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https://en.wikipedia.org/wiki/Rhombicosacron | In geometry, the rhombicosacron (or midly dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform rhombicosahedron, U56. It has 50 vertices, 120 edges, and 60 crossed-quadrilateral faces.
Proportions
Each face has two angles of and two angles of . The diagonals of each antiparallelogram intersect at an angle of . The dihedral angle equals . The ratio between the lengths of the long edges and the short ones equals , which is the square of the golden ratio.
References
External links
Uniform polyhedra and duals
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Great%20icosacronic%20hexecontahedron | In geometry, the great icosacronic hexecontahedron (or great sagittal trisicosahedron) is the dual of the great icosicosidodecahedron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models.
Proportions
Faces have two angles of , one of and one of . Its dihedral angles equal . The ratio between the lengths of the long and short edges is .
References
External links
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Great%20triakis%20octahedron | In geometry, the great triakis octahedron is the dual of the stellated truncated hexahedron (U19). It has 24 intersecting isosceles triangle faces. Part of each triangle lies within the solid, hence is invisible in solid models.
Proportions
The triangles have one angle of and two of . The dihedral angle equals .
References
External links
Mathworld - Great Triakis Octahedron
Mathworld - Small Triakis Octahedron
Uniform polyhedra and duals
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Great%20rhombihexacron | In geometry, the great rhombihexacron (or great dipteral disdodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron (U21). It has 24 identical bow-tie-shaped faces, 18 vertices, and 48 edges.
It has 12 outer vertices which have the same vertex arrangement as the cuboctahedron, and 6 inner vertices with the vertex arrangement of an octahedron.
As a surface geometry, it can be seen as visually similar to a Catalan solid, the disdyakis dodecahedron, with much taller rhombus-based pyramids joined to each face of a rhombic dodecahedron.
Proportions
Each bow-tie has two angles of and two angles of . The diagonals of each bow-tie intersect at an angle of . The dihedral angle equals .
The ratio between the lengths of the long edges and the short ones equals .
Notes
References
uniform polyhedra and duals
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Hemipolyhedron | In geometry, a hemipolyhedron is a uniform star polyhedron some of whose faces pass through its center. These "hemi" faces lie parallel to the faces of some other symmetrical polyhedron, and their count is half the number of faces of that other polyhedron – hence the "hemi" prefix.
The prefix "hemi" is also used to refer to certain projective polyhedra, such as the hemi-cube, which are the image of a 2 to 1 map of a spherical polyhedron with central symmetry.
Wythoff symbol and vertex figure
Their Wythoff symbols are of the form p/(p − q) p/q | r; their vertex figures are crossed quadrilaterals. They are thus related to the cantellated polyhedra, which have similar Wythoff symbols. The vertex configuration is p/q.2r.p/(p − q).2r. The 2r-gon faces pass through the center of the model: if represented as faces of spherical polyhedra, they cover an entire hemisphere and their edges and vertices lie along a great circle. The p/(p − q) notation implies a {p/q} face turning backwards around the vertex figure.
The nine forms, listed with their Wythoff symbols and vertex configurations are:
Note that Wythoff's kaleidoscopic construction generates the nonorientable hemipolyhedra (all except the octahemioctahedron) as double covers (two coincident hemipolyhedra).
In the Euclidean plane, the sequence of hemipolyhedra continues with the following four star tilings, where apeirogons appear as the aforementioned equatorial polygons:
Of these four tilings, only 6/5 6 ∞ is generated as a double cover by Wythoff's construction.
Orientability
Only the octahemioctahedron represents an orientable surface; the remaining hemipolyhedra have non-orientable or single-sided surfaces. This is because proceeding around an equatorial 2r-gon, the p/q-gonal faces alternately point "up" and "down", so any two consecutive ones have opposite senses. This is equivalent to demanding that the p/q-gons in the corresponding quasiregular polyhedra below can be alternatively given positive and negative orientations. But that is only possible for the triangles of the cuboctahedron (corresponding to the triangles of the octahedron, the only regular polyhedron with an even number of faces meeting at a vertex), which are precisely the non-hemi faces of the octahemioctahedron.
Duals of the hemipolyhedra
Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity. In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usu |
https://en.wikipedia.org/wiki/List%20of%20Central%20Hockey%20League%20seasons | This is a list of seasons of the Central Hockey League since its inception.
References
External links
Historic standings and statistics - at Internet Hockey Database
Central Hockey League seasons |
https://en.wikipedia.org/wiki/Noncommutative%20unique%20factorization%20domain | In mathematics, a noncommutative unique factorization domain is a noncommutative ring with the unique factorization property.
Examples
The ring of Hurwitz quaternions, also known as integral quaternions. A quaternion a = a0 + a1i + a2j + a3k is integral if either all the coefficients ai are integers or all of them are half-integers.
All free associative algebras.
References
P.M. Cohn, "Noncommutative unique factorization domains", Transactions of the American Mathematical Society 109:2:313-331 (1963). full text
R. Sivaramakrishnan, Certain number-theoretic episodes in algebra, CRC Press, 2006,
Notes
Ring theory
Number theory |
https://en.wikipedia.org/wiki/Small%20dodecahemicosacron | In geometry, the small dodecahemicosacron is the dual of the small dodecahemicosahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the great dodecahemicosacron.
Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity. In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.
Since the small dodecahemicosahedron has ten hexagonal faces passing through the model center, it can be seen as having ten vertices at infinity.
See also
Hemi-icosahedron - The ten vertices at infinity correspond directionally to the 10 vertices of this abstract polyhedron.
References
(Page 101, Duals of the (nine) hemipolyhedra)
External links
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Great%20dodecahemidodecacron | In geometry, the great dodecahemidodecacron is the dual of the great dodecahemidodecahedron, and is one of nine dual hemipolyhedra. It appears indistinct from the great icosihemidodecacron.
Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity. In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.
The great dodecahemidodecacron can be seen as having six vertices at infinity.
See also
Hemi-dodecahedron - The six vertices at infinity correspond directionally to the six vertices of this abstract polyhedron.
References
(Page 101, Duals of the (nine) hemipolyhedra)
External links
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Great%20icosihemidodecacron | In geometry, the great icosihemidodecacron is the dual of the great icosihemidodecahedron, and is one of nine dual hemipolyhedra. It appears indistinct from the great dodecahemidodecacron.
Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity. In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.
The great icosihemidodecacron can be seen as having six vertices at infinity.
See also
Hemi-dodecahedron - The six vertices at infinity correspond directionally to the six vertices of this abstract polyhedron.
References
(Page 101, Duals of the (nine) hemipolyhedra)
External links
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Presumed%20security | Presumed security is a principle in security engineering that a system is safe from attack due to an attacker assuming, on the basis of probability, that it is secure. Presumed security is the opposite of security through obscurity. A system relying on security through obscurity may have actual security vulnerabilities, but its owners or designers deliberately make the system more complex in the hope that attackers are unable to find a flaw. Conversely a system relying on presumed security makes no attempt to address its security flaws, which may be publicly known, but instead relies upon potential attackers simply assuming that the target is not worth attacking. The reasons for an attacker to make this assumption may range from personal risk (the attacker believes the system owners can easily identify, capture and prosecute them) to technological knowledge (the attacker believes the system owners have sufficient knowledge of security techniques to ensure no flaws exist, rendering an attack moot).
Although this approach to security is implicitly understood by security professionals, it is rarely discussed or documented. The phrase "presumed security" appears to have been first coined by the security commentary website Zero Flaws. The article uses the Royal Military Academy Sandhurst as an example, focusing on the apparent lack of entry security and contrasting it against the presumed security a military installation will have. The article also details the flaws inherent in a trust seal such as the Verisign Secure Site seal, and explains why this presumed security approach is actually detrimental to an overall security posture.
References & notes
Security engineering |
https://en.wikipedia.org/wiki/2009%20CD%20Universidad%20San%20Mart%C3%ADn%20season | The 2009 season is the 6th season of competitive football by Universidad San Martín de Porres.
Statistics
Appearances and goals
Competition Overload
Copa Libertadores 2009
Group stage
Knockout stage
Primera División Peruana 2009
Regular season
Liguilla Final – Group B
Preseason friendlies
Transfers
In
Out
References
External links
Everything about Deportivo Universidad San Martín
Deportivo Universidad San Martín de Porres – season 2009
Deportivo Universidad San Martín de Porres – Copa Libertadores 2009
Club Deportivo Universidad de San Martín de Porres seasons
2009 in Peruvian football |
https://en.wikipedia.org/wiki/Peter%20Aczel | Peter Henry George Aczel (; 31 October 1941 – 1 August 2023) was a British mathematician, logician and Emeritus joint Professor in the Department of Computer Science and the School of Mathematics at the University of Manchester. He is known for his work in non-well-founded set theory, constructive set theory, and Frege structures.
Education
Aczel completed his Bachelor of Arts in Mathematics in 1963 followed by a DPhil at the University of Oxford in 1966 under the supervision of John Crossley.
Career and research
After two years of visiting positions at the University of Wisconsin–Madison and Rutgers University Aczel took a position at the University of Manchester. He has also held visiting positions at the University of Oslo, California Institute of Technology, Utrecht University, Stanford University and Indiana University Bloomington. He was a visiting scholar at the Institute for Advanced Study in 2012.
Aczel was on the editorial board of the Notre Dame Journal of Formal Logic and the Cambridge Tracts in Theoretical Computer Science, having previously served on the editorial boards of the Journal of Symbolic Logic and the Annals of Pure and Applied Logic.
References
External links
1941 births
2023 deaths
British logicians
People associated with the Department of Computer Science, University of Manchester
Institute for Advanced Study visiting scholars
Set theorists
British philosophers
Alumni of the University of Oxford |
https://en.wikipedia.org/wiki/Heinrich%20Guggenheimer | Heinrich Walter Guggenheimer (July 21, 1924 – March 4, 2021) was a German-born Swiss-American mathematician who has contributed to knowledge in differential geometry, topology, algebraic geometry, and convexity. He has also contributed volumes on Jewish sacred literature.
Guggenheimer was born in Nuremberg, Germany. He is the son of Marguerite Bloch and the physicist Dr. Siegfried Guggenheimer. He studied in Zürich, Switzerland at the , receiving his diploma in 1947 and a D.Sc. in 1951. His dissertation was titled "On complex analytic manifolds with Kahler metric". It was published in Commentarii Mathematici Helvetici (in German).
Guggenheimer began his teaching career at the Hebrew University as a lecturer, 1954–56. He was a professor at the Bar Ilan University, 1956–59. In 1959, he immigrated to the United States, becoming a naturalized citizen in 1965. Washington State University was his first American post, where he was an associate professor. After one year he moved to University of Minnesota where he was raised to a full professor in 1962. While in Minnesota, he wrote Differential Geometry (1963), a textbook treating "classical problems with modern methods". According to Robert Hermann in 1979, "Among today's treatises, the best one from the point of view of the Erlangen Program is Differential Geometry by H. Guggenheimer, Dover Publications, 1977."
In 1967 Guggenheimer published Plane Geometry and its Groups (Holden Day), and moved to New York City to teach at Polytechnic University, now called New York University Tandon School of Engineering. In 1977, he published Applicable Geometry: Global and Local Convexity.
Until 1995 Guggenheimer produced a steady stream of papers in mathematical journals. As a supervisor of graduate study in Minnesota and New York, he had six students proceed to Ph.D.s with theses supervised by him, two in Minnesota and four in New York. See the link to the Mathematics Genealogy Project below.
Guggenheimer has also contributed to literature on Judaism. In 1966, he wrote "Logical problems in Jewish tradition". The next year he contributed "Magic and Dialect" to Diogenes where he examines the supposition that "knowledge of the right name gives power over the bearer of that name". In 1995 Guggenheimer presented his A Scholar's Haggadah, which makes a bilingual comparison of variances in the traditions of Passover observance. It includes Ashkenazic, Sephardic, and Oriental sources. His study of the Jerusalem Talmud provided text and commentary.
He died in March 2021 at the age of 96.
Family
On June 6, 1947, Guggenheimer married Eva Auguste Horowitz. Together they wrote Jewish Family Names and their Origins: an Etymological Dictionary (1992). They have two sons, Michael, a professor of Arabic, and Tobias I. S., an architect, and two daughters Dr. Esther Furman, a biochemist, and Hanna Y. Guggenheimer, an artist.
Notes
References
Allen G. Debus, "Heinrich Walter Guggenheimer", Who's Who in Science, 1968.
Hein |
https://en.wikipedia.org/wiki/Compound%20of%20three%20tetrahedra | In geometry, a compound of three tetrahedra can be constructed by three tetrahedra rotated by 60 degree turns along an axis of the middle of an edge. It has dihedral symmetry, D3d, order 12. It is a uniform prismatic compound of antiprisms, UC23.
It is similar to the compound of two tetrahedra with 90 degree turns. It has the same vertex arrangement as the convex hexagonal antiprism.
Related polytopes
A subset of edges of this compound polyhedron can generate a compound regular skew polygon, with 3 skew squares. Each tetrahedron contains one skew square. This regular compound polygon containing the same symmetry as the uniform polyhedral compound.
References
.
External links
Polyhedral compounds |
https://en.wikipedia.org/wiki/Compound%20of%20four%20tetrahedra | In geometry, a compound of four tetrahedra can be constructed by four tetrahedra in a number of different symmetry positions.
Uniform compounds
A uniform compound of four tetrahedra can be constructed by rotating tetrahedra along an axis of symmetry C2 (that is the middle of an edge) in multiples of . It has dihedral symmetry, D8h, and the same vertex arrangement as the convex octagonal prism.
This compound can also be seen as two compounds of stella octangulae fit evenly on the same C2 plane of symmetry, with one pair of tetrahedra shifted . It is a special case of a p/q-gonal prismatic compound of antiprisms, where in this case the component p/q = 2 is a digonal antiprism, or tetrahedron.
Below are two perspective viewpoints of the uniform compound of four tetrahedra, with each color representing one regular tetrahedron:
Four tetrahedra that are not spread equally in angles over C2 can still hold uniform symmetry when allowed rotational freedom. In this case, these tetrahedra share a symmetric arrangement over the common axis of symmetry C2 that is rotated by equal and opposite angles. This compound is indexed as UC22, with parameters p/q = 2 and n = 4 as well.
Other compounds
A nonuniform compound can be generated by rotating tetrahedra about lines extending from the center of each face and through the centroid (as altitudes), with varying degrees of rotation.
A model for this compound polyhedron was first published by Robert Webb, using his program Stella, in 2004, following studies of polyhedron models:
With edge-length as a unit, it has a surface area equal to
.
This compound is self-dual, meaning its dual polyhedron is the same compound polyhedron.
References
(Figure 6.a "Compounds")
See also
Compound of three tetrahedra
Compound of five tetrahedra
Compound of six tetrahedra
External links
Tetrahedron 4-Compound (nonuniform) with adjustable angles at GeoGebra
Polyhedral compounds |
https://en.wikipedia.org/wiki/Bloch%20function | In mathematics, Bloch function may refer to:
Named after Swiss physicist Felix Bloch
a periodic function which appears in the solution of the Schrödinger equation with periodic potential; see Bloch's theorem.
Named after French mathematician André Bloch
an analytic function in the unit disc which is an element of the Bloch space. |
https://en.wikipedia.org/wiki/2008%20CD%20Universidad%20San%20Mart%C3%ADn%20season | The 2008 season was the 5th season of competitive football by Universidad San Martín de Porres.
Statistics
Appearances and goals
Last updated on January, 2008.
Competition Overload
Copa Libertadores 2008
Group stage
Primera División Peruana 2008
Apertura 2008
Clausura 2008
Pre-season friendlies
Transfers
In
Out
References
External links
Everything about Deportivo Universidad San Martín
Deportivo Universidad San Martín de Porres - season 2008
Deportivo Universidad San Martín de Porres - Copa Libertadores 2008
2008
2008 in Peruvian football |
https://en.wikipedia.org/wiki/2007%20CD%20Universidad%20San%20Mart%C3%ADn%20season | The 2007 season was the 4th season of competitive football by Universidad San Martín de Porres.
Statistics
Appearances and goals
Competition Overload
Primera División Peruana 2007
Apertura 2007
Clausura 2007
Mid-season friendlies
Pre-season friendlies
Transfers
In
Out
External links
Everything about Deportivo Universidad San Martín
Deportivo Universidad San Martín de Porres - season 2007
2007
2007 in Peruvian football |
https://en.wikipedia.org/wiki/2006%20CD%20Universidad%20San%20Mart%C3%ADn%20season | The 2007 season was the 3rd season of competitive football by Universidad San Martín de Porres.
Statistics
Appearances and goals
Last updated on January, 2006.
Competition Overload
Copa Sudamericana 2006
Preliminary Chile/Peru
Primera División Peruana 2006
Apertura 2006
Clausura 2006
Pre-season friendlies
Transfers
In
Out
References
External links
Everything about Deportivo Universidad San Martín
Deportivo Universidad San Martín de Porres - Copa Sudamericana 2006
Deportivo Universidad San Martín de Porres - season 2006
2006
2006 in Peruvian football |
https://en.wikipedia.org/wiki/2005%20CD%20Universidad%20San%20Mart%C3%ADn%20season | The 2005 season was the 2nd season of competitive football by Universidad San Martín de Porres.
Statistics
Appearances and goals
Competition Overload
Primera División Peruana 2005
Apertura 2005
Clausura 2005
Pre-season friendlies
Transfers
In
Out
External links
Everything about Deportivo Universidad San Martín
Deportivo Universidad San Martín de Porres – season 2005
2005
2005 in Peruvian football |
https://en.wikipedia.org/wiki/2004%20CD%20Universidad%20San%20Mart%C3%ADn%20season | The 2005 season was the 1st season of competitive football by Universidad San Martín de Porres.
Statistics
Appearances and goals
Competition Overload
Primera División Peruana 2004
Apertura 2004
Clausura 2004
Pre-season friendlies
Transfers
In
Out
External links
Everything about Deportivo Universidad San Martín
Deportivo Universidad San Martín de Porres - season 2004
2004
2004 in Peruvian football |
https://en.wikipedia.org/wiki/Cartesian%20product | In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A×B, is the set of all ordered pairs where a is in A and b is in B. In terms of set-builder notation, that is
A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form .
One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets.
The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product.
Set-theoretic definition
A rigorous definition of the Cartesian product requires a domain to be specified in the set-builder notation. In this case the domain would have to contain the Cartesian product itself. For defining the Cartesian product of the sets and , with the typical Kuratowski's definition of a pair (a,b) as , an appropriate domain is the set where denotes the power set. Then the Cartesian product of the sets and would be defined as
Examples
A deck of cards
An illustrative example is the standard 52-card deck. The standard playing card ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13-element set. The card suits } form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.
returns a set of the form {(A, ♠), (A, ), (A, ), (A, ♣), (K, ♠), …, (3, ♣), (2, ♠), (2, ), (2, ), (2, ♣)}.
returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), …, (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}.
These two sets are distinct, even disjoint, but there is a natural bijection between them, under which (3, ♣) corresponds to (♣, 3) and so on.
A two-dimensional coordinate system
The main historical example is the Cartesian plane in analytic geometry. In order to represent geometrical shapes in a numerical way, and extract numerical information from shapes' numerical representations, René Descartes assigned to each point in the plane a pair of real numbers, called its coordinates. Usually, such a pair's first and second components are called its x and y coordinates, respectively (see picture). The set of all such pairs (i.e., the Cartesian product , with ℝ denoting the real numbers) is thus assigned to the set of all points in the plane.
Most common implementation (set theory)
A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, Kuratowski's definition, is . Under this definition, is an element of , and is a subset of that set, where re |
https://en.wikipedia.org/wiki/Conditional%20probability | In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) is already known to have occurred. This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditional probability with respect to A. If the event of interest is and the event is known or assumed to have occurred, "the conditional probability of given ", or "the probability of under the condition ", is usually written as or occasionally . This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the "given" one happening (how many times A occurs rather than not assuming B has occurred): .
For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person is sick, then they are much more likely to be coughing. For example, the conditional probability that someone unwell (sick) is coughing might be 75%, in which case we would have that = 5% and = 75 %. Although there is a relationship between and in this example, such a relationship or dependence between and is not necessary, nor do they have to occur simultaneously.
may or may not be equal to , i.e., the unconditional probability or absolute probability of . If , then events and are said to be independent: in such a case, knowledge about either event does not alter the likelihood of each other. (the conditional probability of given ) typically differs from . For example, if a person has dengue fever, the person might have a 90% chance of being tested as positive for the disease. In this case, what is being measured is that if event (having dengue) has occurred, the probability of (tested as positive) given that occurred is 90%, simply writing = 90%. Alternatively, if a person is tested as positive for dengue fever, they may have only a 15% chance of actually having this rare disease due to high false positive rates. In this case, the probability of the event (having dengue) given that the event (testing positive) has occurred is 15% or = 15%. It should be apparent now that falsely equating the two probabilities can lead to various errors of reasoning, which is commonly seen through base rate fallacies.
While conditional probabilities can provide extremely useful information, limited information is often supplied or at hand. Therefore, it can be useful to reverse or convert a conditional probability using Bayes' theorem: . Another option is to display conditional probabilities in a conditional probability table to illuminate the relationship between events.
Definition
Conditioning on an event
Kolmogorov definition
Given two events and from the sigma-field of a probability space, with the unconditional probability of being greater than zero |
https://en.wikipedia.org/wiki/Goursat%20tetrahedron | In geometry, a Goursat tetrahedron is a tetrahedral fundamental domain of a Wythoff construction. Each tetrahedral face represents a reflection hyperplane on 3-dimensional surfaces: the 3-sphere, Euclidean 3-space, and hyperbolic 3-space. Coxeter named them after Édouard Goursat who first looked into these domains. It is an extension of the theory of Schwarz triangles for Wythoff constructions on the sphere.
Graphical representation
A Goursat tetrahedron can be represented graphically by a tetrahedral graph, which is in a dual configuration of the fundamental domain tetrahedron. In the graph, each node represents a face (mirror) of the Goursat tetrahedron. Each edge is labeled by a rational value corresponding to the reflection order, being π/dihedral angle.
A 4-node Coxeter-Dynkin diagram represents this tetrahedral graph with order-2 edges hidden. If many edges are order 2, the Coxeter group can be represented by a bracket notation.
Existence requires each of the 3-node subgraphs of this graph, (p q r), (p u s), (q t u), and (r s t), must correspond to a Schwarz triangle.
Extended symmetry
An extended symmetry of the Goursat tetrahedron is a semidirect product of the Coxeter group symmetry and the fundamental domain symmetry (the Goursat tetrahedron in these cases). Coxeter notation supports this symmetry as double-brackets like [Y[X]] means full Coxeter group symmetry [X], with Y as a symmetry of the Goursat tetrahedron. If Y is a pure reflective symmetry, the group will represent another Coxeter group of mirrors. If there is only one simple doubling symmetry, Y can be implicit like [[X]] with either reflectional or rotational symmetry depending on the context.
The extended symmetry of each Goursat tetrahedron is also given below. The highest possible symmetry is that of the regular tetrahedron as [3,3], and this occurs in the prismatic point group [2,2,2] or [2[3,3]] and the paracompact hyperbolic group [3[3,3]].
See Tetrahedron#Isometries of irregular tetrahedra for 7 lower symmetry isometries of the tetrahedron.
Whole number solutions
The following sections show all of the whole number Goursat tetrahedral solutions on the 3-sphere, Euclidean 3-space, and Hyperbolic 3-space. The extended symmetry of each tetrahedron is also given.
The colored tetrahedal diagrams below are vertex figures for omnitruncated polytopes and honeycombs from each symmetry family. The edge labels represent polygonal face orders, which is double the Coxeter graph branch order. The dihedral angle of an edge labeled 2n is π/n. Yellow edges labeled 4 come from right angle (unconnected) mirror nodes in the Coxeter diagram.
3-sphere (finite) solutions
The solutions for the 3-sphere with density 1 solutions are: (Uniform polychora)
Euclidean (affine) 3-space solutions
Density 1 solutions: Convex uniform honeycombs:
Compact hyperbolic 3-space solutions
Density 1 solutions: (Convex uniform honeycombs in hyperbolic space) (Coxeter diagram#Compact (Lannér simplex |
https://en.wikipedia.org/wiki/Popoviciu%27s%20inequality%20on%20variances | In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance σ2 of any bounded probability distribution. Let M and m be upper and lower bounds on the values of any random variable with a particular probability distribution. Then Popoviciu's inequality states:
This equality holds precisely when half of the probability is concentrated at each of the two bounds.
Sharma et al. have sharpened Popoviciu's inequality:
If one additionally assumes knowledge of the expectation, then the stronger Bhatia–Davis inequality holds
where μ is the expectation of the random variable.
In the case of an independent sample of n observations from a bounded probability distribution, the von Szokefalvi Nagy inequality gives a lower bound to the variance of the sample mean:
Proof via the Bhatia–Davis inequality
Let be a random variable with mean , variance , and . Then, since ,
.
Thus,
.
Now, applying the Inequality of arithmetic and geometric means, , with and , yields the desired result:
.
References
Theory of probability distributions
Statistical inequalities
Statistical deviation and dispersion |
https://en.wikipedia.org/wiki/FanGraphs | FanGraphs.com is a website run by Fangraphs Inc., located in Arlington, Virginia, and created and owned by David Appelman that provides statistics for every player in Major League Baseball history.
On September 18, 2009, Fangraphs Inc. launched an iPhone app in partnership with Hawk Ridge Consulting, which has since been discontinued. Fangraphs has a number of content partners including ESPN, SB Nation and Fanhouse.
FanGraphs products
FanGraphs creates several products:
Web sites
The FanGraphs homepage, which contains articles, statistical reports and also covers baseball history as well as current issues and events, including games and series, injuries, forecasts, player profiles, baseball finance, and the player marketplace.
RotoGraphs is FanGraphs' fantasy baseball advice and analysis section. It originally featured David Golebiewski of ESPN Inside Edge, Marc Hulet, Brian Joura of newyorkmetsdaily.com, and Peter Bendix. Currently managed by Eno Sarris.
NotGraphs provides "a place to put things that would otherwise not have a place on FanGraphs". NotGraphs was managed by Carson Cistulli but ceased publishing new content at the conclusion of the 2014 season.
The Community Blog is an opportunity for readers to share their writing through FanGraphs. In May 2010 an editorial staff was put in place to guarantee an "approval process" within 48 hours of submission.
The FanGraphs Library is an encyclopedia of Sabermetric statistics and principles run by Neil Weinberg.
Publishing
FanGraphs releases an annual book The FanGraphs Second Opinion: Fantasy Companion that contains statistics and analysis of the past season, in-depth player profiles, team previews, articles on fantasy strategy and forecasts of the upcoming season.
Podcasts
Hosted by FanGraphs editor Meg Rowley, FanGraphs Audio is a weekly program archived at FanGraphs Audio.
Hosted by Ben Lindbergh and Meg Rowley, Effectively Wild is a thrice weekly podcast. Effectively Wild was initially created as a part of Baseball Prospectus, but moved over to FanGraphs at the beginning of 2017.
Fantasy Baseball
FanGraphs works with Ottoneu baseball to offer a fantasy baseball program with prizes.
Regular writers
FanGraphs
Carson Cistulli – Senior writer of FanGraphs, former editor of NotGraphs and host of the FanGraphs Audio podcast. Joined in August, 2009. Cistulli is responsible for the Daily Notes and writes the "Fringe Five" prospect feature. He previously worked for The Hardball Times.
Paul Swydan – Senior writer of FanGraphs. Joined in January, 2011. Swydan is the co-managing editor of The Hardball Times and is a writer for ESPN Insider. He previously worked 7 years for the Colorado Rockies and wrote for Baseball Prospectus, MLB.com, Rockies Magazine and the Biz Of Baseball.
Eric Longenhagen – FanGraphs lead prospect analyst. Contributed in 2014 and 2015, but re-joined in May, 2016. Longenhagen previously wrote for Crashburn Alley, Sports on Earth, Prospect Insider and ESPN.
Ch |
https://en.wikipedia.org/wiki/P-box | P-box may refer to:
permutation box
probability box
privacy box, used by the Winston Smith Project#P-Box project
P. Box (band) |
https://en.wikipedia.org/wiki/Unit%20doublet | In mathematics, the unit doublet is the derivative of the Dirac delta function. It can be used to differentiate signals in electrical engineering: If u1 is the unit doublet, then
where is the convolution operator.
The function is zero for all values except zero, where its behaviour is interesting. Its integral over any interval enclosing zero is zero. However, the integral of its absolute value over any region enclosing zero goes to infinity. The function can be thought of as the limiting case of two rectangles, one in the second quadrant, and the other in the fourth. The length of each rectangle is k, whereas their breadth is 1/k2, where k tends to zero.
References
Generalized functions |
https://en.wikipedia.org/wiki/D%C3%B6rarp | Dörarp is a small locality (according to the definition of Statistics Sweden) in Ljungby Municipality, Sweden. In 2005, Dörarp had 145 inhabitants.
Dörarp is also the site of heavy metal band Metallica's tour bus accident during the Damage, Inc. Tour on September 27, 1986. Vocalist James Hetfield, guitarist Kirk Hammett and drummer Lars Ulrich survived with minor injuries, but bassist Cliff Burton was pinned under the bus and pronounced dead. Former Flotsam and Jetsam bassist Jason Newsted was recruited as Metallica's new bassist in 1986 until his departure in 2001. The site of the crash has been marked with Burton's commemorative memorial stone.
References
Populated places in Kronoberg County
Finnveden |
https://en.wikipedia.org/wiki/Dual%20identity | Dual identity can refer to:
A secret identity, such as Clark Kent and Superman
In mathematics, the coidentity of a dual group object or the counit of a coalgebra
In sociology, double consciousness |
https://en.wikipedia.org/wiki/Smoothness%20%28probability%20theory%29 | In probability theory and statistics, smoothness of a density function is a measure which determines how many times the density function can be differentiated, or equivalently the limiting behavior of distribution’s characteristic function.
Formally, we call the distribution of a random variable X ordinary smooth of order β if its characteristic function satisfies
for some positive constants d0, d1, β. The examples of such distributions are gamma, exponential, uniform, etc.
The distribution is called supersmooth of order β if its characteristic function satisfies
for some positive constants d0, d1, β, γ and constants β0, β1. Such supersmooth distributions have derivatives of all orders. Examples: normal, Cauchy, mixture normal.
References
Theory of probability distributions |
https://en.wikipedia.org/wiki/Eutactic%20star | In Euclidean geometry, a eutactic star is a geometrical figure in a Euclidean space. A star is a figure consisting of any number of opposing pairs of vectors (or arms) issuing from a central origin. A star is eutactic if it is the orthogonal projection of plus and minus the set of standard basis vectors (i.e., the vertices of a cross-polytope) from a higher-dimensional space onto a subspace. Such stars were called "eutactic" – meaning "well-situated" or "well-arranged" – by because, for a common scalar multiple, their vectors are projections of an orthonormal basis.
Definition
A star is here defined as a set of 2s vectors A = ±a1, ..., ±as issuing from a particular origin in a Euclidean space of dimension n ≤ s. A star is eutactic if the ai are the projections onto n dimensions of a set of mutually perpendicular equal vectors b1, ..., bs issuing from a particular origin in Euclidean s-dimensional space. The configuration of 2s vectors in the s-dimensional space B = ±b1, ... , ±bs is known as a cross. Given these definitions, a eutactic star is, concisely, a star produced by the orthogonal projection of a cross.
An equivalent definition, first mentioned by Schläfli, stipulates that a star is eutactic if a constant ζ exists such that
for every vector v. The existence of such a constant requires that the sum of the squares of the orthogonal projections of A on a line be equal in all directions. In general,
A normalised eutactic star is a projected cross composed of unit vectors. Eutactic stars are often considered in n = 3 dimensions because of their connection with the study of regular polyhedra.
Hadwiger's principal theorem
Let T be the symmetric linear transformation defined for vectors x by
where the aj form any collection of s vectors in the n-dimensional Euclidean space. Hadwiger's principal theorem states that the vectors ±a1, ..., ±as form a eutactic star if and only if there is a constant ζ such that Tx = ζx for every x. The vectors form a normalized eutactic star precisely when T is the identity operator – when ζ = 1.
Equivalently, the star is normalized eutactic if and only if the matrix A = [a1 ... as], whose columns are the vectors ak, has orthonormal rows. A proof may be given in one direction by completing the rows of this matrix to an orthonormal basis of , and in the other by orthogonally projecting onto the n-dimensional subspace spanned by the first n Cartesian coordinate vectors.
Hadwiger's theorem implies the equivalence of Schläfli's stipulation and the geometrical definition of a eutactic star, by the polarization identity. Furthermore, both Schläfli's identity and Hadwiger's theorem give the same value of the constant ζ.
Applications
Eutactic stars are useful largely because of their relationship with the geometry of polytopes and groups of orthogonal transformations. Schläfli showed early on that the vectors from the center of any regular polytope to its vertices form a eutactic star. Brauer and C |
https://en.wikipedia.org/wiki/Poincar%C3%A9%20series%20%28modular%20form%29 | In number theory, a Poincaré series is a mathematical series generalizing the classical theta series that is associated to any discrete group of symmetries of a complex domain, possibly of several complex variables. In particular, they generalize classical Eisenstein series. They are named after Henri Poincaré.
If Γ is a finite group acting on a domain D and H(z) is any meromorphic function on D, then one obtains an automorphic function by averaging over Γ:
However, if Γ is a discrete group, then additional factors must be introduced in order to assure convergence of such a series. To this end, a Poincaré series is a series of the form
where Jγ is the Jacobian determinant of the group element γ, and the asterisk denotes that the summation takes place only over coset representatives yielding distinct terms in the series.
The classical Poincaré series of weight 2k of a Fuchsian group Γ is defined by the series
the summation extending over congruence classes of fractional linear transformations
belonging to Γ. Choosing H to be a character of the cyclic group of order n, one obtains the so-called Poincaré series of order n:
The latter Poincaré series converges absolutely and uniformly on compact sets (in the upper halfplane), and is a modular form of weight 2k for Γ. Note that, when Γ is the full modular group and n = 0, one obtains the Eisenstein series of weight 2k. In general, the Poincaré series is, for n ≥ 1, a cusp form.
Notes
References
.
.
Automorphic forms
Modular forms
Mathematical series |
https://en.wikipedia.org/wiki/Jucys%E2%80%93Murphy%20element | In mathematics, the Jucys–Murphy elements in the group algebra of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula:
They play an important role in the representation theory of the symmetric group.
Properties
They generate a commutative subalgebra of . Moreover, Xn commutes with all elements of .
The vectors constituting the basis of Young's "seminormal representation" are eigenvectors for the action of Xn. For any standard Young tableau U we have:
where ck(U) is the content b − a of the cell (a, b) occupied by k in the standard Young tableau U.
Theorem (Jucys): The center of the group algebra of the symmetric group is generated by the symmetric polynomials in the elements Xk.
Theorem (Jucys): Let t be a formal variable commuting with everything, then the following identity for polynomials in variable t with values in the group algebra holds true:
Theorem (Okounkov–Vershik): The subalgebra of generated by the centers
is exactly the subalgebra generated by the Jucys–Murphy elements Xk.
See also
Representation theory of the symmetric group
Young symmetrizer
References
Permutation groups
Representation theory
Symmetry
Representation theory of finite groups
Symmetric functions |
https://en.wikipedia.org/wiki/Tadao%20Tannaka | was a Japanese mathematician who worked in algebraic number theory.
Biography
Tannaka was born in Matsuyama, Ehime Prefecture on December 27, 1908. After receiving a Bachelor of Science in mathematics from Tohoku Imperial University in 1932, he was appointed a lecturer in the university in 1934 and received a Doctor of Science degree from the university in 1941. He was promoted to assistant professor in 1942 and full professor in 1945. Tannaka was a member at the Institute for Advanced Study from September 1955 to April 1957. Tannaka retired from Tohoku University in 1972, after which he served as a full professor at Tohoku Gakuin University until 1981.
Tannaka was an editor of the Tohoku Mathematical Journal and a member of the board of directors of the Mathematical Society of Japan. Tannaka was also in charge of the "Mathematics Chat" article series in the monthly magazine from 1960 onwards.
Tannaka died in Tokyo on October 25, 1986.
Research
Tannaka is known for developing the theory of Tannaka–Krein duality, which generalizes Pontryagin duality to noncommutative compact groups and led to the development of Tannakian formalism.
Awards
Tannaka was a recipient of the Order of the Rising Sun (3rd Class) in 1980.
See also
Tannaka–Artin problem
References
20th-century Japanese mathematicians
1908 births
1986 deaths
Academic staff of Tohoku University
Tohoku University alumni
Institute for Advanced Study visiting scholars
Academic staff of Tohoku Gakuin University
People from Matsuyama, Ehime |
https://en.wikipedia.org/wiki/Boole%27s%20rule | In mathematics, Boole's rule, named after George Boole, is a method of numerical integration.
Formula
Simple Boole's Rule
It approximates an integral:
by using the values of at five equally spaced points:
It is expressed thus in Abramowitz and Stegun:
where the error term is
for some number between and where .
It is often known as Bode's rule, due to a typographical error that propagated from Abramowitz and Stegun.
The following constitutes a very simple implementation of the method in Common Lisp which ignores the error term:
(defun integrate-booles-rule (f x1 x5)
"Calculates the Boole's rule numerical integral of the function F in
the closed interval extending from inclusive X1 to inclusive X5
without error term inclusion."
(declare (type (function (real) real) f))
(declare (type real x1 x5))
(let ((h (/ (- x5 x1) 4)))
(declare (type real h))
(let* ((x2 (+ x1 h))
(x3 (+ x2 h))
(x4 (+ x3 h)))
(declare (type real x2 x3 x4))
(* (/ (* 2 h) 45)
(+ (* 7 (funcall f x1))
(* 32 (funcall f x2))
(* 12 (funcall f x3))
(* 32 (funcall f x4))
(* 7 (funcall f x5)))))))
Composite Boole's Rule
In cases where the integration is permitted to extend over equidistant sections of the interval , the composite Boole's rule might be applied. Given divisions, the integrated value amounts to:
where the error term is similar to above. The following Common Lisp code implements the aforementioned formula:
(defun integrate-composite-booles-rule (f a b n)
"Calculates the composite Boole's rule numerical integral of the
function F in the closed interval extending from inclusive A to
inclusive B across N subintervals."
(declare (type (function (real) real) f))
(declare (type real a b))
(declare (type (integer 1 *) n))
(let ((h (/ (- b a) n)))
(declare (type real h))
(flet ((f[i] (i)
(declare (type (integer 0 *) i))
(let ((xi (+ a (* i h))))
(declare (type real xi))
(the real (funcall f xi)))))
(* (/ (* 2 h) 45)
(+ (* 7 (+ (f[i] 0) (f[i] n)))
(* 32 (loop for i from 1 to (- n 1) by 2 sum (f[i] i)))
(* 12 (loop for i from 2 to (- n 2) by 4 sum (f[i] i)))
(* 14 (loop for i from 4 to (- n 4) by 4 sum (f[i] i))))))))
See also
Newton–Cotes formulas
Simpson's rule
Romberg's method
Notes
References
Integral calculus
Numerical analysis
Numerical integration (quadrature)
Articles with example Lisp (programming language) code |
https://en.wikipedia.org/wiki/Goldman%20domain | In mathematics, a Goldman domain or G-domain is an integral domain A whose field of fractions is a finitely generated algebra over A. They are named after Oscar Goldman.
An overring (i.e., an intermediate ring lying between the ring and its field of fractions) of a Goldman domain is again a Goldman domain. There exists a Goldman domain where all nonzero prime ideals are maximal although there are infinitely many prime ideals.
An ideal I in a commutative ring A is called a Goldman ideal if the quotient A/I is a Goldman domain. A Goldman ideal is thus prime, but not necessarily maximal. In fact, a commutative ring is a Jacobson ring if and only if every Goldman ideal in it is maximal.
The notion of a Goldman ideal can be used to give a slightly sharpened characterization of a radical of an ideal: the radical of an ideal I is the intersection of all Goldman ideals containing I.
Alternative definition
An integral domain is a G-domain if and only if:
Its field of fractions is a simple extension of
The intersection of its nonzero prime ideals (not to be confused with nilradical) is nonzero
There is a nonzero element such that for any nonzero ideal , for some .
A G-ideal is defined as an ideal such that is a G-domain. Since a factor ring is an integral domain if and only if the ring is factored by a prime ideal, every G-ideal is also a prime ideal. G-ideals can be used as a refined collection of prime ideals in the following sense: the radical of an ideal can be characterized as the intersection of all prime ideals containing the ideal, and in fact we still get the radical even if we take the intersection over the G-ideals.
Every maximal ideal is a G-ideal, since quotient by maximal ideal is a field, and a field is trivially a G-domain. Therefore, maximal ideals are G-ideals, and G-ideals are prime ideals. G-ideals are the only maximal ideals in a Jacobson ring, and in fact this is an equivalent characterization of Jacobson rings: a ring is a Jacobson ring when all G-ideals are maximal ideals. This leads to a simplified proof of the Nullstellensatz.
It is known that given , a ring extension of a G-domain, is algebraic over if and only if every ring extension between and is a G-domain.
A Noetherian domain is a G-domain if and only if its Krull dimension is at most one, and has only finitely many maximal ideals (or equivalently, prime ideals).
Notes
References
Ring theory |
https://en.wikipedia.org/wiki/Overring | In mathematics, an overring of an integral domain contains the integral domain, and the integral domain's field of fractions contains the overring. Overrings provide an improved understanding of different types of rings and domains.
Definition
In this article, all rings are commutative rings, and ring and overring share the same identity element.
Let represent the field of fractions of an integral domain . Ring is an overring of integral domain if is a subring of and is a subring of the field of fractions ; the relationship is .
Properties
Ring of fractions
The rings are the rings of fractions of rings by multiplicative set . Assume is an overring of and is a multiplicative set in . The ring is an overring of . The ring is the total ring of fractions of if every nonunit element of is a zero-divisor. Every overring of contained in is a ring , and is an overring of . Ring is integrally closed in if is integrally closed in .
Noetherian domain
Definitions
A Noetherian ring satisfies the 3 equivalent finitenss conditions i) every ascending chain of ideals is finite, ii) every non-empty family of ideals has a maximal element and iii) every ideal has a finite basis.
An integral domain is a Dedekind domain if every ideal of the domain is a finite product of prime ideals.
A ring's restricted dimension is the maximum rank among the ranks of all prime ideals that contain a regular element.
A ring is locally nilpotentfree if every ring with maximal ideal is free of nilpotent elements or a ring with every nonunit a zero divisor.
An affine ring is the homomorphic image of a polynomial ring over a field.
Properties
Every overring of a Dedekind ring is a Dedekind ring.
Every overrring of a direct sum of rings whose non-unit elements are all zero-divisors is a Noetherian ring.
Every overring of a Krull 1-dimensional Noetherian domain is a Noetherian ring.
These statements are equivalent for Noetherian ring with integral closure .
Every overring of is a Noetherian ring.
For each maximal ideal of , every overring of is a Noetherian ring.
Ring is locally nilpotentfree with restricted dimension 1 or less.
Ring is Noetherian, and ring has restricted dimension 1 or less.
Every overring of is integrally closed.
These statements are equivalent for affine ring with integral closure .
Ring is locally nilpotentfree.
Ring is a finite module.
Ring is Noetherian.
An integrally closed local ring is an integral domain or a ring whose non-unit elements are all zero-divisors.
A Noetherian integral domain is a Dedekind ring if every overring of the Noetherian ring is integrally closed.
Every overring of a Noetherian integral domain is a ring of fractions if the Noetherian integral domain is a Dedekind ring with a torsion class group.
Coherent rings
Definitions
A coherent ring is a commutative ring with each finitely generated ideal finitely presented. Noetherian domains and Prüfer domains are cohere |
https://en.wikipedia.org/wiki/Regular%20ideal | In mathematics, especially ring theory, a regular ideal can refer to multiple concepts.
In operator theory, a right ideal in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that for every .
In commutative algebra a regular ideal refers to an ideal containing a non-zero divisor. This article will use "regular element ideal" to help distinguish this type of ideal.
A two-sided ideal of a ring R can also be called a (von Neumann) regular ideal if for each element x of there exists a y in such that xyx=x.
Finally, regular ideal has been used to refer to an ideal J of a ring R such that the quotient ring R/J is von Neumann regular ring. This article will use "quotient von Neumann regular" to refer to this type of regular ideal.
Since the adjective regular has been overloaded, this article adopts the alternative adjectives modular, regular element, von Neumann regular, and quotient von Neumann regular to distinguish between concepts.
Properties and examples
Modular ideals
The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring to non-unital settings.
A two-sided ideal is modular if and only if is unital. In a unital ring, every ideal is modular since choosing e=1 works for any right ideal. So, the notion is more interesting for non-unital rings such as Banach algebras. From the definition it is easy to see that an ideal containing a modular ideal is itself modular.
Somewhat surprisingly, it is possible to prove that even in rings without identity, a modular right ideal is contained in a maximal right ideal. However, it is possible for a ring without identity to lack modular right ideals entirely.
The intersection of all maximal right ideals which are modular is the Jacobson radical.
Examples
In the non-unital ring of even integers, (6) is regular () while (4) is not.
Let M be a simple right A-module. If x is a nonzero element in M, then the annihilator of x is a regular maximal right ideal in A.
If A is a ring without maximal right ideals, then A cannot have even a single modular right ideal.
Regular element ideals
Every ring with unity has at least one regular element ideal: the trivial ideal R itself. Regular element ideals of commutative rings are essential ideals. In a semiprime right Goldie ring, the converse holds: essential ideals are all regular element ideals.
Since the product of two regular elements (=non-zerodivisors) of a commutative ring R is again a regular element, it is apparent that the product of two regular element ideals is again a regular element ideal. Clearly any ideal containing a regular element ideal is again a regular element ideal.
Examples
In an integral domain, every nonzero element is a regular element, and so every nonzero ideal is a regular element ideal.
The nilradical of a commutative ring is composed entirely of nilpotent elements, and therefore no element can be regular. This giv |
https://en.wikipedia.org/wiki/Small%20stellapentakis%20dodecahedron | In geometry, the small stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great dodecahedron. It has 60 intersecting triangular faces.
Proportions
The triangles have two acute angles of and one obtuse angle of . The dihedral angle equals . Part of each triangle lies within the solid, hence is invisible in solid models.
References
External links
Uniform polyhedra and duals
Nonconvex polyhedra
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Medial%20deltoidal%20hexecontahedron | In geometry, the medial deltoidal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the rhombidodecadodecahedron. Its 60 intersecting quadrilateral faces are kites.
Proportions
The kites have two angles of , one of and one of . The dihedral angle equals . The ratio between the lengths of the long and short edges is . Part of each kite lies inside the solid, hence is invisible in solid models.
References
External links
Uniform polyhedra and duals
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Medial%20pentagonal%20hexecontahedron | In geometry, the medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.
Proportions
Denote the golden ratio by , and let be the smallest (most negative) real zero of the polynomial . Then each face has three equal angles of , one of and one of . Each face has one medium length edge, two short and two long ones. If the medium length is , then the short edges have length
,
and the long edges have length
.
The dihedral angle equals . The other real zero of the polynomial plays a similar role for the medial inverted pentagonal hexecontahedron.
References
External links
Uniform polyhedra and duals
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Great%20stellapentakis%20dodecahedron | In geometry, the great stellapentakis dodecahedron (or great astropentakis dodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces.
Proportions
The triangles have one angle of and two of . The dihedral angle equals . Part of each triangle lies within the solid, hence is invisible in solid models.
References
External links
Uniform polyhedra and duals
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Great%20deltoidal%20hexecontahedron | In geometry, the great deltoidal hexecontahedron (or great sagittal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the nonconvex great rhombicosidodecahedron. It is visually identical to the great rhombidodecacron. It has 60 intersecting cross quadrilateral faces, 120 edges, and 62 vertices. Its faces are darts. Part of each dart lies inside the solid, hence is invisible in solid models.
It is also called a great strombic hexecontahedron.
Proportions
The darts have two angles of , one of and one of . The dihedral angle equals . The ratio between the lengths of the long and short edges is .
References
External links
Uniform polyhedra and duals
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Noncommutative%20measure%20and%20integration | Noncommutative measure and integration refers to the theory of weights, states, and traces on von Neumann algebras (Takesaki 1979 v. 2 p. 141).
References
I. E. Segal. A noncommutative extension of abstract integration. Ann. of Math. (2), 57:401–457, 1953. MR # 14:991f, JSTOR collection. 2.0(2)
.
Operator algebras
Noncommutative geometry |
https://en.wikipedia.org/wiki/2007%20Tajik%20League | Tajik League is the top division of the Tajikistan Football Federation, it was created in 1992. These are the statistics of the Tajik League in the 2007 season.
Table
Top scorers
References
Season at RSSSF
Tajikistan Higher League seasons
1
Tajik
Tajik |
https://en.wikipedia.org/wiki/Lennox%20Mathematics%2C%20Science%20%26%20Technology%20Academy | Lennox Mathematics, Science & Technology Academy (LMSTA) is a charter high school located in Lennox, California, USA. It specialises in mathematics, science and technology for ninth to twelfth grade pupils. In its 2009 rankings, U.S. News & World Report ranked it 21st out of 21,000 US High Schools. The school has continued to perform highly in subsequent editions of the rankings, scoring 25th and making the Gold Medal List in the most recent version of the report.
References
External links
Lennox Mathematics, Science & Technology Academy
High schools in Los Angeles County, California
Charter high schools in California |
https://en.wikipedia.org/wiki/Lukacs%27s%20proportion-sum%20independence%20theorem | In statistics, Lukacs's proportion-sum independence theorem is a result that is used when studying proportions, in particular the Dirichlet distribution. It is named after Eugene Lukacs.
The theorem
If Y1 and Y2 are non-degenerate, independent random variables, then the random variables
are independently distributed if and only if both Y1 and Y2 have gamma distributions with the same scale parameter.
Corollary
Suppose Y i, i = 1, ..., k be non-degenerate, independent, positive random variables. Then each of k − 1 random variables
is independent of
if and only if all the Y i have gamma distributions with the same scale parameter.
References
page 64. Lukacs's proportion-sum independence theorem and the corollary with a proof.
Probability theorems
Characterization of probability distributions |
https://en.wikipedia.org/wiki/Terry%20Ryan%20%28ice%20hockey%2C%20born%201952%29 | Terry Ryan (born September 10, 1952) is a Canadian former professional ice hockey centre who played 76 games in the World Hockey Association for the Minnesota Fighting Saints.
Career statistics
External links
1952 births
Living people
People from Grand Falls-Windsor
Hamilton Red Wings (OHA) players
Ice hockey people from Newfoundland and Labrador
Kalamazoo Wings (1974–2000) players
Minnesota Fighting Saints players
Minnesota North Stars draft picks
Muskegon Mohawks players
Suncoast Suns (SHL) players
Winston-Salem Polar Twins (SHL) players
Canadian ice hockey centres |
https://en.wikipedia.org/wiki/Polyhedral%20group | In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids.
Groups
There are three polyhedral groups:
The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron. It is isomorphic to A4.
The conjugacy classes of T are:
identity
4 × rotation by 120°, order 3, cw
4 × rotation by 120°, order 3, ccw
3 × rotation by 180°, order 2
The octahedral group of order 24, rotational symmetry group of the cube and the regular octahedron. It is isomorphic to S4.
The conjugacy classes of O are:
identity
6 × rotation by ±90° around vertices, order 4
8 × rotation by ±120° around triangle centers, order 3
3 × rotation by 180° around vertices, order 2
6 × rotation by 180° around midpoints of edges, order 2
The icosahedral group of order 60, rotational symmetry group of the regular dodecahedron and the regular icosahedron. It is isomorphic to A5.
The conjugacy classes of I are:
identity
12 × rotation by ±72°, order 5
12 × rotation by ±144°, order 5
20 × rotation by ±120°, order 3
15 × rotation by 180°, order 2
These symmetries double to 24, 48, 120 respectively for the full reflectional groups. The reflection symmetries have 6, 9, and 15 mirrors respectively. The octahedral symmetry, [4,3] can be seen as the union of 6 tetrahedral symmetry [3,3] mirrors, and 3 mirrors of dihedral symmetry Dih2, [2,2]. Pyritohedral symmetry is another doubling of tetrahedral symmetry.
The conjugacy classes of full tetrahedral symmetry, Td≅S4, are:
identity
8 × rotation by 120°
3 × rotation by 180°
6 × reflection in a plane through two rotation axes
6 × rotoreflection by 90°
The conjugacy classes of pyritohedral symmetry, Th, include those of T, with the two classes of 4 combined, and each with inversion:
identity
8 × rotation by 120°
3 × rotation by 180°
inversion
8 × rotoreflection by 60°
3 × reflection in a plane
The conjugacy classes of the full octahedral group, Oh≅S4 × C2, are:
inversion
6 × rotoreflection by 90°
8 × rotoreflection by 60°
3 × reflection in a plane perpendicular to a 4-fold axis
6 × reflection in a plane perpendicular to a 2-fold axis
The conjugacy classes of full icosahedral symmetry, Ih≅A5 × C2, include also each with inversion:
inversion
12 × rotoreflection by 108°, order 10
12 × rotoreflection by 36°, order 10
20 × rotoreflection by 60°, order 6
15 × reflection, order 2
Chiral polyhedral groups
Full polyhedral groups
See also
Wythoff symbol
List of spherical symmetry groups
References
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. (The Polyhedral Groups. §3.5, pp. 46–47)
External links
Polyhedra |
https://en.wikipedia.org/wiki/Market-implied%20rating | A market-implied rating estimates the market observed default probability of an individual, corporation, or even a country. Indeed, a credit rating is simply a probability of default. The methodology used by Moodys consists in a median piecewise fit of the ratings to the credit defaut swap data observed on the market. S&P however uses a log regression between the log cds and the ratings equivalent number, adjusted to firm specifics, continent, and outlook.
See also
Alternative data
Credit risk
Default (finance)
Credit history
Credit score
Risk-based pricing
References
Credit |
https://en.wikipedia.org/wiki/H%C3%A1jek%E2%80%93Le%20Cam%20convolution%20theorem | In statistics, the Hájek–Le Cam convolution theorem states that any regular estimator in a parametric model is asymptotically equivalent to a sum of two independent random variables, one of which is normal with asymptotic variance equal to the inverse of Fisher information, and the other having arbitrary distribution.
The obvious corollary from this theorem is that the “best” among regular estimators are those with the second component identically equal to zero. Such estimators are called efficient and are known to always exist for regular parametric models.
The theorem is named after Jaroslav Hájek and Lucien Le Cam.
Statement
Let ℘ = {Pθ | θ ∈ Θ ⊂ ℝk} be a regular parametric model, and q(θ): Θ → ℝm be a parameter in this model (typically a parameter is just one of the components of vector θ). Assume that function q is differentiable on Θ, with the m × k matrix of derivatives denoted as q̇θ. Define
— the information bound for q,
— the efficient influence function for q,
where I(θ) is the Fisher information matrix for model ℘, is the score function, and ′ denotes matrix transpose.
Theorem . Suppose Tn is a uniformly (locally) regular estimator of the parameter q. Then
<li> There exist independent random m-vectors and Δθ such that
where d denotes convergence in distribution. More specifically,
<li> If the map θ → q̇θ is continuous, then the convergence in (A) holds uniformly on compact subsets of Θ. Moreover, in that case Δθ = 0 for all θ if and only if Tn is uniformly (locally) asymptotically linear with influence function ψq(θ)
References
Theorems in statistics |
https://en.wikipedia.org/wiki/Trygve%20Nagell | Trygve Nagell or Trygve Nagel (July 13, 1895 in Oslo – January 24, 1988 in Uppsala) was a Norwegian mathematician, known for his works on Diophantine equations in number theory.
Education and career
He was born Nagel and adopted the spelling Nagell later in life.
He received his doctorate at the University of Oslo in 1926, where his advisor was Axel Thue. He continued to lecture at the University until 1931. He was a professor at the University of Uppsala from 1931 to 1962. His doctoral students include Harald Bergström.
Contributions
Nagell proved a conjecture of Srinivasa Ramanujan that there are only five numbers that are both triangular numbers and Mersenne numbers. They are the numbers 0, 1, 3, 15, and 4095. The formula expressing the equality of a triangular number and a Mersenne number can be simplified to the equivalent form
which likewise has five solutions in natural numbers and , with solutions for .
In honor of Nagell's solution, this equation is called the Ramanujan–Nagell equation.
The Nagell–Lutz theorem is a result in the Diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers. It was published independently by Nagell and by Élisabeth Lutz.
In 1952, Nagell independently formulated the torsion conjecture for elliptic curves over the rationals after it was originally formulated by Beppo Levi in 1908.
Awards and honors
Nagell was appointed Commander of the Royal Norwegian Order of St. Olav in 1951, and of the Swedish Order of the Polar Star in 1952.
References
1895 births
1988 deaths
Scientists from Oslo
Norwegian mathematicians
Commanders of the Order of the Polar Star
University of Oslo alumni
Academic staff of the University of Oslo
Number theorists
Royal Norwegian Society of Sciences and Letters |
https://en.wikipedia.org/wiki/Dick%20Paradise | Dick Paradise (April 21, 1945) is a retired American ice hockey player who played 144 games in the World Hockey Association for the Minnesota Fighting Saints.
Career statistics
Awards and honors
References
External links
1945 births
Living people
American men's ice hockey defensemen
Buffalo Bisons (AHL) players
Ice hockey people from Saint Paul, Minnesota
Johnstown Jets players
Minnesota Fighting Saints players
Omaha Knights (CHL) players
Seattle Totems (WHL) players
Tidewater Wings players |
https://en.wikipedia.org/wiki/Robert%20Bryant%20%28mathematician%29 | Robert Leamon Bryant (born August 30, 1953, Kipling) is an American mathematician. He works at Duke University and specializes in differential geometry.
Education and career
Bryant grew up in a farming family in Harnett County and was a first-generation college student. He obtained a bachelor's degree at North Caroline State University at Raleigh in 1974 and a PhD at University of North Carolina at Chapel Hill in 1979. His thesis was entitled "Some Aspects of the Local and Global Theory of Pfaffian Systems" and was written under the supervision of Robert Gardner.
He worked at Rice University for seven years, as assistant professor (1979–1981), associate professor (1981–1982) and full professor (1982–1986). He then moved to Duke University, where he worked for twenty years as J. M. Kreps Professor.
Between 2007 and 2013 he worked as full professor at University of California, Berkeley, where he served as the director of the Mathematical Sciences Research Institute (MSRI). In 2013 he returned to Duke University as Phillip Griffiths Professor of Mathematics.
Bryant was awarded in 1982 a Sloan Research Fellowship. In 1986 he was invited speaker at the International Congress of Mathematicians in Berkeley.
He was elected in 2002 a fellow of the American Academy of Arts and Sciences, in 2007 a member of the National Academy of Sciences, in 2013 a fellow of the American Mathematical Society and in 2022 a fellow of the American Association for the Advancement of Science. He is also a member of the Association for Women in Mathematics, the National Association of Mathematicians and the Mathematical Association of America.
He served as the president of the American Mathematical Society for the 2-years term 2015–2016, for which he was the first openly gay president.
Bryant is on the board of directors of EDGE, a transition program for women entering graduate studies in the mathematical sciences. He is also a board member of Spectra, an association for LGBT mathematicians that he helped to create.
Research
Bryant's research has been influenced by Élie Cartan, Shiing-Shen Chern, and Phillip Griffiths. His research interests cover many areas in Riemannian geometry, geometry of PDEs, Finsler geometry and mathematical physics.
In 1987 he proved several properties of surfaces of unit constant mean curvature in hyperbolic space, which are now called Bryant surfaces in his honour. In 2001 he contributed many advancements to the theory of Bochner-Kähler metrics, the class of Kähler metrics whose Bochner curvature vanishes.
In 1987 he produced the first examples of Riemannian metrics with exceptional holonomy (i.e. whose holonomy groups are G2 or Spin(7)); this showed that every group in Marcel Berger's classification can arise as a holonomy group. Later, he also contributed to the classification of exotic holonomy groups of arbitrary (i.e. non-Riemannian) torsion-free affine connections.
Together with Phillip Griffiths and others co-authors, Bryant develo |
https://en.wikipedia.org/wiki/2009%20Universitario%20de%20Deportes%20season | The 2009 season is Universitario de Deportes' 81st season in the Peruvian Primera División and 44th in the Campeonato Descentralizado. This article shows player statistics and all matches (official and friendly) that the club played during the 2009 season. The season's biggest highlight was the signing of Nolberto Solano.
Players
Summer and winter transfers correspond to Southern Hemisphere seasons.
Squad information
Players in/out
In
Out
Goalscorers
Competitions
Overall
Torneo Descentralizado
First stage
Standings
Summary
Results by round
Second stage
Standings
Summary
Results by round
Third stage
Copa Libertadores
Group 8
Matches
Competitive
Copa Libertadores
Torneo Descentralizado
First stage
Second stage
Finals
Friendly
External links
Universitario.pe Official website
2009
Universitario De Deportes |
https://en.wikipedia.org/wiki/Albertson%20conjecture | In combinatorial mathematics, the Albertson conjecture is an unproven relationship between the crossing number and the chromatic number of a graph. It is named after Michael O. Albertson, a professor at Smith College, who stated it as a conjecture in 2007; it is one of his many conjectures in graph coloring theory. The conjecture states that, among all graphs requiring colors, the complete graph is the one with the smallest crossing number.
Equivalently, if a graph can be drawn with fewer crossings than , then, according to the conjecture, it may be colored with fewer than colors.
A conjectured formula for the minimum crossing number
It is straightforward to show that graphs with bounded crossing number have bounded chromatic number: one may assign distinct colors to the endpoints of all crossing edges and then 4-color the remaining planar graph. Albertson's conjecture replaces this qualitative relationship between crossing number and coloring by a more precise quantitative relationship. Specifically,
a different conjecture of states that the crossing number of the complete graph is
It is known how to draw complete graphs with this many crossings, by placing the vertices in two concentric circles; what is unknown is whether there exists a better drawing with fewer crossings. Therefore, a strengthened formulation of the Albertson conjecture is that every -chromatic graph has crossing number at least as large as the right hand side of this formula. This strengthened conjecture would be true if and only if both Guy's conjecture and the Albertson conjecture are true.
Asymptotic bounds
A weaker form of the conjecture, proven by M. Schaefer, states that every graph with chromatic number has crossing number (using big omega notation), or equivalently that every graph with crossing number has chromatic number . published a simple proof of these bounds, by combining the fact that every minimal -chromatic graph has minimum degree at least (because otherwise greedy coloring would use fewer colors) together with the crossing number inequality according to which every graph with has crossing number . Using the same reasoning, they show that a counterexample to Albertson's conjecture for the chromatic number (if it exists) must have fewer than vertices.
Special cases
The Albertson conjecture is vacuously true for . In these cases, has crossing number zero, so the conjecture states only that the -chromatic graphs have crossing number greater than or equal to zero, something that is true of all graphs. The case of Albertson's conjecture is equivalent to the four color theorem, that any planar graph can be colored with four or fewer colors, for the only graphs requiring fewer crossings than the one crossing of are the planar graphs, and the conjecture implies that these should all be at most 4-chromatic. Through the efforts of several groups of authors the conjecture is now known to hold for all . For every integer , Luiz and Richter presente |
https://en.wikipedia.org/wiki/%C3%81d%C3%A1m%20Holczer | Ádám Holczer (born 28 March 1988) is a Hungarian football player who plays for Soroksár.
Club statistics
Updated to games played as of 15 May 2021.
References
References
HLSZ
1988 births
People from Ajka
Footballers from Veszprém County
21st-century Hungarian people
Living people
Hungarian men's footballers
Men's association football goalkeepers
Vecsési FC footballers
Ferencvárosi TC footballers
Kecskeméti TE players
Nyíregyháza Spartacus FC players
Pécsi MFC players
Gyirmót FC Győr players
Kozármisleny SE footballers
Soroksár SC players
Paksi FC players
Tiszakécske FC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/Noel%20F%C3%BCl%C3%B6p | Noel Fülöp (born 29 January 1988) is a Hungarian football player who plays for MTK II.
Career statistics
.
External links
HLSZ
Ferencvarosi Torna Club Official Website
1988 births
Living people
People from Százhalombatta
Hungarian men's footballers
Men's association football defenders
Mosonmagyaróvári TE footballers
Ferencvárosi TC footballers
Szigetszentmiklósi TK footballers
BFC Siófok players
Soroksár SC players
Monori SE players
MTK Budapest FC players
Tiszakécske FC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Nemzeti Bajnokság III players
Footballers from Pest County |
https://en.wikipedia.org/wiki/2002%E2%80%9303%20Real%20Madrid%20CF%20season | The 2002–03 season was Real Madrid's 72nd season in La Liga. This article lists all matches that the club played in the 2002–03 season, and also shows statistics of the club's players. This season marked the return of their purple away kits, and a new shirt sponsor, Siemens Mobile.
Real Madrid returned to domestic league glory under Vicente del Bosque after a 3–1 victory against Athletic Bilbao in the last game of the season, but the club surprisingly sacked del Bosque shortly after winning the La Liga title after he had not been offered a new contract. He was replaced by a surprise candidate Carlos Queiroz. Madrid were also on course to retain their Champions League title (as well as winning La Decima) before being eliminated by Juventus in the semi-finals. In the domestic cup, the club was eliminated by eventual champions Mallorca with a 1–5 aggregate loss.
Players
In
Total spending: €45 million
Out
Total income: €0 million
Squad
Left club during season
Pre-season and friendlies
Competitions
La Liga
League table
Results by round
Matches
Copa del Rey
UEFA Champions League
First group stage
Second group stage
Knockout phase
Quarter-finals
Semi-finals
UEFA Super Cup
Intercontinental Cup
Statistics
Player statistics
Goalscorers
External links
Realmadrid.com Official Site
ARCHIVO HISTÓRICO DEL REAL MADRID CF
Real Madrid (Spain) profile
uefa.com - UEFA Champions League
Web Oficial de la Liga de Fútbol Profesional
FIFA
Real Madrid CF seasons
Real Madrid
Spanish football championship-winning seasons |
https://en.wikipedia.org/wiki/2000%E2%80%9301%20Real%20Madrid%20CF%20season | The 2000–01 season was Real Madrid Club de Fútbol's 70th season in La Liga. This article lists all matches that the club played in the 2000–01 season, and also shows statistics of the club's players.
Summary
This was the season where the club won its 28th La Liga title, having begun a new policy of signing the world's greatest players under a new president, Florentino Pérez, with a goal of making Real Madrid the most fashionable club in the world. Luís Figo was the arrival of the year, along with Claude Makélélé, and they helped a team of stars, dubbed the galácticos, win the league under Vicente del Bosque, as well as reaching the UEFA Champions League semi-finals as defending champions, where they were narrowly knocked out by Bayern Munich. The arrival of Luís Figo in July 2000 was controversial due to his move from Barcelona to Real Madrid, thus generating furious reactions from Barcelona fans and also Boixos Nois hooligans.
Overview
Transfers
In
Total spending: €122 million
Competitions
La Liga
Classification
Results summary
Results by round
Matches
Copa del Rey
Champions League
First group stage
Group A
Second group stage
Group D
Knockout stage
Quarter-finals
Semi-finals
UEFA Super Cup
Intercontinental Cup
Statistics
Players statistics
External links
Real Madrid CF seasons
Real Madrid
2000–01 |
https://en.wikipedia.org/wiki/Circular%20algebraic%20curve | In geometry, a circular algebraic curve is a type of plane algebraic curve determined by an equation F(x, y) = 0, where F is a polynomial with real coefficients and the highest-order terms of F form a polynomial divisible by x2 + y2. More precisely, if
F = Fn + Fn−1 + ... + F1 + F0, where each Fi is homogeneous of degree i, then the curve F(x, y) = 0 is circular if and only if Fn is divisible by x2 + y2.
Equivalently, if the curve is determined in homogeneous coordinates by G(x, y, z) = 0, where G is a homogeneous polynomial, then the curve is circular if and only if G(1, i, 0) = G(1, −i, 0) = 0. In other words, the curve is circular if it contains the circular points at infinity, (1, i, 0) and (1, −i, 0), when considered as a curve in the complex projective plane.
Multicircular algebraic curves
An algebraic curve is called p-circular if it contains the points (1, i, 0) and (1, −i, 0) when considered as a curve in the complex projective plane, and these points are singularities of order at least p. The terms bicircular, tricircular, etc. apply when p = 2, 3, etc. In terms of the polynomial F given above, the curve F(x, y) = 0 is p-circular if Fn−i is divisible by (x2 + y2)p−i when i < p. When p = 1 this reduces to the definition of a circular curve. The set of p-circular curves is invariant under Euclidean transformations. Note that a p-circular curve must have degree at least 2p.
When k is 1 this says that the set of lines (0-circular curves of degree 1) together with the set of circles (1-circular curves of degree 2) form a set which is invariant under inversion.
Examples
The circle is the only circular conic.
Conchoids of de Sluze (which include several well-known cubic curves) are circular cubics.
Cassini ovals (including the lemniscate of Bernoulli), toric sections and limaçons (including the cardioid) are bicircular quartics.
Watt's curve is a tricircular sextic.
Footnotes
References
"Courbe Algébrique Circulaire" at Encyclopédie des Formes Mathématiques Remarquables
"Courbe Algébrique Multicirculaire" at Encyclopédie des Formes Mathématiques Remarquables
Definition at 2dcurves.com
Curves
Analytic geometry |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Iceland | As a candidate country of the European Union, Iceland (IS) is included in the Nomenclature of Territorial Units for Statistics (NUTS). The three NUTS levels are:
NUTS-1: IS0 Iceland
NUTS-2: IS00 Iceland
NUTS-3: Capital area / Rest of country
IS001 Höfuðborgarsvæðið (Capital Region)
IS002 Landsbyggð (rest of country)
Below the NUTS levels, there are two Local Administrative Unitary levels (LAU-1: regions, LAU-2: municipalities).
See also
Administrative divisions of Iceland
ISO 3166-2 codes of Iceland
FIPS region codes of Iceland
Sources
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EFTA countries - Statistical regions at level 1
ÍSLAND - Statistical regions at level 2
ÍSLAND - Statistical regions at level 3
Correspondence between the regional levels and the national administrative units
Regions of Iceland, Statoids.com
Iceland
Subdivisions of Iceland |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Norway | As a member of EFTA, Norway (NO) is not included in the Classification of Territorial Units for Statistics (NUTS), but in a similar classification used for coding statistical regions of countries that are not part of the EU but are candidate countries, potential candidates or EFTA countries. The three levels are:
level 1 (equivalent to NUTS level 1): Norway
level 2 (equivalent to NUTS level 2): 7 Regions
level 3 (equivalent to NUTS level 3): 19 Counties
The codes are as follows:
NO0 Norway
NO01 Oslo og Akershus
NO011 Oslo
NO012 Akershus
NO02 Hedmark og Oppland
NO021 Hedmark
NO022 Oppland
NO03 Sør-Østlandet
NO031 Østfold
NO032 Buskerud
NO033 Vestfold
NO034 Telemark
NO04 Agder og Rogaland
NO041 Aust-Agder
NO042 Vest-Agder
NO043 Rogaland
NO05 Vestlandet
NO051 Hordaland
NO052 Sogn og Fjordane
NO053 Møre og Romsdal
NO06 Trøndelag
NO061 Sør-Trøndelag
NO062 Nord-Trøndelag
NO07 Nord-Norge
NO071 Nordland
NO072 Troms
NO073 Finnmark
Below these levels, there are two LAU levels (LAU-1: economic regions; LAU-2: municipalities).
The LAU codes of Norway can be downloaded here:
See also
Subdivisions of Norway
ISO 3166-2 codes of Norway
FIPS region codes of Norway
References
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EFTA countries - Statistical regions at level 1
NORGE - Statistical regions at level 2
NORGE - Statistical regions at level 3
Correspondence between the regional levels and the national administrative units
Counties of Norway, Statoids.com
Norway
Subdivisions of Norway |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Germany | The Nomenclature of Territorial Units for Statistics (NUTS) is a geocode standard for referencing the subdivisions of Germany for statistical purposes. The standard is developed and regulated by the European Union. The NUTS standard is instrumental in delivering the European Union's Structural Funds. The NUTS code for Germany is DE and a hierarchy of three levels is established by Eurostat. Below these is a further levels of geographic organisation – the local administrative unit (LAU). In Germany the LAUs 1 is collective municipalities, and the LAU 2 is municipalities.
Overall
NUTS codes
Older Version
In the 2003 version, before the abolition of the government regions in Sachsen-Anhalt, the codes were as follows
DEE1 Dessau
DEE11 Dessau, Kreisfreie Stadt
DEE12 Anhalt-Zerbst
DEE13 Bernburg
DEE14 Bitterfeld
DEE15 Köthen
DEE16 Wittenberg
DEE2 Halle
DEE21 Halle (Saale), Kreisfreie Stadt
DEE22 Burgenlandkreis
DEE23 Mansfelder Land
DEE24 Merseburg-Querfurt
DEE25 Saalkreis
DEE26 Sangerhausen
DEE27 Weißenfels
DEE3 Magdeburg
DEE31 Magdeburg, Kreisfreie Stadt
DEE32 Aschersleben-Staßfurt
DEE33 Bördekreis
DEE34 Halberstadt
DEE35 Jerichower Land
DEE36 Ohrekreis
DEE37 Stendal
DEE38 Quedlinburg
DEE39 Schönebeck
DEE3A Wernigerode
DEE3B Altmarkkreis Salzwedel
Changes
Changes from 2006 to 2010
BRANDENBURG merged from two regions
SAXONY restructured after 2008 redistricting legislation
Changes from 2010 to 2013
MECKLENBURG-VORPOMMERN completely restructured and merged districts
Changes from 2013 to 2016:
DE91C created from merger of old DE915 Göttingen and DE919 Osterode am Harz
Boundary changes for DEB1C (old DEB16) and DEB1D (old DEB19)
See also
ISO 3166-2 codes of Germany
FIPS region codes of Germany
Subdivisions of Germany
References
Sources
Hierarchical list of the Nomenclature of territorial units for statistics – NUTS and the Statistical regions of Europe
Overview map of EU Countries – NUTS level 1
DEUTSCHLAND – NUTS level 2
DEUTSCHLAND North – NUTS level 3
DEUTSCHLAND East – NUTS level 3
DEUTSCHLAND South – NUTS level 3
DEUTSCHLAND West – NUTS level 3
Correspondence between the NUTS levels and the national administrative units
List of current NUTS codes
Download current NUTS codes (ODS format)
States of Germany, Statoids.com
Districts of Germany, Statoids.com
External links
The LAU codes of Germany:
Germany
Nuts |
https://en.wikipedia.org/wiki/%C4%90or%C4%91e%20Zafirovi%C4%87 | Đorđe Zafirović (Serbian Cyrillic: Ђорђе Зафировић; born 26 February 1978) is a retired Serbian professional football player.
Statistics
External links
1978 births
Living people
Serbian men's footballers
Men's association football midfielders
FK Milicionar players
FK Partizan players
FK Teleoptik players
FK Zvezdara players
FK Vojvodina players
FK Proleter Zrenjanin players
FK Budućnost Banatski Dvor players
FK Banat Zrenjanin players
FK Smederevo 1924 players
FK Alfa Modriča players
Serbian SuperLiga players
Serbian expatriate men's footballers
Expatriate men's footballers in Bosnia and Herzegovina |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Estonia | The Nomenclature of Territorial Units for Statistics (NUTS) is a geocode standard for referencing the subdivisions of Estonia for statistical purposes. The standard is developed and regulated by the European Union. The NUTS standard is instrumental in delivering the European Union's Structural Funds. The NUTS code for Estonia is EE and a hierarchy of three levels is established by Eurostat. Below these is a further levels of geographic organisation - the local administrative unit (LAU). In Estonia, the LAU 1 is counties and the LAU 2 are municipalities.
Overall
NUTS Codes
Local administrative units
Below the NUTS levels, the two LAU (Local Administrative Units) levels are:
The LAU codes of Estonia can be downloaded here:
NUTS codes
NUTS 3
EE0 Estonia
EE00 Estonia
EE001 Põhja-Eesti (Harju County)
EE004 Lääne-Eesti (Hiiu County, Lääne County, Pärnu County, Saare County)
EE006 Kesk-Eesti (Järva County, Lääne-Viru County, Rapla County)
EE007 Kirde-Eesti (Ida-Viru County)
EE008 Lõuna-Eesti (Jõgeva County, Põlva County, Tartu County, Valga County, Viljandi County, Võru County)
See also
Subdivisions of Estonia
ISO 3166-2 codes of Estonia
FIPS region codes of Estonia
References
Sources
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EU Countries - NUTS level 1
EESTI - NUTS level 2
EESTI - NUTS level 3
Correspondence between the NUTS levels and the national administrative units
List of current NUTS codes
Download current NUTS codes (ODS format)
Counties of Estonia, Statoids.com
Estonia
Nuts
Reform in Estonia |
https://en.wikipedia.org/wiki/Super%20Tonks%E2%80%93Girardeau%20gas | In physics, the super-Tonks–Girardeau gas represents an excited quantum gas phase with strong attractive interactions in a one-dimensional spatial geometry.
Usually, strongly attractive quantum gases are expected to form dense particle clusters and lose all gas-like properties. But in 2005, it was proposed by Stefano Giorgini and co-workers that there is a many-body state of attractively interacting bosons that does not decay in one-dimensional systems. If prepared in a special way, this lowest gas-like state should be stable and show new quantum mechanical properties.
Particles in a super-Tonks gas should be strongly correlated and show long range order with a Luttinger Liquid parameter K<1. Since each particle occupies a certain volume, the gas properties are similar to a classical gas of hard rods. Despite the mutual attraction, the single particle wave functions separate and the bosons behave similar to fermions with repulsive, long-range interaction.
To prepare the super-Tonks–Girardeau phase it is necessary to increase the repulsive interaction strength all the way through the Tonks–Girardeau regime up to infinity. Sudden switching from infinitely strong repulsive to infinitely attractive interactions stabilizes the gas against collapse and connects the ground state of the Tonks gas to the excited state of the super-Tonks gas.
Experimental realization
The super-Tonks–Girardeau gas was experimentally observed in Ref. using an ultracold gas of cesium atoms. Reducing the magnitude of the attractive interactions caused the gas to became unstable to collapse into cluster-like bound states. Repulsive dipolar interactions stabilize the gas when instead using highly magnetic dysprosium atoms. This enabled the creation of prethermal quantum many-body scar states via the topological pumping of these super-Tonks-Girardeau gases.
See also
Tonks–Girardeau gas
Quantum many-body scar states
References
Condensed matter physics |
https://en.wikipedia.org/wiki/Edwin%20Power |
Edwin Albert Power (12 February 1928 – 31 January 2004) was an English physicist and an emeritus professor of applied mathematics at University College London. He made several contributions to the field of non-relativistic quantum electrodynamics.
Life
Power was born in Honiton, England on 12 February 1928. He obtained his B.Sc and M.Sc in mathematics from University College London in 1948 and 1949 respectively. He obtained his Ph.D under the supervision of John Currie Gunn at the University of Glasgow, for which he obtained the Kelvin Prize in 1951 (the prize recognizes the best physics thesis of the year). His doctoral work concerned meson production from proton–proton collisions.
After his Ph.D, he worked at University College, where became professor of applied mathematics in 1967, and fellow in 1991. In 1953, he became a Commonwealth Fund Fellow. He then spent two years in the United States, one at Cornell University, one at Princeton University. While at Princeton, he and John Wheeler worked on electromagnetism and gravity, resulting in the proposition of "thermal geons" in a paper published in Reviews of Modern Physics in 1957.
Power then researched non-relativistic quantum electrodynamics, particularly the interactions between radiation fields and particles, and developed several techniques. In 1959, he and Sigurd Zienau published a paper on the Coulomb gauge and its relation to the shape of spectral lines, non-relativistic Lamb shift, and other phenomena in the Philosophical Transactions of the Royal Society of London A. Power also studied the relation between quantum electrodynamics and various optical and molecular phenomena. In 1964, he published a book, Introductory Quantum Electrodynamics, based on a series of lectures he gave in Chile and the US.
Power retired as a professor in 1993, but remained active in research until his death following a short illness. He died on 31 January 2004, in London, England.
Selected publications
References
English physicists
1928 births
2004 deaths
Alumni of the University of London
British expatriates in the United States
Alumni of the University of Glasgow |
https://en.wikipedia.org/wiki/2010%E2%80%9311%20Real%20Madrid%20CF%20season | The 2010–11 season was Real Madrid Club de Fútbol's 80th season in La Liga. This article shows player statistics and all matches (official and friendly) that the club played during the 2010–11 season.
The rebuilt Madrid under star manager José Mourinho successfully fought on all fronts, going toe to toe with a brilliant Barcelona side which some regard as the greatest team in football history. Ultimately, Madrid finished second in the league, with 92 points and four behind their perennial rivals, defeated them in the Copa del Rey final, and lost to Barça in the Champions League semi-finals, where Real progressed to for the first time since 2002–03. Moreover, from 16 April through 3 May, a rare occurrence happened when, for the first time ever, four Clásicos were to be played in a span of just 18 days. The first fixture was in the league campaign on 16 April (which ended 1–1 with penalty goals for both sides), the second one was in the Copa del Rey final (which was won by Madrid 1–0 a.e.t., bringing them their first trophy in the second Galáctico era) on 20 April and the third and fourth ones in the two-legged Champions League semi-finals on 27 April and 3 May (Barcelona won on aggregate with a 2–0 away victory and a 1–1 home draw). The matches in the Champions League proved the most controversial, as multiple refereeing decisions were harshly criticized by Mourinho and Madrid players who accused UEFA of favoring the Catalan side. Namely, Pepe's red card in the 61st minute of the first leg was questioned, after which Barcelona scored two goals, with Mourinho being ejected and subsequently banned for the second leg for protesting, and several controversial offside calls were made, as well as Real having a goal disallowed in the second leg, when the score was tied 0–0. Madrid again became the highest scoring team in La Liga, with 102 goals, repeating its output from the previous season, with Cristiano Ronaldo scoring a record 40 and winning the European Golden Shoe.
This season was the first since 1993–94 without Raúl, who departed to join Schalke 04 after his contract was terminated, having stayed at the club for sixteen years and the first since 1994–95 without Guti who departed to join Beşiktaş.
Club
Coaching staff
Kit
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|
|
Other information
Players
Squad information
|
|
Transfers
In
Total spending: €74.5 million
Out
Total income: €16 million, (€5M from Antonio Cassano to Milan transfer)
Long-term injury list
Last updated: 1 April 2011
Source: Realmadrid.com
Pre-season and friendlies
Last updated: 24 May 2011
Source: Preseason, US Tour, US & European Tour, Santiago Bernabéu Trophy, Real Madrid in support of Murcia and Lorca
Competitions
La Liga
League table
Results by round
Matches
Last updated: 21 May 2011
Source: realmadrid.com, LFP, La Liga Schedule, La Liga
Copa del Rey
Round of 32
Round of 16
Quarter-finals
Semi-finals
Final
Last updated: 20 April 2011
Source: Sorteo deciseisavos Copa del Rey
UEF |
https://en.wikipedia.org/wiki/Patos | Patos is a municipality of the state of Paraíba in the Northeast Region of Brazil. It is classified by the Brazilian Institute of Geography and Statistics as a sub-regional center A.
It is located in the Espinharas River valley, surrounded by the Borborema Plateau to east and south, and by the pediplain Sertanejo to the west. It originated from the village of Patos, spun off from the Parish of Nossa Senhora do Bom Sucesso de Pombal on October 6, 1788.
The city is 306 km from the city of João Pessoa, the center of its immediate and intermediate geographic regions. It stands out as an educational, commercial, banking, religious and health center, both in the back country of Paraíba, and in areas of Pernambuco and Rio Grande do Norte. It is the third most important municipality in the state considering the economic, political and social aspects (behind João Pessoa and Campina Grande). According to IBGE estimates for 2021, it is the fourth most populous municipality in the state with 108,766 inhabitants.
Esporte and Nacional are the city's two football (soccer) clubs. They play at the José Cavalcanti Municipal Stadium. There are four multisport arenas: Rivaldão, AABB, SESC and SESI.
The city is served by Brig. Firmino Ayres Airport.
Climate
The city of Patos experiences a tropical hot semi-arid climate (Köppen: BSh) with a short rainy season from February to April. During this period, when the equatorial rainband associated with the highly seasonalized positioning of the Intertropical Convergence Zone is over the city, warm to hot temperatures and abundant equatorial rainfall prevail. Conversely, the dry season dominates the remaining majority of the year, with abundant sunshine prevalent from May to January. The wettest month is March, with an average monthly precipitation of 213 mm (8.38 in), while the driest month is September, with an average precipitation of only 1 mm (0.04 in).
See also
List of municipalities in Paraíba
References
External links
CPRM – Serviço Geológico do Brasil
Prefeitura Municipal de Patos
FAMUP
Municipalities in Paraíba |
https://en.wikipedia.org/wiki/Octagrammic%20prism | In geometry, the octagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two octagrams.
Prismatoid polyhedra |
https://en.wikipedia.org/wiki/Soci%C3%A9t%C3%A9%20de%20Math%C3%A9matiques%20Appliqu%C3%A9es%20et%20Industrielles | The Société de Mathématiques Appliquées et Industrielles (SMAI) is a French scientific society aiming at promoting applied mathematics, similarly to the Society for Industrial and Applied Mathematics (SIAM).
SMAI was founded in 1983 to contribute to the development of applied mathematics for research, commercial applications, publications, teaching, and industrial training. As of 2009, the society has nearly 1300 members, including both individuals and institutions.
SMAI is directed by an administration elected by the general assembly. Its chief activities are:
to organize conferences and workshops,
to publish the thrice-yearly bulletin Matapli, which contains overviews, book reviews, and information about theses and upcoming conferences,
to publish scholarly journals including Modélisation Mathématique et Analyse Numérique (M2AN), Contrôle Optimisation et Calcul des Variations (COCV), Probabilités et Statistiques (P&S), Recherche opérationnelle (RO), ESAIM: Proceedings and Surveys, and the cross-disciplinary journal MathematicS in Action (MathS in A.).
Prizes
Prix Jacques-Louis Lions, established in 2003 with INRIA et le CNES, recognized by the Académie des Sciences
Prix Blaise Pascal, established in 1984 with GAMNI, recognized by the Académie des Sciences
Grand Prix Louis Bachelier established in 2007 with the NATIXIS Foundation for Quantitative Research", awarded by the French Academy of Sciences until 2012 and jointly with the London Mathematical Society since 2014.
Prix Lagrange de l'ICIAM, established in 1998 with SEMA (Spain) and SIMAI (Italy)
Prix Maurice Audin, awarded by SMF and SMAI
Interest groups
SMAI contains five special interest groups, organized by specific mathematical areas, as follows:
SMAI-GAMNI (Groupe thématique pour l’Avancement des Méthodes Numériques de l’Ingénieur) which promotes the use of numerical analysis in industry.
SMAI-MAIRCI (Mathématiques Appliquées, Informatique, Réseaux, Calcul, Industrie) which is at the frontiers of interdisciplinary applied mathematics.
SMAI-MAS (Modélisation Aléatoire et Statistique) which promotes the use of statistics and probability in industry.
SMAI-MODE (Mathématiques de l’Optimisation et de la Décision) which is dedicated to fields such as nonlinear analysis, optimization, discrete mathematics, operational research, and mathematical modeling in economics, finance, and the social sciences.
SMAI-SIGMA (Signal - Image - Géométrie - Modélisation - Approximation; formerly SMAI-AFA Association Française d’Approximation'') to promote the study and use of approximations in general.
External links
Société de Mathématiques Appliquées et Industrielles (SMAI)
www.conferencealerts.in
www.worldconferencealerts.com
Mathematical societies
1983 establishments in France
Learned societies of France |
https://en.wikipedia.org/wiki/Richland%20Collegiate%20High%20School | Richland Collegiate High School (RCHS) of Mathematics, Science, and Engineering is a charter high school opened in 2006 at Dallas College in Dallas, Texas.
Students can complete their last two years of high school at Dallas College, Richland Campus, taking college courses and earning college credits with a focus on mathematics, science, or engineering. RCHS students can graduate from Dallas College with an Associate of Arts or Associate of Science degree and a high school diploma, ready to transfer or enter as a freshman to a four-year university.
In the fall semester of 2010, the Richland Collegiate High School of Visual, Performing, and Digital Arts opened.
References
External links
Dallas College Website
Richland Collegiate High School Website
Public high schools in Dallas
Charter high schools in Texas
University-affiliated schools in the United States
Dallas College |
https://en.wikipedia.org/wiki/Andrew%20M.%20Stuart | Andrew M. Stuart is a British and American mathematician, working in applied and computational mathematics. In particular, his research has focused on the numerical analysis of dynamical systems, applications of stochastic differential equations and stochastic partial differential equations, the Bayesian approach to inverse problems, data assimilation, and machine learning.
Education
Andrew Stuart graduated in Mathematics from Bristol University in 1983, and then obtained his DPhil from the Oxford University Computing Laboratory in 1986.
Career
After postdoctoral research in applied mathematics at Oxford and MIT, Stuart held permanent positions at the University of Bath (1989–1992), in mathematics, at Stanford University (1991–1999), in engineering, and at Warwick University (1999–2016), in mathematics. He is currently Bren Professor of Computing and Mathematical Sciences at the California Institute of Technology.
Honors and awards
Stuart has been honored with several awards, including the 1989 Leslie Fox Prize for Numerical Analysis, the Monroe H. Martin Prize from the Institute for Physical Science and Technology at the University of Maryland, College Park, the SIAM James Wilkinson Prize, the SIAM Germund Dahlquist Prize in 1997, the Whitehead Prize from the London Mathematical Society in 2000, and the SIAM J.D. Crawford Prize in 2007. He was an invited speaker at the International Council for Industrial and Applied Mathematics (ICIAM) in Zurich in 2007 and Tokyo in 2023, and at the International Congress of Mathematicians (ICM) in Seoul, 2014. In 2009 he was elected an inaugural fellow of the Society for Industrial and Applied Mathematics (SIAM), and in 2020 he was elected a Fellow of the Royal Society. In 2022, he was named a Vannevar Bush Faculty Fellow.
Publications
The majority of Stuart's published work is in archived journals. In addition to mathematics research published in archival journals, Stuart is also the author of several books in mathematics, including a research monograph concerning Dynamical Systems and Numerical Analysis, a research text on Multiscale Methods, a graduate text on Continuum Mechanics, a text on Data Assimilation, and a text on Inverse Problems.
Stuart, A., Humphries, A.R. (1998). Dynamical Systems and Numerical Analysis. Cambridge University Press.
G. A. Pavliotis, Andrew Stuart (2008). Multiscale Methods: Averaging and Homogenization. Springer Science & Business Media.
Gonzalez, O., & Stuart, A. (2008). A First Course in Continuum Mechanics (Cambridge Texts in Applied Mathematics). Cambridge: Cambridge University Press.
Kody Law, Andrew Stuart, Konstantinos Zygalakis (2015). Data Assimilation: A Mathematical Introduction. Springer.
Sanz-Alonso, D., Stuart, A., & Taeb, A. (2023). Inverse Problems and Data Assimilation (London Mathematical Society Student Texts). Cambridge: Cambridge University Press.
References
External links
Andrew Stuart Caltech site
1962 births
Living people
Alumni of th |
https://en.wikipedia.org/wiki/Andy%20Murray%20career%20statistics | Andy Murray is a professional tennis player who has been ranked world number 1 for 41 weeks. He is the only player, male or female, to win two Olympic gold medals in singles, which he did at the 2012 and 2016 Summer Olympics (since tennis was re-introduced to the Olympics in 1988). He has reached eleven grand slam finals in total, winning the 2016 Wimbledon Championships, 2013 Wimbledon Championships and the 2012 US Open, and finished as runner-up at the 2008 US Open, the 2010, 2011, 2013, 2015 and 2016 Australian Open, at Wimbledon in 2012 and the 2016 French Open.
Murray made his professional tennis debut on the main tour in Barcelona in 2005. Murray has won 46 singles titles. This includes three Grand Slam titles, 14 Masters 1000 Series titles (the fifth-most since 1990), two gold medals at the Olympics, and a title at the ATP Finals. He also has two exhibition titles, two doubles titles with his brother Jamie Murray and an Olympic silver medal in the mixed doubles with Laura Robson.
Below is a list of career achievements and titles won by Andy Murray.
Career achievements
Murray reached his first Major semi-final and final at the 2008 US Open, where he lost in the final to Roger Federer in straight sets. He reached his second Major final at the 2010 Australian Open, again losing to Federer in straight sets. At the 2011 Australian Open, Murray's third Major final appearance ended in another straight sets defeat, this time at the hands of Novak Djokovic. He made his fourth appearance in a Major final at the 2012 Wimbledon Championships, becoming the first male British player since Bunny Austin in 1938 to make it to a Wimbledon final. He lost to Federer, who recovered from losing the first set to prevail in four sets. This meant that Murray matched Ivan Lendl's record of losing his first four Major finals.
A month after this defeat, however, at the same venue, Murray won the gold medal at the 2012 London Olympics, defeating Federer in three sets in the final, losing only 7 games. This was Murray's first victory over Federer in the best of five sets format. Later the same day, he and Laura Robson won the silver medal in the mixed doubles. In his fifth Major final appearance, at the 2012 US Open, he defeated Djokovic in five sets. By winning his first Major final at the fifth attempt, he again emulated his coach Ivan Lendl, who also needed five Major final appearances to win his maiden Grand Slam tournament. His victory over Djokovic took four hours and fifty-four minutes, equal to the 1988 US Open final between Ivan Lendl and Mats Wilander as the longest U.S. Open singles final in terms of time.
In addition, Murray has appeared in 21 Masters 1000 Series finals, winning 14. He qualified for the ATP World Tour Finals every year from 2008 to 2016, with his best result coming in the 2016 event in which he went undefeated in round-robin play and then defeated Milos Raonic in the semi-finals. En route to the final, he played the two longest 3-set |
https://en.wikipedia.org/wiki/Stefan%20E.%20Warschawski | Stefan Emanuel "Steve" Warschawski (April 18, 1904 – May 5, 1989) was a mathematician, a professor and department chair at the University of Minnesota and the founder of the mathematics department at the University of California, San Diego.
Early life and education
Warschawski was born in Lida, now in Belarus; at the time of his birth Lida was part of the Russian Empire. His father was a Russian medical doctor, and his mother was ethnically German; the family spoke German at home. In 1915, his family moved to Königsberg, in Prussia (now Kaliningrad, Russia), the home of his mother's family.
Warschawski studied at the University of Königsberg until 1926 and then moved to the University of Göttingen for his doctoral studies under the supervision of Alexander Ostrowski. Ostrowski moved to the University of Basel and Warschawski followed him there to complete his studies.
Career
After receiving his Ph.D., Warschawski took a position at Göttingen in 1930 but, due to the rise of Hitler and his own Jewish ancestry, he soon moved to Utrecht University in Utrecht, Netherlands and then Columbia University in New York City.
After a sequence of temporary positions, he found a permanent faculty position at Washington University in St. Louis in 1939. During World War II he moved to Brown University and then the University of Minnesota, where he remained until his 1963 move to San Diego, where he was the founding chair of the mathematics department. Warschawski stepped down as chair in 1967, and retired in 1971, but remained active in research: approximately one third of his research publications were written after his retirement. Over the course of his career, he advised 19 Ph.D. students, all but one at either Minnesota or San Diego. Vernor Vinge is among Warschawski's doctoral students.
Research
Warschawski was known for his research on complex analysis and in particular on conformal maps. He also made contributions to the theory of minimal surfaces and harmonic functions.
The Noshiro–Warschawski theorem is named after Warschawski and Noshiro, who discovered it independently; it states that, if f is an analytic function on the open unit disk such that the real part of its first derivative is positive, then f is one-to-one.
In 1980, he solved the Visser–Ostrowski problem for derivatives of conformal mappings at the boundary.
Legacy
Warschawski was honored in 1978 by the creation of the Stefan E. Warschawski Assistant Professorship at San Diego. The Stephen E. Warschawski Memorial Scholarship was also given in his name in 1999–2000 to four UCSD undergraduates as a one-time award. His wife, Ilse, died in 2009 and left a US$1 million bequest to UCSD, part of which went towards endowing a professorship in the mathematics department.
References
1904 births
1989 deaths
People from Lida
People from Lidsky Uyezd
Mathematical analysts
20th-century American mathematicians
University of Königsberg alumni
University of Göttingen alumni
University of Basel alum |
https://en.wikipedia.org/wiki/Solinas%20prime | In mathematics, a Solinas prime, or generalized Mersenne prime, is a prime number that has the form , where is a low-degree polynomial with small integer coefficients. These primes allow fast modular reduction algorithms and are widely used in cryptography. They are named after Jerome Solinas.
This class of numbers encompasses a few other categories of prime numbers:
Mersenne primes, which have the form ,
Crandall or pseudo-Mersenne primes, which have the form for small odd .
Modular reduction algorithm
Let be a monic polynomial of degree with coefficients in and suppose that is a Solinas prime. Given a number with up to bits, we want to find a number congruent to mod with only as many bits as – that is, with at most bits.
First, represent in base :
Next, generate a -by- matrix by stepping times the linear-feedback shift register defined over by the polynomial : starting with the -integer register , shift right one position, injecting on the left and adding (component-wise) the output value times the vector at each step (see [1] for details). Let be the integer in the th register on the th step and note that the first row of is given by . Then if we denote by the integer vector given by:
,
it can be easily checked that:
.
Thus represents an -bit integer congruent to .
For judicious choices of (again, see [1]), this algorithm involves only a relatively small number of additions and subtractions (and no divisions!), so it can be much more efficient than the naive modular reduction algorithm ().
Examples
Four of the recommended primes in NIST's document "Recommended Elliptic Curves for Federal Government Use" are Solinas primes:
p-192
p-224
p-256
p-384
Curve448 uses the Solinas prime
See also
Mersenne prime
References
Classes of prime numbers
Eponymous numbers in mathematics
fr:Nombre de Mersenne premier#Généralisations |
https://en.wikipedia.org/wiki/Christian%20Ludwig%20Gersten | Christian Ludwig Gersten (7 February 1701 – 13 August 1762) was a German scientist.
He was born in Gießen, a town in the German federal state of Hessen. He studied law and mathematics at the University of Gießen and in the beginning of the 1730s he travelled to London, England, to improve his mathematical knowledge. In London in 1733 he became a fellow of the English Royal Society. In the same 1733 he return to Gießen, to accept the position of a professor of mathematics at the university of his home town.
Christian Ludwig Gersten is primarily known by his book for a series of experiments, using the barometer, entitled "Tentamina Systematis Novi ad Mutationes Barometri ex Natura elateris Aerei demonstrandas, cui adjecta sub finem Dissertatio Roris decidui errorem Antiquum et vulgarem per Observationes et Experimenta Nova excutiens". Gersten actually was the first scientist to find out, based on observations, that dew did not fall from the heavens, but ascends from earth, especially from plants.
Gersten is also known to present a very interesting calculating device (see the link in the External Links section).
References
Scientists from Hesse
1701 births
1762 deaths
Fellows of the Royal Society
University of Giessen alumni
18th-century German scientists |
https://en.wikipedia.org/wiki/A.E.K.%20Athens%20F.C.%20in%20international%20football%20competitions | AEK Athens F.C. history and statistics in the UEFA competitions.
Notable European Campaigns
1976–77 UEFA Cup semi-finals campaign
The club's most memorable moment in European competitions was the campaign to the semi-final of the UEFA Cup during the 1976–77 season under František Fadrhonc's management. On the way to the semi-final, Athens AEK managed to eliminate four clubs. In the first round they faced Soviet champions Dynamo Moscow. In Athens, AEK won 2–0 with goals by Takis Nikoloudis and Mimis Papaioannou. In Moscow, Dynamo paid them back by winning 2–0 and leading the match to extra time. In the last minute of extra time, AEK managed to score thanks to a penalty kick by Tasos Konstantinou and proceeded to the second round. They were drawn against English 4th placed side Derby County. In Athens, a goal by Walter Wagner and an own goal by Rod Thomas gave AEK the 2–0 win. At Derby, AEK found themselves behind in the score line but responded by scoring three times with Takis Nikoloudis, Tasos Konstantinou and Walter Wagner. Derby Country only managed to score a consolation goal and the match ended in a 2–3 win for AEK. In the third round AEK had to oppose Yugoslav giants Red Star Belgrade. In Athens, AEK was once again victorious by winning 2–0. Mimis Papaioannou and Thomas Mavros were the goal-scorers. In Belgrade Red Star took the lead with a goal by Petar Baralić but Walter Wagner quickly equalised. The two additional goals scored by Zoran Filipović and Dušan Savić were not enough and AEK won on away goals. In the quarter-final AEK faced their greatest challenge to that moment, the English league's runners-up QPR. The first leg was played in London. The two penalty kick goals in the first ten minutes scored by Gerry Francis and another one scored by Stan Bowles gave QPR the 3–0 win and what looked like a certain qualification. Nevertheless, AEK made the impossible possible. With two goals by Thomas Mavros and one more by Mimis Papaioannou AEK sent the match to extra time and eventually to a penalty shootout. Three minutes before the final whistle, František Fadrhonc had Nikos Christidis substitute Lakis Stergioudas, the team's regular goalkeeper. His move proved vital as Nikos Christidis saved two penalties and gave AEK a 7–6 win. In the semi-finals draw, AEK were to play either the Italian league's runner-up side Juventus or Spanish league's third-placed side Athletic Bilbao. Ultimately AEK had to face the Italians. In Turin, Juventus scored first with Antonello Cuccureddu but AEK responded with a goal by Lazaros Papadopoulos. Two goals by Roberto Bettega and one by Franco Causio followed, giving Juventus a 4–1 victory. Juventus also won in Nikos Goumas Stadium thanks to a goal scored by Roberto Boninsegna end went on to win their first European title.
2002–03 UEFA Champions League unbeaten campaign
Another unforgettable feat was the unbeaten run in the UEFA Champions League was the highlight of the season. The club played against Cypr |
https://en.wikipedia.org/wiki/Hyperoctahedral%20group | In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a parameter , the dimension of the hypercube.
As a Coxeter group it is of type , and as a Weyl group it is associated to the symplectic groups and with the orthogonal groups in odd dimensions. As a wreath product it is where is the symmetric group of degree . As a permutation group, the group is the signed symmetric group of permutations π either of the set or of the set such that for all . As a matrix group, it can be described as the group of orthogonal matrices whose entries are all integers. Equivalently, this is the set of matrices with entries only 0, 1, or –1, which are invertible, and which have exactly one non-zero entry in each row or column. The representation theory of the hyperoctahedral group was described by according to .
In three dimensions, the hyperoctahedral group is known as where is the octahedral group, and is a symmetric group (here a cyclic group) of order 2. Geometric figures in three dimensions with this symmetry group are said to have octahedral symmetry, named after the regular octahedron, or 3-orthoplex. In 4-dimensions it is called a hexadecachoric symmetry, after the regular 16-cell, or 4-orthoplex. In two dimensions, the hyperoctahedral group structure is the abstract dihedral group of order eight, describing the symmetry of a square, or 2-orthoplex.
By dimension
Hyperoctahedral groups can be named as Bn, a bracket notation, or as a Coxeter group graph:
Subgroups
There is a notable index two subgroup, corresponding to the Coxeter group Dn and the symmetries of the demihypercube. Viewed as a wreath product, there are two natural maps from the hyperoctahedral group to the cyclic group of order 2: one map coming from "multiply the signs of all the elements" (in the n copies of ), and one map coming from the parity of the permutation. Multiplying these together yields a third map . The kernel of the first map is the Coxeter group In terms of signed permutations, thought of as matrices, this third map is simply the determinant, while the first two correspond to "multiplying the non-zero entries" and "parity of the underlying (unsigned) permutation", which are not generally meaningful for matrices, but are in the case due to the coincidence with a wreath product.
The kernels of these three maps are all three index two subgroups of the hyperoctahedral group, as discussed in H1: Abelianization below, and their intersection is the derived subgroup, of index 4 (quotient the Klein 4-group), which corresponds to the rotational symmetries of the demihypercube.
In the other direction, the center is the subgroup of scalar matrices, {±1}; geometrically, quotienting out by this corresponds to passing to the projective orthogonal group.
In dimension 2 these groups completely de |
https://en.wikipedia.org/wiki/Wall-crossing | In algebraic geometry and string theory, the phenomenon of wall-crossing describes the discontinuous change of a certain quantity, such as an integer geometric invariant, an index or a space of BPS state, across a codimension-one wall in a space of stability conditions, a so-called wall of marginal stability.
References
Kontsevich, M. and Soibelman, Y. "Stability structures, motivic Donaldson–Thomas invariants and cluster transformations" (2008). .
M. Kontsevich, Y. Soibelman, "Motivic Donaldson–Thomas invariants: summary of results",
Joyce, D. and Song, Y. "A theory of generalized Donaldson–Thomas invariants," (2008). .
Gaiotto, D. and Moore, G. and Neitzke, A. "Four-dimensional wall-crossing via three-dimensional field theory" (2008). .
Mina Aganagic, Hirosi Ooguri, Cumrun Vafa, Masahito Yamazaki, "Wall crossing and M-theory",
Kontsevich, M. and Soibelman, Y., "Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry",
String theory
Algebraic geometry |
https://en.wikipedia.org/wiki/John%20Oliver%20%28footballer%2C%20born%201913%29 | John Oliver (1913 – 10 February 1991) was an English professional footballer who played in the Football League for Burnley.
Career statistics
Source:
References
1913 births
1991 deaths
Footballers from Gateshead
English men's footballers
Men's association football defenders
Gateshead A.F.C. players
Walker Celtic F.C. players
Stoke City F.C. players
Spennymoor United A.F.C. players
Burnley F.C. players
English Football League players |
https://en.wikipedia.org/wiki/%C3%89tale%20algebra | In commutative algebra, an étale algebra over a field is a special type of algebra, one that is isomorphic to a finite product of finite separable field extensions. An étale algebra is a special sort of commutative separable algebra.
Definitions
Let be a field. Let be a commutative unital associative -algebra. Then is called an étale -algebra if any one of the following equivalent conditions holds:
Examples
The -algebra is étale because it is a finite separable field extension.
The -algebra is not étale, since .
Properties
Let denote the absolute Galois group of . Then the category of étale -algebras is equivalent to the category of finite -sets with continuous -action. In particular, étale algebras of dimension are classified by conjugacy classes of continuous homomorphisms from to the symmetric group . These globalize to e.g. the definition of étale fundamental groups and categorify to Grothendieck's Galois theory.
Notes
References
http://www.jmilne.org/math/CourseNotes/FT.pdf
Commutative algebra |
https://en.wikipedia.org/wiki/%C3%89tale%20group%20scheme | In mathematics, more precisely in algebra, an étale group scheme is a certain kind of group scheme.
Definition
A finite group scheme over a field is called an étale group scheme if it is represented by an étale K-algebra , i.e. if is isomorphic to .
References
Algebraic groups
Scheme theory |
https://en.wikipedia.org/wiki/John%20Pearson%20%28footballer%2C%20born%201896%29 | John Cecil Pearson (14 March 1896 – December 1979) was an English professional footballer who played as a full back in the Football League for Burnley, Brentford and Grimsby Town.
Career statistics
References
1896 births
1979 deaths
Footballers from Dudley
English men's footballers
Men's association football fullbacks
Cradley Heath F.C. players
Halesowen Town F.C. players
Burnley F.C. players
Brentford F.C. players
Grimsby Town F.C. players
English Football League players |
https://en.wikipedia.org/wiki/Heptagonal%20prism | In geometry, the heptagonal prism is a prism with heptagonal base. This polyhedron has 9 faces (2 bases and 7 sides), 21 edges, and 14 vertices.
Area
The area of a right heptagonal prism with height and with a side length of and apothem is given by:
Volume
The volume is found by taking the area of the base, with a side length of and apothem , and multiplying it by the height , giving the formula:
This formula also works for the oblique prism due to the Cavalieri's principle.
Images
The heptagonal prism can also be seen as a tiling on a sphere:
Related polyhedra
References
External links
Prismatoid polyhedra |
https://en.wikipedia.org/wiki/Enneagonal%20prism | In geometry, the enneagonal prism (or nonagonal prism) is the seventh in an infinite set of prisms, formed by square sides and two regular enneagon caps.
If faces are all regular, it is a semiregular polyhedron.
Related polyhedra
Prismatoid polyhedra |
https://en.wikipedia.org/wiki/Timeline%20of%20manifolds | This is a timeline of manifolds, one of the major geometric concepts of mathematics. For further background see history of manifolds and varieties.
Background
Manifolds in contemporary mathematics come in a number of types. These include:
smooth manifolds, which are basic in calculus in several variables, mathematical analysis and differential geometry;
piecewise-linear manifolds;
topological manifolds.
There are also related classes, such as homology manifolds and orbifolds, that resemble manifolds. It took a generation for clarity to emerge, after the initial work of Henri Poincaré, on the fundamental definitions; and a further generation to discriminate more exactly between the three major classes. Low-dimensional topology (i.e., dimensions 3 and 4, in practice) turned out to be more resistant than the higher dimension, in clearing up Poincaré's legacy. Further developments brought in fresh geometric ideas, concepts from quantum field theory, and heavy use of category theory.
Participants in the first phase of axiomatization were influenced by David Hilbert: with Hilbert's axioms as exemplary, by Hilbert's third problem as solved by Dehn, one of the actors, by Hilbert's fifteenth problem from the needs of 19th century geometry. The subject matter of manifolds is a strand common to algebraic topology, differential topology and geometric topology.
Timeline to 1900 and Henri Poincaré
1900 to 1920
1920 to the 1945 axioms for homology
1945 to 1960
Terminology: By this period manifolds are generally assumed to be those of Veblen-Whitehead, so locally Euclidean Hausdorff spaces, but the application of countability axioms was also becoming standard. Veblen-Whitehead did not assume, as Kneser earlier had, that manifolds are second countable. The term "separable manifold", to distinguish second countable manifolds, survived into the late 1950s.
1961 to 1970
1971–1980
1981–1990
1991–2000
2001–present
See also
differentiable stack
factorization homology
Kuranishi theory
Floer homology
Glossary of algebraic topology
Timeline of bordism
Notes
Manifolds
Historical timelines
Manifolds |
https://en.wikipedia.org/wiki/D%C3%A1vid%20Kulcs%C3%A1r | Dávid Kulcsár (born 25 February 1988, in Miskolc) is a Hungarian football player who plays for III. Kerületi TVE.
Club statistics
Updated to games played as of 15 May 2021.
References
HLSZ
Ferencvarosi TC Official Website
1988 births
Living people
Footballers from Miskolc
Hungarian men's footballers
Hungary men's youth international footballers
Men's association football midfielders
Diósgyőri VTK players
Ferencvárosi TC footballers
Vecsési FC footballers
Vasas SC players
Paksi FC players
III. Kerületi TVE footballers
Nemzeti Bajnokság I players |
https://en.wikipedia.org/wiki/Applied%20mathematics | Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.
In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.
History
Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis); and applied probability. These areas of mathematics related directly to the development of Newtonian physics, and in fact, the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. This history left a pedagogical legacy in the United States: until the early 20th century, subjects such as classical mechanics were often taught in applied mathematics departments at American universities rather than in physics departments, and fluid mechanics may still be taught in applied mathematics departments. Engineering and computer science departments have traditionally made use of applied mathematics.
Divisions
Today, the term "applied mathematics" is used in a broader sense. It includes the classical areas noted above as well as other areas that have become increasingly important in applications. Even fields such as number theory that are part of pure mathematics are now important in applications (such as cryptography), though they are not generally considered to be part of the field of applied mathematics per se.
There is no consensus as to what the various branches of applied mathematics are. Such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees.
Many mathematicians distinguish between "applied mathematics", which is concerned with mathematical methods, and the "applications of mathematics" within science and engineering. A biologist using a population model and applying known mathematics would not be doing applied mathematics, but rather using it; however, mathematical biologists have posed problems that have stimulated the growth of pure mathematics. Mathematicians such as Poincaré and Arnold deny the existence of "applied mathematics" and claim that there are only "applications of mathematics." Similarly, non-mathematicians blend applied mathematics and applications of mathematics. The use |
https://en.wikipedia.org/wiki/Roberto%20Delgado%20%28footballer%29 | Roberto Alfonso Delgado (born 7 May 1986) is a Spanish footballer who plays for Italian Eccellenza club Fiano Romano.
Statistics accurate as of match played 3 August 2010
Career Honours
Cupa României
Runner-up: 2010
External links
Roberto Delgado at TuttoCampo
1986 births
Living people
Spanish men's footballers
Spanish expatriate men's footballers
People from Tenerife
Footballers from the Province of Santa Cruz de Tenerife
Men's association football forwards
FC Delta Tulcea players
SS Lazio players
SPAL players
CS Sporting Vaslui players
FC Universitatea Cluj players
Potenza SC players
US Pergolettese 1932 players
US Palestrina 1919 players
Serie A players
Serie D players
Eccellenza players
Liga I players
Spanish expatriate sportspeople in Italy
Spanish expatriate sportspeople in Romania
Expatriate men's footballers in Italy
Expatriate men's footballers in Romania |
https://en.wikipedia.org/wiki/List%20of%20FC%20Seoul%20records%20and%20statistics | Below are statistics and records related to FC Seoul.
Honours
Domestic competitions
League
K League 1
Winners (6): 1985, 1990, 2000, 2010, 2012, 2016
Runners-up (5): 1986, 1989, 1993, 2001, 2008
Cups
FA Cup
Winners (2): 1998, 2015
Runners-up (3): 2014, 2016, 2022
League Cup
Winners (2): 2006, 2010
Runners-up (4): 1992, 1994, 1999, 2007
Super Cup
Winners (1): 2001
Runners-up (1): 1999
National Football Championship
Winners (1): 1988
International competitions
Asian
AFC Champions League
Runners-up (2): 2001–02, 2013
Friendly competitions
Saitama City Cup
Winners (1): 2017
Doubles
Domestic double
K League and League Cup Champions (1): 2010
Team
Season-by-season records
※ K League: Only regular season results are counted. Postseason (League Championship and Promotion-relegation PO) results are not included.
※ 1993, 1998, 1999, 2000 seasons had penalty shoot-outs instead of draws.
※ A: Adidas Cup, P: Prospecs Cup, PM: Philip Morris Cup, D: Daehan Fire Insurance Cup
K League Championship records
K League promotion-relegation playoffs
All-Time Competitions Records
※ As of 31 December 2016
※ Walkover results are counted.
※ Penalty shoot-outs results are counted as a drawn match.
All-Time K League 1 Results by Opponents
※ As of 1984–2013 seasons
All-Time K League 1 Records
※ 1984-2012 seasons
※ Penalty shoot-outs results are not counted as a drawn match
Firsts
Mosts in Goals
Mosts in Goal difference
Most Consecutive Records
※ 1984-2013 seasons
Historic Victory
※ Only matches in K League 1 and League Cup are counted.
Historic Goal
※ Only matches in K League 1 and League Cup are counted.
Players
※ Statistics correct as of last match played in 2015 season.
※ Bold denotes players still playing in the K League
※ Appearance, Goals, Clean Sheets, Goals conceded per Match records included FA Cup and Asian Club Competitions, Other competitive competitions.
※ Other competitive competitions are as below:
K League unofficial matches: 1986 K League Championship - 2 matches, 1992 League Cup Final - 2 matches
1999 Korean Super Cup - 1 match, 2001 Korean Super Cup - 1 match
1988 Korean National Football Championship - 4 matches, 1989 Korean National Football Championship - 3 matches
2018 K League promotion-relegation playoffs - 2 matches
※ Assists and Attack Points records only includes K League 1 and Korean League Cup.
※ Attack Points are the aggregate number of goals assists in K League 1 and Korean League Cup.
Appearances
※ FC Seoul players with at least 200 appearances.
※ As of May 3, 2019
Goals
Assists
Attack Points
Clean Sheets
Goals conceded per Match
※ FC Seoul goalkeepers with at least 50 appearances.
Mosts in a Single Match
Mosts in a Single Season
Most Goals in a Single Season by Competitions
Most Clean Sheets in a Single Season by Competitions
Most Consecutive Records
Managers
Managerial History
※ For details on all-time manager statistics, see List of FC Seoul coaching staffs.
Match Results
※ Win |
https://en.wikipedia.org/wiki/Pentakis%20icosidodecahedron | In geometry, the pentakis icosidodecahedron or subdivided icosahedron is a convex polyhedron with 80 triangular faces, 120 edges, and 42 vertices. It is a dual of the truncated rhombic triacontahedron (chamfered dodecahedron).
Construction
Its name comes from a topological construction from the icosidodecahedron with the kis operator applied to the pentagonal faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general icosahedral symmetry can be maintained even with the 12 order-5 vertices at a different distance from the center as the other 30.
It can also be topologically constructed from the icosahedron, dividing each triangular face into 4 triangles by adding mid-edge vertices. From this construction, all 80 triangles will be equilateral, but faces will be coplanar.
Related polyhedra
]]== Related polytopes ==
It represents the exterior envelope of a vertex-centered orthogonal projection of the 600-cell, one of six convex regular 4-polytopes, into 3 dimensions.
See also
Tetrakis cuboctahedron
References
George W. Hart, Sculpture based on Propellorized Polyhedra, Proceedings of MOSAIC 2000, Seattle, WA, August, 2000, pp. 61–70
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008,
Chapter 21: Naming the Archimedean and Catalan polyhedra and Tilings (p 284)
Dover 1999
External links
VTML polyhedral generator Try "k5aD" (Conway polyhedron notation)
Geodesic polyhedra |
https://en.wikipedia.org/wiki/Tetrakis%20cuboctahedron | In geometry, the tetrakis cuboctahedron is a convex polyhedron with 32 triangular faces, 48 edges, and 18 vertices. It is a dual of the truncated rhombic dodecahedron.
Its name comes from a topological construction from the cuboctahedron with the kis operator applied to the square faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general octahedral symmetry can be maintain even with the 6 order-4 vertices at a different distance from the center as the other 12.
Related polyhedra
It can also be topologically constructed from the octahedron, dividing each triangular face into 4 triangles by adding mid-edge vertices (an ortho operation). From this construction, all 32 triangles will be equilateral.
This polyhedron can be confused with a slightly smaller Catalan solid, the tetrakis hexahedron, which has only 24 triangles, 32 edges, and 14 vertices.
See also
Pentakis icosidodecahedron
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008,
Chapter 21: Naming the Archimedean and Catalan polyhedra and Tilings (p284)
External links
VTML polyhedral generator Try "k4aC" (Conway polyhedron notation)
Polyhedra |
https://en.wikipedia.org/wiki/Sergei%20Gukov | Sergei Gukov (; born 1977) is a professor of mathematics and theoretical physicist. Gukov graduated from Moscow Institute of Physics and Technology (MIPT) in Moscow, Russia before obtaining a doctorate in physics from Princeton University under the supervision of Edward Witten.
He held a Long-term Prize fellowship of Clay Mathematics Institute at Harvard University (2001-2006) and during 2007-2008 was a member of the school of mathematics at the Institute for Advanced Study, Princeton. Since 2007, he has been professor of mathematics and theoretical physics at the California Institute of Technology (Caltech). Starting 2010, Gukov was elected as an external scientific member of the Max Planck Society at the MPIM, Bonn.
Sergei Gukov is a member of the Scientific Board of the American Institute of Mathematics (AIM) and a member of the International Advisory Board of the Centre for Quantum Mathematics (QM). He has served on numerous other scientific committees and advisory boards. He is editor of the journal Communications in Mathematical Physics, Journal of Knot Theory and Its Ramifications, and Letters in Mathematical Physics.
In 2010, along with Alain Connes, Gukov was invited to deliver the 8th Takagi Lectures, the only named lecture series of the Mathematical Society of Japan. In 2019, Gukov was invited to give the Whittemore Lectures at Yale University.
Known for Gukov-Vafa-Witten superpotential, Gukov-Witten surface operators, and Gukov-Pei-Putrov-Vafa (GPPV) invariants. In recent years, he teamed up with Boris Feigin, Hiraku Nakajima and other mathematicians to explore hidden algebraic structures in topology and in quantum field theory.
References
External links
https://www.ias.edu/scholars/sergei-gukov
Faculty page at Caltech
Clay Mathematics Institute
Interview at MIPT (in Russian)
1977 births
Living people
California Institute of Technology faculty
Scientists from Moscow
Princeton University alumni
Russian physicists
American string theorists
University of California, Santa Barbara faculty
Moscow Institute of Physics and Technology alumni |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Greece | The NUTS codes of Greece are part of the Nomenclature of Territorial Units for Statistics, an official nomenclature of the European Commission used by Eurostat for statistical purposes.
Changes
In 2011, the NUTS1 code of Greece was changed from GR to EL. GR1 was changed to EL5, GR2 to EL6, GR3 to EL3 and GR4 to EL4. The change became official per European Commission regulation No. 31/2011. With regard to the transmission of data to Eurostat, the new codes entered into force by 1 January 2012.
Following the Kallikratis territorial reform, the NUTS regions of Greece were redefined. With the region of Epirus being reclassified as part of Voreia Ellada ("Northern Greece", former EL1), and the region of Thessaly in exchange going to Kentriki Ellada ("Central Greece", former EL2), new NUTS1 codes have been assigned to both regions. Apart from that, a number of third-level divisions have been changed. The changes became official in December 2013 by European Commission regulation No. 1319/2013. With regard to the transmission of data to Eurostat, the new codes entered into force on 1 January 2015.
NUTS levels
The three NUTS levels are as follows:
NUTS codes
Per 1 January 2015, the NUTS codes for Greece are as follows:
Local administrative units
Below the NUTS levels, the two LAU (Local Administrative Units) levels are:
The LAU codes of Greece can be downloaded here: ''
See also
Subdivisions of Greece
ISO 3166-2 codes of Greece
FIPS region codes of Greece
References
External links
Hierarchical list of the Nomenclature of territorial units for statistics – NUTS and the Statistical regions of Europe
Overview map of EU Countries – NUTS level 1
ELLADA – NUTS level 2
ELLADA – NUTS level 3
Correspondence between the NUTS levels and the national administrative units
List of current NUTS codes
Download current NUTS codes (ODS format)
Departments of Greece, Statoids.com
Greece
Nuts |
https://en.wikipedia.org/wiki/Mass%20point%20geometry | Mass point geometry, colloquially known as mass points, is a problem-solving technique in geometry which applies the physical principle of the center of mass to geometry problems involving triangles and intersecting cevians. All problems that can be solved using mass point geometry can also be solved using either similar triangles, vectors, or area ratios, but many students prefer to use mass points. Though modern mass point geometry was developed in the 1960s by New York high school students, the concept has been found to have been used as early as 1827 by August Ferdinand Möbius in his theory of homogeneous coordinates.
Definitions
The theory of mass points is defined according to the following definitions:
Mass Point - A mass point is a pair , also written as , including a mass, , and an ordinary point, on a plane.
Coincidence - We say that two points and coincide if and only if and .
Addition - The sum of two mass points and has mass and point where is the point on such that . In other words, is the fulcrum point that perfectly balances the points and . An example of mass point addition is shown at right. Mass point addition is closed, commutative, and associative.
Scalar Multiplication - Given a mass point and a positive real scalar , we define multiplication to be . Mass point scalar multiplication is distributive over mass point addition.
Methods
Concurrent cevians
First, a point is assigned with a mass (often a whole number, but it depends on the problem) in the way that other masses are also whole numbers.
The principle of calculation is that the foot of a cevian is the addition (defined above) of the two vertices (they are the endpoints of the side where the foot lie).
For each cevian, the point of concurrency is the sum of the vertex and the foot.
Each length ratio may then be calculated from the masses at the points. See Problem One for an example.
Splitting masses
Splitting masses is the slightly more complicated method necessary when a problem contains transversals in addition to cevians. Any vertex that is on both sides the transversal crosses will have a split mass. A point with a split mass may be treated as a normal mass point, except that it has three masses: one used for each of the two sides it is on, and one that is the sum of the other two split masses and is used for any cevians it may have. See Problem Two for an example.
Other methods
Routh's theorem - Many problems involving triangles with cevians will ask for areas, and mass points does not provide a method for calculating areas. However, Routh's theorem, which goes hand in hand with mass points, uses ratios of lengths to calculate the ratio of areas between a triangle and a triangle formed by three cevians.
Special cevians - When given cevians with special properties, like an angle bisector or an altitude, other theorems may be used alongside mass point geometry that determine length ratios. One very common theorem used likewis |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Lithuania | In the NUTS (Nomenclature of Territorial Units for Statistics) codes of Lithuania (LT), the three levels are:
NUTS codes
LT0 Lithuania
LT01 Sostinės regionas
LT011 Vilnius County
LT02 Vidurio ir vakarų Lietuvos regionas
LT021 Alytus County
LT022 Kaunas County
LT023 Klaipėda County
LT024 Marijampolė County
LT025 Panevėžys County
LT026 Šiauliai County
LT027 Tauragė County
LT028 Telšiai County
LT029 Utena County
Local administrative units
Below the NUTS levels, the two LAU (Local Administrative Units) levels are:
The LAU codes of Lithuania can be downloaded here: ''
See also
Subdivisions of Lithuania
ISO 3166-2 codes of Lithuania
FIPS region codes of Lithuania
Sources
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EU Countries - NUTS level 1
LIETUVA - NUTS level 2
LIETUVA - NUTS level 3
Correspondence between the NUTS levels and the national administrative units
List of current NUTS codes
Download current NUTS codes (ODS format)
Counties of Lithuania, Statoids.com
References
Lithuania
Nuts
Lithuania geography-related lists |
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