia.org/wiki/Order-7%20tetrahedral%20honeycomb | In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,7}. It has seven tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.
Images
Related polytopes and honeycombs
It is a part of a sequence of regular polychora and honeycombs with tetrahedral cells, {3,3,p}.
It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, {p,3,7}.
It is a part of a sequence of hyperbolic honeycombs, {3,p,7}.
Order-8 tetrahedral honeycomb
In the geometry of hyperbolic 3-space, the order-8 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,8}. It has eight tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-8 triangular tiling vertex arrangement.
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(3,4,3)}, Coxeter diagram, , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,8,1+] = [3,((3,4,3))].
Infinite-order tetrahedral honeycomb
In the geometry of hyperbolic 3-space, the infinite-order tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,∞}. It has infinitely many tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(3,∞,3)}, Coxeter diagram, = , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,∞,1+] = [3,((3,∞,3))].
See also
Convex uniform honeycombs in hyperbolic space
List of regular polytopes
References
Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, , (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
Jeffrey R. Weeks The Shape of Space, 2nd edition (Chapters 16–17: Geometries on Three-manifolds I,II)
George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982)
Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)
Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
External links
John Baez, Visual insights: {7,3,3} Honeycomb (2014/08/01) {7,3,3} Honeycomb Meets Plane at Infinity (2014/08/14)
Danny Calegari, Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination 4 M |
https://en.wikipedia.org/wiki/The%20Manga%20Guides | is a series of educational Japanese manga books. Each volume explains a particular subject in science or mathematics. The series is published in Japan by Ohmsha, in America by No Starch Press, in France by H&K, in Italy by L'Espresso, in Malaysia by Pelangi, and in Taiwan by 世茂出版社. Different volumes are written by different authors.
Volume list
The series to date of February 18, 2023 consists of 50 volumes in Japan. Fourteen of them have been published in English and six in French so far, with more planned, including one on sociology. In contrast, 49 of them have been published and translated in Chinese. One of the books has been translated into Swedish.
The Manga Guide to Electricity
This 207-page guide consists of five chapters, excluding the preface, prologue, and epilogue. It explains fundamental concepts in the study of electricity, including Ohm's law and Fleming's rules. There are written explanations after each manga chapter. An index and two pages to write notes on are provided.
The story begins with Rereko, an average high-school student who lives in Electopia (the land of electricity), failing her final electricity exam. She was forced to skip her summer vacation and go to Earth for summer school. The high school teacher Teteka sensei gave her a “transdimensional walkie-talkie and observation robot” named Yonosuke, which she will use later for going back and forth to Earth. Rereko then met her mentor Hikaru sensei, who did Electrical Engineering Research at a university in Tokyo, Japan. Hikaru sensei explained to Rereko the basic components of electricity with occasional humorous moments.
In the fifth chapter, Hikaru sensei told Rereko her studies are over. Yonosuke soon received Electopia’s call to pick Rereko up. Hikaru sensei told her that he learned a lot from teaching her, and she should keep at it, even back on Electopia. Rereko told Hikaru sensei to keep working on his research and clean his room often. Her sentence was interrupted, and she was transported back to her hometown.
A year later, Hikaru sensei was waiting at the university bus stop. Suddenly, lightning struck his laboratory. He ran to it and found Rereko waiting inside. Rereko told him she graduated, and Teteka sensei assigned her to work at the university as a research assistant, which makes Hikaru sensei and Rereko lab partners.
The Manga Guide to Physics
This 232-page book covers the physics of common objects. It consists of 4 chapters, excluding the preface, prologue, epilogue, appendix, and index. The artist is Keita Takatsu, and the scenario writer is re_akino. The plot revolves around Megumi Ninomiya, an athletic girl, and Ryota Nonomura, a physics Olympics silver medalist.
Megumi was bothered by physics. On the test, she circled an incorrect answer on a question involving Newton's Third Law. The question bothered her during her tennis match with her competitor Sayaka, causing her to be unable to concentrate.
When Megumi was cleaning up after the matc |
https://en.wikipedia.org/wiki/2014%20Lao%20League | Statistics of Lao League in the 2014 season. The league is composed of 10 clubs starts on 22 February 2014. SHB Champasak are the defending champions, having won their first league title in 2013.
Teams
SHB Champasak
Ezra
Hoang Anh Attapeu
Lanexang Intra FC
Lao Police Club
Eastern Star FC
Yotha FC
Relegation to Lao First Division
Pheuanphatthana FC
Promotion from Lao First Division
Lao Army
Lao Toyota FC
Savan FC
League table
References
External links
Lao League at rsssf.com
Lao Premier League seasons
1
Laos
Laos |
https://en.wikipedia.org/wiki/Order-4%20square%20hosohedral%20honeycomb | In geometry, the order-4 square hosohedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {2,4,4}. It has 4 square hosohedra {2,4} around each edge. In other words, it is a packing of infinitely tall square columns. It is a degenerate honeycomb in Euclidean space, but can be seen as a projection onto the sphere. Its vertex figure, a square tiling is seen on each hemisphere.
Images
Stereographic projections of spherical projection, with all edges being projected into circles.
Related honeycombs
It is a part of a sequence of honeycombs with a square tiling vertex figure:
Truncated order-4 square hosohedral honeycomb
The {2,4,4} honeycomb can be truncated as t{2,4,4} or {}×{4,4}, Coxeter diagram , seen as a layer of cubes, partially shown here with alternately colored cubic cells. Thorold Gosset identified this semiregular infinite honeycomb as a cubic semicheck.
The alternation of this honeycomb, , consists of infinite square pyramids and infinite tetrahedrons, between 2 square tilings.
See also
Order-6 triangular hosohedral honeycomb
Order-7 tetrahedral honeycomb
List of regular polytopes
References
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, , (Chapter 10, Regular Honeycombs in Hyperbolic Space)
Honeycombs (geometry) |
https://en.wikipedia.org/wiki/Mohammad%20Hosseinpour | Mohammad Hosseinpour (, May 26, 1993 in Iran) is an Iranian football midfielder, who currently plays for Khoneh Be Khoneh in Azadegan League.
Club career
Club career statistics
Assist Goals
International career
U23
He invited to Iran U-23 training camp by Nelo Vingada to preparation for Incheon 2014 and 2016 AFC U-22 Championship (Summer Olympic qualification).
References
External links
Mohammad Hosseinpour at Persian League
1993 births
Iranian men's footballers
Living people
Esteghlal F.C. players
Rah Ahan Tehran F.C. players
Malavan F.C. players
Mes Rafsanjan F.C. players
Men's association football midfielders
Sportspeople from Babol
Footballers from Mazandaran province |
https://en.wikipedia.org/wiki/Nova%20Methodus%20pro%20Maximis%20et%20Minimis | "Nova Methodus pro Maximis et Minimis" is the first published work on the subject of calculus. It was published by Gottfried Leibniz in the Acta Eruditorum in October 1684. It is considered to be the birth of infinitesimal calculus.
Full title
The full title of the published work is "Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus." In English, the full title can be translated as "A new method for maxima and minima, and for tangents, that is not hindered by fractional or irrational quantities, and a singular kind of calculus for the above mentioned." It is from this title that this branch of mathematics takes the name calculus.
Influence
Although calculus was independently co-invented by Isaac Newton, most of the notation in modern calculus is from Leibniz. Leibniz's careful attention to his notation makes some believe that "his contribution to calculus was much more influential than Newton's."
See also
Leibniz–Newton calculus controversy
References
External links
Mathematical Treasure: Leibniz's Papers on Calculus: "Nova Methodus pro Maximis et Minimis..." (Latin original)
English translation
Mathematics papers
Calculus
Works by Gottfried Wilhelm Leibniz
Texts in Latin |
https://en.wikipedia.org/wiki/L%27Interm%C3%A9diaire%20des%20math%C3%A9maticiens | L'Intermédiaire des mathématiciens was a peer-reviewed scientific journal covering mathematics published by Gauthier-Villars et fils. It was established in 1894 by Émile Lemoine and Charles-Ange Laisant and was published until 1920. A second series started in 1922 and was published until 1925.
External links
L'Intermédiaire des mathématiciens at the HathiTrust Digital Library
Academic journals established in 1894
Publications disestablished in 1925
Mathematics journals
French-language journals |
https://en.wikipedia.org/wiki/North%20American%20Soccer%20League%20records%20and%20statistics | The following is a compilation of notable records and statistics for teams and players in and seasons of North American Soccer League.
All-Time Regular Season Successes
Through completion of 2013 regular season.
All-Time Regular Season Records
Through completion of 2013 regular season.
Highest scoring games
Biggest winning margin
Streaks
Winning streaks
Undefeated streaks
italics streaks still active, as of April 19, 2014
Losing streaks
Winless streaks
Average Season Attendances
Highest single attendance - 13,151 San Antonio Scorpions vs Puerto Rico Islanders, 15 April 2012.
Lowest single attendance - 520 Puerto Rico Islanders vs FC Edmonton, 18 April 2012.
Highest average attendance - 11,507 Montreal Impact, 2011 Season.
Lowest average attendance - 1,525 FC Edmonton, 2012 Season.
Trivia
The longest unbeaten end to a season was achieved by the New York Cosmos in 2013, who closed the campaign thirteen matches unbeaten (with eight wins and three draws).
The highest number of points achieved by a team during a regular season (inclusive of Spring / Fall) was 54 (17–3–8), by Carolina RailHawks in 2011.
The lowest number of points achieved by a team during a regular season (inclusive of Spring / Fall) was 16 (4-4-20), by Atlanta Silverbacks in 2011.
The highest number of points achieved by a team at home during a regular season (inclusive of Spring / Fall) was 34 (11-1-2), by Carolina RailHawks in 2011.
The highest number of points achieved by a team on the road during a regular season (inclusive of Spring / Fall) was 20 (5-5-4), by the San Antonio Scorpions in 2012.
The highest number of points achieved by a team on in a split season 31 (9-4-1), by the New York Cosmos in Fall 2013.
The lowest number of points achieved by a team on in a split season 8 (2-2-8), by the Fort Lauderdale Strikers in Spring 2013.
See also
Major League Soccer records and statistics
Notes
Records and statistics
All-time football league tables
Soccer records and statistics in the United States
Association football league records and statistics |
https://en.wikipedia.org/wiki/Hook%20length%20formula | In combinatorial mathematics, the hook length formula is a formula for the number of standard Young tableaux whose shape is a given Young diagram.
It has applications in diverse areas such as representation theory, probability, and algorithm analysis; for example, the problem of longest increasing subsequences. A related formula gives the number of semi-standard Young tableaux, which is a specialization of a Schur polynomial.
Definitions and statement
Let be a partition of .
It is customary to interpret graphically as a Young diagram, namely a left-justified array of square cells with rows of lengths .
A (standard) Young tableau of shape is a filling of the cells of the Young diagram with all the integers , with no repetition, such that each row and each column form increasing sequences.
For the cell in position , in the th row and th column, the hook is the set of cells such that and or and .
The hook length is the number of cells in .
The hook length formula expresses the number of standard Young tableaux of shape , denoted by or , as
where the product is over all cells of the Young diagram.
Examples
The figure on the right shows hook lengths for the cells in the Young diagram , corresponding to the partition 9 = 4 + 3 + 1 + 1. The hook length formula gives the number of standard Young tableaux as:
A Catalan number counts Dyck paths with steps going up (U) interspersed with steps going down (D), such that at each step there are never more preceding D's than U's. These are in bijection with the Young tableaux of shape : a Dyck path corresponds to the tableau whose first row lists the positions of the U-steps, while the second row lists the positions of the D-steps. For example, UUDDUD correspond to the tableaux with rows 125 and 346.
This shows that , so the hook formula specializes to the well-known product formula
History
There are other formulas for , but the hook length formula is particularly simple and elegant.
A less convenient formula expressing in terms of a determinant was deduced independently by Frobenius and Young in 1900 and 1902 respectively using algebraic methods.
MacMahon found an alternate proof for the Young–Frobenius formula in 1916 using difference methods.
The hook length formula itself was discovered in 1953 by Frame, Robinson, and Thrall as an improvement to the Young–Frobenius formula. Sagan describes the discovery as follows.
Despite the simplicity of the hook length formula, the Frame–Robinson–Thrall proof is not very insightful and does not provide any intuition for the role of the hooks. The search for a short, intuitive explanation befitting such a simple result gave rise to many alternate proofs.
Hillman and Grassl gave the first proof that illuminates the role of hooks in 1976 by proving a special case of the Stanley hook-content formula, which is known to imply the hook length formula.
Greene, Nijenhuis, and Wilf found a probabilistic proof using the hook walk in which the hook len |
https://en.wikipedia.org/wiki/Order-6%20triangular%20hosohedral%20honeycomb | In geometry, the order-6 triangular hosohedral honeycomb a regular space-filling tessellation (or honeycomb) with Schläfli symbol {2,3,6}. It has 6 triangular hosohedra {2,3} around each edge. It is a degenerate honeycomb in Euclidean space, but can be seen as a projection onto the sphere. Its vertex figure, a triangular tiling is seen on each hemisphere.
Images
Stereographic projections of central spherical projection, with all edges being projected into circles. Seen below triangular tiling edges are colored into 3 parallel sets for each hemisphere.
Related honeycombs
This honeycomb can be truncated as t{2,3,6} or {}×{3,6}, Coxeter diagram , seen as one layer of triangular prisms, within a triangular prismatic honeycomb, .
See also
Order-7 tetrahedral honeycomb
List of regular polytopes
References
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, , (Chapter 10, Regular Honeycombs in Hyperbolic Space)
Honeycombs (geometry) |
https://en.wikipedia.org/wiki/Krzysztof%20Kami%C5%84ski | Krzysztof Kamiński (born 26 November 1990) is a Polish professional footballer who plays as a goalkeeper for Ruch Chorzów, on loan from Wisła Płock.
Career statistics
References
External links
Profile at Júbilo Iwata
Living people
1990 births
Polish men's footballers
Polish expatriate men's footballers
Poland men's under-21 international footballers
Men's association football goalkeepers
Footballers from Masovian Voivodeship
People from Nowy Dwór Mazowiecki
MKP Pogoń Siedlce players
Wisła Płock players
Ruch Chorzów players
Ekstraklasa players
I liga players
II liga players
III liga players
J1 League players
J2 League players
Júbilo Iwata players
Polish expatriate sportspeople in Japan
Expatriate men's footballers in Japan |
https://en.wikipedia.org/wiki/M.%20B.%20W.%20Tent | Margaret Bunting Wyman Tent (born Margaret Bunting Wyman; November 2, 1944 – September 20, 2014), also known by her pen name M. B. W. Tent, was an American mathematics educator and writer. She was the author of several bestselling books.
Life
Tent was born in Westfield, Massachusetts and grew up in Amherst, Massachusetts. She worked as a teacher at Altamont high school in Birmingham, Alabama beginning in 1985. In 2010, she and her husband moved to Frederick, Maryland to retire from teaching and continue writing.
On September 20, 2014 she died of breast cancer at a hospice in Mount Airy, Maryland at age 69.
Career
Tent attended Amherst Regional High School and is a graduate of Mount Holyoke College. She received her bachelor's and master's degree from the University of Alabama, Birmingham.
Before teaching post-secondary education, she taught adult education at business colleges and for two years for the American Military in Berlin, Germany.
She has been teaching and writing about mathematics for many years.
Critical reception
Most of her books have received good reviews from organizations such as the Mathematical Association of America, Association for Computing Machinery and Goodreads.
Her books have received hundreds of citations in the academic press.
Her books have also received praise from other authors and mathematicians like William Dunham, Peter Lax, Cathleen Synge Morawetz, Charles Ashbacher and Peter M. Neumann.
Bibliography
Some of her best known works are:
The Prince of Mathematics: Carl Friedrich Gauss
Emmy Noether: The Mother of Modern Algebra
Gottfried Wilhelm Leibniz: The Polymath Who Brought Us Calculus
Leonhard Euler and the Bernoullis: Mathematicians from Basel
References
External links
Goodreads author page
1944 births
2014 deaths
Schoolteachers from Alabama
American women educators
Mount Holyoke College alumni
University of Alabama at Birmingham alumni
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Geometric%20mechanics | Geometric mechanics is a branch of mathematics applying particular geometric methods to many areas of mechanics, from mechanics of particles and rigid bodies to fluid mechanics and control theory.
Geometric mechanics applies principally to systems for which the configuration space is a Lie group, or a group of diffeomorphisms, or more generally where some aspect of the configuration space has this group structure. For example, the configuration space of a rigid body such as a satellite is the group of Euclidean motions (translations and rotations in space), while the configuration space for a liquid crystal is the group of diffeomorphisms coupled with an internal state (gauge symmetry or order parameter).
Momentum map and reduction
One of the principal ideas of geometric mechanics is reduction, which goes back to Jacobi's elimination of the node in the 3-body problem, but in its modern form is due to K. Meyer (1973) and independently J.E. Marsden and A. Weinstein (1974), both inspired by the work of Smale (1970). Symmetry of a Hamiltonian or Lagrangian system gives rise to conserved quantities, by Noether's theorem, and these conserved quantities are the components of the momentum map J. If P is the phase space and G the symmetry group, the momentum map is a map , and the reduced spaces are quotients of the level sets of J by the subgroup of G preserving the level set in question: for one defines , and this reduced space is a symplectic manifold if is a regular value of J.
Variational principles
Hamilton's principle
Lagrange d'Alembert principle
Maupertuis' principle of least action
Euler–Poincaré
Vakonomic
Geometric integrators
One of the important developments arising from the geometric approach to mechanics is the incorporation of the geometry into numerical methods.
In particular symplectic and variational integrators are proving particularly accurate for long-term integration of Hamiltonian and Lagrangian systems.
History
The term "geometric mechanics" occasionally refers to 17th-century mechanics.
As a modern subject, geometric mechanics has its roots in four works written in the 1960s. These were by Vladimir Arnold (1966), Stephen Smale (1970) and Jean-Marie Souriau (1970), and the first edition of Abraham and Marsden's Foundation of Mechanics (1967). Arnold's fundamental work showed that Euler's equations for the free rigid body are the equations for geodesic flow on the rotation group SO(3) and carried this geometric insight over to the dynamics of ideal fluids, where the rotation group is replaced by the group of volume-preserving diffeomorphisms. Smale's paper on Topology and Mechanics investigates the conserved quantities arising from Noether's theorem when a Lie group of symmetries acts on a mechanical system, and defines what is now called the momentum map (which Smale calls angular momentum), and he raises questions about the topology of the energy-momentum level surfaces and the effect on the dynamics. In h |
https://en.wikipedia.org/wiki/Eknath%20Prabhakar%20Ghate | Eknath Prabhakar Ghate is a mathematician specialising in number theory and working at the School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India. He was awarded the Shanti Swarup Bhatnagar Prize for science and technology, the highest science award in India, for the year 2013 in the mathematical sciences category.
Early life and education
Ghate was schooled at Mayo College, Ajmer and at the International School Manila. He studied at St. Stephen's College, Delhi and obtained his bachelor's degree from the College of Arts & Sciences, University of Pennsylvania in 1991. He earned his Ph.D. from the University of California, Los Angeles, in 1996; his doctoral advisor was Haruzo Hida.
Career
Ghate is a professor at the Tata Institute of Fundamental Research. In number theory, Ghate is mostly interested in problems connected to automorphic forms, Galois representations, and the special values of L-functions.
Awards
Ghate was awarded the Bhatnagar Award in 2013. Ghate was elected a Fellow of the Indian Academy of Sciences in 2014. Ghate was awarded the JTM Gibson Award for Excellence by Mayo College in 2019. He was elected Fellow of the Indian National Science Academy in 2021.
References
External links
Eknath Ghate's Homepage
1969 births
Living people
Indian number theorists
Academic staff of Tata Institute of Fundamental Research
St. Stephen's College, Delhi alumni
University of California, Los Angeles alumni
University of Pennsylvania alumni
20th-century Indian mathematicians
Recipients of the Shanti Swarup Bhatnagar Award in Mathematical Science |
https://en.wikipedia.org/wiki/Schur%20algorithm | In mathematics, the Schur algorithm may be:
The Schur algorithm for expanding a function in the Schur class as a continued fraction
The Lehmer–Schur algorithm for finding complex roots of a polynomial |
https://en.wikipedia.org/wiki/Martin%20Konczkowski | Martin Konczkowski (born 14 September 1993) is a Polish professional footballer who plays as a right-back for Ekstraklasa side Śląsk Wrocław.
Career statistics
Club
Honours
Club
Piast Gliwice
Ekstraklasa: 2018–19
References
External links
Living people
1993 births
Polish men's footballers
Poland men's youth international footballers
Men's association football defenders
Ruch Chorzów players
Piast Gliwice players
Śląsk Wrocław players
Ekstraklasa players
Footballers from Ruda Śląska |
https://en.wikipedia.org/wiki/Dirac%20structure | In mathematics a Dirac structure is a geometric construction generalizing both symplectic structures and Poisson structures, and having several applications to mechanics. It is based on the notion of constraint introduced by Paul Dirac and was first introduced by Ted Courant and Alan Weinstein.
In more detail, let V be a real vector space, and V* its dual. A (linear) Dirac structure on V is a linear subspace D of satisfying
for all one has ,
D is maximal with respect to this property.
In particular, if V is finite dimensional then the second criterion is satisfied if . (Similar definitions can be made for vector spaces over other fields.)
An alternative (equivalent) definition often used is that satisfies , where orthogonality is with respect to the symmetric bilinear form on given by
Examples
If is a vector subspace, then is a Dirac structure on , where is the annihilator of ; that is, .
Let be a skew-symmetric linear map, then the graph of is a Dirac structure.
Similarly, if is a skew-symmetric linear map, then its graph is a Dirac structure.
A Dirac structure on a manifold M is an assignment of a (linear) Dirac structure on the tangent space to M at m, for each . That is,
for each , a Dirac subspace of the space .
Many authors, in particular in geometry rather than the mechanics applications, require a Dirac structure to satisfy an extra integrability condition as follows:
suppose are sections of the Dirac bundle () then
In the mechanics literature this would be called a closed or integrable Dirac structure.
Examples
Let be a smooth distribution of constant rank on a manifold M, and for each let
,
then the union of these subspaces over m forms a Dirac structure on M.
Let be a symplectic form on a manifold , then its graph is a (closed) Dirac structure. More generally this is true for any closed 2-form. If the 2-form is not closed then the resulting Dirac structure is not closed (integrable).
Let be a Poisson structure on a manifold , then its graph is a (closed) Dirac structure.
Applications
Port-Hamiltonian systems
Nonholonomic constraints
Thermodynamics
References
H. Bursztyn, A brief introduction to Dirac manifolds. Geometric and topological methods for quantum field theory, 4–38, Cambridge Univ. Press, Cambridge, 2013.
Classical mechanics
Differential geometry
Symplectic geometry |
https://en.wikipedia.org/wiki/2005%E2%80%9306%20Trabzonspor%20season | In the 2005–06 season, Trabzonspor finished in fourth place in the Süper Lig. The top scorer of the team was Fatih Tekke, who scored 28 goals.
This article shows statistics of the club's players and matches during the season.
Sponsor
Avea
01 Jefferson
03 Fabiano Eller
05 Hüseyin Cimsir
07 Miroslaw Szymkowiak
09 Fatih Tekke
10 Mehmet Hilmi Yilmaz
11 Ibrahima Yattara
13 Eul Young Lee
18 Tayfun Cora
19 Hasan Üçünçü
20 Ufuk Bayraktar
25 Ibrahim Ege
29 Tolga Zengin
34 Emrah Eren
38 Erdinç Yavuz
47 Volkan Bekiroglu
61 Gökdeniz Karadeniz
66 Adem Koçak
99 Celaleddin Koçak
Süper Lig
Turkish Cup
See also
2005–06 Süper Lig
2005–06 Turkish Cup
References
Turkish football clubs 2005–06 season
Trabzonspor seasons |
https://en.wikipedia.org/wiki/2006%E2%80%9307%20Trabzonspor%20season | In the 2006–07 season, Trabzonspor finished in fourth place in the Süper Lig. The top scorer of the team was Umut Bulut, who scored twenty goals.
This article shows statistics of the club's players and matches during the season.
Sponsor
Avea
Players
Süper Lig
Turkish Cup
See also
2006–07 Süper Lig
2006–07 Turkish Cup
Notes
External links
Turkish football clubs 2006–07 season
Trabzonspor seasons |
https://en.wikipedia.org/wiki/Christy%20Ring%20Cup%20records%20and%20statistics | This page details statistics of the Christy Ring Cup.
General performances
Wins by team
By province
Number of participating counties in the Christy Ring Cup
A total of 17 counties have played in the Christy Ring Cup. Season in bold represents teams qualified for the final that season.
Team records and statistics
Team results
Legend
– Champions
– Runners-up
– Semi-finals/Quarter-finals/Group Stage
– Relegated
AI – All-Ireland Senior Hurling Championship
JM – Joe McDonagh Cup
NR – Nicky Rackard Cup
For each year, the number of counties (in brackets) are shown.
Debut of teams
Seasons in Christy Ring Cup
The number of years that each county has played in the Christy Ring Cup between 2005 and 2024. A total of 17 counties have competed in at least one season of the Christy Ring Cup. Wicklow have participated in the most seasons. The counties in bold participate in the 2024 Christy Ring Cup.
List of Christy Ring Cup counties
Christy Ring Cup all-time table
All-time table
Legend
As of 28 June 2023 20:05. Includes Relegation Playoffs.
Teams
By Semi-Final Appearances
By decade
The most successful team of each decade, judged by number of Christy Ring Cup titles, is as follows:
2000s: 2 each for Westmeath (2005, 2007) and Carlow (2008, 2009)
2010s: 2 each for Kerry (2011, 2015), Kildare (2014, 2018) and Meath (2016, 2019)
2020s: 2 for Kildare (2020, 2022)
Match records
Most matches played
72, Kildare
Most wins
46, Kildare
Most losses
40, Mayo
Most draws
4, Meath
Other records
Finishing positions
Most championships
4, Kildare (2014, 2018, 2020, 2022)
Most second-place finishes
4 Down (2005, 2009, 2019, 2020)
Most third-place finishes
1, Derry (2022)
1, Sligo (2023)
Most fourth-place finishes
2, London (2022, 2023)
Most fifth-place finishes
1, Sligo (2022)
1, Tyrone (2023)
Most sixth-place finishes
1, Wicklow (2022)
1, Mayo (2023)
Most semi-final finishes
7, Kildare (2005, 2006, 2010, 2011, 2013, 2015, 2016)
Most quarter-final finishes
5, Derry (2009, 2011, 2012, 2013, 2014)
Most group stage finishes
3, Meath (2005, 2006, 2008)
3, London (2006, 2007, 2008)
3, Mayo (2006, 2007, 2018)
3, Wicklow (2006, 2007, 2019)
Unbeaten sides
Ten teams have won the Christy Ring Cup unbeaten:
Westmeath had 5 wins and 1 draw in 2007.
Carlow had 4 wins and 1 draw in 2009.
Kerry had 4 wins in 2011.
Kerry had 4 wins in 2015.
Meath had 4 wins and 1 draw in 2016.
Kildare had 5 wins in 2018.
Meath had 5 wins in 2019.
Kildare had 3 wins in 2020.
Offaly had 3 wins in 2021.
Kildare had 6 wins in 2022.
Beaten sides
The group stage of the cup has resulted in 9 'back-door' Christy Ring Cup champions:
Westmeath (2005) were beaten by Meath in round 2.
Antrim (2006) were beaten by Down in round 1.
Carlow (2008) were beaten by Down in round 3.
Westmeath (2010) were beaten by Kerry in round 2.
London (2012) were beaten by Meath in round 1.
Down (2013 |
https://en.wikipedia.org/wiki/The%20Taming%20of%20Chance | The Taming of Chance is a 1990 book about the history of probability by the philosopher Ian Hacking. First published by Cambridge University Press, it is a sequel to Hacking's The Emergence of Probability (1975). The book received positive reviews.
Summary
Hacking discusses the history of probability. He draws on the work of the philosopher Michel Foucault.
Publication history
The Taming of Chance was first published in the United Kingdom by Cambridge University Press in 1990. It is part of the series Ideas in Context.
Reception
The Taming of Chance has been described as ground-breaking. The book received positive reviews from the statistician Dennis Lindley in Nature, the philosopher Stephen P. Turner in the American Journal of Sociology, the historian of science Theodore M. Porter in American Scientist and in Poetics Today, and Timothy L. Alborn in Isis. The book received mixed reviews from the philosopher Margaret Schabas in Science and Bruce Kuklick in American Historical Review.
Lindley credited Hacking with presenting a careful and entertaining discussion of the development of the idea of chance, successfully showing that the laws of chance developed from "collections of data." He noted that, "Hacking's argument is supported by a vast number of references to statistical work and the interpretations put upon it." However, he criticized Hacking's style for being sometimes "overwhelming in its complexity", and questioned whether Hacking's thesis was original. Turner wrote that the book was useful for both sociologists of science and historians of social science, and that while Hacking's arguments were open to objections, Hacking was "too sophisticated" to be caught by them. Porter wrote in American Scientist that Hacking made "outstanding use" of Foucault's insights. He believed that Hacking's perspective was "especially fitting as an approach to the history of probability and statistics." Although he was not entirely satisfied with Hacking's arguments, he concluded that the book was "eminently worth reading." In Poetics Today, Porter described the book as "exceptionally illuminating on the issue of statistics and control" and credited Hacking with suggesting a "suitably subtle way of understanding social statistics." Alborn wrote that Hacking had a "vibrant writing style" and presented a "wealth of material". However, he also wrote that the book left many questions unanswered.
Schabas complimented Hacking for his discussion of "the debate over free will and determinism." However, she wrote that because the book built on previous works by Hacking such as The Emergence of Probability, it "does not exhilarate quite as much." She disputed the novelty of parts of Hacking's argument, noting that Porter had dealt with much of the same subject matter in The Rise of Statistical Thinking (1986). Kuklick noted that the book was a sequel to Hacking's earlier work The Emergence of Probability. Kuklick praised Hacking for the "richness of his ideas" |
https://en.wikipedia.org/wiki/2007%E2%80%9308%20Trabzonspor%20season | In the 2007–08 season, Trabzonspor finished in sixth place in the Süper Lig. The top scorer of the team was Umut Bulut, who scored nineteen goals.
This article shows statistics of the club's players and matches during the season.
Sponsor
Avea
Players
Süper Lig
Turkish Cup
See also
2007–08 Süper Lig
2007–08 Turkish Cup
Turkish football clubs 2007–08 season
Trabzonspor seasons |
https://en.wikipedia.org/wiki/2008%E2%80%9309%20Trabzonspor%20season | In the 2008–09 season, Trabzonspor finished in third place in the Süper Lig. The top scorer of the team was Gökhan Ünal, who scored sixteen goals.
This article shows statistics of the club's players and matches during the season.
Sponsor
Avea
Players
Süper Lig
Turkish Cup
|}
See also
2008–09 Süper Lig
2008–09 Turkish Cup
References
Turkish football clubs 2008–09 season
Trabzonspor seasons |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20Trabzonspor%20season | In the 2012–13 season, Trabzonspor finished in ninth place in the Süper Lig. The top scorer of the team was Adrian Mierzejewski, who scored thirteen goals.
This article shows statistics of the club's players and matches during the season.
Sponsor
Türk Telekom
Players
Super Lig
Turkish Cup
Fourth round
Fifth round
Group stage
Semi-final
Second leg
Final
UEFA Europa League
Play-off round
See also
2012–13 Turkish Cup
References
External links
Trabzonspor
Trabzonspor seasons |
https://en.wikipedia.org/wiki/2011%E2%80%9312%20Trabzonspor%20season | In the 2011–12 season, Trabzonspor finished in third place in the Süper Lig. The top scorer of the team was Burak Yılmaz, who scored 35 goals.
This article shows statistics of the club's players and matches during the season.
Sponsor
Türk Telekom
Players
Süper Lig
Turkish Playoff Championship Group
Turkish Cup
Third round
Fourth round
References
Trabzonspor
Trabzonspor seasons
Trabzonspor
Trabzonspor |
https://en.wikipedia.org/wiki/Assemble-to-order%20system | In applied probability, an assemble-to-order system is a model of a warehouse operating a build to order policy where products are assembled from components only once an order has been made.
The time to assemble a product from components is negligible, but the time to create components is significant (for example, they must be ordered from a supplier).
Research typically focuses on finding good policies for inventory levels and on the impact of different configurations (such as having more shared parts). The special case of only one product is an assembly system, the case of just once component is a distribution system.
Model definition
Single period model
This case is a generalisation of the newsvendor model (which has only one component and one product). The problem involves three stages and we give one formation of the problem below
components acquired
demand realized
components allocated, products produced
We use the following notation
In the final stage when demands are known the optimization problem faced is to
and we can therefore write the optimization problem at the first stage as
with x0 representing the starting inventory vector and c the cost function for acquiring the components.
Continuous time
In continuous time orders for products arrive according to a Poisson process and the time required to produce components are independent and identically distributed for each component. Two problems typically studied in this system are to minimize the expected backlog of orders subject to a constraint on the component inventory, and to minimize the expected component inventory subject to constraints on the rate at which orders must be completed.
References
Inventory optimization |
https://en.wikipedia.org/wiki/Riaz%20Mohammad%20Khan | Riaz Mohammad Khan holds a master's degree in mathematics and a B.A. (honors) from Punjab University, Lahore.
Prior to joining Pakistan's Foreign Service in 1969, Khan taught quantum physics from 1965 to 1969 as assistant professor in the Mathematics Department at Punjab University, Lahore.
His diplomatic career began with a posting to Beijing in 1970. He then served seven years at Pakistan's Mission to the United Nations in New York City from 1979 to 1986. Khan remained director general of Afghanistan and Soviet affairs at the Foreign Office, during which time he took a sabbatical to serve as a diplomat-in-residence at Georgetown University's Institute for the Study of Diplomacy. His other assignments include:
served as Pakistan's first ambassador to Kazakhstan and Kyrgyzstan (1992–1995).
Ambassador to Belgium, Luxembourg and the European Union (1995–1998).
Additional Secretary in charge of international organizations and arms control issues for Pakistan's Ministry of Foreign Affairs (1998–2002).
Spokesman of the Foreign Office (2000–2001).
Khan's last field assignment was as Ambassador of Pakistan to China from 2002 to 2005. He served as Pakistan's foreign secretary from 2005 to 2008. He retired from service in 2008.
After his retirement from the Foreign Service, he spent a year as a scholar at the Woodrow Wilson Center in Washington, D.C. He also served as Pakistan's envoy for "back channel" diplomacy with India from 2009 to 2012.
Books written by Khan
Untying the Afghan Knot: Negotiating Soviet Withdrawal (Duke University Press, 1991)
Afghanistan and Pakistan: Conflict, Extremism and Resistance to Modernity (Woodrow Wilson Center, Johns Hopkins URiaz Mohammad Khan | Wilson Centerniversity Press, Oxford University Press, 2011).
See also
Foreign Secretary (Pakistan)
References
Ambassadors of Pakistan to Belgium
Ambassadors of Pakistan to China
Ambassadors of Pakistan to the European Union
Ambassadors of Pakistan to Kazakhstan
Ambassadors of Pakistan to Kyrgyzstan
Ambassadors of Pakistan to Luxembourg
Foreign Secretaries of Pakistan
Living people
University of the Punjab alumni
Academic staff of the University of the Punjab
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Hrvoje%20Kraljevi%C4%87 | Hrvoje Kraljević (born 16 March 1944) is Croatian mathematician and a former politician.
He was born in Zagreb. He graduated theoretical physics in 1966 and received his PhD in mathematics in 1973 at the Faculty of Science, University of Zagreb. He worked at the Faculty since 1966, becoming a full professor in 1982. He was awarded fellowships in the United States (Princeton, 1974/75), France (Palaiseau, 1978) and Italy (Trento in 1979, 1981).
As a mathematician, he investigates representation theory, functional analysis and algebra. His most significant result is the research and of unitary and nonunitary dual group SU(n, 1), index generalization for semisimple Banach algebras, contributions to the research of Landau-type inequalities for infinitesimal generators to the theory of almost-summability.
He was the director of Department of Mathematics at the Faculty of Science (1983–89), dean of the Faculty (1991–98), Editor-in-Chief of the journal Glasnik matematički (1990–1992) and editor of the conference proceedings Functional Analysis (1982, 1986, 1990, 1994). He was awarded Ruđer Bošković Science Award in 1983. He became a corresponding member of the Croatian Academy of Sciences and Arts in 1992.
He served as the Minister of Science and Technology in Croatia in 2000–2002, as a Member of Parliament and Chairman of the Parliamentary Committee for Science, Higher Education and Culture in 2002–2003.
References
Croatian mathematicians
1944 births
Living people
Scientists from Zagreb
Party of Liberal Democrats politicians
Croatian Social Liberal Party politicians
Representatives in the modern Croatian Parliament
Mathematical analysts
Functional analysts |
https://en.wikipedia.org/wiki/2003%E2%80%9304%20Trabzonspor%20season | This article shows statistics of the club's players in the season.
In the 2003-2004 season Trabzonspor arrived second in Süper Lig.
The top goalscorer of the team was Gökdeniz Karadeniz who scored 14 goals.
Sponsor
Fly Air
Players
Fatih Tekke
Hüseyin Çimşir
Hasan Üçüncü
Gökdeniz Karadeniz
Mehmet Kahriman
Tayfun Cora
Alişen Kandil
Mehmet Yılmaz
Erdinç Yavuz
Michael Petkovic
Mustafa Yalçınkaya
Oumar Dieng
Tolga Zengin
Ibrahima Yattara
Emrah Eren
İbrahim Ege
Augustine Ahinful
Maxim Romashchenko
Karel D'Haene
Göksel Yaman
Hasan Sönmez
Recep Asil
Super Lig
Turkish Cup
First round
|}
Second round
|}
Quarterfinals
|}
Semifinals
|}
Final
|}
See also
Süper Lig 2003–04
2003–04 Turkish Cup
External links
Turkish football clubs 2003–04 season
Trabzonspor seasons |
https://en.wikipedia.org/wiki/Michal%20Bojnansk%C3%BD | Michal Bojnanský (born 20 June 1995) is a Slovak professional footballer who plays for DAC Dunajská Streda.
Club statistics
Updated to games played as of 4 January 2014.
References
1995 births
Living people
Slovak men's footballers
Men's association football goalkeepers
FC DAC 1904 Dunajská Streda players
Slovak First Football League players |
https://en.wikipedia.org/wiki/Spain%20national%20football%20team%20records%20and%20statistics | The following details the Spain national football team records.
Individual records
Player records
Nationwide
Most hat-tricks scored: 3 – Fernando Torres & David Villa
Most consecutive games with at least one goal: 6 – David Villa
Top scorer in World Cup finals: 9 – David Villa
Most goals scored in one World Cup: 5 – Emilio Butragueño (1986) & David Villa (2010)
Most consecutive matches scored in at World Cup: 4 – David Villa (2010)
Top scorer in European Championship finals: 6 – Álvaro Morata
Most goals scored in one European Championship: 4 – David Villa (2008)
Top scorer in Confederations Cup finals: 8 – Fernando Torres
Most goals scored in one Confederations Cup: 5 – Fernando Torres (2013)
Top scorer in UEFA Nations League finals: 6 – Ferran Torres
Most goals scored in one UEFA Nations League: 6 – Ferran Torres (2020–21)
Most caps
As of 12 October 2023, the players with the most caps for Spain are:
Bold denotes players still active at international level for the national team.
Most goals
As of 15 October 2023, the ten highest scorers for Spain are:
Bold denotes players still active at international level for the national team.
Most assists
As of 14 June 2021, the highest assist-providers for Spain are:
Bold denotes players still active at international level for the national team.
These are Opta defined assists.
Most penalty goals
As of 29 March 2022.
Bold denotes players still active at international level for the national team.
Hat-tricks
As of 8 September 2023.
4 Player scored 4 goals
5 Player scored 5 goals
6 Player scored 6 goals
Manager records
Most manager appearances
Vicente del Bosque: 114
Team records
Worldwide
World Cup winners: 2010
Most consecutive wins including friendlies: 15 (2008–2009)
Most consecutive wins achieved by an international coach from debut: 13 – Vicente del Bosque
Longest streak without conceding a goal: 9 matches (1992–1993)
Most penalty shoot-outs in one World Cup by one team: 2 at the 2002 FIFA World Cup (shared with in 1990 and 2022, in 2014, in 2014 and 2022, in 2018 and in 2018 and 2022)
Highest maximum number of points in World Cup qualification: 30 out of 30 (2010) (shared with for 2018)
Biggest wins
Heaviest defeats
Scores from 4–0 and up
FIFA Rankings
Last update was on 6 December 2022.
Source:
References
National association football team records and statistics |
https://en.wikipedia.org/wiki/Vijayan%20Nair | Vijayan (Vijay) N. Nair is currently Head of the Statistical Learning and Advanced Computing Group in Corporate Model Risk at Wells Fargo. He was Donald A. Darling Collegiate Professor of Statistics and Professor of Industrial and Operations Engineering at the University of Michigan, Ann Arbor from 1993 to 2017. He served as Chair of the Statistics Department at Michigan from 1998 to 2010 (sabbatical in 2003–04). Vijay was instrumental in launching the Michigan Institute for Data Science (MIDAS) and was recognized as a Distinguished Scientist by MIDAS. Prior to joining Michigan, he spent 15 years as a research scientist in the Mathematical Sciences Research Center at Bell Laboratories in New Jersey.
Vijay received his undergraduate degree in Economics from the University of Malaya, Malaysia and his PhD in Statistics from the University of California, Berkeley. He has published on a wide range of topics in statistical methodology and inference, engineering statistics, network tomography, reliability, design and analysis of experiments, behavioral intervention, and quality Improvement. His current research interests include risk modeling and machine learning.
Professional recognition
Vijay served as President of the International Statistical Institute from 2013 to 2015 and President of the International Society for Business and Industrial Statistics during 2011–2013. He has been elected as Fellow of American Statistical Association, Institute of Mathematical Statistics, American Society for Quality and American Association for the Advancement of Science. He was co-editor-in-chief of the International Statistical Review from 2010 to 2015 and 1994 to 1998. He was also editor of Technometrics from 1990 to 1992. He was the Gosset Lecturer at the World Statistics Congress in Marrakech in 2017, the Deming Lecturer at the Joint Statistical Meetings in 2013, the Isabel Loutit Lecturer at the Statistical Society of Canada Meetings in 2008, and the Youden Memorial Lecturer at the Fall Technical Conference in 2007. Vijay served as Chair of the Board of Trustees of the National Institute of Statistical Sciences from 2004 to 2008. The conference "From Industrial Statistics to Data Science" was held during October 1–3, 2015 in Ann Arbor to recognize his contributions.
References
Vijay Nair's home page
University of Michigan Faculty
International Statistical Institute
International Society for Business and Industrial Statistics
Vijay Nair on Mathematics Genealogy Project
University of Michigan faculty
American statisticians
Year of birth missing (living people)
Living people
Fellows of the American Statistical Association
University of Malaya alumni
University of California, Berkeley alumni
Presidents of the International Statistical Institute
Fellows of the Institute of Mathematical Statistics
Fellows of the American Association for the Advancement of Science
21st-century American mathematicians
Scientists at Bell Labs |
https://en.wikipedia.org/wiki/England%20national%20football%20team%20all-time%20record | The following tables show the England national football team's all-time international record. The statistics are composed of FIFA World Cup, UEFA European Football Championship, UEFA Nations League and British Home Championship (1883–1984) matches, as well as numerous international friendly tournaments and matches.
England played the world's first international fixture against Scotland on 30 November 1872, which ended in a 0–0 draw. England and Scotland have since contested 116 official matches: England have won 49, Scotland have won 41 and 26 have been drawn.
England have contested matches against more than 80 other national teams. England are unbeaten against 54 of them, having earned a perfect winning percentage against 30 of these teams. England have never beaten five teams that they have played at least once: Algeria, Ghana, Honduras, Saudi Arabia and South Korea. England have played all of these teams only once, with the exception of Saudi Arabia (two matches), and all of their meetings have been draws.
England have a negative record (more losses than wins) against only four countries: Brazil, Italy, the Netherlands and Uruguay. England have never lost to an African or Asian country.
Performances
Last match updated on 17 October 2023
Performance by competition
Performance by manager
Performance by venue
Competition records
FIFA World Cup
UEFA European Championship
UEFA Nations League
Minor tournaments
Head-to-head record
Last match updated on 17 October 2023
Combined predecessor and successor records
List of FIFA members who have never played against England
AFC
CAF
CONCACAF
CONMEBOL
OFC
UEFA
FIFA Rankings
Last update was on 21 September 2023.
Source:
Notes
References
England national football team records and statistics
National association football team all-time records |
https://en.wikipedia.org/wiki/Janet%20basis | In mathematics, a Janet basis is a normal form for systems of linear homogeneous partial differential equations (PDEs) that removes the inherent arbitrariness of any such system. It was introduced in 1920 by Maurice Janet. It was first called the Janet basis by Fritz Schwarz in 1998.
The left hand sides of such systems of equations may be considered as differential polynomials of a ring, and Janet's normal form as a special basis of the ideal that they generate. By abuse of language, this terminology will be applied both to the original system and the ideal of differential polynomials generated by the left hand sides. A Janet basis is the predecessor of a Gröbner basis introduced by Bruno Buchberger for polynomial ideals. In order to generate a Janet basis for any given system of linear PDEs a ranking of its derivatives must be provided; then the corresponding Janet basis is unique. If a system of linear PDEs is given in terms of a Janet basis its differential dimension may easily be determined; it is a measure for the degree of indeterminacy of its general solution. In order to generate a Loewy decomposition of a system of linear PDEs its Janet basis must be determined first.
Generating a Janet basis
Any system of linear homogeneous PDEs is highly non-unique, e.g. an arbitrary linear combination of its elements may be added to the system without changing its solution set. A priori it is not known whether it has any nontrivial solutions. More generally, the degree of arbitrariness of its general solution is not known, i.e. how many undetermined constants or functions it may contain. These questions were the starting point of Janet's work; he considered systems of linear PDEs in any number of dependent and independent variables and generated a normal form for them. Here mainly linear PDEs in the plane with the coordinates and will be considered; the number of unknown functions is one or two. Most results described here may be generalized in an obvious way to any number of variables or functions.
In order to generate a unique representation for a given system of linear PDEs, at first a ranking of its derivatives must be defined.
Definition:
A ranking of derivatives is a total ordering such that for any two derivatives , and
, and any derivation operator the relations and
are valid.
A derivative is called higher than if . The highest
derivative in an equation is called its leading derivative. For the derivatives up to order two of a single function depending on and with two possible order are
the LEX order and the GRLEX order .
Here the usual notation is used. If the number of functions is higher than one, these orderings have to be generalized appropriately, e.g. the orderings or may be applied.
The first basic operation to be applied in generating a Janet basis is the reduction of an equation w.r.t. another one . In colloquial terms this means the following: Whenever a derivative of may be
obtained from the leading deri |
https://en.wikipedia.org/wiki/Kiril%20Tenekedjiev | Kiril Ivanov Tenekedjiev (Bulgarian: Кирил Иванов Тенекеджиев; born 18 December 1960 in Varna, Bulgaria) is a professor in quantitative decision analysis and subjective statistics. His research achievements are in statistical pattern recognition, as well as his work on fuzzy-rational quantitative decision analysis and ribbon-risk analysis.
Personal details and aspirations
Kiril Tenekedjiev is the first of two children to Ivan Kirilov Tenekedjiev (a mechanical engineer), and Tatiana Museevna Tenekedjieva (a French language high school teacher).
Education and academic career
Kiril Tenekedjiev obtained his mechanical engineering degree (both at Bachelor and Master's degree) in ship machines and equipment from Technical University of Varna, Bulgaria in 1986. He was awarded a PhD degree in Engineering from the Higher Attestation Commission of Bulgaria in 1994, undertaking his studies with Technical University - Varna. His Doctor of Sciences degree from 2004, awarded again by the Higher Attestation Committee of Bulgaria, was on a thesis entitled Decision Analysis: Utility Theory and Subjective Statistics. He was promoted to Full Professor again by the Higher Attestation Committee of Bulgaria while still employed by the Technical University of Varna, in 2008.
Kiril started his academic career in 1986 right after graduation from university. He worked as a Researcher (operator) with the Ship propellers and cavitation unit of the Bulgarian Ship Hydrodynamics Center, Varna, Bulgaria (a research institute with industrial research orientation in mechanical engineering, hydrodynamics and cavitation). He worked on several research tasks there, related to his interests in mechanical engineering. After a year he moved to Technical University of Varna in 1987, where he was a lecturer in the Department of Resistance of Materials. Following was a series of academic positions with other sections of the Technical University of Varna from 1987 to 2011 (such as the Department of Ship Machines and Equipment, Department of Technique and Technology for Water and Air Protection and some others), eventually spending some 12 years with the Department of Economics and Management, at that time part of the Faculty of Marine Sciences and Ecology. In 2011, he took a role as a Professor with the Faculty of Engineering, Department of Information Technologies of the Nikola Vaptsarov Naval Academy-Varna (Bulgaria). Since 2016, he is affiliated with the Australian Maritime College, University of Tasmania (Australia) as a professor in Systems Engineering in various capacities and roles.
Kiril has research interests spanning over quantitative decision analysis, subjective/applied statistics, simulation modelling, data analytics, mathematical modelling of biochemical reactions, statistical pattern recognition and technical diagnostics. His overarching research topic is intelligent systems and data analytics. He has found application areas to his novel works in medical research (bioch |
https://en.wikipedia.org/wiki/Decomposable%20measure | In mathematics, a decomposable measure (also known as a strictly localizable measure) is a measure that is a disjoint union of finite measures. This is a generalization of σ-finite measures, which are the same as those that are a disjoint union of countably many finite measures. There are several theorems in measure theory such as the Radon–Nikodym theorem that are not true for arbitrary measures but are true for σ-finite measures. Several such theorems remain true for the more general class of decomposable measures. This extra generality is not used much as most decomposable measures that occur in practice are σ-finite.
Examples
Counting measure on an uncountable measure space with all subsets measurable is a decomposable measure that is not σ-finite. Fubini's theorem and Tonelli's theorem hold for σ-finite measures but can fail for this measure.
Counting measure on an uncountable measure space with not all subsets measurable is generally not a decomposable measure.
The one-point space of measure infinity is not decomposable.
References
Bibliography
Second printing.
Measures (measure theory) |
https://en.wikipedia.org/wiki/OpenFlight | OpenFlight (or .flt) is a 3d geometry model file format originally developed by Software Systems Inc. for its MultiGen real-time 3d modeling package in 1988. Originally called Flight, the format was designed as a nonproprietary 3d model format for use by real-time 3d visual simulation image generators. The format was later renamed to OpenFlight to denote its nonproprietary image generation (IG) usage. The MultiGen modeling package (known now as Creator) and the OpenFlight format were rapidly adopted by the early commercial flight simulation industry in the later 1980s and early 1990s. NASA Ames was the first customer for the MultiGen modeling package.
The early advantage OpenFlight held over many 3d geometry model file formats (.obj, .dxf, .3ds) was its specific real-time 3d graphics industry design. This means that the format is polygon based (rather than NURB surfaces), and provides a real-time tree structure essential for real-time IG systems. Most early graphics file formats, such as Wavefront Technologies, or Alias Systems Corporation, tried to focus more on visual aesthetics for non-real-time based rendering graphics packages.
The OpenFlight file format is still widely used today in the high end real-time visual simulation industry as the standard interchange format between different IG systems, and is currently administrated by Presagis.
File format
Associated File Formats
OpenFlight models can have several associated files in different formats, that define elements such as material characteristics or shaders.
Versions and History
API
Modeling Tools
There are several modeling tools currently on the market that both read and write the OpenFlight file format. The standard bearer of the file format Presagis Creator offers the widest compatibility with the file format. Another modeling tool using OpenFlight as its native format in Remo 3D from Remograph. Blender previously had integrated support for importing models in OpenFlight format. However, it appears as though this functionality has been abandoned in newer versions and there is currently no support for exporting to this format. Autodesk 3DS Max still supports exporting to OpenFlight format as of the 2024 version of their software.
Vendor specific alterations
Because the OpenFlight file format allows for vendor specific data field additions, some modeling and simulation tools might not fully support vendor specific additions to the file format.
References
External links
OpenFlight Specification
3D computer graphics
Graphics file formats
Graphics standards
Virtual reality
IRIX software |
https://en.wikipedia.org/wiki/Sampling%20in%20order | In statistics, some Monte Carlo methods require independent observations in a sample to be drawn from a one-dimensional distribution in sorted order. In other words, all n order statistics are needed from the n observations in a sample. The naive method performs a sort and takes O(n log n) time. There are also O(n) algorithms which are better suited for large n. The special case of drawing n sorted observations from the uniform distribution on [0,1] is equivalent to drawing from the uniform distribution on an n-dimensional simplex; this task is a part of sequential importance resampling.
Further reading
Monte Carlo methods |
https://en.wikipedia.org/wiki/Scan%20statistic | In statistics, a scan statistic or window statistic is a problem relating to the clustering of randomly positioned points. An example of a typical problem is the maximum size of a cluster of points on a line or the longest series of successes recorded by a moving window of fixed length.
Joseph Naus first published on the problem in the 1960s, and has been called the "father of the scan statistic" in honour of his early contributions. The results can be applied in epidemiology, public health and astronomy to find unusual clusters of events.
It was extended by Martin Kulldorff to multidimensional settings and varying window sizes in a 1997 paper, which is () the most cited article in its journal, Communications in Statistics – Theory and Methods. This work lead to the creation of the software SaTScan, a program trademarked by Martin Kulldorff that applies his methods to data.
Recent results have shown that using scale-dependent critical values for the scan statistic allows to attain asymptotically optimal detection simultaneously for all signal lengths, thereby improving on the traditional scan, but this procedure has been criticized for losing too much power for short signals. Walther and Perry (2022) considered the problem of detecting an elevated mean on an interval with unknown location and length in the univariate Gaussian sequence model. They explain this discrepancy by showing that these asymptotic optimality results will necessarily be too imprecise to discern the performance of scan statistics in a practically relevant way, even in a large sample context. Instead, they propose to assess the performance with a new finite sample criterion. They presented three new calibration techniques for scan statistics that perform well across a range of relevant signal lengths to optimally increase performance of short signals.
The scan-statistic-based methods have been specifically developed to detect rare variant associations in the noncoding genome, especially for the intergenic region. Compared with fixed-size sliding window analysis, scan-statistic-based methods use data-adaptive size dynamic window to scan the genome continuously, and increase the analysis power by flexibly selecting the locations and sizes of the signal regions. Some examples of these methods are Q-SCAN, SCANG,
WGScan.
References
External links
SaTScan free software for the spatial, temporal and space-time scan statistics
Summary statistics |
https://en.wikipedia.org/wiki/National%20Premier%20Soccer%20League%20records%20and%20statistics | The following is a compilation of notable records and statistics for teams and players in and seasons of National Premier Soccer League n the United States.
Regular season
All-Time Championship Game Performance
All-Time Regular Season Records
Through completion of 2013 regular season.
* Missing GF/GA/GD for 2006 & 2007 Seasons, Missing W/L/T & GF/GA/GD for 2008 Season with the exception of Atlanta Silverbacks Reserves, Performance FC Phoenix, Rocket City United, San Diego United, Arizona Sahuaros and Albuquerque Asylum for which W/L/T is included for 2008 Season but GF/GA/GD is missing.
** Alabama Spirit though scheduled to complete in 2008 never played a game
Best Regular Season Record by Year
Best regular season record determined by average points per game since the number of games played varies
* Team did not score the most overall points during the year but had best average points per game
All-Time Regular Season Division / Conference Championships
All-Time Regular Season Attendance Bests
U.S. Open Cup
U.S. Open Cup Performance by Year
Through completion of 2013 regular season.
Automatic entries granted to NPSL teams starting with 2011 tournament
* Chico Rooks qualified for 2003 tournament before the MPSL officially began play
† Arizona Sahuaros competed in the USASA from 2005 thru 2007 as well as 2009 and 2010 seasons but were officially on hiatus from the NPSL due to the lack of a Southwest Conference''''‡ Indois USA joined the NPSL for the 2007 season and qualified for the U.S. Open Cup before competing within the NPSL♣ Milwaukee Bavarians completed in 2009 USASA while on hiatus from NPSL¶ Brooklyn Italians joined the NPSL for the 2010 season and qualified for the U.S. Open Cup before competing within the NPSL All-Time U.S. Open Cup Appearances and Performance
*Atlanta FC now known as Atlanta Silverbacks Reserves and Fullerton Rangers now known as Orange County Pateadores''
See also
Major League Soccer records and statistics
North American Soccer League records and statistics
References
Records and statistics
All-time football league tables
Association football league records and statistics |
https://en.wikipedia.org/wiki/Spherical%20contact%20distribution%20function | In probability and statistics, a spherical contact distribution function, first contact distribution function, or empty space function is a mathematical function that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both. More specifically, a spherical contact distribution function is defined as probability distribution of the radius of a sphere when it first encounters or makes contact with a point in a point process. This function can be contrasted with the nearest neighbour function, which is defined in relation to some point in the point process as being the probability distribution of the distance from that point to its nearest neighbouring point in the same point process.
The spherical contact function is also referred to as the contact distribution function, but some authors define the contact distribution function in relation to a more general set, and not simply a sphere as in the case of the spherical contact distribution function.
Spherical contact distribution functions are used in the study of point processes as well as the related fields of stochastic geometry and spatial statistics, which are applied in various scientific and engineering disciplines such as biology, geology, physics, and telecommunications.
Point process notation
Point processes are mathematical objects that are defined on some underlying mathematical space. Since these processes are often used to represent collections of points randomly scattered in space, time or both, the underlying space is usually d-dimensional Euclidean space denoted here by , but they can be defined on more abstract mathematical spaces.
Point processes have a number of interpretations, which is reflected by the various types of point process notation. For example, if a point belongs to or is a member of a point process, denoted by , then this can be written as:
and represents the point process being interpreted as a random set. Alternatively, the number of points of located in some Borel set is often written as:
which reflects a random measure interpretation for point processes. These two notations are often used in parallel or interchangeably.
Definitions
Spherical contact distribution function
The spherical contact distribution function is defined as:
where b(o,r) is a ball with radius r centered at the origin o. In other words, spherical contact distribution function is the probability there are no points from the point process located in a hyper-sphere of radius r.
Contact distribution function
The spherical contact distribution function can be generalized for sets other than the (hyper-)sphere in . For some Borel set with positive volume (or more specifically, Lebesgue measure), the contact distribution function (with respect to ) for is defined by the equation:
Examples
Poisson p |
https://en.wikipedia.org/wiki/Ahmed%20Nasri | Ahmad H. Nasri is the current President of Fahd bin Sultan University.
Professor Nasri was educated at the Lebanese University (BS Mathematics, 1978) and the University of East Anglia (PhD, 1985). He was formerly a Professor at the American University of Beirut.
References
Year of birth missing (living people)
Living people
Lebanese University alumni
Alumni of the University of East Anglia
Academic staff of the American University of Beirut |
https://en.wikipedia.org/wiki/William%20C.%20Davidon | William Cooper Davidon (March 18, 1927 – November 8, 2013) was an American professor of physics and mathematics, and a peace activist. As the mastermind of the March 8, 1971, FBI office break-in, in Media, Pennsylvania, Davidon was the informal leader of the Citizens' Commission to Investigate the FBI. The Media break-in resulted in the disclosure of COINTELPRO, which in turn led to subsequent investigations and reforms of the FBI.
Life
Davidon was born in Fort Lauderdale, Florida, in 1927.
He attended Purdue University and graduated from the University of Chicago with a B.S. (1947), masters (1950), and Ph.D. (1954) in physics. From 1954 to 1956, Davidon was a research associate at the Enrico Fermi Institute. From 1956 to 1961, he was an associate physicist at the Argonne National Laboratory, where he developed the first quasi-Newton algorithm, now known as the Davidon–Fletcher–Powell formula.
Davidon was professor of physics at Haverford College from 1961 to 1981, and then Professor of Mathematics (1981–1991), as his interests shifted to include mathematical logic, set theory and non-standard analysis. Davidon was a 1966 Fulbright Scholar. He retired in 1991.
Davidon moved to Highlands Ranch, Colorado, in 2010. He died November 8, 2013, of Parkinson's disease.
Activism
In 1966, Davidon traveled to South Vietnam, with A. J. Muste, sponsored by the Committee for Non-Violent Action. He also announced that year that he would be refusing to pay his federal income tax in protest against the Vietnam War. Later, he became a sponsor of the War Tax Resistance project, which practiced and advocated tax resistance as a form of anti-war protest.
In 1971, he was named an "unindicted co-conspirator" in the Harrisburg Seven case. During much of this time, he served on the board of directors of the American Civil Liberties Union, Philadelphia affiliate.
As the leader of the Citizens' Commission to Investigate the FBI, Davidon was instrumental in planning and organizing a break-in of the FBI's Media, Pennsylvania office. The documents stolen there led to the disclosure of COINTELPRO. According to The Burglary, a book published shortly after his death, Davidon also had engaged in draft board raids, stealing or destroying files, and subsequent to Media participated in two acts of sabotage against military materiel intended for use in Vietnam. Due at least in part to his exceptionally careful planning and his co-conspirators' total commitment to secrecy and discretion, neither he nor anyone else was ever charged in any of those actions, despite an intense, five-year FBI investigation.
Family
In 1963, Davidon married Ann Morrissett (1925–2004), a noted pacifist/feminist essayist and activist. They had two daughters Sarah Davidon and Ruth Rodgers. Davidon and Morrissett divorced in 1978. He subsequently married Maxine Libros, who died in 2010. Davidon had a son, Alan (1949– ), from his first marriage to Phyllis Wise (1927– ).. Davidon also had a child with Zu |
https://en.wikipedia.org/wiki/Ben%20Reichert | Ben Reichert (; born 4 March 1994) is an Israeli footballer who plays as a midfielder for Hapoel Nir Ramat HaSharon.
Career statistics
Club
References
1994 births
Living people
Israeli men's footballers
Hapoel Nir Ramat HaSharon F.C. players
Maccabi Tel Aviv F.C. players
Hapoel Tel Aviv F.C. players
S.V. Zulte Waregem players
Hapoel Acre F.C. players
F.C. Ashdod players
Hapoel Kfar Saba F.C. players
Maccabi Petah Tikva F.C. players
Bnei Yehuda Tel Aviv F.C. players
Israeli Premier League players
Belgian Pro League players
Liga Leumit players
Footballers from Ramat HaSharon
Israeli people of Polish-Jewish descent
Israel men's under-21 international footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Serpentine%20geometry%20plasma%20actuator | The serpentine plasma actuator represents a broad class of plasma actuator. The actuators vary from the standard type in that their electrode geometry has been modified in to be periodic across its span.
History
This class of plasma actuators was developed at the Applied Physics Research Group (APRG) at the University of Florida in 2008 by Subrata Roy for the purpose of controlling laminar and turbulent boundary layer flows. Since then, APRG has continued to characterize and develop uses for this class of plasma actuators. Several patents resulted from the early work on serpentine geometry plasma actuators.
In 2013, these actuators started to get broader attention in the scientific press, and several articles were written about these actuators, including articles in AIP's EurekAlert, Inside Science and various blogs.
Current Research and Operating Mechanisms
Serpentine plasma actuators (like other Dielectric Barrier Discharge actuators, i.e. plasma actuators) are able to induce an atmospheric plasma and introduce an electrohydrodynamic body force to a fluid. This body force can be used to implement flow control, and there are a range of potential applications, including drag reduction for aircraft and flow stabilization in combustion chambers.
The important distinction between serpentine plasma actuators and more traditional geometries is that the geometry of the electrodes has been modified in order to be periodic across its span. As the electrode has been made periodic, the resulting plasma and body force are also spanwise periodic. With this spanwise periodicity, three-dimensional flow effects can be induced in the flow, which cannot be done with more traditional plasma actuator geometries.
It is thought that the introduction of three-dimensional flow effects allow for the plasma actuators to apply much greater levels of control authority as they allow for the plasma actuators to project onto a greater range of physical mechanisms (such as boundary layer streaks or secondary instabilities of the Tollmien-Schlichting wave). Recent work indicate that these plasma actuators may have a significant impact on controlling laminar and transitional flows on a flat plate. In addition, the serpentine actuator has been experimentally demonstrated to increase lift, decrease drag and generate controlling rolling moments when applied to aircraft wing geometries.
With the greater level of control authority that these plasma actuators may potentially possess, there is currently research being performed at several labs in the United States and in the United Kingdom looking to apply these actuators for real world applications. Recent numerical work predicted significant
turbulent drag reduction by collocating serpentine plasma actuators in a pattern to modify energetic modes of transitional flow.
See also
Plasma actuator
Wingless Electromagnetic Air Vehicle
Applied Physics Research Group
University of Florida
University of Florida College of Engin |
https://en.wikipedia.org/wiki/Poisson%20point%20process | In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The Poisson point process is also called a Poisson random measure, Poisson random point field or Poisson point field. When the process is defined on the real line, it is often called simply the Poisson process.
This point process has convenient mathematical properties, which has led to its being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy, biology, ecology, geology, seismology, physics, economics, image processing, and telecommunications.
The process is named after French mathematician Siméon Denis Poisson despite Poisson's never having studied the process. Its name derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution. The process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics.
The Poisson point process is often defined on the real line, where it can be considered as a stochastic process. In this setting, it is used, for example, in queueing theory to model random events, such as the arrival of customers at a store, phone calls at an exchange or occurrence of earthquakes, distributed in time. In the plane, the point process, also known as a spatial Poisson process, can represent the locations of scattered objects such as transmitters in a wireless network, particles colliding into a detector, or trees in a forest. In this setting, the process is often used in mathematical models and in the related fields of spatial point processes, stochastic geometry, spatial statistics and continuum percolation theory. The Poisson point process can be defined on more abstract spaces. Beyond applications, the Poisson point process is an object of mathematical study in its own right. In all settings, the Poisson point process has the property that each point is stochastically independent to all the other points in the process, which is why it is sometimes called a purely or completely random process. Modeling a system as a Poisson Process is insufficient when the point-to-point interactions are too strong (i.e. the points are not stochastically independent). Such a system may be better modeled with a different point process.
The point process depends on a single mathematical object, which, depending on the context, may be a constant, a locally integrable function or, in more general settings, a Radon measure. In the first case, the constant, known as the rate or intensity, is the average density of the points in the Poisson process located in some regio |
https://en.wikipedia.org/wiki/Filipinos%20in%20Belgium | Filipinos in Belgium comprise migrants from the Philippines to Belgium and their descendants living there. While the Belgian National Institute of Statistics has 3,067 Filipinos officially registered, the Commission on Filipinos Overseas (CFO) estimated that there are 12,224 Filipinos in Belgium in December 2013.
Demographics
Filipinos in Belgium work primarily as tradesmen, in the hospitality industry, as domestic workers, or as seamen on Belgian-flagged ships. A number of Filipino international students also attend Belgian institutions of higher education, but are considered "temporary migrants." Gender-wise, most Filipinos in Belgium are female, amount to roughly 60% of the population.
The Philippine Embassy in Belgium considers "limited employment opportunities, illegal residency status and fraudulent documentation" to be the largest issues facing the Filipino community in Belgium. Of the 12,224 Filipinos estimated to be living in Belgium, the CFO estimates that 5,000 (40.9%) are "irregular" or living without legal residency status, while 6,840 (56.0%) and 384 (3.1%) Filipinos have "permanent" and "temporary" status, respectively.
Economics
In 2012, Filipinos in Belgium officially sent a total of approximately $62.0 million USD in remittances back to the Philippines (US$32.3 land-based and US$19.8 sea-based), after a peak of $63.4 million USD in remittances in 2010. This figure accounts for 0.28% of all remittances sent to the Philippines. Three Filipino banks have correspondent accounts with banks in Norway to allow for remittance transfers.
Society and culture
In addition to the Philippine Embassy in Brussels and a consulate in Antwerp, there are about seventy Filipino associations in Belgium to serve different needs of the Filipino-Belgian community, many of which are coordinated by the Council of Filipino Associations in Belgium (COFAB) or the Council of Filipino Associations in Flanders (COFAF). There are general regional-based, sporting, service-oriented, and cultural organizations, as well as organizations for native Belgians married to Filipino spouses. In addition, there are four chapters of the Knights of Rizal, a fraternal organization.
The Philippine Embassy in Brussels organizes events around major Filipino holidays, including the Philippine Independence Day and Christmas, that attract thousands of Filipinos from Belgium and neighboring nations such as Luxembourg.
Notable people
José Alejandrino, Filipino general; studied at the University of Ghent
Jeffrey Christiaens, footballer
Angeline Flor Pua, Miss Belgium 2018
Racso Jugarap, Wire artist
José Rizal, Filipino revolutionary; lived for a period in Belgium, where he published El filibusterismo
Roxanne Allison Baeyens, actress, model and Miss Philippines Earth 2020
Glen Ramaekers, Chef
Pauline Cucharo Amelinckx, model and Miss Supranational Philippines 2023
References
B
Ethnic groups in Belgium
Belgian people of Filipino descent |
https://en.wikipedia.org/wiki/Nearest%20neighbour%20distribution | In probability and statistics, a nearest neighbor function, nearest neighbor distance distribution, nearest-neighbor distribution function or nearest neighbor distribution is a mathematical function that is defined in relation to mathematical objects known as point processes, which are often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both. More specifically, nearest neighbor functions are defined with respect to some point in the point process as being the probability distribution of the distance from this point to its nearest neighboring point in the same point process, hence they are used to describe the probability of another point existing within some distance of a point. A nearest neighbor function can be contrasted with a spherical contact distribution function, which is not defined in reference to some initial point but rather as the probability distribution of the radius of a sphere when it first encounters or makes contact with a point of a point process.
Nearest neighbor function are used in the study of point processes as well as the related fields of stochastic geometry and spatial statistics, which are applied in various scientific and engineering disciplines such as biology, geology, physics, and telecommunications.
Point process notation
Point processes are mathematical objects that are defined on some underlying mathematical space. Since these processes are often used to represent collections of points randomly scattered in space, time or both, the underlying space is usually d-dimensional Euclidean space denoted here by , but they can be defined on more abstract mathematical spaces.
Point processes have a number of interpretations, which is reflected by the various types of point process notation. For example, if a point belongs to or is a member of a point process, denoted by , then this can be written as:
and represents the point process being interpreted as a random set. Alternatively, the number of points of located in some Borel set is often written as:
which reflects a random measure interpretation for point processes. These two notations are often used in parallel or interchangeably.
Definitions
Nearest neighbor function
The nearest neighbor function, as opposed to the spherical contact distribution function, is defined in relation to some point of a point process already existing in some region of space. More precisely, for some point in the point process , the nearest neighbor function is the probability distribution of the distance from that point to the nearest or closest neighboring point.
To define this function for a point located in at, for example, the origin , the -dimensional ball of radius centered at the origin o is considered. Given a point of existing at , then the nearest neighbor function is defined as:
where denotes the conditional probability that there is one point of located in given there is a point of loc |
https://en.wikipedia.org/wiki/Intersection%20%28geometry%29 | In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point (sometimes called a vertex) or does not exist (if the lines are parallel). Other types of geometric intersection include:
Line–plane intersection
Line–sphere intersection
Intersection of a polyhedron with a line
Line segment intersection
Intersection curve
Determination of the intersection of flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of a system of linear equations. In general the determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton iteration. Intersection problems between a line and a conic section (circle, ellipse, parabola, etc.) or a quadric (sphere, cylinder, hyperboloid, etc.) lead to quadratic equations that can be easily solved. Intersections between quadrics lead to quartic equations that can be solved algebraically.
On a plane
Two lines
For the determination of the intersection point of two non-parallel lines
one gets, from Cramer's rule or by substituting out a variable, the coordinates of the intersection point :
(If the lines are parallel and these formulas cannot be used because they involve dividing by 0.)
Two line segments
For two non-parallel line segments and there is not necessarily an intersection point (see diagram), because the intersection point of the corresponding lines need not to be contained in the line segments. In order to check the situation one uses parametric representations of the lines:
The line segments intersect only in a common point of the corresponding lines if the corresponding parameters fulfill the condition .
The parameters are the solution of the linear system
It can be solved for s and t using Cramer's rule (see above). If the condition is fulfilled one inserts or into the corresponding parametric representation and gets the intersection point .
Example: For the line segments and one gets the linear system
and . That means: the lines intersect at point .
Remark: Considering lines, instead of segments, determined by pairs of points, each condition can be dropped and the method yields the intersection point of the lines (see above).
A line and a circle
For the intersection of
line and circle
one solves the line equation for or and substitutes it into the equation of the circle and gets for the solution (using the formula of a quadratic equation) with
if If this condition holds with strict inequality, there are two intersection points; in this case the line is called a secant line of the circle, and the line segment connecting the intersection points is called a chord of the circle.
If holds, there exists only one intersection point and |
https://en.wikipedia.org/wiki/2004%E2%80%9305%20FC%20Thun%20season | This article covers the results and statistics of FC Thun during the 2004–05 season. During the season Thun competed in the Swiss Super League and the Swiss Cup.
Season summary
FC Thun finished second in Swiss Super League.
Squad
Fabio Coltorti
Alain Portmann
Pascal Cerrone
Armand Deumi
Sandro Galli
Selver Hodzic
Ljubo Miličević
David Pallas
Lukas Schenkel
Sehid Sinani
Silvan Aegerter
Andres Gerber
Baykal Kulaksizoglu
Nelson Ferreira
Mario Raimondi
Michel Renggli
Nenad Savić
Fabian Stoller
Gelson Rodrigues
Mauro Lustrinelli
Adrian Moser
Samuel Ojong
Swiss cup
Round 1
Teams from Super League and Challenge League were seeded in this round. In a match, the home advantage was granted to the team from the lower league, if applicable.
|colspan="3" style="background-color:#99CCCC"|17 September 2004
|-
|colspan="3" style="background-color:#99CCCC"|18 September 2004
|-
|colspan="3" style="background-color:#99CCCC"|19 September 2004
|}
Round 2
|colspan="3" style="background-color:#99CCCC"|22 October 2004
|-
|colspan="3" style="background-color:#99CCCC"|23 October 2004
|-
|colspan="3" style="background-color:#99CCCC"|24 October 2004
|}
Round 3
|colspan="3" style="background-color:#99CCCC"|20 November 2004
|-
|colspan="3" style="background-color:#99CCCC"|21 November 2004
|}
Quarterfinals
References
External links
FC Thun seasons
Thun |
https://en.wikipedia.org/wiki/Intersection%20curve | In geometry, an intersection curve is a curve that is common to two geometric objects. In the simplest case, the intersection of two non-parallel planes in Euclidean 3-space is a line. In general, an intersection curve consists of the common points of two transversally intersecting surfaces, meaning that at any common point the surface normals are not parallel. This restriction excludes cases where the surfaces are touching or have surface parts in common.
The analytic determination of the intersection curve of two surfaces is easy only in simple cases; for example: a) the intersection of two planes, b) plane section of a quadric (sphere, cylinder, cone, etc.), c) intersection of two quadrics in special cases. For the general case, literature provides algorithms, in order to calculate points of the intersection curve of two surfaces.
Intersection line of two planes
Given: two planes linearly independent, i.e. the planes are not parallel.
Wanted: A parametric representation of the intersection line.
The direction of the line one gets from the crossproduct of the normal vectors: .
A point of the intersection line can be determined by intersecting the given planes with the plane , which is perpendicular to and . Inserting the parametric representation of into the equations of und yields the parameters and .
Example:
The normal vectors are and the direction of the intersection line is . For point , one gets from the formula above Hence
is a parametric representation of the line of intersection.
Remarks:
In special cases, the determination of the intersection line by the Gaussian elimination may be faster.
If one (or both) of the planes is given parametrically by , one gets as normal vector and the equation is: .
Intersection curve of a plane and a quadric
In any case, the intersection curve of a plane and a quadric (sphere, cylinder, cone,...) is a conic section. For details, see. An important application of plane sections of quadrics is contour lines of quadrics. In any case (parallel or central projection), the contour lines of quadrics are conic sections. See below and Umrisskonstruktion.
Intersection curve of a cylinder or cone and a quadric
It is an easy task to determine the intersection points of a line with a quadric (i.e. line-sphere); one only has to solve a quadratic equation. So, any intersection curve of a cone or a cylinder (they are generated by lines) with a quadric consists of intersection points of lines and the quadric (see pictures).
The pictures show the possibilities which occur when intersecting a cylinder and a sphere:
In the first case, there exists just one intersection curve.
The second case shows an example where the intersection curve consists of two parts.
In the third case, the sphere and cylinder touch each other at one singular point. The intersection curve is self-intersecting.
If the cylinder and sphere have the same radius and the midpoint of the sphere is located on the axis of th |
https://en.wikipedia.org/wiki/List%20of%20FK%20Genclerbirliyi%20Sumqayit%20records%20and%20statistics | Genclerbirliyi Sumqayit was an Azerbaijani professional football club based in Sumqayit from 2003 to 2008.
This list encompasses the major records set by the club and their players in the Azerbaijan Premier League. The player records section includes details of the club's goalscorers and those who have made more than 50 appearances in first-team competitions.
Player
Most appearances
Players played over 50 competitive, professional matches only. Appearances as substitute (goals in parentheses) included in total.
Overall scorers
Competitive, professional matches only, appearances including substitutes appear in brackets.
Team
Goals
Most Premier League goals scored in a season: 32 – 2004–05
Fewest League goals scored in a season: 16 – 2006-07
Most League goals conceded in a season: 68 – 2007–08
Fewest League goals conceded in a season: 28 – 2004-05
Points
Most points in a season:
33 in 34 matches, Azerbaijan Premier League, 2004–05
Fewest points in a season:
14 in 26 matches, Azerbaijan Premier League, 2007–08
International representatives
References
FK Genclerbirliyi Sumqayit
records and statistics |
https://en.wikipedia.org/wiki/Klara%20L%C3%B6benstein | Klara Löbenstein (15 February 1883 – 10 June 1968) was a German mathematician. She was among the first women to obtain a doctorate in Germany. Her doctoral work was on the topology of algebraic curves.
Life and work
Löbenstein was born in Hildesheim, Prussia on 15 February 1883 to merchant Lehmann Löbenstein and his wife Sofie (née Schönfeld).
In 1904, Löbenstein was given permission to take her Abitur at the Realgymnasium I Gymnasium in Hanover. Thus she belonged to a small group of talented young women in Germany at the beginning of the 20th century who were allowed to take the Abitur externally at boys' schools.
Since Prussia began to allow women to formally attend university only from the winter semester of 1908–09, Löbenstein and her friend Margarete Kahn first attended the universities of Berlin and Göttingen as guest students. They studied mathematics, physics, and propaedeutics at Berlin and Göttingen. Löbenstein's field of expertise was algebraic geometry. Together with Kahn she made a contribution to Hilbert's sixteenth problem. Hilbert's sixteenth problem concerned the topology of algebraic curves in the complex projective plane; as a difficult special case in his formulation of the problem Hilbert proposed that there are no algebraic curves of degree 6 consisting of 11 separate ovals. Löbenstein and Kahn developed methods to address this problem.
Löbenstein obtained a doctorate in 1910 under David Hilbert in Göttingen, with a dissertation titled Über den Satz, daß eine ebene, algebraische Kurve 6. Ordnung mit 11 sich einander ausschließenden Ovalen nicht existiert [On the proposition that no plane algebraic curve of degree 6 with 11 mutually exclusive ovals exists], and was therefore one of the first German women to obtain a doctorate in mathematics (the mathematics division was part of the faculty of philosophy then). She took her oral examination – again, along with Kahn – on 30 June 1909. Afterwards Löbenstein worked as a high school teacher in Metz and Landsberg. She was dismissed on 1 January 1936 due to the Nazi racial laws. In 1941 she emigrated to Argentina. She died on 10 June 1968 in Buenos Aires, where she rests in the Cementerio alemán.
Publications
References
External links
1883 births
1968 deaths
20th-century German mathematicians
Algebraic geometers
University of Göttingen alumni
German emigrants to Argentina |
https://en.wikipedia.org/wiki/Marian%20Ewurama%20Addy | Marian Ewurama Addy (née Cole; 7 February 1942 – 14 January 2014) was a Ghanaian biochemist and the first Host of the National Science and Maths Quiz. The first Ghanaian woman to attain the rank of full professor of natural science, Addy became a role model for school girls and budding female scientists on the limitless opportunities in science, technology, engineering and mathematics (STEM) disciplines. Marian Addy was also a Fellow of the Ghana Academy of Arts and Sciences, elected in 1999. In the same year, she was awarded the UNESCO Kalinga Prize for the Popularization of Science.
Early life and education
Ewurama Addy was born 7 February 1942 in Nkawkaw in the Eastern Region of Ghana, to Samuel Joseph Cole and Angelina Kwofie Cole. She was educated at St Monica's Secondary School in Mampong-Ashanti from January 1956 to June 1960 where she excelled in sports and obtained her 'O' and 'A' level certificates. She also attended the Holy Child Girls' School in Cape Coast. She earned her bachelor's degree with first class honours in botany with chemistry from the University of Ghana, Legon. She later obtained a master's degree and a doctorate in biochemistry from the Pennsylvania State University.
Career
Addy reached the rank of full professor of biochemistry at the University of Ghana, where she was not only the first female professor in the sciences at the university, but also in Ghana as a whole country. At the same university, she became the Head of the Department for Biochemistry, Cell and Molecular Biology from 1988 to 1991 and 1994 to 1997. Addy retired in 2002 as a Professor of Biochemistry.
During her time as a professor and department head, she was a chair for the Policy Committee on Developing Countries (PCDC) and chaired the National Board for Professional and Technicians Examinations (NABPTEX). She served as the program director for the Accra-based Science Education Programme for Africa (SEPA), a Pan African programme for pre-tertiary science education in the 1970s. She served on the Kwami Committee, a technical committee on polytechnic education set up by the National Council for Tertiary Education (NCTE), to study and recommend policies to assist the Ghanaian government in supporting polytechnic education. In 1994, she was also a member of a 4-member UNDP team of consultants in Ghana tasked with formulating a National Action Program for Science and Technology Development. She was a board member of the Ghana Atomic Energy Commission from 1996 to 1998. She served as a member of WHO Regional Expert Committee on Traditional Medicine, and worked as an advisor to the International Foundation for Science, in Stockholm, Sweden. She was the Founder and First Executive Secretary of Western Africa Network of Natural Products Research Scientists (WANNPRES), which was established in February 2002.
She had extensive experience in both basic and applied science, lecturing to undergraduate, post-graduate, dental and medical students at the Un |
https://en.wikipedia.org/wiki/Fan%20chart%20%28statistics%29 | A fan chart is made of a group of dispersion fan diagrams,
which may be positioned according to two categorising dimensions.
A dispersion fan diagram is a circular diagram which
reports the same information about a dispersion as a box plot:
namely median, quartiles, and two extreme values.
Elements
The elements of a dispersion fan diagram
are:
a circular line as scale
a diameter which indicates the median
a fan (a segment of a circle) which indicates the quartiles
two feathers which indicate the extreme values.
The scale on the circular line begins at the left
with the starting value (e. g. with zero).
The following values are applicated clockwise.
The white tail of diameter indicates the median.
The dark fan indicates the dispersion of the middle half of the observed
values; thus it encompasses the values from the first to the third quartile.
The white feathers indicate the dispersion of the middle 90% of the
observed values.
The length of the white part of the diameter corresponds with the number
of observations.
Application
A fan chart gives a quick summary of observed values which depend from two variables. This is possible thanks of a dense representation
and a constant size which does not depend on the size of the single dispersion fan diagrams.
An essential advantage compared to a sequence of box plots
is the possibility to compare dispersion fan diagrams not only within one direction
but within two directions (horizontally and vertically).
Example
The following example presents data from the data set MathAchieve
which is part of the R package
nlme
of José Pinheiro et al.
It contains mathematics achievement scores of 7185 students.
The students are categorised
according to sex and membership of a minority ethnic group.
The graphics show the mathematics achievement scores in dependency
on the socio-economic status of the students (x axis)
and on the average socio-economic status of all students
at the same school (y axis).
The four graphic panels differentiate the students
according to sex and membership of a minority ethnic group.
The fan charts reveals clearly how the median value
is partially following a big main tendency
while the values of the single subgroups (with the cells) scatter largely
what could lead to doubts about a possible correlation.
See also
Box plot
References
External links
Fischer, Wolfram (2012): Streuungsfächerkarten und pseudogeografische Anordnungen. Mit Beispielen zum verfügbaren Einkommen und zu Krankenkassenprämien in der Schweiz. Swiss Days of Official Statistics, Vaduz LI, 2012.
Fischer, Wolfram (2010): Visualising Twofold Dependencies by Fan Charts. ZIM.
Statistical charts and diagrams |
https://en.wikipedia.org/wiki/Topologies%20on%20spaces%20of%20linear%20maps | In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves.
The article operator topologies discusses topologies on spaces of linear maps between normed spaces, whereas this article discusses topologies on such spaces in the more general setting of topological vector spaces (TVSs).
Topologies of uniform convergence on arbitrary spaces of maps
Throughout, the following is assumed:
is any non-empty set and is a non-empty collection of subsets of directed by subset inclusion (i.e. for any there exists some such that ).
is a topological vector space (not necessarily Hausdorff or locally convex).
is a basis of neighborhoods of 0 in
is a vector subspace of which denotes the set of all -valued functions with domain
𝒢-topology
The following sets will constitute the basic open subsets of topologies on spaces of linear maps.
For any subsets and let
The family
forms a neighborhood basis
at the origin for a unique translation-invariant topology on where this topology is necessarily a vector topology (that is, it might not make into a TVS).
This topology does not depend on the neighborhood basis that was chosen and it is known as the topology of uniform convergence on the sets in or as the -topology.
However, this name is frequently changed according to the types of sets that make up (e.g. the "topology of uniform convergence on compact sets" or the "topology of compact convergence", see the footnote for more details).
A subset of is said to be fundamental with respect to if each is a subset of some element in
In this case, the collection can be replaced by without changing the topology on
One may also replace with the collection of all subsets of all finite unions of elements of without changing the resulting -topology on
Call a subset of -bounded if is a bounded subset of for every
Properties
Properties of the basic open sets will now be described, so assume that and
Then is an absorbing subset of if and only if for all absorbs .
If is balanced (respectively, convex) then so is
The equality
always holds.
If is a scalar then so that in particular,
Moreover,
and similarly
For any subsets and any non-empty subsets
which implies:
if then
if then
For any and subsets of if then
For any family of subsets of and any family of neighborhoods of the origin in
Uniform structure
For any and be any entourage of (where is endowed with its canonical uniformity), let
Given the family of all sets as ranges over any fundamental system of entourages of forms a fundamental system of entourages for a uniform structure on called or simply .
The is the least upper bound of all -convergence uniform structures as ranges over
Nets and uniform convergence
Let and let be a net in Then for any subse |
https://en.wikipedia.org/wiki/Sabine%20Landau | Sabine Landau is Professor of Biostatistics at the Institute of Psychiatry, King's College London. Landau was acting and then head of the Biostatistics Department in 2005–2009 and during 2008–2009 was the head of the Mental Health and Neurosciences Clinical Trials Unit.
Landau is a member of the UK Mental Health Research Network's Methodology Research Group and the Royal Statistical Society's General Applications (GAS) committee. She is a member of the King's Trials Partnership steering committee with the aim of to sharing and expanding clinical trials knowledge.
Selected publications
References
21st-century German mathematicians
21st-century women mathematicians
20th-century German mathematicians
20th-century women mathematicians
Academics of King's College London
British women academics
German statisticians
Women statisticians
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Society%20for%20Mathematics%20and%20Computation%20in%20Music | The Society for Mathematics and Computation in Music (SMCM) was founded in 2006 as an International Forum for researchers and musicians working in the trans-disciplinary field at the intersection of music, mathematics and computation. The SMCM is registered in the USA. At its inaugural meeting in Berlin on May 20, 2007, 13 board members were elected. The board later elected the officers for the society.
Officers
President: Guerino Mazzola, University of Minnesota
Vice President: Moreno Andreatta, IRCAM/CNRS, Paris
Treasurer: David Clampitt, Ohio State University
Secretary: Johanna Devaney, Ohio State University
Past Officers
Treasurer (2007-2013): Ian Quinn, Yale University
Secretary (2007-2013): Elaine Chew, University of Southern California / Queen Mary University of London
Conferences
The Society hosts a biennial meeting, the International Conference on Mathematics and Computation in Music (MCM).
MCM 2007, National Institute for Music Research, Berlin, Germany, May 18–20, 2007
MCM 2009, Yale University, New Haven, Connecticut, USA, June 19–22, 2009
MCM 2011, IRCAM, Paris, France, June 15–17, 2011
MCM 2013, McGill University Schulich School of Music and CIRMMT, Montreal, Canada, June 12–14, 2013
MCM 2015, Queen Mary University of London, London, United Kingdom, June 22–25, 2015
MCM 2017, MCM 2017, Universidad Nacional Autónoma di México (UNAM), Mexico City, Mexico, June 26–29, 2017
Journals
SMCM’s official journal is the Journal of Mathematics and Music.
See also
Music and mathematics
List of music software
External links
SMCM Official website: http://www.smcm-net.info/
SMCM Official Journal: http://www.tandfonline.com/loi/tmam20
References
Music organizations based in the United States
Mathematical societies |
https://en.wikipedia.org/wiki/Cathryn%20Lewis | Cathryn Lewis is Professor of Genetic Epidemiology and Statistics at King's College London. She is Head of Department at the Social, Genetic and Developmental Psychiatry Centre, Institute of Psychiatry, Psychology and Neuroscience.
She completed her BA in mathematics at St. Hilda’s College, University of Oxford, and her PhD in statistics at the University of Sheffield. She then spent five years at the University of Utah working on projects to identify BRCA1 and BRCA2 genes, before joining King’s College London in 1996. She now leads the Statistical Genetics Unit at King’s College London, a multi-disciplinary research group that develops and applies statistical methods to human genetics, to identify and characterise genes contributing to common, complex disorders.
In the Psychiatric Genomics Consortium, the international collaboration for sharing and analysing genetic data, she co-chairs the Major Depressive Disorder Working Group, with Professor Andrew McIntosh. She is an editor for journal Biological Psychiatry: Global Open Science. She was also an academic editor for journals PLOS Medicine and an Associate Editor for Human Heredity. She leads the Medical Research Council's Skills Development Fellowship programme at King's College London.
She featured in a BBC news feature broadcast on the Victoria Derbyshire (TV programme), where James Longman asked the question “Do you inherit your parent's mental illness?” and a mental health special issue of the BBC’s Trust Me I’m a Doctor.
In 2010, Lewis was part of the "Music from the Genome" team which analysed the DNA from 40 members of the New London Chamber Choir. These gene patterns were used to create a choral work, "Allele". The music subsequently won Michael Zev Gordon the 2011 British Academy of Songwriters, Composers and Authors Composer of the Year award.
Selected publications
References
21st-century British biologists
21st-century British women scientists
Academics of King's College London
Alumni of the University of Oxford
British women biologists
British geneticists
Genetic epidemiologists
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Domenico%20Marchiori | Domenico Marchiori (Lendinara, Province of Rovigo; 1828 – 1905) was an Italian painter. He was attracted to Neo-Pompeian subject matter.
Biography
Marchiori studied mathematics at the University of Padua, and came late to painting. He was also active as a parliamentarian (1878-1880), in addition to a dilettante poet. He studied in Rome at the Accademia del Nudo.
In 1881 at Milan, he exhibited a canvas depicting a Priest of the Ancient Bacchus. In 1884 at Turin, he displayed a portrait in watercolor. In 1887 at Venice, he exhibited a watercolor titled: Dal triclinio al cubicolo, and a canvas Aspettilo anca ti.
He also painted frescoes for the Palazzo Marchiori in Lendinara, portraits, and an altarpiece for the church in Cavazzana, Rovigo.
References
19th-century Italian painters
Italian male painters
20th-century Italian painters
1828 births
1905 deaths
Neo-Pompeian painters
People from the Province of Rovigo
19th-century Italian male artists
20th-century Italian male artists |
https://en.wikipedia.org/wiki/List%20of%20Greek%20mathematicians | In historical times, Greek civilization has played one of the major roles in the history and development of Greek mathematics. To this day, a number of Greek mathematicians are considered for their innovations and influence on mathematics.
Ancient Greek mathematicians
Anaxagoras
Anthemius of Tralles
Antiphon
Apollonius of Perga
Archimedes
Archytas
Aristaeus the Elder
Aristarchus of Samos
Aristotle
Asclepius of Tralles
Attalus of Rhodes
Autolycus of Pitane
Bion of Abdera
Bryson of Heraclea
Callippus
Carpus of Antioch
Chrysippus
Cleomedes
Conon of Samos
Ctesibius
Democritus
Dicaearchus
Dinostratus
Diocles
Dionysodorus
Diophantus
Domninus of Larissa
Eratosthenes
Euclid
Eudoxus of Cnidus
Eutocius of Ascalon
Geminus
Heliodorus of Larissa
Hero of Alexandria
Hipparchus
Hippasus
Hippias
Hippocrates of Chios
Hypatia
Hypsicles
Leodamas of Thasos
Marinus of Neapolis
Menaechmus
Menelaus of Alexandria
Meton of Athens
Metrodorus
Nicomachus
Nicomedes
Nicoteles of Cyrene
Oenopides
Pappus of Alexandria
Perseus (geometer)
Philolaus
Philon
Philonides of Laodicea
Polyaenus of Lampsacus
Posidonius
Proclus
Ptolemy
Pythagoras
Serenus of Antinouplis
Simplicius of Cilicia
Sporus of Nicaea
Thales
Theaetetus
Theano
Theodorus of Cyrene
Theodosius of Bithynia
Theon of Alexandria
Theon of Smyrna
Theudius
Thrasyllus of Mendes
Thymaridas
Xenocrates
Zeno of Elea
Zenodorus
Byzantine mathematicians
Stephanus of Alexandria
Maximus Planudes
Isaac Argyros
Isidore of Miletus
John Philoponus
Anthemius of Tralles
Modern Greek mathematicians
Leonidas Alaoglu (1914–1981) - Known for Banach- Alaoglu theorem.
Charalambos D. Aliprantis (1946–2009) - Founder and Editor-in-Chief of the journals Economic Theory as well as Annals of Finance.
Roger Apéry (1916–1994) - Professor of mathematics and mechanics at the University of Caen Proved the irrationality of zeta(3).
Tom M. Apostol (1923–2016) - Professor of mathematics in California Institute of Technology, he has authored a number of books about mathematics.
Dimitri Bertsekas (born 1942) - Member of the National Academy of Engineering professor with the Department of Electrical Engineering and Computer Science. Author of fifteen books and research monographs, and coauthor of an introductory probability textbook
Giovanni Carandino (1784–1834)
Constantin Carathéodory (1873–1950) - Mathematician who pioneered the Axiomatic Formulation of Thermodynamics.
Demetrios Christodoulou (born 1951) - Mathematician-physicist who has contributed in the field of general relativity.
Constantine Dafermos (born 1941) - Usually notable for hyperbolic conservation laws and control theory.
Mihalis Dafermos (born 1976) - Professor of Mathematics at Princeton University and Lowndean Chair of Astronomy and Geometry at the University of Cambridge
Apostolos Doxiadis (born 1953) - Australian born Mathematician.
Athanassios S. Fokas (born 1952) - Contributor in the field of integrable nonlinear partial differential equations.
Michael Katehakis (born 1952) - Professo |
https://en.wikipedia.org/wiki/Tail%20dependence | In probability theory, the tail dependence of a pair of random variables is a measure of their comovements in the tails of the distributions. The concept is used in extreme value theory. Random variables that appear to exhibit no correlation can show tail dependence in extreme deviations. For instance, it is a stylized fact of stock returns that they commonly exhibit tail dependence.
Definition
The lower tail dependence is defined as
where
,
that is, the inverse of the cumulative probability distribution function for q.
The upper tail dependence is defined analogously as
See also
Correlation
Dependence
References
Covariance and correlation
Independence (probability theory)
Theory of probability distributions |
https://en.wikipedia.org/wiki/Institute%20of%20Applied%20Mathematics | Th Institute of Applied Mathematics could refer to:
The Keldysh Institute of Applied Mathematics in Moscow
The Institute of Informatics and Applied Mathematics in Tirana, Albania
The Institute for Pure and Applied Mathematics (IPAM), an American mathematics institute
The Instituto de Investigaciones en Matemáticas Aplicadas y Sistemas (IIMAS, “Applied Mathematics and Systems Research Institute”) in Mexico City
The Institute of Applied Mathematics at Heidelberg University
The Instituto Nacional de Matemática Pura e Aplicada (“National Institute for Pure and Applied Mathematics”) in Brazil |
https://en.wikipedia.org/wiki/H%C3%B6lder%20summation | In mathematics, Hölder summation is a method for summing divergent series introduced by .
Definition
Given a series
define
If the limit
exists for some k, this is called the Hölder sum, or the (H,k) sum, of the series.
Particularly, since the Cesàro sum of a convergent series always exists, the Hölder sum of a series (that is Hölder summable) can be written in the following form:
See also
Cesàro summation
References
Summability methods |
https://en.wikipedia.org/wiki/Maximum%20disjoint%20set | In computational geometry, a maximum disjoint set (MDS) is a largest set of non-overlapping geometric shapes selected from a given set of candidate shapes.
Every set of non-overlapping shapes is an independent set in the intersection graph of the shapes. Therefore, the MDS problem is a special case of the maximum independent set (MIS) problem. Both problems are NP complete, but finding a MDS may be easier than finding a MIS in two respects:
For the general MIS problem, the best known exact algorithms are exponential. In some geometric intersection graphs, there are sub-exponential algorithms for finding a MDS.
The general MIS problem is hard to approximate and doesn't even have a constant-factor approximation. In some geometric intersection graphs, there are polynomial-time approximation schemes (PTAS) for finding a MDS.
Finding an MDS is important in applications such as automatic label placement, VLSI circuit design, and cellular frequency division multiplexing.
The MDS problem can be generalized by assigning a different weight to each shape and searching for a disjoint set with a maximum total weight.
In the following text, MDS(C) denotes the maximum disjoint set in a set C.
Greedy algorithms
Given a set C of shapes, an approximation to MDS(C) can be found by the following greedy algorithm:
INITIALIZATION: Initialize an empty set, S.
SEARCH: For every shape x in C:
Calculate N(x) - the subset of all shapes in C that intersect x (including x itself).
Calculate the largest independent set in this subset: MDS(N(x)).
Select an x such that |MDS(N(x))| is minimized.
Add x to S.
Remove x and N(x) from C.
If there are shapes in C, go back to Search.
END: return the set S.
For every shape x that we add to S, we lose the shapes in N(x), because they are intersected by x and thus cannot be added to S later on. However, some of these shapes themselves intersect each other, and thus in any case it is not possible that they all be in the optimal solution MDS(S). The largest subset of shapes that can all be in the optimal solution is MDS(N(x)). Therefore, selecting an x that minimizes |MDS(N(x))| minimizes the loss from adding x to S.
In particular, if we can guarantee that there is an x for which |MDS(N(x))| is bounded by a constant (say, M), then this greedy algorithm yields a constant M-factor approximation, as we can guarantee that:
Such an upper bound M exists for several interesting cases:
1-dimensional intervals: exact polynomial algorithm
When C is a set of intervals on a line, M=1, and thus the greedy algorithm finds the exact MDS. To see this, assume w.l.o.g. that the intervals are vertical, and let x be the interval with the highest bottom endpoint. All other intervals intersected by x must cross its bottom endpoint. Therefore, all intervals in N(x) intersect each other, and MDS(N(x)) has a size of at most 1 (see figure).
Therefore, in the 1-dimensional case, the MDS can be found exactly in time O(n log n):
Sort the |
https://en.wikipedia.org/wiki/Pseudo-functor | In mathematics, a pseudofunctor F is a mapping between 2-categories, or from a category to a 2-category, that is just like a functor except that and do not hold as exact equalities but only up to coherent isomorphisms.
The Grothendieck construction associates to a pseudofunctor a fibered category.
See also
Lax functor
Prestack (an example of pseudofunctor)
Fibered category
References
C. Sorger, Lectures on moduli of principal G-bundles over algebraic curves
External links
http://ncatlab.org/nlab/show/pseudofunctor
Functors |
https://en.wikipedia.org/wiki/Bundle%20theorem | In Euclidean geometry, the bundle theorem is a statement about six circles and eight points in the Euclidean plane. In general incidence geometry, it is a similar property that a Möbius plane may or may not satisfy. According to Kahn's Theorem, it is fulfilled by "ovoidal" Möbius planes only; thus, it is the analog for Möbius planes of Desargues' Theorem for projective planes.
Bundle theorem. If for eight different points five of the six quadruples are concyclic (contained in a cycle) on at least four cycles , then the sixth quadruple is also concyclic.
The bundle theorem should not be confused with Miquel's theorem.
An ovoidal Möbius plane in real Euclidean space may be considered as the geometry of the plane sections of an egglike surface, like a sphere, or an ellipsoid, or half a sphere glued to a suitable half of an ellipsoid, or the surface with equation , etc. If the egglike surface is a sphere one gets the space model of the classical real Möbius plane, which is the "circle geometry" on the sphere.
The essential property of an ovoidal Möbius plane is the existence of a space model via an ovoid. An ovoid in a 3-dimensional projective space is a set of points, which a) is intersected by lines in 0, 1, or 2 points and b) its tangents at an arbitrary point covers a plane (tangent plane). The geometry of an ovoid in projective 3-space is a Möbius plane, called an ovoidal Möbius plane. The point set of the geometry consists of the points of the ovoid and the curves ("cycles") are the plane sections of the ovoid. A suitable stereographical projection shows that for any ovoidal Möbius plane there exists a plane model. In the classical case the plane model is the geometry of the circles and lines (where each line is completed by a point at infinity). The bundle theorem has a planar and a spatial interpretation. In the planar model there may be lines involved. The proof of the bundle theorem is performed within the spatial model.
Theorem. The bundle theorem holds in every ovoidal Möbius plane.
The proof is a consequence of the following considerations, which use essentially the fact that three planes in a 3-dimensional projective space intersect in a single point:
The planes containing the cycles intersect in a point . Hence is the intersection point of the lines (in space !) .
The planes containing the cycles intersect in a point . Hence is the intersection point of the lines , too.
This yields: a) and b) intersect at point , too. The last statement means: are concyclic. The planes involved have point in common, they are elements of a bundle of planes.
The importance of the bundle theorem was shown by Jeff Kahn.
Theorem of Kahn. A Möbius plane is ovoidal if and only if it fulfills the bundle theorem.
The bundle theorem is analogous for Möbius planes to the Theorem of Desargues for projective planes. From the bundle theorem follows the existence of a) a skewfield (division ring) and b) an ovoid. If the more strict theorem of |
https://en.wikipedia.org/wiki/Peter%20Keevash | Peter Keevash (born 30 November 1978) is a British mathematician, working in combinatorics. He is a professor of mathematics at the University of Oxford and a Fellow of Mansfield College.
Early years
Keevash was born in Brighton, England, but mostly grew up in Leeds. He competed in the International Mathematical Olympiad in 1995. He entered Trinity College, University of Cambridge, in 1995 and completed his B.A. in mathematics in 1998. He earned his doctorate from Princeton University with Benny Sudakov as advisor. He took a postdoctoral position at the California Institute of Technology before moving to Queen Mary, University of London as a lecturer, and subsequently professor, before his move to Oxford in September 2013.
Mathematics
Keevash has published many results in combinatorics, particularly in extremal graph and hypergraph theory and Ramsey Theory. In joint work with Tom Bohman he established the best-known lower bound for the off-diagonal Ramsey Number , namely (This result was obtained independently at the same time by Fiz Pontiveros, Griffiths and Morris.)
On 15 January 2014, he released a preprint establishing the existence of block designs with arbitrary parameters, provided only that the underlying set is sufficiently large and satisfies certain obviously necessary divisibility conditions. In particular, his work provides the first examples of Steiner systems with parameter t ≥ 6 (and in fact provides such systems for all t).
In 2018, he was an invited speaker at the International Congress of Mathematicians in Rio de Janeiro.
References
External links
Peter Keevash home page at the University of Oxford
1978 births
Living people
Alumni of Trinity College, Cambridge
20th-century English mathematicians
21st-century English mathematicians
Combinatorialists
International Mathematical Olympiad participants
Whitehead Prize winners
Academics of the University of Oxford |
https://en.wikipedia.org/wiki/Cantor%27s%20paradise | Cantor's paradise is an expression used by in describing set theory and infinite cardinal numbers developed by Georg Cantor. The context of Hilbert's comment was his opposition to what he saw as L. E. J. Brouwer's reductive attempts to circumscribe what kind of mathematics is acceptable; see Brouwer–Hilbert controversy.
References
Saharon Shelah. You can enter Cantor's paradise! Paul Erdős and his mathematics, II (Budapest, 1999), 555–564, Bolyai Soc. Math. Stud., 11, János Bolyai Math. Soc., Budapest, 2002.
Peckhaus, Volker. Fixing Cantor's paradise: the prehistory of Ernst Zermelo's axiomatization of set theory. New approaches to classes and concepts, 11–22, Stud. Log. (Lond.), 14, Coll. Publ., London, 2008.
"About Cantor's Paradise". Medium. A Medium Corporation. Retrieved 24 January 2021.
Set theory
David Hilbert |
https://en.wikipedia.org/wiki/Webbed%20space | In mathematics, particularly in functional analysis, a webbed space is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains are webbed spaces. A space is called webbed if there exists a collection of sets, called a web that satisfies certain properties. Webs were first investigated by de Wilde.
Web
Let be a Hausdorff locally convex topological vector space. A is a stratified collection of disks satisfying the following absorbency and convergence requirements.
Stratum 1: The first stratum must consist of a sequence of disks in such that their union absorbs
Stratum 2: For each disk in the first stratum, there must exists a sequence of disks in such that for every : and absorbs The sets will form the second stratum.
Stratum 3: To each disk in the second stratum, assign another sequence of disks in satisfying analogously defined properties; explicitly, this means that for every : and absorbs The sets form the third stratum.
Continue this process to define strata That is, use induction to define stratum in terms of stratum
A is a sequence of disks, with the first disk being selected from the first stratum, say and the second being selected from the sequence that was associated with and so on. We also require that if a sequence of vectors is selected from a strand (with belonging to the first disk in the strand, belonging to the second, and so on) then the series converges.
A Hausdorff locally convex topological vector space on which a web can be defined is called a .
Examples and sufficient conditions
All of the following spaces are webbed:
Fréchet spaces.
Projective limits and inductive limits of sequences of webbed spaces.
A sequentially closed vector subspace of a webbed space.
Countable products of webbed spaces.
A Hausdorff quotient of a webbed space.
The image of a webbed space under a sequentially continuous linear map if that image is Hausdorff.
The bornologification of a webbed space.
The continuous dual space of a metrizable locally convex space endowed with the strong dual topology is webbed.
If is the strict inductive limit of a denumerable family of locally convex metrizable spaces, then the continuous dual space of with the strong topology is webbed.
So in particular, the strong duals of locally convex metrizable spaces are webbed.
If is a webbed space, then any Hausdorff locally convex topology weaker than this (webbed) topology is also webbed.
Theorems
If the spaces are not locally convex, then there is a notion of web where the requirement of being a disk is replaced by the requirement of being balanced. For such a notion of web we have the following results:
See also
Citations
References
Functional analysis
Topological vector spaces |
https://en.wikipedia.org/wiki/2014%20Sydney%20Swans%20season | The 2014 AFL season was the 118th season in the Australian Football League contested by the Sydney Swans.
Squad for 2014
Statistics are correct as of end of 2013 season.
Flags represent the state of origin, i.e. the state in which the player played his Under-18s football.
For players: (c) denotes captain, (vc) denotes vice-captain, (lg) denotes leadership group.
For coaches: (s) denotes senior coach, (cs) denotes caretaker senior coach, (a) denotes assistant coach, (d) denotes development coach.
Playing list changes
The following summarises all player changes between the conclusion of the 2013 season and the beginning of the 2014 season.
In
Out
List management
Season summary
Pre-season matches
Home and away season
Finals matches
Ladder
Team awards and records
Season records
Sydney broke the 40,000 members milestone for the first time.
Between Round 5 and Round 17 the Swans record 12 wins in a row, equaling the club record last set in 1935.
Sydney's score of 19.22 (136) in the Preliminary Final broke the club record for biggest score in a final.
Sydney's winning margin of 71 points in the Preliminary Final broke the club record for biggest victory in a final.
Individual awards and records
Bob Skilton Medal
Rising Star Award - Harry Cunningham
Dennis Carroll Trophy for Most Improved Player – Ben McGlynn
Barry Round Shield for Best Clubman – Jarrad McVeigh
Paul Kelly Players’ Player – Luke Parker
Paul Roos Award for Best Player in a Finals Series – Lance Franklin
Coleman Medal
Lance Franklin won the 2014 Coleman medal with 67 goals, from Jay Schulz with 64 goals.
Milestones
Round 1 - Rhyce Shaw (100 club games)
Round 2 - Nick Smith (100 career games), Josh Kennedy (100 club games)
Round 7 - Rhyce Shaw (200 career games), Tom Derickx (first goal), Jake Lloyd (first goal)
Round 11 - Lance Franklin (600 career goals)
Round 12 - Heath Grundy (150 career games)
Round 13 - Kieren Jack (150 career games)
Round 15 - Zak Jones (first goal)
Round 16 - Adam Goodes (341 games, most by an Indigenous footballer)
Round 21 - Lance Franklin (200 career games), Dean Towers (first goal)
Round 23 - Lewis Jetta (100 career games)
Preliminary Final - Adam Goodes (350 career games)
Grand Final - Ben McGlynn (100 club games)
Debuts
Round 1 - Lance Franklin (club debut), Jeremy Laidler (club debut)
Round 2 - Tom Derickx (club debut)
Round 5 - Jake Lloyd (debut)
Round 14 - Zak Jones (debut)
Round 17 - Dean Towers (debut)
Round 19 - Tim Membrey (debut)
All-Australian Team
Lance Franklin - Full-forward
Josh Kennedy - Centre
Nick Malceski - Half-back flank
Nick Smith - Back pocket
Luke Parker (nominated)
AFL Rising Star
The following Sydney players were nominated for the 2014 NAB AFL Rising Star award:
Round 15 – Harry Cunningham (nominated)
Round 21 – Jake Lloyd (nominated)
22 Under 22 team
Luke Parker - Wing
Reserves results
Regular season
Finals series
Ladder
References
2014
2014 Australian Football League season
2014 in New South Wales |
https://en.wikipedia.org/wiki/Octahedral%20pyramid | In 4-dimensional geometry, the octahedral pyramid is bounded by one octahedron on the base and 8 triangular pyramid cells which meet at the apex. Since an octahedron has a circumradius divided by edge length less than one, the triangular pyramids can be made with regular faces (as regular tetrahedrons) by computing the appropriate height.
Having all regular cells, it is a Blind polytope. Two copies can be augmented to make an octahedral bipyramid which is also a Blind polytope.
Occurrences of the octahedral pyramid
The regular 16-cell has octahedral pyramids around every vertex, with the octahedron passing through the center of the 16-cell. Therefore placing two regular octahedral pyramids base to base constructs a 16-cell. The 16-cell tessellates 4-dimensional space as the 16-cell honeycomb.
Exactly 24 regular octahedral pyramids will fit together around a vertex in four-dimensional space (the apex of each pyramid). This construction yields a 24-cell with octahedral bounding cells, surrounding a central vertex with 24 edge-length long radii. The 4-dimensional content of a unit-edge-length 24-cell is 2, so the content of the regular octahedral pyramid is 1/12. The 24-cell tessellates 4-dimensional space as the 24-cell honeycomb.
The octahedral pyramid is the vertex figure for a truncated 5-orthoplex, .
The graph of the octahedral pyramid is the only possible minimal counterexample to Negami's conjecture, that the connected graphs with planar covers are themselves projective-planar.
Example 4-dimensional coordinates, 6 points in first 3 coordinates for cube and 4th dimension for the apex.
(±1, 0, 0; 0)
( 0,±1, 0; 0)
( 0, 0,±1; 0)
( 0, 0, 0; 1)
Other polytopes
Cubic pyramid
The dual to the octahedral pyramid is a cubic pyramid, seen as a cubic base and 6 square pyramids meeting at an apex.
Example 4-dimensional coordinates, 8 points in first 3 coordinates for cube and 4th dimension for the apex.
(±1,±1,±1; 0)
( 0, 0, 0; 1)
Square-pyramidal pyramid
The square-pyramidal pyramid, ( ) ∨ [( ) ∨ {4}], is a bisected octahedral pyramid. It has a square pyramid base, and 4 tetrahedrons along with another one more square pyramid meeting at the apex. It can also be seen in an edge-centered projection as a square bipyramid with four tetrahedra wrapped around the common edge. If the height of the two apexes are the same, it can be given a higher symmetry name [( ) ∨ ( )] ∨ {4} = { } ∨ {4}, joining an edge to a perpendicular square.
The square-pyramidal pyramid can be distorted into a rectangular-pyramidal pyramid, { } ∨ [{ } × { }] or a rhombic-pyramidal pyramid, { } ∨ [{ } + { }], or other lower symmetry forms.
The square-pyramidal pyramid exists as a vertex figure in uniform polytopes of the form , including the bitruncated 5-orthoplex and bitruncated tesseractic honeycomb.
Example 4-dimensional coordinates, 2 coordinates for square, and axial points for pyramidal points.
(±1,±1; 0; 0)
( 0, 0; 1; 0)
( 0, 0; 0; 1)
References
External links
|
https://en.wikipedia.org/wiki/Cuboctahedral%20pyramid | In 4-dimensional geometry, the cuboctahedral pyramid is bounded by one cuboctahedron on the base, 6 square pyramid, and 8 triangular pyramid cells which meet at the apex. It has 38 faces: 32 triangles and 6 squares. It has 32 edges, and 13 vertices.
Since a cuboctahedron's circumradius is equal to its edge length, the triangles must be taller than equilateral to create a positive height.
The dual to the cuboctahedral pyramid is a rhombic dodecahedral pyramid, seen as a rhombic dodecahedral base, and 12 rhombic pyramids meeting at an apex.
References
External links
Richard Klitzing, Axial-Symmetrical Edge Facetings of Uniform Polyhedra
4-polytopes |
https://en.wikipedia.org/wiki/Gillies%27%20conjecture | In number theory, Gillies' conjecture is a conjecture about the distribution of prime divisors of Mersenne numbers and was made by Donald B. Gillies in a 1964 paper in which he also announced the discovery of three new Mersenne primes. The conjecture is a specialization of the prime number theorem and is a refinement of conjectures due to I. J. Good and Daniel Shanks. The conjecture remains an open problem: several papers give empirical support, but it disagrees with the widely accepted (but also open) Lenstra–Pomerance–Wagstaff conjecture.
The conjecture
He noted that his conjecture would imply that
The number of Mersenne primes less than is .
The expected number of Mersenne primes with is .
The probability that is prime is .
Incompatibility with Lenstra–Pomerance–Wagstaff conjecture
The Lenstra–Pomerance–Wagstaff conjecture gives different values:
The number of Mersenne primes less than is .
The expected number of Mersenne primes with is .
The probability that is prime is with a = 2 if p = 3 mod 4 and 6 otherwise.
Asymptotically these values are about 11% smaller.
Results
While Gillie's conjecture remains open, several papers have added empirical support to its validity, including Ehrman's 1964 paper.
References
Conjectures
Unsolved problems in number theory
Hypotheses
Mersenne primes |
https://en.wikipedia.org/wiki/Heinz-Herbert%20Noll | Heinz-Herbert Noll (born 1 January 1949) is a German sociologist.
Education
Noll studied sociology, economics, social policy and statistics at the Goethe University Frankfurt. After the diploma in sociology he wrote a dissertation at the chair of Wolfgang Zapf at the University of Mannheim and received his philosophers degree (Dr. phil.) in 1981 with a thesis on "Occupational Chances and Working Conditions: A Social Report for the Federal Republic of Germany 1950–1980“ („Beschäftigungschancen und Arbeitsbedingungen: Ein Sozialbericht für die Bundesrepublik 1950–1980“) (1982).
Profession
Research
From 1981 to 1987 Noll was scientific member of the SPES-project and the Special Research Group 3 „Microanalytic Foundations of Societal Policies“ (Sonderforschungsbereich 3 „Mikroanalytische Grundlagen der Gesellschaftspolitik“) (Sfb3); afterwards he became project director and director of a research area of the Sfb 3. During this time he specialized in the field of labour market sociology and wrote a dissertation within this realm.
In 1987 he became head of the newly created „Department for Social Indicators“ („Abteilung Soziale Indikatoren“) of ZUMA (today GESIS – Leibniz Institute for the Social Sciences). The main task of this research group consists in the permanent institutionalization of welfare research and of social reporting for Germany as a scientific basic service. Over several years, diverse activities developed from this general purpose. These consist in:
to continue and update the SPES „Social Indicator Table“ for Germany
to organize seminaries and workshops in social indicators research
to contribute to the „Data Report“ („Datenreport“)
to edit a Newsletter or „Information Service Social Indicators“ („Informationsdienst soziale Indikatoren“)
scientific counselling and writing of expert opinions
international co-operation, as e.g. in the International Society for Quality of Life Studies (ISQLS)
to organize research projects, as, for example of a project to develop a system of European social indicators
Teaching
Noll taught at the Universities of Mannheim and Heidelberg, the University of Tartu, the Université Fribourg, the Università degli Studi di Firenze and the École Nationale de la Statistique et de l'Administration Économique in Paris. In addition he was offered guest professorships from several foreign research institutes and universities.
Honorary offices
Noll was called into several scientific and policy advisory bodies, for example as a speaker of the "Section of Social Indicators" ("Sektion Soziale Indikatoren") of the German Sociological Association (Deutsche Gesellschaft für Soziologie, DGS) or president of the ISA-Research Committee 55 "Social Indicators". Finally until 2012 he was president of the International Society for Quality of Life Studies (ISQLS).
Noll furthermore held positions as a councillor in various research projects and was member of the Editorial boards of journals like Social Indicators Res |
https://en.wikipedia.org/wiki/Pollock%27s%20conjectures | Pollock's conjectures are two closely related unproven conjectures in additive number theory. They were first stated in 1850 by Sir Frederick Pollock, better known as a lawyer and politician, but also a contributor of papers on mathematics to the Royal Society. These conjectures are a partial extension of the Fermat polygonal number theorem to three-dimensional figurate numbers, also called polyhedral numbers.
Pollock tetrahedral numbers conjecture: Every positive integer is the sum of at most five tetrahedral numbers.
The numbers that are not the sum of at most 4 tetrahedral numbers are given by the sequence 17, 27, 33, 52, 73, ..., of 241 terms, with 343867 being almost certainly the last such number.
Pollock octahedral numbers conjecture: Every positive integer is the sum of at most seven octahedral numbers. This conjecture has been proven for all but finitely many positive integers.
Polyhedral numbers conjecture: Every positive integer is the sum of at most 5 tetrahedral numbers, or the sum of at most 9 cube numbers, or the sum of at most 7 octahedral numbers, or the sum of at most 22 dodecahedral numbers, or the sum of at most 15 icosahedral numbers. The cube numbers case was established from 1909 to 1912 by Wieferich and A. J. Kempner.
References
Conjectures
Unsolved problems in number theory
Figurate numbers
Additive number theory |
https://en.wikipedia.org/wiki/Unemployment%20in%20Ontario | Unemployment in Ontario is the measure indicating the number of Ontarians "without work, are available for work, and are actively seeking work". The rate of unemployment is measured by Statistics Canada using a Labour Force Survey. In September 2018 approximately 452,900 people were deemed unemployed in Ontario. With an Unemployment rate of roughly 5.9% Ontario is even with the Canada's overall unemployment level. The Unemployment rate is quite stable from month to month with an approximate 0.2% fluctuation. Since 2013 Ontario's Unemployment rate has dropped 2.0%.
Unemployment by demographic group
Age
As of 2018, the majority of individuals unemployed in Ontario were between the ages of 25 and 54. This reflects the fact that most of Ontario's workers - 64% of the overall labour force as of 2018 - are in the 25 to 54 demographic. Yet while 25-54 year-olds make up the majority of the unemployed, as of 2018 they had the lowest unemployment rate of any demographic, at 5.0 percent. The 15 to 24 year-old age group has the highest unemployment rate at 12.2% (2018).
Ontario's Unemployment rate by Age Groupings (Statistics Canada estimates)
Gender
As of 2018, men and women in Ontario experienced comparable unemployment rates of approximately six percent. Unemployment rose between September, 2017 and September, 2018, with a larger increase in unemployment being experienced by females.
Unemployment Rates (Statistics Canada estimates)
Recent Immigrants and First Nations
Immigrants, especially recent immigrants, face challenges with unemployment. Inuit and First Nations people also have higher rates of unemployment.
Unemployment by region
In recent years following the Great Recession, regions or cities such as Windsor, Oshawa, London and Peterborough, heavily dependent on auto manufacturing have been severely impacted by unemployment. In December 2013 Toronto proper unemployment rate deteriorated to 10.1%.
Solutions
Issues related to creating employment involve many socio-political-economic-environmental factors and complexities. In Ontario, center and left leaning governments have supported strong infrastructure building and social safety net policies while right leaning governments have pursued lower taxes and government spending policies. Regardless of government policies external factors such as Global recession, change in technologies, lower labour costs and lack of strong regulations in developing countries impact unemployment in Ontario.
Social safety net
Canada has a federal Employment Insurance system which covers workers for several months immediately after they lose work. Ontario also provides various social assistance services for those in need.
Infrastructure
Stable political and social systems, corruption free and efficient government operations, quality education, health, transportation, energy, information and communication, water and sanitation and financial systems help create employment and improve private sector produ |
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