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https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s%20existence%20theorem | In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand side of the differential equation be continuous, while Carathéodory's theorem shows existence of solutions (in a more general sense) for some discontinuous equations. The theorem is named after Constantin Carathéodory.
Introduction
Consider the differential equation
with initial condition
where the function ƒ is defined on a rectangular domain of the form
Peano's existence theorem states that if ƒ is continuous, then the differential equation has at least one solution in a neighbourhood of the initial condition.
However, it is also possible to consider differential equations with a discontinuous right-hand side, like the equation
where H denotes the Heaviside function defined by
It makes sense to consider the ramp function
as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at , because the function is not differentiable there. This suggests that the idea of a solution be extended to allow for solutions that are not everywhere differentiable, thus motivating the following definition.
A function y is called a solution in the extended sense of the differential equation with initial condition if y is absolutely continuous, y satisfies the differential equation almost everywhere and y satisfies the initial condition. The absolute continuity of y implies that its derivative exists almost everywhere.
Statement of the theorem
Consider the differential equation
with defined on the rectangular domain . If the function satisfies the following three conditions:
is continuous in for each fixed ,
is measurable in for each fixed ,
there is a Lebesgue-integrable function such that for all ,
then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.
A mapping is said to satisfy the Carathéodory conditions on if it fulfills the condition of the theorem.
Uniqueness of a solution
Assume that the mapping satisfies the Carathéodory conditions on and there is a Lebesgue-integrable function , such that
for all Then, there exists a unique solution
to the initial value problem
Moreover, if the mapping is defined on the whole space
and if for any initial condition , there exists a compact rectangular domain such that the mapping satisfies all conditions from above on . Then, the domain of definition of the function is open and is continuous on .
Example
Consider a linear initial value problem of the form
Here, the components of the matrix-valued mapping and of the inhomogeneity are assumed to be integrable on every finite interval. Then, the right hand side of the differential equation satisfies the Carathéodory conditions and there exists a unique solution to the initia |
https://en.wikipedia.org/wiki/Roland%20Zielke | Roland Zielke (born 30 July 1946 in Opladen) is a German politician for the Free Democratic Party.
He studied mathematics in Cologne, Columbus and Konstanz. He received his doctor in mathematics in 1971. He went on to study medicine in Münster and received his doctorate in medicine in 1988. From 1975 to 2003 he was professor for applied mathematics at Osnabrück University.
He was elected to the Landtag of Lower Saxony in 2003, and has been re-elected on one occasion. He was member of the Landtag from 2003 to 2013.
External links
Leverkusen who's who
Free Democratic Party (Germany) politicians
Members of the Landtag of Lower Saxony
1946 births
Living people |
https://en.wikipedia.org/wiki/Rota%E2%80%93Baxter%20algebra | In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map R which satisfies the Rota–Baxter identity. It appeared first in the work of the American mathematician Glen E. Baxter in the realm of probability theory. Baxter's work was further explored from different angles by Gian-Carlo Rota, Pierre Cartier, and Frederic V. Atkinson, among others. Baxter’s derivation of this identity that later bore his name emanated from some of the fundamental results of the famous probabilist Frank Spitzer in random walk theory.
In the 1980s, the Rota-Baxter operator of weight 0 in the context of Lie algebras was rediscovered as the operator form of the classical Yang–Baxter equation, named after the well-known physicists Chen-Ning Yang and Rodney Baxter.
The study of Rota–Baxter algebras experienced a renaissance this century, beginning with several developments, in the algebraic approach to renormalization of perturbative quantum field theory, dendriform algebras, associative analogue of the classical Yang–Baxter equation and mixable shuffle product constructions.
Definition and first properties
Let k be a commutative ring and let be given. A linear operator R on a k-algebra A is called a Rota–Baxter operator of weight if it satisfies the Rota–Baxter relation of weight :
for all . Then the pair or simply A is called a Rota–Baxter algebra of weight . In some literature, is used in which case the above equation becomes
called the Rota-Baxter equation of weight . The terms Baxter operator algebra and Baxter algebra are also used.
Let be a Rota–Baxter of weight . Then is also a Rota–Baxter operator of weight . Further, for in k, is a Rota-Baxter operator of weight .
Examples
Integration by parts
Integration by parts is an example of a Rota–Baxter algebra of weight 0. Let be the algebra of continuous functions from the real line to the real line. Let : be a continuous function. Define integration as the Rota–Baxter operator
Let G(x) = I(g)(x) and F(x) = I(f)(x). Then the formula for integration for parts can be written in terms of these variables as
In other words
which shows that I is a Rota–Baxter algebra of weight 0.
Spitzer identity
The Spitzer identity appeared is named after the American mathematician Frank Spitzer. It is regarded as a remarkable
stepping stone in the theory of sums of independent random variables in fluctuation theory of probability. It can naturally be understood in terms of Rota–Baxter operators.
Bohnenblust–Spitzer identity
Notes
External links
Li Guo. WHAT IS...a Rota-Baxter Algebra? Notices of the AMS, December 2009, Volume 56 Issue 11
Algebras
Combinatorics |
https://en.wikipedia.org/wiki/Non-Desarguesian%20plane | In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective spaces of dimension not 2; in other words, the only projective spaces of dimension not equal to 2 are the classical projective geometries over a field (or division ring). However, David Hilbert found that some projective planes do not satisfy it.
The current state of knowledge of these examples is not complete.
Examples
There are many examples of both finite and infinite non-Desarguesian planes. Some of the known examples of infinite non-Desarguesian planes include:
The Moulton plane.
Moufang planes over alternative division rings that are not associative, such as the projective plane over the octonions. Since all finite alternative division rings are fields (Artin–Zorn theorem), the only non-Desarguesian Moufang planes are infinite.
Regarding finite non-Desarguesian planes, every projective plane of order at most 8 is Desarguesian, but there are three non-Desarguesian examples of order 9, each with 91 points and 91 lines. They are:
The Hughes plane of order 9.
The Hall plane of order 9. Initially discovered by Veblen and Wedderburn, this plane was generalized to an infinite family of planes by Marshall Hall. Hall planes are a subclass of the more general André planes.
The dual of the Hall plane of order 9.
Numerous other constructions of both finite and infinite non-Desarguesian planes are known, see for example . All known constructions of finite non-Desarguesian planes produce planes whose order is a proper prime power, that is, an integer of the form pe, where p is a prime and e is an integer greater than 1.
Classification
Hanfried Lenz gave a classification scheme for projective planes in 1954, which was refined by Adriano Barlotti in 1957. This classification scheme is based on the types of point–line transitivity permitted by the collineation group of the plane and is known as the Lenz–Barlotti classification of projective planes. The list of 53 types is given in and a table of the then known existence results (for both collineation groups and planes having such a collineation group) in both the finite and infinite cases appears on page 126. As of 2007, "36 of them exist as finite groups. Between 7 and 12 exist as finite projective planes, and either 14 or 15 exist as infinite projective planes."
Other classification schemes exist. One of the simplest is based on special types of planar ternary ring (PTR) that can be used to coordinatize the projective plane. These types are fields, skewfields, alternative division rings, semifields, nearfields, right nearfields, quasifields and right quasifields.
Conics and Ovals
In a Desarguesian projective plane a conic can be defined in several different ways that can be proved to be equivalent. In non-Desarguesian planes these proofs are no longer val |
https://en.wikipedia.org/wiki/Glen%20E.%20Baxter | Glen Earl Baxter (March 19, 1930 – March 30, 1983) was an American mathematician.
Baxter's fields of research include probability theory, combinatorial analysis, statistical mechanics and functional analysis. He is known for the Baxter strong limit theorem. Lately, his 1960 work on the derivation of a specific operator identity that later bore his name, the Rota–Baxter identity, and emanated from some of the fundamental results of the famous probabilist Frank Spitzer in random walk theory has received attention in fields as remote as renormalization theory in perturbative quantum field theory.
In 1983 the Glen E. Baxter Memorial Fund was established by family and friends at Purdue University.
See also
Baxter permutation
References
External links
1930 births
1983 deaths
20th-century American mathematicians |
https://en.wikipedia.org/wiki/Top%20%28algebra%29 | In the context of a module M over a ring R, the top of M is the largest semisimple quotient module of M if it exists.
For finite-dimensional k-algebras (k a field) R, if rad(M) denotes the intersection of all proper maximal submodules of M (the radical of the module), then the top of M is M/rad(M). In the case of local rings with maximal ideal P, the top of M is M/PM. In general if R is a semilocal ring (=semi-artinian ring), that is, if R/Rad(R) is an Artinian ring, where Rad(R) is the Jacobson radical of R, then M/rad(M) is a semisimple module and is the top of M. This includes the cases of local rings and finite dimensional algebras over fields.
See also
Projective cover
Radical of a module
Socle (mathematics)
References
David Eisenbud, Commutative algebra with a view toward Algebraic Geometry
Commutative algebra
Module theory |
https://en.wikipedia.org/wiki/Krull%E2%80%93Schmidt%20category | In category theory, a branch of mathematics, a Krull–Schmidt category is a generalization of categories in which the Krull–Schmidt theorem holds. They arise, for example, in the study of finite-dimensional modules over an algebra.
Definition
Let C be an additive category, or more generally an additive -linear category for a commutative ring . We call C a Krull–Schmidt category provided that every object decomposes into a finite direct sum of objects having local endomorphism rings. Equivalently, C has split idempotents and the endomorphism ring of every object is semiperfect.
Properties
One has the analogue of the Krull–Schmidt theorem in Krull–Schmidt categories:
An object is called indecomposable if it is not isomorphic to a direct sum of two nonzero objects. In a Krull–Schmidt category we have that
an object is indecomposable if and only if its endomorphism ring is local.
every object is isomorphic to a finite direct sum of indecomposable objects.
if where the and are all indecomposable, then , and there exists a permutation such that for all .
One can define the Auslander–Reiten quiver of a Krull–Schmidt category.
Examples
An abelian category in which every object has finite length. This includes as a special case the category of finite-dimensional modules over an algebra.
The category of finitely-generated modules over a finite -algebra, where is a commutative Noetherian complete local ring.
The category of coherent sheaves on a complete variety over an algebraically-closed field.
A non-example
The category of finitely-generated projective modules over the integers has split idempotents, and every module is isomorphic to a finite direct sum of copies of the regular module, the number being given by the rank. Thus the category has unique decomposition into indecomposables, but is not Krull-Schmidt since the regular module does not have a local endomorphism ring.
See also
Quiver
Karoubi envelope
Notes
References
Michael Atiyah (1956) On the Krull-Schmidt theorem with application to sheaves Bull. Soc. Math. France 84, 307–317.
Henning Krause, Krull-Remak-Schmidt categories and projective covers, May 2012.
Irving Reiner (2003) Maximal orders. Corrected reprint of the 1975 original. With a foreword by M. J. Taylor. London Mathematical Society Monographs. New Series, 28. The Clarendon Press, Oxford University Press, Oxford. .
Claus Michael Ringel (1984) Tame Algebras and Integral Quadratic Forms, Lecture Notes in Mathematics 1099, Springer-Verlag, 1984.
Category theory
Representation theory |
https://en.wikipedia.org/wiki/Krull%E2%80%93Schmidt%20theorem | In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.
Definitions
We say that a group G satisfies the ascending chain condition (ACC) on subgroups if every sequence of subgroups of G:
is eventually constant, i.e., there exists N such that GN = GN+1 = GN+2 = ... . We say that G satisfies the ACC on normal subgroups if every such sequence of normal subgroups of G eventually becomes constant.
Likewise, one can define the descending chain condition on (normal) subgroups, by looking at all decreasing sequences of (normal) subgroups:
Clearly, all finite groups satisfy both ACC and DCC on subgroups. The infinite cyclic group satisfies ACC but not DCC, since (2) > (2)2 > (2)3 > ... is an infinite decreasing sequence of subgroups. On the other hand, the -torsion part of (the quasicyclic p-group) satisfies DCC but not ACC.
We say a group G is indecomposable if it cannot be written as a direct product of non-trivial subgroups G = H × K.
Statement
If is a group that satisfies either ACC or DCC on normal subgroups, then there is exactly one way of writing as a direct product of finitely many indecomposable subgroups of . Here, uniqueness means direct decompositions into indecomposable subgroups have the exchange property. That is: suppose is another expression of as a product of indecomposable subgroups. Then and there is a reindexing of the 's satisfying
and are isomorphic for each ;
for each .
Proof
Proving existence is relatively straightforward: let be the set of all normal subgroups that can not be written as a product of indecomposable subgroups. Moreover, any indecomposable subgroup is (trivially) the one-term direct product of itself, hence decomposable. If Krull-Schmidt fails, then contains ; so we may iteratively construct a descending series of direct factors; this contradicts the DCC. One can then invert the construction to show that all direct factors of appear in this way.
The proof of uniqueness, on the other hand, is quite long and requires a sequence of technical lemmas. For a complete exposition, see.
Remark
The theorem does not assert the existence of a non-trivial decomposition, but merely that any such two decompositions (if they exist) are the same.
Remak decomposition
A Remak decomposition, introduced by Robert Remak, is a decomposition of an abelian group or similar object into a finite direct sum of indecomposable objects. The Krull–Schmidt theorem gives conditions for a Remak decomposition to exist and for its factors to be unique.
Krull–Schmidt theorem for modules
If is a module that satisfies the ACC and DCC on submodules (that is, it is both Noetherian and Artinian or – equivalently – of finite length), then is a direct sum of indecomposable modules. Up to a permutation, the indecomposable components in such a direct sum are uniquely determ |
https://en.wikipedia.org/wiki/Visual%20comfort%20probability | Visual comfort probability (VCP), also known as Guth Visual Comfort Probability, is a metric used to rate lighting scenes.
VCP is defined as the percentage of people that will find a certain scene (viewpoint and direction) comfortable with regard to visual glare.
It was defined by Sylvester K. Guth in 1963.
References
Lighting
Architectural lighting design
Vision |
https://en.wikipedia.org/wiki/List%20of%20Valencia%20CF%20records%20and%20statistics | Valencia Club de Fútbol (also known as Los Che) are a professional football club based in Valencia, Spain. This article contains honours won, and statistics and records pertaining to the club.
Honours
National titles
La Liga
Winners (6): 1941–42, 1943–44, 1946–47, 1970–71, 2001–02, 2003–04
Runners-up (6): 1947–48, 1948–49, 1952–53, 1971–72, 1989–90, 1995–96
Third place (10): 1940–41, 1949–50, 1950–51, 1953–54, 1988–89, 1999–2000, 2005–06, 2009–10, 2010–11, 2011–12
Copa del Rey
Winners (8): 1941, 1948–49, 1954, 1966–67, 1978–79, 1998–99, 2007–08, 2018–19
Runners-up (10): 1934, 1944, 1944–45, 1946, 1952, 1969–70, 1970–71, 1971–72, 1994–95, 2021–22
Supercopa de España
Winners: 1999
Runners-up (3): 2002, 2004, 2008
Copa Eva Duarte (Predecessor to the Supercopa de España)
Winners: 1949
Runners-up: 1947
Copa Presidente FEF (Predecessor to the Supercopa de España)
Runners-up: 1947
Segunda División
Winners: 1930–31, 1986–87
European titles
UEFA Champions League
Runners-up (2): 1999–2000, 2000–01
UEFA Cup Winners' Cup
Winners: 1979–80
UEFA Cup / UEFA Europa League
Winners: 2003–04
Semi-finals (3): 2011–12, 2013–14, 2018–19
Inter-cities Fairs Cup:Winners: 1961–62, 1962–63
Runners-up: 1963–64
European Super Cup / UEFA Super CupWinners: 1980, 2004
UEFA Intertoto CupWinners: 1998
Runners-up: 2005
Regional titles
Levante Championship / Valencian Championship (10): 1922–23, 1924–25, 1925–26, 1926–27, 1930–31, 1931–32, 1932–33, 1933–34, 1936–37, 1939–40
Friendly competitions
Naranja Trophy (27): 1961, 1962, 1970, 1975, 1978, 1979, 1980, 1983, 1984, 1988, 1989, 1991, 1993, 1994, 1996, 1998, 1999, 2001, 2002, 2006, 2008, 2009, 2010, 2011, 2012, 2014, 2016.
Ciudad de Valencia Trophy (4): 1988, 1990, 1993, 1994
Copa Generalitat Trophy (3): 1999, 2001, 2002
Ciudad de la Línea Trophy (2): 1970, 1993
Martini Rossi Trophy (2): 1948–49, 1949–50
Teresa Herrera Trophy (1): 1952
Concepción Arenal Trophy (1): 1954
Ciudad de México Trophy (1): 1966
Ramón de Carranza Trophy (1): 1967
Bodas de Oro Trophy (1): 1969
Tournoi de Paris (1): 1975
Ibérico Trophy (1): 1975
Comunidad Valenciana Trophy (1): 1982
75 Aniversario Levante UD Trophy (1): 1984
Festa d'Elx Trophy (1): 1991
Groningen Trophy (1): 1992
La Laguna Trophy (1): 1992
Ciudad de Palma Trophy (1): 1993
Villa de Benidorm Trophy (1): 1993
80 Aniversario San Mamés Trophy (1): 1993
Joan Gamper Trophy (1): 1994
Ciudad de Alicante Trophy (1): 1994
Ciudad de Benidorm Trophy (1): 1994
Ciudad de Mérida Trophy (1): 1995
Trofeu Ciutat de Barcelona (1): 1995
Copa Fuji Trophy (1): 1997
Trofeo de la Cerámica (1): 2001
Ladbrokes.com cup Trophy (1): 2003
Thomas Cook Trophy (1): 2007
Borussia Dortmund 100th Anniversary tournament trophy (1): 2009
Sparkasse Emsland Cup (1): 2009
CD Acero 90th Anniversary Trophy (1): 2009
Kärnten Soccer Cup (1): 2011
Emirates Cup (1): 2014
Recent seasons
Statistics in La Liga
Average Attendance: 46,894
Socios: 45,116
Seasons in First Division: 77
Seasons in Second Division: 4
H |
https://en.wikipedia.org/wiki/Bayesian%20approaches%20to%20brain%20function | Bayesian approaches to brain function investigate the capacity of the nervous system to operate in situations of uncertainty in a fashion that is close to the optimal prescribed by Bayesian statistics. This term is used in behavioural sciences and neuroscience and studies associated with this term often strive to explain the brain's cognitive abilities based on statistical principles. It is frequently assumed that the nervous system maintains internal probabilistic models that are updated by neural processing of sensory information using methods approximating those of Bayesian probability.
Origins
This field of study has its historical roots in numerous disciplines including machine learning, experimental psychology and Bayesian statistics. As early as the 1860s, with the work of Hermann Helmholtz in experimental psychology, the brain's ability to extract perceptual information from sensory data was modeled in terms of probabilistic estimation. The basic idea is that the nervous system needs to organize sensory data into an accurate internal model of the outside world.
Bayesian probability has been developed by many important contributors. Pierre-Simon Laplace, Thomas Bayes, Harold Jeffreys, Richard Cox and Edwin Jaynes developed mathematical techniques and procedures for treating probability as the degree of plausibility that could be assigned to a given supposition or hypothesis based on the available evidence. In 1988 Edwin Jaynes presented a framework for using Bayesian Probability to model mental processes. It was thus realized early on that the Bayesian statistical framework holds the potential to lead to insights into the function of the nervous system.
This idea was taken up in research on unsupervised learning, in particular the Analysis by Synthesis approach, branches of machine learning. In 1983 Geoffrey Hinton and colleagues proposed the brain could be seen as a machine making decisions based on the uncertainties of the outside world. During the 1990s researchers including Peter Dayan, Geoffrey Hinton and Richard Zemel proposed that the brain represents knowledge of the world in terms of probabilities and made specific proposals for tractable neural processes that could manifest such a Helmholtz Machine.
Psychophysics
A wide range of studies interpret the results of psychophysical experiments in light of Bayesian perceptual models. Many aspects of human perceptual and motor behavior can be modeled with Bayesian statistics. This approach, with its emphasis on behavioral outcomes as the ultimate expressions of neural information processing, is also known for modeling sensory and motor decisions using Bayesian decision theory. Examples are the work of Landy, Jacobs, Jordan, Knill, Kording and Wolpert, and Goldreich.
Neural coding
Many theoretical studies ask how the nervous system could implement Bayesian algorithms. Examples are the work of Pouget, Zemel, Deneve, Latham, Hinton and Dayan. George and Hawkins published a paper that |
https://en.wikipedia.org/wiki/Office%20of%20Immigration%20Statistics | The Office of Immigration Statistics (OIS) is an agency of the United States Department of Homeland Security under the Office of Strategy, Policy, and Plans.
Since the passage of the Homeland Security Act of 2002, the Department of Homeland Security's Office of Immigration Statistics has had the responsibility to carry out two statutory requirements: 1) To collect and disseminate to Congress and the public data and information useful in evaluating the social, economic, environmental, and demographic impact of immigration laws; and 2) To establish standards of reliability and validity for immigration statistics collected by the department's operational Components.
Introduction
Located within the Department of Homeland Security's Office of Strategy, Policy, and Plans and focused on data collection and analysis, the Office of Immigration Statistics (OIS) is positioned to gather information from across the department and the entire federal government and to perform these centralizing analytic and dissemination functions.
The department's operational components need cross-cutting data to perform certain enforcement activities and to adjudicate certain claims. Congress requires regular and comprehensive reporting on immigration benefits, immigration enforcement, border security, and migration inflows. Under the department's analytic agenda, senior leaders use detailed operational data and outcome metrics for strategic planning. Other executive branch actors also rely on empirical analysis to monitor and refine immigration policy. Most importantly, a firm commitment to transparency based on the collection and dissemination of clear and credible immigration data is essential to combat widespread confusion and misinformation about immigration trends, and to contribute to a more informed policy debate.
History
Immigration reporting began with the "Steerage" (or "Passenger") Act of March 2, 1819, which required the Secretary of State to report to each session of Congress on the age, sex, occupation, and origins of passengers on arriving vessels. The Department of State performed these duties until the early 1870s, after which responsibility shifted to the Treasury Department's Bureau of Statistics, followed by the Department of Commerce and Labor in 1903, and then the Department of Labor in 1913. The Immigration and Naturalization Service (INS) was transferred from the Department of Labor to the Department of Justice in 1940, where it resided until 2003, when the components of INS were subsumed under the Department of Homeland Security (DHS).
Section 103 of the Immigration and Nationality Act of 1952 establishes OIS's modern mandate: in consultation with interested academics, government agencies, and other parties, to provide Congress and the public, on an annual basis, with information about immigration and the impact of immigration laws. Section 701 of the Homeland Security Act of 2002 transferred these duties from the Statistics Branch of the Off |
https://en.wikipedia.org/wiki/Hughes%20plane | In mathematics, a Hughes plane is one of the non-Desarguesian projective planes found by .
There are examples of order p2n for every odd prime p and every positive integer n.
Construction
The construction of a Hughes plane is based on a nearfield N of order p2n for p an odd prime whose kernel K has order pn and coincides with the center of N.
Properties
A Hughes plane H:
is a non-Desarguesian projective plane of odd square prime power order of Lenz-Barlotti type I.1,
has a Desarguesian Baer subplane H0,
is a self-dual plane in which every orthogonal polarity of H0 can be extended to a polarity of H,
every central collineation of H0 extends to a central collineation of H, and
the full collineation group of H has two point orbits (one of which is H0), two line orbits, and four flag orbits.
The smallest Hughes Plane (order 9)
The Hughes plane of order 9 was actually found earlier by Veblen and Wedderburn in 1907. A construction of this plane can be found in where it is called the plane Ψ.
Notes
References
Projective geometry
Finite geometry |
https://en.wikipedia.org/wiki/Generalized%20map | In mathematics, a generalized map is a topological model which allows one to represent and to handle subdivided objects. This model was defined starting from combinatorial maps in order to represent non-orientable and open subdivisions, which is not possible with combinatorial maps. The main advantage of generalized map is the homogeneity of one-to-one mappings in any dimensions, which simplifies definitions and algorithms comparing to combinatorial maps. For this reason, generalized maps are sometimes used instead of combinatorial maps, even to represent orientable closed partitions.
Like combinatorial maps, generalized maps are used as efficient data structure in image representation and processing, in geometrical modeling, they are related to simplicial set and to combinatorial topology, and this is a boundary representation model (B-rep or BREP), i.e. it represents object by its boundaries.
General definition
The definition of generalized map in any dimension is given in and:
A nD generalized map (or nG-map) is an (n + 2)-tuple G = (D, α0, ..., αn) such that:
D is a finite set of darts;
α0, ..., αn are involutions on D;
αi o αj is an involution if i + 2 ≤ j (i, j ∈ { 0, ,..., n }).
An nD generalized map represents the subdivision of an open or closed orientable or not nD space.
See also
Boundary representation
Combinatorial map
Quad-edge data structure
Rotation system
Simplicial set
Winged edge
References
Algebraic topology
Topological graph theory
Data structures |
https://en.wikipedia.org/wiki/Kenneth%20L.%20Clarkson | Kenneth Lee Clarkson is an American computer scientist known for his research in computational geometry. He is a researcher at the IBM Almaden Research Center, and co-editor-in-chief of the Journal of Computational Geometry.
Biography
Clarkson received his Ph.D. from Stanford University in 1984, under the supervision of Andrew Yao. Until 2007 he worked for Bell Labs.
In 1998 he was co-chair of the ACM Symposium on Computational Geometry.
Research
Clarkson's primary research interests are in computational geometry.
His most highly cited paper, with Peter Shor, uses random sampling to devise optimal randomized algorithms for several problems of constructing geometric structures, following up on an earlier singly-authored paper by Clarkson on the same subject.
It includes algorithms for finding all intersections among a set of line segments in the plane in expected time , finding the diameter of a set of points in three dimensions in expected time , and constructing the convex hull of points in -dimensional Euclidean space in expected time . The same paper also uses random sampling to prove bounds in discrete geometry, and in particular to give tight bounds on the number of ≤k-sets.
Clarkson has also written highly cited papers on the complexity of arrangements of curves and surfaces, nearest neighbor search, motion planning, and low-dimensional linear programming and LP-type problems.
Awards and honors
In 2008 Clarkson was elected a Fellow of the ACM for his "contributions to computational geometry."
References
External links
Clarkson's web page at IBM
Year of birth missing (living people)
Living people
Researchers in geometric algorithms
Stanford University alumni
Fellows of the Association for Computing Machinery |
https://en.wikipedia.org/wiki/Sphere%20spectrum | In stable homotopy theory, a branch of mathematics, the sphere spectrum S is the monoidal unit in the category of spectra. It is the suspension spectrum of S0, i.e., a set of two points. Explicitly, the nth space in the sphere spectrum is the n-dimensional sphere Sn, and the structure maps from the suspension of Sn to Sn+1 are the canonical homeomorphisms. The k-th homotopy group of a sphere spectrum is the k-th stable homotopy group of spheres.
The localization of the sphere spectrum at a prime number p is called the local sphere at p and is denoted by .
See also
Chromatic homotopy theory
Adams-Novikov spectral sequence
Framed cobordism
References
Algebraic topology
Homotopy theory |
https://en.wikipedia.org/wiki/METeOR | METeOR (Metadata Online Registry), Australia’s repository for national metadata standards for health, housing and community services statistics and information. METeOR is a Metadata registry based on the 2003 version of the ISO/IEC 11179 Information technology - Metadata registries standard. The development of METeOR was commissioned by the Australian Institute of Health and Welfare to store, manage and disseminate metadata in the Australian health, community services and housing assistance sectors. Development of METeOR was performed by Aggmedia and Synop, and based on the open-source Sytadel CMS.
In August 2018, the AIHW entered an agreement with the CSIRO for the supply of a new platform for the METeOR registry, this was due to be delivered in 2020. As of August 2020, the AIHW has committed to replacing METeOR with an internally developed solution.
On 29 April 2022, the new METEOR platform was launched. By launch, over AUD$1.7 million was spent between July 2019 and June 2022 on the update of the Meteor platform, with $771,000 spent during 2021 on internal software development, and a further $267,000 spent on additional work in-progress in previous years.
Awards
In 2011, the AIHW won a FutureGov 2011 international award for innovation and modernisation.
References
External links
METeOR web site
See also
Aristotle Metadata Registry (a commercial ISO/IEC 11179 metadata registry)
Metadata
Metadata registry
ISO/IEC 11179
Content management system
Web content management system
XML
Commonwealth Government agencies of Australia
Metadata registry |
https://en.wikipedia.org/wiki/Andr%C3%A9%20plane | In mathematics, André planes are a class of finite translation planes found by André. The Desarguesian plane and the Hall planes are examples of André planes; the two-dimensional regular nearfield planes are also André planes.
Construction
Let be a finite field, and let be a degree extension field of . Let be the group of field automorphisms of over , and let be an arbitrary mapping from to such that . Finally, let be the norm function from to .
Define a quasifield with the same elements and addition as K, but with multiplication defined via , where denotes the normal field multiplication in . Using this quasifield to construct a plane yields an André plane.
Properties
André planes exist for all proper prime powers with prime and a positive integer greater than one.
Non-Desarguesian André planes exist for all proper prime powers except for where is prime.
Small Examples
For planes of order 25 and below, classification of Andrè planes is a consequence of either theoretical calculations or computer searches which have determined all translation planes of a given order:
The smallest non-Desarguesian André plane has order 9, and it is isomorphic to the Hall plane of that order.
The translation planes of order 16 have all been classified, and again the only non-Desarguesian André plane is the Hall plane.
There are three non-Desarguesian André planes of order 25. These are the Hall plane, the regular nearfield plane, and a third plane not constructible by other techniques.
There is a single non-Desarguesian André plane of order 27.
Enumeration of Andrè planes specifically has been performed for other small orders:
References
Finite geometry |
https://en.wikipedia.org/wiki/Preconditioning%20%28disambiguation%29 | Preconditioning is a concept in numerical linear algebra.
Preconditioning may also refer to:
Preconditioning (adaptation), a general concept in which an entity is exposed to a form of some stress or stimulus in order to prepare that entity to be more resilient against the stimulus when and if the stimulus is encountered in the future.
Ischemic preconditioning, an experimental technique for producing resistance to the loss of blood supply to tissues of many types
Sensory preconditioning, a phenomenon of classical conditioning that demonstrates learning of an association between two conditioned stimuli
Conditioning in stem cell transplantation before the stem cell transfer.
See also
Precondition |
https://en.wikipedia.org/wiki/A%20Certain%20Ambiguity | A Certain Ambiguity: A Mathematical Novel is a mathematical fiction by Indian authors Gaurav Suri and Hartosh Singh Bal. It is a story about finding certainty in mathematics and philosophy. In a certain ambiguity we meet Ravi Kapoor, who travels to America to further his education, and is fascinated both by mathematics and philosophy. There he finds about his grandfather being jailed in the year 1919. The book talks about Ravi's experience in the college and his quest to uncover the reason for his grandfather's arrest.
The book is the winner of the 2007 Award for Best Professional/Scholarly Book in Mathematics, Association of American Publishers.
External links
Review at Popular Science
2007 Indian novels
Philosophical novels
Novels about mathematics |
https://en.wikipedia.org/wiki/Bence%20Horv%C3%A1th%20%28footballer%29 | Bence Horváth (born 12 June 1986) is a Hungarian football (forward) player who currently plays for Dél-Balaton FC.
Club statistics
Updated to games played as of 1 March 2014.
References
HLSZ
1986 births
Living people
Footballers from Miskolc
Hungarian men's footballers
Men's association football forwards
BFC Siófok players
SKN St. Pölten players
Nemzeti Bajnokság I players
Hungarian expatriate men's footballers
Expatriate men's footballers in Austria
Hungarian expatriate sportspeople in Austria |
https://en.wikipedia.org/wiki/Vectorial%20addition%20chain | In mathematics, for positive integers k and s, a vectorial addition chain is a sequence V of k-dimensional vectors of nonnegative integers vi for −k + 1 ≤ i ≤ s together with a sequence w,
such that
⋮
⋮
vi =vj+vr for all 1≤i≤s with -k+1≤j, r≤i-1
vs = [n0,...,nk-1]
w = (w1,...ws), wi=(j,r).
For example, a vectorial addition chain for [22,18,3] is
V=([1,0,0],[0,1,0],[0,0,1],[1,1,0],[2,2,0],[4,4,0],[5,4,0],[10,8,0],[11,9,0],[11,9,1],[22,18,2],[22,18,3])
w=((-2,-1),(1,1),(2,2),(-2,3),(4,4),(1,5),(0,6),(7,7),(0,8))
Vectorial addition chains are well suited to perform multi-exponentiation:
Input: Elements x0,...,xk-1 of an abelian group G and a vectorial addition chain of dimension k computing [n0,...,nk-1]
Output:The element x0n0...xk-1nr-1
for i =-k+1 to 0 do yi → xi+k-1
for i = 1 to s do yi →yj×yr
return ys
Addition sequence
An addition sequence for the set of integer S ={n0, ..., nr-1} is an addition chain v that contains every element of S.
For example, an addition sequence computing
{47,117,343,499}
is
(1,2,4,8,10,11,18,36,47,55,91,109,117,226,343,434,489,499).
It's possible to find addition sequence from vectorial addition chains and vice versa, so they are in a sense dual.
See also
Addition chain
Addition-chain exponentiation
Exponentiation by squaring
Non-adjacent form
References
Addition chains |
https://en.wikipedia.org/wiki/Jehovah%27s%20Witnesses%20by%20country | Jehovah's Witnesses have an active presence in most countries. These are the most recent statistics by continent, based on active members, or "publishers" as reported by the Watch Tower Society of Pennsylvania. The Watch Tower Society reports its activity in various dependencies and constituent states as separate 'lands', as noted.
Bible study figures indicate the average number of Bible studies conducted each month by Jehovah's Witness members with non-members. This includes studies conducted by Jehovah's Witness parents with their unbaptized children, which can be counted as one hour per week and one Bible study per month, per child.
Africa
North America
Caribbean
South America
Asia
Europe
Oceania
Other
In addition to the published figures for individual countries, statistics are also published collectively for countries where Jehovah's Witnesses operate covertly under ban, including several Islamic and communist states.
There are 37 sovereign states not specifically listed in the Watch Tower Society's reported statistics:
Afghanistan
Algeria
Bahrain
Bhutan
Brunei
China
Comoros
Djibouti
Egypt
Eritrea
Iran
Iraq
Jordan
Kuwait
Laos
Lebanon
Libya
Maldives
Mauritania
Monaco
Morocco
North Korea
Oman
Qatar
Russia
Saudi Arabia
Singapore
Somalia
Syria
Tajikistan
Tunisia
Turkmenistan
United Arab Emirates
Uzbekistan
Vatican City
Vietnam
Yemen
Total
See also
Demographics of Jehovah's Witnesses
Notes
Sources
2022 Country and Territory Reports, Watch Tower Bible & Tract Society of Pennsylvania.
References
External links
Jehovah's Witnesses Worldwide
ml:യഹോവയുടെ സാക്ഷികളുടെ സ്ഥിതിവിവര കണക്ക് |
https://en.wikipedia.org/wiki/Conductor-discriminant%20formula | In mathematics, the conductor-discriminant formula or Führerdiskriminantenproduktformel, introduced by for abelian extensions and by for Galois extensions, is a formula calculating the relative discriminant of a finite Galois extension of local or global fields from the Artin conductors of the irreducible characters of the Galois group .
Statement
Let be a finite Galois extension of global fields with Galois group . Then the discriminant equals
where equals the global Artin conductor of .
Example
Let be a cyclotomic extension of the rationals. The Galois group equals . Because is the only finite prime ramified, the global Artin conductor equals the local one . Because is abelian, every non-trivial irreducible character is of degree . Then, the local Artin conductor of equals the conductor of the -adic completion of , i.e. , where is the smallest natural number such that . If , the Galois group is cyclic of order , and by local class field theory and using that one sees easily that if factors through a primitive character of , then whence as there are primitive characters of we obtain from the formula , the exponent is
Notes
References
Algebraic number theory |
https://en.wikipedia.org/wiki/Gerg%C5%91%20Goh%C3%A9r | Gergő Gohér (born 16 June 1987) is a Hungarian footballer who plays for Szolnok.
Club career
In July 2021, Gohér returned to Szolnok.
Club statistics
Updated to games played as of 30 June 2020.
Club honours
Szolnoki MÁV FC
Hungarian National Championship II: Runners-up: 2007–08
Diósgyőri VTK
Hungarian National Championship II (1): Winner: 2010–11
Hungarian League Cup (1): 2013–14
References
References
Gergő Gohér at mezokovesdzsory
Gergő Gohér at HLSZ
Gergő Gohér at MLSZ
Gergő Gohér at worldfootball.net
1987 births
Living people
People from Hatvan
Hungarian men's footballers
Men's association football central defenders
Ferencvárosi TC footballers
III. Kerületi TVE footballers
Szolnoki MÁV FC footballers
Diósgyőri VTK players
Budapest Honvéd FC players
Puskás Akadémia FC players
Soroksár SC players
Mezőkövesdi SE footballers
Budafoki MTE footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Nemzeti Bajnokság III players
Footballers from Heves County |
https://en.wikipedia.org/wiki/P%C3%A9ter%20Tak%C3%A1cs%20%28footballer%29 | Péter Takács (born 25 January 1990 in Miskolc) is a Hungarian football player who currently plays for Dorogi FC.
Club statistics
Updated to games played as of 1 June 2014.
Honours
FIFA U-20 World Cup:
Third place: 2009
References
External links
Profile
1990 births
Living people
Footballers from Miskolc
Hungarian men's footballers
Hungary men's youth international footballers
Hungary men's under-21 international footballers
Men's association football midfielders
Diósgyőri VTK players
Pápai FC footballers
Mezőkövesdi SE footballers
Békéscsaba 1912 Előre footballers
Ceglédi VSE footballers
Kazincbarcikai SC footballers
Dorogi FC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/41%20%28number%29 | 41 (forty-one) is the natural number following 40 and preceding 42.
In mathematics
the 13th smallest prime number. The next is 43, making both twin primes.
the sum of the first six prime numbers (2 + 3 + 5 + 7 + 11 + 13).
the 12th supersingular prime
a Newman–Shanks–Williams prime.
the smallest Sophie Germain prime to start a Cunningham chain of the first kind of three terms, {41, 83, 167}.
an Eisenstein prime, with no imaginary part and real part of the form 3n − 1.
a Proth prime as it is 5 × 23 + 1.
the largest lucky number of Euler: the polynomial yields primes for all the integers k with .
the sum of two squares, 42 + 52.
the sum of the sum of the divisors of the first 7 positive integers.
the smallest integer whose reciprocal has a 5-digit repetend. That is a consequence of the fact that 41 is a factor of 99999.
the smallest integer whose square root has a continued fraction with period 3.
a centered square number.
a prime index prime, as 13 is prime.
In science
The atomic number of niobium.
In astronomy
Messier object M41, a magnitude 5.0 open cluster in the constellation Canis Major.
The New General Catalogue object NGC 41, a spiral galaxy in the constellation Pegasus.
In music
"#41", a song by Dave Matthews Band.
"American Skin (41 Shots)" is a song by Bruce Springsteen about an immigrant murder victim who was shot at 41 times by the NYPD.
In film
The name of an independent documentary about Nicholas O'Neill, the youngest victim of the Station nightclub fire.
2012 documentary on the life of the 41st President of the United States George H. W. Bush.
In other fields
The international direct dialing (IDD) code for Switzerland.
Bush 41, George H. W. Bush, the 41st President of the United States.
In Mexico "cuarenta y uno" (41) is slang referring to a homosexual. This is due to the 1901 arrest of 41 homosexuals at a hotel in Mexico City during the government of Porfirio Díaz (1876–1911). See: Dance of the Forty-One
Number of ballistic missile submarines of the George Washington class and its successors, collectively known as the "41 for Freedom".
The 41st season of CBS's reality program Survivor is simply subtitled Survivor 41.
See also
List of highways numbered 41
References
Integers |
https://en.wikipedia.org/wiki/Complete%20field | In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the p-adic numbers).
Constructions
Real and complex numbers
The real numbers are the field with the standard euclidean metric . Since it is constructed from the completion of with respect to this metric, it is a complete field. Extending the reals by its algebraic closure gives the field (since its absolute Galois group is ). In this case, is also a complete field, but this is not the case in many cases.
p-adic
The p-adic numbers are constructed from by using the p-adic absolute valuewhere Then using the factorization where does not divide its valuation is the integer . The completion of by is the complete field called the p-adic numbers. This is a case where the field is not algebraically closed. Typically, the process is to take the separable closure and then complete it again. This field is usually denoted
Function field of a curve
For the function field of a curve every point corresponds to an absolute value, or place, . Given an element expressed by a fraction the place measures the order of vanishing of at minus the order of vanishing of at Then, the completion of at gives a new field. For example, if at the origin in the affine chart then the completion of at is isomorphic to the power-series ring
References
See also
Field (mathematics) |
https://en.wikipedia.org/wiki/Cross-covariance%20matrix | In probability theory and statistics, a cross-covariance matrix is a matrix whose element in the i, j position is the covariance between the i-th element of a random vector and j-th element of another random vector. A random vector is a random variable with multiple dimensions. Each element of the vector is a scalar random variable. Each element has either a finite number of observed empirical values or a finite or infinite number of potential values. The potential values are specified by a theoretical joint probability distribution. Intuitively, the cross-covariance matrix generalizes the notion of covariance to multiple dimensions.
The cross-covariance matrix of two random vectors and is typically denoted by or .
Definition
For random vectors and , each containing random elements whose expected value and variance exist, the cross-covariance matrix of and is defined by
where and are vectors containing the expected values of and . The vectors and need not have the same dimension, and either might be a scalar value.
The cross-covariance matrix is the matrix whose entry is the covariance
between the i-th element of and the j-th element of . This gives the following component-wise definition of the cross-covariance matrix.
Example
For example, if and are random vectors, then
is a matrix whose -th entry is .
Properties
For the cross-covariance matrix, the following basic properties apply:
If and are independent (or somewhat less restrictedly, if every random variable in is uncorrelated with every random variable in ), then
where , and are random vectors, is a random vector, is a vector, is a vector, and are matrices of constants, and is a matrix of zeroes.
Definition for complex random vectors
If and are complex random vectors, the definition of the cross-covariance matrix is slightly changed. Transposition is replaced by Hermitian transposition:
For complex random vectors, another matrix called the pseudo-cross-covariance matrix is defined as follows:
Uncorrelatedness
Two random vectors and are called uncorrelated if their cross-covariance matrix matrix is a zero matrix.
Complex random vectors and are called uncorrelated if their covariance matrix and pseudo-covariance matrix is zero, i.e. if .
References
Covariance and correlation
Matrices |
https://en.wikipedia.org/wiki/G%C3%A1bor%20Horv%C3%A1th%20%28footballer%2C%20born%201985%29 | Gábor Horváth (born 4 July 1985) is a retired Hungarian football player.
Career statistics
International statistics
Updated 13 August 2009
Honours
Hungarian Player of the Year: 2009–10
External links
Profile
1985 births
Living people
Footballers from Székesfehérvár
Hungarian men's footballers
Men's association football defenders
Hungary men's international footballers
Fehérvár FC players
NAC Breda players
ADO Den Haag players
Paksi FC players
Nemzeti Bajnokság I players
Eredivisie players
Hungarian expatriate men's footballers
Expatriate men's footballers in the Netherlands
Hungarian expatriate sportspeople in the Netherlands |
https://en.wikipedia.org/wiki/D%C3%A1vid%20Mohl | Dávid Mohl (; born 22 April 1985) is a Hungarian footballer who plays as a left back for Fehérvár II.
Club statistics
Updated to games played as of 19 May 2019.
References
External links
Profile
1985 births
Living people
Footballers from Székesfehérvár
Hungarian men's footballers
Hungary men's youth international footballers
Men's association football defenders
FC Admira Wacker Mödling players
Fehérvár FC players
Debreceni VSC players
Kecskeméti TE players
Pécsi MFC players
Újpest FC players
Szombathelyi Haladás footballers
Szeged-Csanád Grosics Akadémia footballers
Austrian Football Bundesliga players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Nemzeti Bajnokság III players
Hungarian expatriate men's footballers
Expatriate men's footballers in Austria
Hungarian expatriate sportspeople in Austria |
https://en.wikipedia.org/wiki/Utansj%C3%B6 | Utansjö is a settlement in Härnösand Municipality, Västernorrland County, Sweden. Until 2015, it was classified by Statistics Sweden as a locality (tätort), with 209 inhabitants in 2010. Since 2015, it is regarded as two smaller localities (småorter), with 92 and 112 inhabitants each in 2015. It is located close to the Höga Kusten Bridge.
References
External links
http://www.utansjö.se/ (only in swedish so far)
Populated places in Härnösand Municipality
Ångermanland |
https://en.wikipedia.org/wiki/Expectation | Expectation or Expectations may refer to:
Science
Expectation (epistemic)
Expected value, in mathematical probability theory
Expectation value (quantum mechanics)
Expectation–maximization algorithm, in statistics
Music
Expectation (album), a 2013 album by Girl's Day
Expectation, a 2006 album by Matt Harding
Expectations (Keith Jarrett album), 1971
Expectations (Dance Exponents album), 1985
Expectations (Hayley Kiyoko album), 2018
"Expectations/Overture", a song from the album
Expectations (Bebe Rexha album), 2018
Expectations (Katie Pruitt album), 2020
"Expectations", a song from the album
"Expectation" (waltz), a 1980 waltz composed by Ilya Herold Lavrentievich Kittler
"Expectation" (song), a 2010 song by Tame Impala
"Expectations" (song), a 2018 song by Lauren Jauregui
"Expectations", a song by Three Days Grace from Transit of Venus, 2012
See also
Great Expectations, a novel by Charles Dickens
Xpectation, 2003 studio album by Prince |
https://en.wikipedia.org/wiki/Euler%20sequence | In mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an -fold sum of the dual of the Serre twisting sheaf.
The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter article for this generalization.)
Statement
Let be the n-dimensional projective space over a commutative ring A. Let be the sheaf of 1-differentials on this space, and so on. The Euler sequence is the following exact sequence of sheaves on :
The sequence can be constructed by defining a homomorphism with and in degree 1, surjective in degrees , and checking that locally on the standard charts, the kernel is isomorphic to the relative differential module.
Geometric interpretation
We assume that A is a field k.
The exact sequence above is dual to the sequence
,
where is the tangent sheaf of .
Let us explain the coordinate-free version of this sequence, on for an -dimensional vector space V over k:
This sequence is most easily understood by interpreting sections of the central term as 1-homogeneous vector fields on V. One such section, the Euler vector field, associates to each point of the variety the tangent vector . This vector field is radial in the sense that it vanishes uniformly on 0-homogeneous functions, that is, the functions that are invariant by homothetic rescaling, or "independent of the radial coordinate".
A function (defined on some open set) on gives rise by pull-back to a 0-homogeneous function on V (again partially defined). We obtain 1-homogeneous vector fields by multiplying the Euler vector field by such functions. This is the definition of the first map, and its injectivity is immediate.
The second map is related to the notion of derivation, equivalent to that of vector field. Recall that a vector field on an open set U of the projective space can be defined as a derivation of the functions defined on this open set. Pulled-back in V, this is equivalent to a derivation on the preimage of U that preserves 0-homogeneous functions. Any vector field on can be thus obtained, and the defect of injectivity of this mapping consists precisely of the radial vector fields.
Therefore the kernel of the second morphism equals the image of the first one.
The canonical line bundle of projective spaces
By taking the highest exterior power, one sees that the canonical sheaf of a projective space is given by In particular, projective spaces are Fano varieties, because the canonical bundle is anti-ample and this line bundle has no non-zero global sections, so the geometric genus is 0. This can be found by looking at the Euler sequence and plugging it into the determinant formula for any short exact sequence of the form .
Chern classes
The Euler sequence can be used to compute the Chern classes of projective space. Recall that given a short exact sequence of coherent |
https://en.wikipedia.org/wiki/List%20of%20Sporting%20CP%20records%20and%20statistics | This List of Sporting CP records and statistics provides information about Sporting Clube de Portugal, which is a Portuguese sports club based in Lisbon. The club is particularly renowned for its football branch. With more than 100,000 registered club members, Sporting CP is one of the most successful and popular sports clubs in Portugal. Its teams, athletes, and supporters are often nicknamed Os Leões (The Lions).
Honours
Domestic competitions
Primeira Liga
Winners (19): 1940–41, 1943–44, 1946–47, 1947–48, 1948–49, 1950–51, 1951–52, 1952–53, 1953–54, 1957–58, 1961–62, 1965–66, 1969–70, 1973–74, 1979–80, 1981–82, 1999–2000, 2001–02, 2020–21
Runners-up (22): 1934–35, 1938–39, 1939–40, 1941–42, 1942–43, 1944–45, 1949–50, 1959–60, 1960–61, 1967–68, 1970–71, 1976–77, 1984–85, 1994–95, 1996–97, 2005–06, 2006–07, 2007–08, 2008–09, 2013–14, 2015–16, 2021–22
Campeonato de Portugal (*defunct)
Winners (4): 1922–23, 1933–34, 1935–36, 1937–38
Runners-up (6): 1922, 1924–25, 1927–28, 1932–33, 1934–35, 1936–37Taça de Portugal
Winners (17): 1940–41, 1944–45, 1945–46, 1947–48, 1953–54, 1962–63, 1970–71, 1972–73, 1973–74, 1977–78, 1981–82, 1994–95, 2001–02, 2006–07, 2007–08, 2014–15, 2018–19
Runners-up (12): 1951–52, 1954–55, 1959–60, 1969–70, 1971–72, 1978–79, 1986–87, 1993–94, 1995–96, 1999–2000, 2011–12, 2017–18Taça da Liga
Winners (4): 2017–18, 2018–19, 2020–21, 2021–22
Runners-up (3): 2007–08, 2008–09, 2022–23Supertaça Cândido de Oliveira
Winners (9): 1982, 1987, 1995, 2000, 2002, 2007, 2008, 2015, 2021
Runners-up (2): 1980, 2019Taça Império Winners (1): 1943–44Taça Monumental "O Século" Winners (2): 1948, 1953
Regional competitionsCampeonato de Lisboa Winners (19): 1915, 1919, 1922, 1923, 1925, 1928, 1931, 1934, 1935, 1936, 1937, 1938, 1939, 1941, 1942, 1943, 1945, 1947, 1948Taça de Honra Winners (13): 1915, 1916, 1917, 1948, 1962, 1964, 1966, 1971, 1985, 1991, 1992, 2014, 2015Campeonato de reservas Winners (16): 1934–35, 1937–38, 1939–40, 1941–42, 1958–59, 1959–60, 1960–61, 1961–62, 1963–64, 1966–67, 1967–68, 1968–69, 1973–74, 1983–84, 1984–85, 1985–86Campeonatos da 2ª Categoria Winners (3): 1934–35, 1940–41, 1945–46Campeonatos da 3ª CategoriaWinners (2): 1923–24, 1930–31Campeonatos da 4ª CategoriaWinners (2): 1911–12, 1912–13Taça Mutilados de GuerraWinners (1): 1917–18 (Provisional)Taça LisboaWinners (1): 1930–31
European competitionsEuropean Cup Winners' Cup Winners (1): 1963–64Intertoto Cup Winners (1): 1968 Intertoto CupIberian Cup Winners (1): 2000Small club World Cup Winners (1): 1981UEFA Cup Runners-up (1): 2004–05Latin Cup Runners-up (1): 1949
Friendly competitionsTeresa Herrera Trophy Winners (1): 1961Iberian Trophy (Badajoz, Spain) Winners (2): 1967, 1970
Runners-up (1): 2005Trofeo Internacional Montilla-Moriles (Córdoba, Spain) Winners (1): 1969Tournament of Bulgaria Winners (1): 1981Tournament City San Sebastián Winners (1): 1991Trofeo Ciudad de Vigo Winners (1): 2001
Runners-up (1): 1977Guadiana Trophy Winners (3): 2005, |
https://en.wikipedia.org/wiki/Rado%C5%A1%20Proti%C4%87 | Radoš Protić (; born 31 January 1987) is a Serbian footballer who plays for Radnički Sremska Mitrovica.
Club career
On 28 June 2021, he signed with Železničar Pančevo.
Career statistics
Honours
Sarajevo
Premier League of Bosnia and Herzegovina: 2014–15
References
External links
Radoš Protić Stats at Utakmica.rs
1987 births
Sportspeople from Sremska Mitrovica
Footballers from Srem District
Living people
Serbian men's footballers
Men's association football defenders
FK Rad players
FK Teleoptik players
FK Leotar players
FK Mačva Šabac players
FK Jagodina players
FC Oleksandriya players
FK Novi Pazar players
FK Sarajevo players
FK Mladost Lučani players
Kisvárda FC players
FK Inđija players
FK Železničar Pančevo players
FK Radnički Sremska Mitrovica players
Serbian SuperLiga players
Premier League of Bosnia and Herzegovina players
Ukrainian Premier League players
Ukrainian First League players
Nemzeti Bajnokság I players
Serbian First League players
Serbian expatriate men's footballers
Expatriate men's footballers in Bosnia and Herzegovina
Serbian expatriate sportspeople in Bosnia and Herzegovina
Expatriate men's footballers in Ukraine
Serbian expatriate sportspeople in Ukraine
Expatriate men's footballers in Hungary
Serbian expatriate sportspeople in Hungary |
https://en.wikipedia.org/wiki/Lumpability | In probability theory, lumpability is a method for reducing the size of the state space of some continuous-time Markov chains, first published by Kemeny and Snell.
Definition
Suppose that the complete state-space of a Markov chain is divided into disjoint subsets of states, where these subsets are denoted by ti. This forms a partition of the states. Both the state-space and the collection of subsets may be either finite or countably infinite.
A continuous-time Markov chain is lumpable with respect to the partition T if and only if, for any subsets ti and tj in the partition, and for any states n,n’ in subset ti,
where q(i,j) is the transition rate from state i to state j.
Similarly, for a stochastic matrix P, P is a lumpable matrix on a partition T if and only if, for any subsets ti and tj in the partition, and for any states n,n’ in subset ti,
where p(i,j) is the probability of moving from state i to state j.
Example
Consider the matrix
and notice it is lumpable on the partition t = {(1,2),(3,4)} so we write
and call Pt the lumped matrix of P on t.
Successively lumpable processes
In 2012, Katehakis and Smit discovered the Successively Lumpable processes for which the stationary probabilities can be obtained by successively computing the stationary probabilities of a propitiously constructed sequence of Markov chains. Each of the latter chains has a (typically much) smaller state space and this yields significant computational improvements. These results have many applications reliability and queueing models and problems.
Quasi–lumpability
Franceschinis and Muntz introduced quasi-lumpability, a property whereby a small change in the rate matrix makes the chain lumpable.
See also
Nearly completely decomposable Markov chain
References
Markov processes |
https://en.wikipedia.org/wiki/Markov%20kernel | In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite state space.
Formal definition
Let and be measurable spaces. A Markov kernel with source and target is a map with the following properties:
For every (fixed) , the map is -measurable
For every (fixed) , the map is a probability measure on
In other words it associates to each point a probability measure on such that, for every measurable set , the map is measurable with respect to the -algebra .
Examples
Simple random walk on the integers
Take , and (the power set of ). Then a Markov kernel is fully determined by the probability it assigns to singletons for each :
.
Now the random walk that goes to the right with probability and to the left with probability is defined by
where is the Kronecker delta. The transition probabilities for the random walk are equivalent to the Markov kernel.
General Markov processes with countable state space
More generally take and both countable and .
Again a Markov kernel is defined by the probability it assigns to singleton sets for each
,
We define a Markov process by defining a transition probability where the numbers define a (countable) stochastic matrix i.e.
We then define
.
Again the transition probability, the stochastic matrix and the Markov kernel are equivalent reformulations.
Markov kernel defined by a kernel function and a measure
Let be a measure on , and a measurable function with respect to the product -algebra such that
,
then i.e. the mapping
defines a Markov kernel. This example generalises the countable Markov process example where was the counting measure. Moreover it encompasses other important examples such as the convolution kernels, in particular the Markov kernels defined by the heat equation. The latter example includes the Gaussian kernel on with standard Lebesgue measure and
.
Measurable functions
Take and arbitrary measurable spaces, and let be a measurable function. Now define i.e.
for all .
Note that the indicator function is -measurable for all iff is measurable.
This example allows us to think of a Markov kernel as a generalised function with a (in general) random rather than certain value. That is, it is a multivalued function where the values are not equally weighted.
Galton–Watson process
As a less obvious example, take , and the real numbers with the standard sigma algebra of Borel sets. Then
where is the number of element at the state , are i.i.d. random variables (usually with mean 0) and where is the indicator function. For the simple case of coin flips this models the different levels of a Galton board.
Composition of Markov Kernels and the Markov Category
Given measurable spaces , we consider a Markov kernel as a morphism . Intuitively, ra |
https://en.wikipedia.org/wiki/Homogeneous%20system | Homogeneous system:
Homogeneous system of linear algebraic equations
System of homogeneous differential equations
System of homogeneous first-order differential equations
System of homogeneous linear differential equations
in physics |
https://en.wikipedia.org/wiki/Imaginary%20hyperelliptic%20curve | A hyperelliptic curve is a particular kind of algebraic curve.
There exist hyperelliptic curves of every genus . If the genus of a hyperelliptic curve equals 1, we simply call the curve an elliptic curve. Hence we can see hyperelliptic curves as generalizations of elliptic curves. There is a well-known group structure on the set of points lying on an elliptic curve over some field , which we can describe geometrically with chords and tangents. Generalizing this group structure to the hyperelliptic case is not straightforward. We cannot define the same group law on the set of points lying on a hyperelliptic curve, instead a group structure can be defined on the so-called Jacobian of a hyperelliptic curve. The computations differ depending on the number of points at infinity. Imaginary hyperelliptic curves are hyperelliptic curves with exactly 1 point at infinity: real hyperelliptic curves have two points at infinity.
Formal definition
Hyperelliptic curves can be defined over fields of any characteristic. Hence we consider an arbitrary field and its algebraic closure . An (imaginary) hyperelliptic curve of genus over is given by an equation of the form
where is a polynomial of degree not larger than and is a monic polynomial of degree . Furthermore, we require the curve to have no singular points. In our setting, this entails that no point satisfies both and the equations and . This definition differs from the definition of a general hyperelliptic curve in the fact that can also have degree in the general case. From now on we drop the adjective imaginary and simply talk about hyperelliptic curves, as is often done in literature. Note that the case corresponds to being a cubic polynomial, agreeing with the definition of an elliptic curve. If we view the curve as lying in the projective plane with coordinates , we see that there is a particular point lying on the curve, namely the point at infinity denoted by . So we could write .
Suppose the point not equal to lies on the curve and consider . As can be simplified to , we see that is also a point on the curve. is called the opposite of and is called a Weierstrass point if , i.e. . Furthermore, the opposite of is simply defined as .
Alternative definition
The definition of a hyperelliptic curve can be slightly simplified if we require that the characteristic of is not equal to 2. To see this we consider the change of variables and , which makes sense if char. Under this change of variables we rewrite to which, in turn, can be rewritten to . As we know that and hence is a monic polynomial of degree . This means that over a field with char every hyperelliptic curve of genus is isomorphic to one given by an equation of the form where is a monic polynomial of degree and the curve has no singular points. Note that for curves of this form it is easy to check whether the non-singularity criterion is met. A point on the curve is singular if and only if and . As an |
https://en.wikipedia.org/wiki/Juhor%20ad-Dik | Juhor ad-Dik () is a Palestinian village in the Gaza Governorate, south of Gaza City, in the central Gaza Strip. According to the Palestinian Central Bureau of Statistics (PCBS), the village had a population of 4,586 inhabitants in 2017. In the 1997 census by the PCBS, Palestinian refugees made up 72.3% of the population which at the time was 2,275.
Thirteen residents, including at least two children were killed and 18 others wounded, some seriously, when Israeli tanks fired on a mosque and two houses at the village on April 16, 2008. During Israel's invasion of the Gaza Strip in 2008-09, Israeli troops and Palestinian militiamen battled frequently in Juhor ad-Dik. Gaza City's main garbage dump is in the Juhor ad-Dik area. But because of the Operation Swords of Iron bombing, the garbage men were unable to pick up thousands of tons of garbage. On 28 October, 2023, (when nighttime lasts an hour longer because daylight savings ends) an IDF batallion or brigade maneuvered into Beit Hanoun in the northeast corner of the Gaza Strip & another raid maneuvered into the Strip, east of al-Bureij refugee camp, in the Juhor ad-Dik area.
References
Villages in the Gaza Strip
Municipalities of the State of Palestine |
https://en.wikipedia.org/wiki/Buekenhout%20geometry | In mathematics, a Buekenhout geometry or diagram geometry is a generalization of projective spaces, Tits buildings, and several other geometric structures, introduced by .
Definition
A Buekenhout geometry consists of a set X whose elements are called "varieties", with a symmetric reflexive relation on X called "incidence", together with a function τ called the "type map" from X to a set Δ whose elements are called "types" and whose size is called the "rank". Two distinct varieties of the same type cannot be incident.
A flag is a subset of X such that any two elements of the flag are incident.
The Buekenhout geometry has to satisfy the following axiom:
Every flag is contained in a flag with exactly one variety of each type.
Example: X is the linear subspaces of a projective space with two subspaces incident if one is contained in the other, Δ is the set of possible dimensions of linear subspaces, and the type map takes a linear subspace to its dimension. A flag in this case is a chain of subspaces, and each flag is contained in a so-called complete flag.
If F is a flag, the residue of F consists of all elements of X that are not in F but are incident with all elements of F. The residue of a flag forms a Buekenhout geometry in the obvious way, whose type are the types of X that are not types of F. A geometry is said to have some property residually if every residue of rank at least 2 has the property. In particular a geometry is called residually connected if every residue of rank at least 2 is connected (for the incidence relation).
Diagrams
The diagram of a Buekenhout geometry has a point for each type, and two points x, y are connected with a line labeled to indicate what sort of geometry the rank 2 residues of type {x,y} have as follows.
If the rank 2 residue is a digon, meaning any variety of type x is incident with every variety of type y, then the line from x to y is omitted. (This is the most common case.)
If the rank 2 residue is a projective plane, then the line from x to y is not labelled. This is the next most common case.
If the rank 2 residue is a more complicated geometry, the line is labelled by some symbol, which tends to vary from author to author.
References
External links
Incidence geometry
Group theory
Algebraic combinatorics
Geometric group theory |
https://en.wikipedia.org/wiki/Buekenhout | Buekenhout may refer to:
Francis Buekenhout, a Belgian mathematician;
Buekenhout geometry, or Buekenhout–Tits geometry, a generalization of projective spaces, Tits buildings, and several other geometric structures, introduced in 1979 by Francis Buekenhout. |
https://en.wikipedia.org/wiki/P%C3%A1l%20L%C3%A1z%C3%A1r | Pál Lázár (born 11 March 1988) is a Hungarian former football player.
Club statistics
Updated to games played as of 7 April 2018.
External links
Profile at magyarfutball.hu
Profile
1988 births
Living people
Footballers from Debrecen
Romanian sportspeople of Hungarian descent
Hungarian men's footballers
Romanian men's footballers
Men's association football defenders
Nemzeti Bajnokság I players
Süper Lig players
FC Sopron players
CF Liberty Oradea players
Fehérvár FC players
Samsunspor footballers
Pécsi MFC players
Debreceni VSC players
Diósgyőri VTK players
Mezőkövesdi SE footballers
Hungarian expatriate men's footballers
Expatriate men's footballers in Romania
Expatriate men's footballers in Turkey
Hungarian expatriate sportspeople in Romania
Hungarian expatriate sportspeople in Turkey
Hungary men's international footballers
Hungary men's under-21 international footballers |
https://en.wikipedia.org/wiki/MATHLAB | MATHLAB is a computer algebra system created in 1964 by Carl Engelman at MITRE and written in Lisp.
"MATHLAB 68" was introduced in 1967 and became rather popular in university environments running on DECs PDP-6 and PDP-10 under TOPS-10 or TENEX. In 1969 this version was included in the DECUS user group's library (as 10-142) as royalty-free software.
Carl Engelman left MITRE for Symbolics where he contributed his expert knowledge in the development of Macsyma.
Features
Abstract from DECUS Library Catalog:
MATHLAB is an on-line system providing machine aid for the mechanical symbolic processes encountered in analysis. It is capable of performing, automatically and symbolically, such common procedures as simplification, substitution, differentiation, polynomial factorization, indefinite integration, direct and inverse Laplace transforms, the solution of linear differential equations with constant coefficients, the solution of simultaneous linear equations, and the inversion of matrices. It also supplies fairly elaborate bookkeeping facilities appropriate to its on-line operation.
Applications
MATHLAB 68 has been used to solve electrical linear circuits using an acausal modeling approach for symbolic circuit analysis. This application was developed as a plug-in for MATHLAB 68 (open-source), building on MATHLAB's linear algebra facilities (Laplace transforms, inverse Laplace transforms and linear algebra manipulation).
Print publications
References
Computer algebra systems
Notebook interface |
https://en.wikipedia.org/wiki/List%20of%20Port%20Vale%20F.C.%20records%20and%20statistics | Port Vale F.C. is an English professional association football club based in Burslem, Stoke-on-Trent, Staffordshire, who play in , as of the season. The club was formed in the 1870s, in 1884 they took the name Burslem Port Vale F.C., dropping the 'Burslem' in 1907. They played their home matches at The Old Recreation Ground between 1912 and 1950 and at Vale Park from 1950 to the present day. The club joined the English Football League in 1892 as founder members of the Football League Second Division, resigning in 1907, only to return in 1919. Vale's highest league finish was fifth in the Second Division in 1930–31, whilst they were FA Cup semi-finalists in 1953–54. They competed in Europe in the Anglo-Italian Cup on one occasion and would go on to reach the final, losing 5–2 to Italian club Genoa on 17 March 1996.
Port Vale have won four promotions out of the third tier, going up as champions in 1929–30 and 1953–54, and have won five promotions out of the fourth tier, being crowned champions in 1958–59. They have lifted the Football League Trophy twice, in 1993 and 2001. Two club records are also Football League records: most clean sheets in a season (30 in 46 Third Division North matches in the 1953–54 season) and biggest league defeat (0–10 against Sheffield United on 10 December 1892). Roy Sproson made 842 appearances (760 in the league) for Vale between 1950 and 1972, later becoming manager from January 1974 to October 1977. Wilf Kirkham is the club's record goalscorer with 164 goals in all competitions over two spells between 1923 and 1933, and set the record for most Football League goals in a single season with 38 in the 1926–27 campaign. Gareth Ainsworth is the player Vale have both received and spent the highest sum on in the transfer market: £500,000 was given to Lincoln City in September 1997 and £2 million received from Wimbledon as he departed in October 1998. All top five transfers either in or out were made in the 1990s, before the Bosman ruling and the departure of highly successful manager John Rudge. Chris Birchall is the club record international cap holder with three goals in 27 appearances playing for Trinidad and Tobago between 2001 and 2006, including three appearances in the 2006 World Cup.
Honours and achievements
Football League
Football League Third Division / Third Division North / League One (3rd tier)
Champions: 1929–30, 1953–54
2nd place promotion: 1993–94
Play-off winners: 1988–89
Football League Fourth Division / Third Division / League Two (4th tier)
Champions: 1958–59
3rd place promotion: 1982–83, 2012–13
4th place promotion: 1969–70, 1985–86
Play-off winners: 2021–22
Football League Trophy
Winners: 1993, 2001
Others
North Staffordshire & District League
Champions: 1909–10
The Central League
Runners–up: 1911–12
Anglo-Italian Cup
Runners–up: 1996
Debenhams Cup
Runners–up: 1977
Staffordshire Senior Cup
Winners: 1898, 1912, 1920, 1947, 1949, 1953, 2001
Runners–up: 1900, 1928, 1930, 1948, 1973, 2010, 201 |
https://en.wikipedia.org/wiki/1906%E2%80%9307%20Manchester%20United%20F.C.%20season | The 1906–07 season was Manchester United's 15th season in the Football League.
First Division
FA Cup
Squad statistics
References
Manchester United F.C. seasons
Manchester United |
https://en.wikipedia.org/wiki/List%20of%20national%20capitals%20by%20population | This is a list of national capitals, ordered according to population. Capitals of dependent territories and disputed territories are marked in italics. The population statistics given refer only to the official capital area, and do not include the wider metropolitan/urban district.
Table
* indicates "Cities of COUNTRY or TERRITORY" links.
See also
Capital city
List of countries whose capital is not their largest city
List of capitals outside the territories they serve
List of national capitals by latitude
List of countries and dependencies by population
List of towns and cities with 100,000 or more inhabitants
List of population concern organizations
List of national capitals
List of national capitals by area
Notes
References
Population
Capitals, national
Capitals
Capitals, national |
https://en.wikipedia.org/wiki/Engutoto%2C%20Arusha%20District | Engutoto is an administrative ward in the Arusha District of the Arusha Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,287 people in the ward, from 6,970 in 2012.
References
Arusha District
Wards of Arusha Region |
https://en.wikipedia.org/wiki/Cylindrical%20%CF%83-algebra | In mathematics — specifically, in measure theory and functional analysis — the cylindrical σ-algebra or product σ-algebra is a type of σ-algebra which is often used when studying product measures or probability measures of random variables on Banach spaces.
For a product space, the cylinder σ-algebra is the one that is generated by cylinder sets.
In the context of a Banach space the cylindrical σ-algebra is defined to be the coarsest σ-algebra (that is, the one with the fewest measurable sets) such that every continuous linear function on is a measurable function. In general, is not the same as the Borel σ-algebra on which is the coarsest σ-algebra that contains all open subsets of
See also
References
(See chapter 2)
Banach spaces
Functional analysis
Measure theory |
https://en.wikipedia.org/wiki/Blade%20%28geometry%29 | In the study of geometric algebras, a -blade or a simple -vector is a generalization of the concept of scalars and vectors to include simple bivectors, trivectors, etc. Specifically, a -blade is a -vector that can be expressed as the exterior product (informally wedge product) of 1-vectors, and is of grade .
In detail:
A 0-blade is a scalar.
A 1-blade is a vector. Every vector is simple.
A 2-blade is a simple bivector. Sums of 2-blades are also bivectors, but not always simple. A 2-blade may be expressed as the wedge product of two vectors and :
A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors , , and :
In a vector space of dimension , a blade of grade is called a pseudovector or an antivector.
The highest grade element in a space is called a pseudoscalar, and in a space of dimension is an -blade.
In a vector space of dimension , there are dimensions of freedom in choosing a -blade for , of which one dimension is an overall scaling multiplier.
A vector subspace of finite dimension may be represented by the -blade formed as a wedge product of all the elements of a basis for that subspace. Indeed, a -blade is naturally equivalent to a -subspace endowed with a volume form (an alternating -multilinear scalar-valued function) normalized to take unit value on the -blade.
Examples
In two-dimensional space, scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades in this context known as pseudoscalars, in that they are elements of a one-dimensional space distinct from regular scalars.
In three-dimensional space, 0-blades are again scalars and 1-blades are three-dimensional vectors, while 2-blades are oriented area elements. In this case 3-blades are called pseudoscalars and represent three-dimensional volume elements, which form a one-dimensional vector space similar to scalars. Unlike scalars, 3-blades transform according to the Jacobian determinant of a change-of-coordinate function.
See also
Grassmannian
Multivector
Exterior algebra
Differential form
Geometric algebra
Clifford algebra
Notes
References
A Lasenby, J Lasenby & R Wareham (2004) A covariant approach to geometry using geometric algebra Technical Report. University of Cambridge Department of Engineering, Cambridge, UK.
External links
A Geometric Algebra Primer, especially for computer scientists.
Geometric algebra
Vector calculus |
https://en.wikipedia.org/wiki/Richard%20J.%20Wood | Richard J. Wood is a mathematics professor at Dalhousie University in Halifax, Nova Scotia, Canada. He graduated from McMaster University in 1972 with his M.Sc. and then later went on to do his Ph.D. at Dalhousie University. He is interested in category theory and lattice theory.
References
Publications
External links
Year of birth missing (living people)
Living people
Canadian mathematicians
Category theorists
Lattice theorists |
https://en.wikipedia.org/wiki/Line%20integral | In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.
The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as have natural continuous analogues in terms of line integrals, in this case which computes the work done on an object moving through an electric or gravitational field along a path
Vector calculus
In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given tensor field along a given curve. For example, the line integral over a scalar field (rank 0 tensor) can be interpreted as the area under the field carved out by a particular curve. This can be visualized as the surface created by and a curve C in the xy plane. The line integral of f would be the area of the "curtain" created—when the points of the surface that are directly over C are carved out.
Line integral of a scalar field
Definition
For some scalar field where , the line integral along a piecewise smooth curve is defined as
where is an arbitrary bijective parametrization of the curve such that and give the endpoints of and . Here, and in the rest of the article, the absolute value bars denote the standard (Euclidean) norm of a vector.
The function is called the integrand, the curve is the domain of integration, and the symbol may be intuitively interpreted as an elementary arc length of the curve (i.e., a differential length of ). Line integrals of scalar fields over a curve do not depend on the chosen parametrization of .
Geometrically, when the scalar field is defined over a plane , its graph is a surface in space, and the line integral gives the (signed) cross-sectional area bounded by the curve and the graph of . See the animation to the right.
Derivation
For a line integral over a scalar field, the integral can be constructed from a Riemann sum using the above definitions of , and a parametrization of . This can be done by partitioning the interval into sub-intervals of length , then denotes some point, call it a sample point, on the curve . We can use the set of sample points to approximate the curve as a polygonal path by introducing the straight line piece between each of the sample points and . (The approximation of a curve to a polygonal path is called rectification of a curve, see here for more det |
https://en.wikipedia.org/wiki/Line%20coordinates | In geometry, line coordinates are used to specify the position of a line just as point coordinates (or simply coordinates) are used to specify the position of a point.
Lines in the plane
There are several possible ways to specify the position of a line in the plane. A simple way is by the pair where the equation of the line is y = mx + b. Here m is the slope and b is the y-intercept. This system specifies coordinates for all lines that are not vertical. However, it is more common and simpler algebraically to use coordinates where the equation of the line is lx + my + 1 = 0. This system specifies coordinates for all lines except those that pass through the origin. The geometrical interpretations of l and m are the negative reciprocals of the x and y-intercept respectively.
The exclusion of lines passing through the origin can be resolved by using a system of three coordinates to specify the line with the equation lx + my + n = 0. Here l and m may not both be 0. In this equation, only the ratios between l, m and n are significant, in other words if the coordinates are multiplied by a non-zero scalar then line represented remains the same. So is a system of homogeneous coordinates for the line.
If points in the real projective plane are represented by homogeneous coordinates , the equation of the line is lx + my + nz = 0, provided In particular, line coordinate represents the line z = 0, which is the line at infinity in the projective plane. Line coordinates and represent the x and y-axes respectively.
Tangential equations
Just as f(x, y) = 0 can represent a curve as a subset of the points in the plane, the equation φ(l, m) = 0 represents a subset of the lines on the plane. The set of lines on the plane may, in an abstract sense, be thought of as the set of points in a projective plane, the dual of the original plane. The equation φ(l, m) = 0 then represents a curve in the dual plane.
For a curve f(x, y) = 0 in the plane, the tangents to the curve form a curve in the dual space called the dual curve. If φ(l, m) = 0 is the equation of the dual curve, then it is called the tangential equation, for the original curve. A given equation φ(l, m) = 0 represents a curve in the original plane determined as the envelope of the lines that satisfy this equation. Similarly, if φ(l, m, n) is a homogeneous function then φ(l, m, n) = 0 represents a curve in the dual space given in homogeneous coordinates, and may be called the homogeneous tangential equation of the enveloped curve.
Tangential equations are useful in the study of curves defined as envelopes, just as Cartesian equations are useful in the study of curves defined as loci.
Tangential equation of a point
A linear equation in line coordinates has the form al + bm + c = 0, where a, b and c are constants. Suppose (l, m) is a line that satisfies this equation. If c is not 0 then lx + my + 1 = 0, where x = a/c and y = b/c, so every line satisfying the original equation passes through the point |
https://en.wikipedia.org/wiki/Product-form%20solution | In probability theory, a product-form solution is a particularly efficient form of solution for determining some metric of a system with distinct sub-components, where the metric for the collection of components can be written as a product of the metric across the different components. Using capital Pi notation a product-form solution has algebraic form
where B is some constant. Solutions of this form are of interest as they are computationally inexpensive to evaluate for large values of n. Such solutions in queueing networks are important for finding performance metrics in models of multiprogrammed and time-shared computer systems.
Equilibrium distributions
The first product-form solutions were found for equilibrium distributions of Markov chains. Trivially, models composed of two or more independent sub-components exhibit a product-form solution by the definition of independence. Initially the term was used in queueing networks where the sub-components would be individual queues. For example, Jackson's theorem gives the joint equilibrium distribution of an open queueing network as the product of the equilibrium distributions of the individual queues. After numerous extensions, chiefly the BCMP network it was thought local balance was a requirement for a product-form solution.
Gelenbe's G-network model was the first to show that this is not the case. Motivated by the need to model biological neurons which have a point-process like spiking behaviour, he introduced the precursor of G-Networks, calling it the random neural network. By introducing "negative customers" which can destroy or eliminate other customers, he generalised the family of product form networks. Then this was further extended in several steps, first by Gelenbe's "triggers" which are customers which have the power of moving other customers from some queue to another. Another new form of customer that also led to product form was Gelenbe's "batch removal". This was further extended by Erol Gelenbe and Jean-Michel Fourneau with customer types called "resets" which can model the repair of failures: when a queue hits the empty state, representing (for instance) a failure, the queue length can jump back or be "reset" to its steady-state distribution by an arriving reset customer, representing a repair. All these previous types of customers in G-Networks can exist in the same network, including with multiple classes, and they all together still result in the product form solution, taking us far beyond the reversible networks that had been considered before.
Product-form solutions are sometimes described as "stations are independent in equilibrium". Product form solutions also exist in networks of bulk queues.
J.M. Harrison and R.J. Williams note that "virtually all of the models that have been successfully analyzed in classical queueing network theory are models having a so-called product-form stationary distribution" More recently, product-form solutions have been published fo |
https://en.wikipedia.org/wiki/Kodaira%20surface | In mathematics, a Kodaira surface is a compact complex surface of Kodaira dimension 0 and odd first Betti number. The concept is named after Kunihiko Kodaira.
These are never algebraic, though they have non-constant meromorphic functions. They are usually divided into two subtypes: primary Kodaira surfaces with trivial canonical bundle, and secondary Kodaira surfaces which are quotients of these by finite groups of orders 2, 3, 4, or 6, and which have non-trivial canonical bundles. The secondary Kodaira surfaces have the same relation to primary ones that Enriques surfaces have to K3 surfaces, or bielliptic surfaces have to abelian surfaces.
Invariants: If the surface is the quotient of a primary Kodaira surface by a group of order k = 1,2,3,4,6, then the plurigenera Pn are 1 if n is divisible by k and 0 otherwise.
Hodge diamond:
Examples: Take a non-trivial line bundle over an elliptic curve, remove the zero section, then quotient out the fibers by Z acting as multiplication by powers of some complex number z.
This gives a primary Kodaira surface.
References
– the standard reference book for compact complex surfaces
Complex surfaces |
https://en.wikipedia.org/wiki/Gordon%E2%80%93Newell%20theorem | In queueing theory, a discipline within the mathematical theory of probability, the Gordon–Newell theorem is an extension of Jackson's theorem from open queueing networks to closed queueing networks of exponential servers where customers cannot leave the network. Jackson's theorem cannot be applied to closed networks because the queue length at a node in the closed network is limited by the population of the network. The Gordon–Newell theorem calculates the open network solution and then eliminates the infeasible states by renormalizing the probabilities. Calculation of the normalizing constant makes the treatment more awkward as the whole state space must be enumerated. Buzen's algorithm or mean value analysis can be used to calculate the normalizing constant more efficiently.
Definition of a Gordon–Newell network
A network of m interconnected queues is known as a Gordon–Newell network or closed Jackson network if it meets the following conditions:
the network is closed (no customers can enter or leave the network),
all service times are exponentially distributed and the service discipline at all queues is FCFS,
a customer completing service at queue i will move to queue j with probability , with the such that ,
the utilization of all of the queues is less than one.
Theorem
In a closed Gordon–Newell network of m queues, with a total population of K individuals, write (where ki is the length of queue i) for the state of the network and S(K, m) for the state space
Then the equilibrium state probability distribution exists and is given by
where service times at queue i are exponentially distributed with parameter μi. The normalizing constant G(K) is given by
and ei is the visit ratio, calculated by solving the simultaneous equations
See also
BCMP network
References
Probability theorems
Queueing theory |
https://en.wikipedia.org/wiki/Arrival%20theorem | In queueing theory, a discipline within the mathematical theory of probability, the arrival theorem (also referred to as the random observer property, ROP or job observer property) states that "upon arrival at a station, a job observes the system as if in steady state at an arbitrary instant for the system without that job."
The arrival theorem always holds in open product-form networks with unbounded queues at each node, but it also holds in more general networks. A necessary and sufficient condition for the arrival theorem to be satisfied in product-form networks is given in terms of Palm probabilities in Boucherie & Dijk, 1997. A similar result also holds in some closed networks. Examples of product-form networks where the arrival theorem does not hold include reversible Kingman networks and networks with a delay protocol.
Mitrani offers the intuition that "The state of node i as seen by an incoming job has a different distribution from the state seen by a random observer. For instance, an incoming job can never see all 'k jobs present at node i, because it itself cannot be among the jobs already present."
Theorem for arrivals governed by a Poisson process
For Poisson processes the property is often referred to as the PASTA property (Poisson Arrivals See Time Averages) and states that the probability of the state as seen by an outside random observer is the same as the probability of the state seen by an arriving customer. The property also holds for the case of a doubly stochastic Poisson process where the rate parameter is allowed to vary depending on the state.
Theorem for Jackson networks
In an open Jackson network with m queues, write for the state of the network. Suppose is the equilibrium probability that the network is in state . Then the probability that the network is in state immediately before an arrival to any node is also .
Note that this theorem does not follow from Jackson's theorem, where the steady state in continuous time is considered. Here we are concerned with particular points in time, namely arrival times. This theorem first published by Sevcik and Mitrani in 1981.
Theorem for Gordon–Newell networks
In a closed Gordon–Newell network with m queues, write for the state of the network. For a customer in transit to state , let denote the probability that immediately before arrival the customer 'sees' the state of the system to be
This probability, , is the same as the steady state probability for state for a network of the same type with one customer less. It was published independently by Sevcik and Mitrani, and Reiser and Lavenberg, where the result was used to develop mean value analysis.
Notes
Queueing theory
Probability theorems |
https://en.wikipedia.org/wiki/Covering%20problem%20of%20Rado | The covering problem of Rado is an unsolved problem in geometry concerning covering planar sets by squares. It was formulated in 1928 by Tibor Radó and has been generalized to more general shapes and higher dimensions by Richard Rado.
Formulation
In a letter to Wacław Sierpiński, motivated by some results of Giuseppe Vitali, Tibor Radó observed that for every covering of a unit interval, one can select a subcovering consisting of pairwise disjoint intervals with total length at least 1/2 and that this number cannot be improved. He then asked for an analogous statement in the plane.
If the area of the union of a finite set of squares in the plane with parallel sides is one, what is the guaranteed maximum total area of a pairwise disjoint subset?
Radó proved that this number is at least 1/9 and conjectured that it is at least 1/4 a constant which cannot be further improved. This assertion was proved for the case of equal squares independently by A. Sokolin, R. Rado, and V. A. Zalgaller. However, in 1973, Miklós Ajtai disproved Radó's conjecture, by constructing a system of squares of two different sizes for which any subsystem consisting of disjoint squares covers the area at most of the total area covered by the system.
Upper and lower bounds
Problems analogous to Tibor Radó's conjecture but involving other shapes were considered by Richard Rado starting in late 1940s. A typical setting is a finite family of convex figures in the Euclidean space Rd that are homothetic to a given X, for example, a square as in the original question, a disk, or a d-dimensional cube. Let
where S ranges over finite families just described, and for a given family S, I ranges over all subfamilies that are independent, i.e. consist of disjoint sets, and bars denote the total volume (or area, in the plane case). Although the exact value of F(X) is not known for any two-dimensional convex X, much work was devoted to establishing upper and lower bounds in various classes of shapes. By considering only families consisting of sets that are parallel and congruent to X, one similarly defines f(X), which turned out to be much easier to study. Thus, R. Rado proved that if X is a triangle, f(X) is exactly 1/6 and if X is a centrally symmetric hexagon, f(X) is equal to 1/4.
In 2008, Sergey Bereg, Adrian Dumitrescu, and Minghui Jiang established new bounds for various F(X) and f(X) that improve upon earlier results of R. Rado and V. A. Zalgaller. In particular, they proved that
and that for any convex planar X.
References
; preliminary announcement in SWAT 2008,
Covering lemmas
Discrete geometry
Unsolved problems in geometry |
https://en.wikipedia.org/wiki/Commentarii%20Mathematici%20Helvetici | The Commentarii Mathematici Helvetici is a quarterly peer-reviewed scientific journal in mathematics. The Swiss Mathematical Society started the journal in 1929 after a meeting in May of the previous year. The Swiss Mathematical Society still owns and operates the journal; the publishing is currently handled on its behalf by the European Mathematical Society. The scope of the journal includes research articles in all aspects in mathematics. The editors-in-chief have been Rudolf Fueter (1929–1949), J.J. Burckhardt (1950–1981), P. Gabriel (1982–1989), H. Kraft (1990–2005), and Eva Bayer-Fluckiger (2006–present).
Abstracting and indexing
The journal is abstracted and indexed in:
According to the Journal Citation Reports, the journal has a 2019 impact factor of 0.854.
References
External links
Mathematics journals
Academic journals established in 1929
English-language journals
Quarterly journals
European Mathematical Society academic journals |
https://en.wikipedia.org/wiki/End%20extension | In model theory and set theory, which are disciplines within mathematics, a model of some axiom system of set theory in the language of set theory is an end extension of , in symbols , if
is a substructure of , (i.e., and ), and
whenever and hold, i.e., no new elements are added by to the elements of .
The second condition can be equivalently written as for all .
For example, is an end extension of if and are transitive sets, and .
A related concept is that of a top extension (also known as rank extension), where a model is a top extension of a model if and for all and , we have , where denotes the rank of a set.
Existence
Keisler and Morley showed that every countable model of ZF has an end extension of which it is an elementary substructure. If the elementarity requirement is weakened to being elementary for formulae that are on the Lévy hierarchy, every countable structure in which -collection holds has a -elementary end extension.
References
Mathematical logic
Model theory
Set theory |
https://en.wikipedia.org/wiki/Imre%20Z.%20Ruzsa | Imre Z. Ruzsa (born 23 July 1953) is a Hungarian mathematician specializing in number theory.
Life
Ruzsa participated in the International Mathematical Olympiad for Hungary, winning a silver medal in 1969, and two consecutive gold medals with perfect scores in 1970 and 1971. He graduated from the Eötvös Loránd University in 1976. Since then he has been at the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences. He was awarded the Rollo Davidson Prize in 1988. He was elected corresponding member (1998) and member (2004) of the Hungarian Academy of Sciences. He was invited speaker at the European Congress of Mathematics at Stockholm, 2004, and in the Combinatorics section of the International Congress of Mathematicians in Madrid, 2006. In 2012 he became a fellow of the American Mathematical Society.
Work
With Endre Szemerédi he proved subquadratic upper and lower bounds for the Ruzsa–Szemerédi problem on the number of triples of points in which the union of any three triples contains at least seven points. He proved that an essential component has at least (log x)1+ε elements up to x, for some ε > 0. On the other hand, for every ε > 0 there is an essential component that has at most (log x)1+ε elements up to x, for every x. He gave a new proof to Freiman's theorem. Ruzsa also showed the existence of a Sidon sequence which has at least x0.41 elements up to x.
In a result complementing the Erdős–Fuchs theorem he showed that there exists a sequence a0, a1, ... of natural numbers such that for every n the number of solutions of the inequality ai + aj ≤ n is cn + O(n1/4log n) for some c > 0.
Selected publications
See also
Ruzsa triangle inequality
Plünnecke–Ruzsa inequality
References
External links
Some of Ruzsa's papers at the Rényi Institute
1953 births
Living people
Number theorists
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Members of the Hungarian Academy of Sciences
Fellows of the American Mathematical Society
International Mathematical Olympiad participants |
https://en.wikipedia.org/wiki/Portugaliae%20Mathematica | Portugaliae Mathematica is a peer-reviewed scientific journal published by the European Mathematical Society on behalf of the Portuguese Mathematical Society. It covers all branches of mathematics. The journal was established in 1937, by António Aniceto Monteiro, its first editor-in-chief. The journal is abstracted and indexed in Zentralblatt MATH, Mathematical Reviews, the Science Citation Index Expanded, and Current Contents/Physical, Chemical & Earth Sciences. The current editor-in-chief is José Francisco Rodrigues (Universidade de Lisboa).
References
External links
Mathematics journals
Academic journals established in 1937
European Mathematical Society academic journals
Quarterly journals
English-language journals |
https://en.wikipedia.org/wiki/1893%E2%80%9394%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1893–94.
Overview
Celtic became Scottish Division One champions. Renton were relegated, Dundee and Leith Athletic re-elected to Division One.
Clyde were elected to Division One, Hibernian and Cowlairs remained in the Scottish Division Two.
Port Glasgow Athletic were docked seven points for fielding an ineligible player.
Scottish League Division One
Scottish League Division Two
See also
1893–94 in Scottish football
1893-94 |
https://en.wikipedia.org/wiki/Sarawut%20Treephan | Sarawut Treephan (Thai สระราวุฒิ ตรีพันธ์) is a Thai professional football manager and former player.
Managerial statistics
A win or loss by penalty shoot-out is counted as a draw.
Honours
Player
International
Thailand U 23
Sea Games Gold medal (1) : 2001
Club
Bangkok Christian College F.C.
Thai Division 1 League Winners (1) ; 2001
Chula United
Thai Division 2 League Winners (1) ; 2006
Manager
Songkhla
Thai League 3 Southern Region
Winners : 2022–23
Individual
Thai League 1 Coach of the Month: October 2020
References
1979 births
Living people
Sarawut Treephan
Sarawut Treephan
Footballers at the 2002 Asian Games
Sarawut Treephan
SEA Games medalists in football
Men's association football fullbacks
Competitors at the 2001 SEA Games
Sarawut Treephan
Sarawut Treephan
Sarawut Treephan
Sarawut Treephan |
https://en.wikipedia.org/wiki/Pengfei%20Guan | Pengfei Guan is a Canadian mathematician and Canada Research Chair in Geometric Analysis. He is a professor of mathematics at McGill University and a Fellow of the Royal Society of Canada.
Biography
Guan graduated from the Department of Mathematics of Zhejiang University in Hangzhou. From 1982 to 1984, Guan was a graduate student at the Institute of Mathematics of the Chinese Academy of Sciences in Beijing. From 1984 to 1985, Guan studied at the University of North Carolina at Chapel Hill. Guan later continued his studies at Princeton University. Guan obtained his MS in 1986 and his PhD in 1989, both in mathematics from Princeton University.
Guan was an assistant professor (from 1989 to 1993), associate professor (from 1993 to 1997), and professor (from 1997 to 2004), all at the Department of Mathematics at McMaster University. Since 2004, Guan has been a professor of mathematics at McGill University.
Guan was awarded the Alfred P. Sloan Fellowship from 1993 to 1995. Guan has held the Canada Research Chair since 2004. Guan was elected to the Fellow of Royal Society of Canada in 2008.
References
External links
The Mathematics Genealogy Project - Pengfei Guan
管鹏飞当选为加拿大皇家科学院院士
Academic Personnel at McGill University - Pengfei Guan
Home Page of Pengfei Guan at McGill University
Living people
Zhejiang University alumni
Princeton University alumni
Academic staff of McMaster University
Academic staff of McGill University
20th-century Chinese mathematicians
21st-century Canadian mathematicians
Fellows of the Royal Society of Canada
Canada Research Chairs
Canadian people of Chinese descent
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Surgery%20structure%20set | In mathematics, the surgery structure set is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold X. It is a concept which helps to answer the question whether two homotopy equivalent manifolds are diffeomorphic (or PL-homeomorphic or homeomorphic). There are different versions of the structure set depending on the category (DIFF, PL or TOP) and whether Whitehead torsion is taken into account or not.
Definition
Let X be a closed smooth (or PL- or topological) manifold of dimension n. We call two homotopy equivalences from closed manifolds of dimension to () equivalent if there exists a cobordism together with a map such that , and are homotopy equivalences.
The structure set is the set of equivalence classes of homotopy equivalences from closed manifolds of dimension n to X.
This set has a preferred base point: .
There is also a version which takes Whitehead torsion into account. If we require in the definition above the homotopy equivalences F, and to be simple homotopy equivalences then we obtain the simple structure set .
Remarks
Notice that in the definition of resp. is an h-cobordism resp. an s-cobordism. Using the s-cobordism theorem we obtain another description for the simple structure set , provided that n>4: The simple structure set is the set of equivalence classes of homotopy equivalences from closed manifolds of dimension n to X with respect to the following equivalence relation. Two homotopy equivalences (i=0,1) are equivalent if there exists a
diffeomorphism (or PL-homeomorphism or homeomorphism) such that is homotopic to .
As long as we are dealing with differential manifolds, there is in general no canonical group structure on . If we deal with topological manifolds, it is possible to endow with a preferred structure of an abelian group (see chapter 18 in the book of Ranicki).
Notice that a manifold M is diffeomorphic (or PL-homeomorphic or homeomorphic) to a closed manifold X if and only if there exists a simple homotopy equivalence whose equivalence class is the base point in . Some care is necessary because it may be possible that a given simple homotopy equivalence is not homotopic to a diffeomorphism (or PL-homeomorphism or homeomorphism) although M and X are diffeomorphic (or PL-homeomorphic or homeomorphic). Therefore, it is also necessary to study the operation of the group of homotopy classes of simple self-equivalences of X on .
The basic tool to compute the simple structure set is the surgery exact sequence.
Examples
Topological Spheres: The generalized Poincaré conjecture in the topological category says that only consists of the base point. This conjecture was proved by Smale (n > 4), Freedman (n = 4) and Perelman (n = 3).
Exotic Spheres: The classification of exotic spheres by Kervaire and Milnor gives for n > 4 (smooth category).
References
External links
Andrew Ranicki's homepage
Shmuel Weinberger's homepage
Geometric topology
Al |
https://en.wikipedia.org/wiki/Elliptic%20divisibility%20sequence | In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by Morgan Ward
in the 1940s. They attracted only sporadic attention until around 2000, when EDS were taken up as a class of nonlinear recurrences that are more amenable to analysis than most such sequences. This tractability is due primarily to the close connection between EDS and elliptic curves. In addition to the intrinsic interest that EDS have within number theory, EDS have applications to other areas of mathematics including logic and cryptography.
Definition
A (nondegenerate) elliptic divisibility sequence (EDS) is a sequence of integers
defined recursively by four initial values
, , , ,
with ≠ 0 and with subsequent values determined by the formulas
It can be shown that if divides each of , , and if further divides , then every term in the sequence is an integer.
Divisibility property
An EDS is a divisibility sequence in the sense that
In particular, every term in an EDS is divisible by , so
EDS are frequently normalized to have = 1 by dividing every term by the initial term.
Any three integers , ,
with divisible by lead to a normalized EDS on setting
It is not obvious, but can be proven, that the condition | suffices to ensure that every term
in the sequence is an integer.
General recursion
A fundamental property of elliptic divisibility sequences
is that they satisfy the general recursion relation
(This formula is often applied with = 1 and = 1.)
Nonsingular EDS
The discriminant of a normalized EDS is the quantity
An EDS is nonsingular if its discriminant is nonzero.
Examples
A simple example of an EDS is the sequence of natural numbers 1, 2, 3,... . Another interesting example is 1, 3, 8, 21, 55, 144, 377, 987,... consisting of every other term in the Fibonacci sequence, starting with the second term. However, both of these sequences satisfy a linear recurrence and both are singular EDS. An example of a nonsingular EDS is
Periodicity of EDS
A sequence is said to be periodic
if there is a number so
that = for every ≥ 1.
If a nondegenerate EDS
is periodic, then one of its terms vanishes. The smallest ≥ 1 with = 0 is called the rank of apparition of the EDS. A deep theorem of Mazur
implies that if the rank of apparition of an EDS is finite, then it satisfies ≤ 10 or = 12.
Elliptic curves and points associated to EDS
Ward proves that associated to any nonsingular EDS ()
is an elliptic curve /Q and a point
ε (Q) such that
Here ψ is the
division polynomial
of ; the roots of ψ are the
nonzero points of order on . There is
a complicated formula
for and in terms of , , , and .
There is an alternative definition of EDS that directly uses elliptic curves and yields a sequence which, up to sign, almost satisfies the EDS recursion. This definitio |
https://en.wikipedia.org/wiki/Normal%20invariant | In mathematics, a normal map is a concept in geometric topology due to William Browder which is of fundamental importance in surgery theory. Given a Poincaré complex X (more geometrically a Poincaré space), a normal map on X endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular, X has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from a manifold M to X matching the fundamental classes and preserving normal bundle information. If the dimension of X is 5 there is then only the algebraic topology surgery obstruction due to C. T. C. Wall to X actually being homotopy equivalent to a closed manifold. Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by Sergei Novikov.
The cobordism classes of normal maps on X are called normal invariants. Depending on the category of manifolds (differentiable, piecewise-linear, or topological), there are similarly defined, but inequivalent, concepts of normal maps and normal invariants.
It is possible to perform surgery on normal maps, meaning surgery on the domain manifold, and preserving the map. Surgery on normal maps allows one to systematically kill elements in the relative homotopy groups by representing them as embeddings with trivial normal bundle.
Definition
There are two equivalent definitions of normal maps, depending on whether one uses normal bundles or tangent bundles of manifolds. Hence it is possible to switch between the definitions which turns out to be quite convenient.
1. Given a Poincaré complex X (i.e. a CW-complex whose cellular chain complex satisfies Poincaré duality) of formal dimension , a normal map on X consists of
a map from some closed n-dimensional manifold M,
a bundle over X, and a stable map from the stable normal bundle of to , and
usually the normal map is supposed to be of degree one. That means that the fundamental class of should be mapped under to the fundamental class of : .
2. Given a Poincaré complex (i.e. a CW-complex whose cellular chain complex satisfies Poincaré duality) of formal dimension , a normal map on (with respect to the tangent bundle) consists of
a map from some closed -dimensional manifold ,
a bundle over , and a stable map from the stable tangent bundle of to , and
similarly as above it is required that the fundamental class of should be mapped under to the fundamental class of : .
Two normal maps are equivalent if there exists a normal bordism between them.
Role in surgery theory
Surgery on maps versus surgery on normal maps
Consider the question:
Is the Poincaré complex X of formal dimension n homotopy-equivalent to a closed n-manifold?
A naive surgery approach to this question would be: start with some map from some manifold to , and try to do surgery on it to make a homotopy equivalence out of it. Notice the following: Sin |
https://en.wikipedia.org/wiki/Surgery%20obstruction | In mathematics, specifically in surgery theory, the surgery obstructions define a map from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property when :
A degree-one normal map is normally cobordant to a homotopy equivalence if and only if the image in .
Sketch of the definition
The surgery obstruction of a degree-one normal map has a relatively complicated definition.
Consider a degree-one normal map . The idea in deciding the question whether it is normally cobordant to a homotopy equivalence is to try to systematically improve so that the map becomes -connected (that means the homotopy groups for ) for high . It is a consequence of Poincaré duality that if we can achieve this for then the map already is a homotopy equivalence. The word systematically above refers to the fact that one tries to do surgeries on to kill elements of . In fact it is more convenient to use homology of the universal covers to observe how connected the map is. More precisely, one works with the surgery kernels which one views as -modules. If all these vanish, then the map is a homotopy equivalence. As a consequence of Poincaré duality on and there is a -modules Poincaré duality , so one only has to watch half of them, that means those for which .
Any degree-one normal map can be made -connected by the process called surgery below the middle dimension. This is the process of killing elements of for described here when we have such that . After this is done there are two cases.
1. If then the only nontrivial homology group is the kernel . It turns out that the cup-product pairings on and induce a cup-product pairing on . This defines a symmetric bilinear form in case and a skew-symmetric bilinear form in case . It turns out that these forms can be refined to -quadratic forms, where . These -quadratic forms define elements in the L-groups .
2. If the definition is more complicated. Instead of a quadratic form one obtains from the geometry a quadratic formation, which is a kind of automorphism of quadratic forms. Such a thing defines an element in the odd-dimensional L-group .
If the element is zero in the L-group surgery can be done on to modify to a homotopy equivalence.
Geometrically the reason why this is not always possible is that performing surgery in the middle dimension to kill an element in possibly creates an element in when or in when . So this possibly destroys what has already been achieved. However, if is zero, surgeries can be arranged in such a way that this does not happen.
Example
In the simply connected case the following happens.
If there is no obstruction.
If then the surgery obstruction can be calculated as the difference of the signatures of M and X.
If then the surgery obstruction is the Arf-invariant of the associated kernel quadratic form over .
References
Surgery theory |
https://en.wikipedia.org/wiki/David%20A.%20Jones | David Arfon Jones is a senior climatologist at the Australian Bureau of Meteorology.
He initially studied mathematics and chemistry at university but changed to atmospheric studies. Jones obtained his PhD in Earth Science from the University of Melbourne, Australia in 1995. He subsequently completed the postgraduate diploma in weather forecasting in 1995 at the Bureau of Meteorology. In 1995 Jones commenced work in the Climate Analysis Section of the Australian National Climate Centre, focusing on the automation of climate monitoring using objective analysis techniques. Subsequently he moved to the Bureau of Meteorology Research Centre in 1997 undertaking research on the variability and change of Australia's climate.
Jones became the supervisor of Climate Analysis at the Bureau of Meteorology in 2002. In this role he has promoted the automation of analysis, monitoring and forecasting products and the introduction of a range of innovative climate monitoring activities, with a focus on encouraging the interpretation of climate variability in the context of a rapidly changing climate.
Owing to the continued misrepresentation of climate change in the Australian media Jones has written a number of public pieces correcting or explaining climate change including in The Age, and in articles for the Australian Science Media Centre.
In 2006 Jones was awarded the National Australia Day Council Achievement Medallion.
References
External links
Australian Climate Change and Variability
Australian meteorologists
Living people
Australian climatologists
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Local%20parameter | In the geometry of complex algebraic curves, a local parameter for a curve C at a smooth point P is a meromorphic function on C that has a simple zero at P. This concept can be generalized to curves defined over fields other than (or schemes), because the local ring at a smooth point P of an algebraic curve C (defined over an algebraically closed field) is always a discrete valuation ring. This valuation will show a way to count the order (at the point P) of rational functions (which are natural generalizations for meromorphic functions in the non-complex realm) having a zero or a pole at P.
Local parameters, as its name indicates, are used mainly to properly count multiplicities in a local way.
Introduction
If C is a complex algebraic curve, count multiplicities of zeroes and poles of meromorphic functions defined on it. However, when discussing curves defined over fields other than , if there is no access to the power of the complex analysis, a replacement must be found in order to define multiplicities of zeroes and poles of rational functions defined on such curves. In this last case, say that the germ of the regular function vanishes at if . This is in complete analogy with the complex case, in which the maximal ideal of the local ring at a point P is actually conformed by the germs of holomorphic functions vanishing at P.
The valuation function on is given by
This valuation can naturally be extended to K(C) (which is the field of rational functions of C) because it is the field of fractions of . Hence, the idea of having a simple zero at a point P is now complete: it will be a rational function such that its germ falls into , with d at most 1.
This has an algebraic resemblance with the concept of a uniformizing parameter (or just uniformizer) found in the context of discrete valuation rings in commutative algebra; a uniformizing parameter for the DVR (R, m) is just a generator of the maximal ideal m. The link comes from the fact that a local parameter at P will be a uniformizing parameter for the DVR (, ), whence the name.
Definition
Let C be an algebraic curve defined over an algebraically closed field K, and let K(C) be the field of rational functions of C. The valuation on K(C) corresponding to a smooth point is defined as
, where is the usual valuation on the local ring (, ). A local parameter for C at P is a function such that .
References
Commutative algebra
Algebraic geometry |
https://en.wikipedia.org/wiki/List%20of%20barangays%20in%20Valenzuela | This is a list of barangays in Valenzuela in the Philippines based on 2015 census data of the Philippine Statistics Authority.
List of barangays
Alternate names of barangays
Canumay West is the political name for the barangay but it is sometimes called Canumay.
Gen. T. de Leon is sometimes spelled as Hen. T. de Leon ("Hen." being Heneral, the Filipino equivalent for Gen. or General), and sometimes abbreviated as GTDL.
Karuhatan is sometimes spelled as Caruhatan.
Marulas is sometimes called BBB or simply BB. Balintawak Beer Brewery (BBB) used to be located in Marulas before it was acquired by San Miguel Corporation to form San Miguel Polo Beer Brewery.
Paso de Blas is sometimes called Tollgate because of Paso de Blas Exit (also known as Malinta Exit and Valenzuela Exit) at Km. 15 of North Luzon Expressway.
Veinte Reales is sometimes spelled as Viente Reales or Veintereales, with i and e interchanged, without affecting its pronunciation.
Gallery
See also
List of populated places in Metro Manila
External links
References
Valenzuela, Metro Manila
Valenzuela
Valenzuela |
https://en.wikipedia.org/wiki/Assembly%20map | In mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data.
Assembly maps for algebraic K-theory and L-theory play a central role in the topology of high-dimensional manifolds, since their homotopy fibers have a direct geometric
Homotopy-theoretical viewpoint
It is a classical result that for any generalized homology theory on the category of topological spaces (assumed to be homotopy equivalent to CW-complexes), there is a spectrum such that
where .
The functor from spaces to spectra has the following properties:
It is homotopy-invariant (preserves homotopy equivalences). This reflects the fact that is homotopy-invariant.
It preserves homotopy co-cartesian squares. This reflects the fact that has Mayer-Vietoris sequences, an equivalent characterization of excision.
It preserves arbitrary coproducts. This reflects the disjoint-union axiom of .
A functor from spaces to spectra fulfilling these properties is called excisive.
Now suppose that is a homotopy-invariant, not necessarily excisive functor. An assembly map is a natural transformation from some excisive functor to such that is a homotopy equivalence.
If we denote by the associated homology theory, it follows that the induced natural transformation of graded abelian groups is the universal transformation from a homology theory to , i.e. any other transformation from some homology theory factors uniquely through a transformation of homology theories .
Assembly maps exist for any homotopy invariant functor, by a simple homotopy-theoretical construction.
Geometric viewpoint
As a consequence of the Mayer-Vietoris sequence, the value of an excisive functor on a space only depends on its value on 'small' subspaces of , together with the knowledge how these small subspaces intersect. In a cycle representation of the associated homology theory, this means that all cycles must be representable by small cycles. For instance, for singular homology, the excision property is proved by subdivision of simplices, obtaining sums of small simplices representing arbitrary homology classes.
In this spirit, for certain homotopy-invariant functors which are not excisive, the corresponding excisive theory may be constructed by imposing 'control conditions', leading to the field of controlled topology. In this picture, assembly maps are 'forget-control' maps, i.e. they are induced by forgetting the control conditions.
Importance in geometric topology
Assembly maps are studied in geometric topology mainly for the two functors , algebraic L-theory of , and , algebraic K-theory of spaces of . In fact, the homotopy fibers of both assembly maps have a direct geometric interpretation when is a |
https://en.wikipedia.org/wiki/Quantitative%20comparative%20linguistics | Quantitative comparative linguistics is the use of quantitative analysis as applied to comparative linguistics. Examples include the statistical fields of lexicostatistics and glottochronology, and the borrowing of phylogenetics from biology.
History
Statistical methods have been used for the purpose of quantitative analysis in comparative linguistics for more than a century. During the 1950s, the Swadesh list emerged: a standardised set of lexical concepts found in most languages, as words or phrases, that allow two or more languages to be compared and contrasted empirically.
Probably the first published quantitative historical linguistics study was by Sapir in 1916, while Kroeber and Chretien in 1937 investigated nine Indo-European (IE) languages using 74 morphological and phonological features (extended in 1939 by the inclusion of Hittite). Ross in 1950 carried out an investigation into the theoretical basis for such studies. Swadesh, using word lists, developed lexicostatistics and glottochronology in a series of papers published in the early 1950s but these methods were widely criticised though some of the criticisms were seen as unjustified by other scholars. Embleton published a book on "Statistics in Historical Linguistics" in 1986 which reviewed previous work and extended the glottochronological method. Dyen, Kruskal and Black carried out a study of the lexicostatistical method on a large IE database in 1992.
During the 1990s, there was renewed interest in the topic, based on the application of methods of computational phylogenetics and cladistics. Such projects often involved collaboration by linguistic scholars, and colleagues with expertise in information science and/or biological anthropology. These projects often sought to arrive at an optimal phylogenetic tree (or network), to represent a hypothesis about the evolutionary ancestry and perhaps its language contacts. Pioneers in these methods included the founders of CPHL: computational phylogenetics in historical linguistics (CPHL project): Donald Ringe, Tandy Warnow, Luay Nakhleh and Steven N. Evans.
In the mid-1990s a group at Pennsylvania University computerised the comparative method and used a different IE database with 20 ancient languages. In the biological field several software programs were then developed which could have application to historical linguistics. In particular a group at the University of Auckland developed a method that gave controversially old dates for IE languages. A conference on "Time-depth in Historical Linguistics" was held in August 1999 at which many applications of quantitative methods were discussed. Subsequently many papers have been published on studies of various language groups as well as comparisons of the methods.
Greater media attention was generated in 2003 after the publication by anthropologists Russell Gray and Quentin Atkinson of a short study on Indo-European languages in Nature. Gray and Atkinson attempted to quantify, in |
https://en.wikipedia.org/wiki/Exponential%20field | In mathematics, an exponential field is a field with a further unary operation that is a homomorphism from the field's additive group to its multiplicative group. This generalizes the usual idea of exponentiation on the real numbers, where the base is a chosen positive real number.
Definition
A field is an algebraic structure composed of a set of elements, F, two binary operations, addition (+) such that F forms an abelian group with identity 0F and multiplication (·), such that F excluding 0F forms an abelian group under multiplication with identity 1F, and such that multiplication is distributive over addition, that is for any elements a, b, c in F, one has . If there is also a function E that maps F into F, and such that for every a and b in F one has
then F is called an exponential field, and the function E is called an exponential function on F. Thus an exponential function on a field is a homomorphism between the additive group of F and its multiplicative group.
Trivial exponential function
There is a trivial exponential function on any field, namely the map that sends every element to the identity element of the field under multiplication. Thus every field is trivially also an exponential field, so the cases of interest to mathematicians occur when the exponential function is non-trivial.
Exponential fields are sometimes required to have characteristic zero as the only exponential function on a field with nonzero characteristic is the trivial one. To see this first note that for any element x in a field with characteristic p > 0,
Hence, taking into account the Frobenius endomorphism,
And so E(x) = 1 for every x.
Examples
The field of real numbers R, or as it may be written to highlight that we are considering it purely as a field with addition, multiplication, and special constants zero and one, has infinitely many exponential functions. One such function is the usual exponential function, that is , since we have and , as required. Considering the ordered field R equipped with this function gives the ordered real exponential field, denoted .
Any real number gives an exponential function on R, where the map satisfies the required properties.
Analogously to the real exponential field, there is the complex exponential field, .
Boris Zilber constructed an exponential field Kexp that, crucially, satisfies the equivalent formulation of Schanuel's conjecture with the field's exponential function. It is conjectured that this exponential field is actually Cexp, and a proof of this fact would thus prove Schanuel's conjecture.
Exponential rings
The underlying set F may not be required to be a field but instead allowed to simply be a ring, R, and concurrently the exponential function is relaxed to be a homomorphism from the additive group in R to the multiplicative group of units in R. The resulting object is called an exponential ring.
An example of an exponential ring with a nontrivial exponential function is the ring of |
https://en.wikipedia.org/wiki/Foundations%20of%20geometry | Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.
Axiomatic systems
Based on ancient Greek methods, an axiomatic system is a formal description of a way to establish the mathematical truth that flows from a fixed set of assumptions. Although applicable to any area of mathematics, geometry is the branch of elementary mathematics in which this method has most extensively been successfully applied.
There are several components of an axiomatic system.
Primitives (undefined terms) are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are things like points, lines and planes while a fundamental relationship is that of incidence – of one object meeting or joining with another. The terms themselves are undefined. Hilbert once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer mugs. His point being that the primitive terms are just empty shells, place holders if you will, and have no intrinsic properties.
Axioms (or postulates) are statements about these primitives; for example, any two points are together incident with just one line (i.e. that for any two points, there is just one line which passes through both of them). Axioms are assumed true, and not proven. They are the building blocks of geometric concepts, since they specify the properties that the primitives have.
The laws of logic.
The theorems are the logical consequences of the axioms, that is, the statements that can be obtained from the axioms by using the laws of deductive logic.
An interpretation of an axiomatic system is some particular way of giving concrete meaning to the primitives of that system. If this association of meanings makes the axioms of the system true statements, then the interpretation is called a model of the system. In a model, all the theorems of the system are automatically true statements.
Properties of axiomatic systems
In discussing axiomatic systems several properties are often focused on:
The axioms of an axiomatic system are said to be consistent if no logical contradiction can be derived from them. Except in the simplest systems, consistency is a difficult property to establish in an axiomatic system. On the other hand, if a model exists for the axiomatic system, then any contradiction derivable |
https://en.wikipedia.org/wiki/Graph%20structure%20theorem | In mathematics, the graph structure theorem is a major result in the area of graph theory. The result establishes a deep and fundamental connection between the theory of graph minors and topological embeddings. The theorem is stated in the seventeenth of a series of 23 papers by Neil Robertson and Paul Seymour. Its proof is very long and involved. and are surveys accessible to nonspecialists, describing the theorem and its consequences.
Setup and motivation for the theorem
A minor of a graph is any graph that is isomorphic to a graph that can be obtained from a subgraph of by contracting some edges. If does not have a graph as a minor, then we say that is -free. Let be a fixed graph. Intuitively, if is a huge -free graph, then there ought to be a "good reason" for this. The graph structure theorem provides such a "good reason" in the form of a rough description of the structure of . In essence, every -free graph suffers from one of two structural deficiencies: either is "too thin" to have as a minor, or can be (almost) topologically embedded on a surface that is too simple to embed upon. The first reason applies if is a planar graph, and both reasons apply if is not planar. We first make precise these notions.
Tree width
The tree width of a graph is a positive integer that specifies the "thinness" of . For example, a connected graph has tree width one if and only if it is a tree, and has tree width two if and only if it is a series–parallel graph. Intuitively, a huge graph has small tree width if and only if takes the structure of a huge tree whose nodes and edges have been replaced by small graphs. We give a precise definition of tree width in the subsection regarding clique-sums. It is a theorem that if is a minor of , then the tree width of is not greater than that of . Therefore, one "good reason" for to be -free is that the tree width of is not very large. The graph structure theorem implies that this reason always applies in case is planar.
Corollary 1. For every planar graph , there exists a positive integer such that every -free graph has tree width less than .
It is unfortunate that the value of in Corollary 1 is generally much larger than the tree width of (a notable exception is when , the complete graph on four vertices, for which ). This is one reason that the graph structure theorem is said to describe the "rough structure" of -free graphs.
Surface embeddings
Roughly, a surface is a set of points with a local topological structure of a disc. Surfaces fall into two infinite families: the orientable surfaces include the sphere, the torus, the double torus and so on; the nonorientable surfaces include the real projective plane, the Klein bottle and so on. A graph embeds on a surface if the graph can be drawn on the surface as a set of points (vertices) and arcs (edges) that do not cross or touch each other, except where edges and vertices are incident or adjacent. A graph is planar if |
https://en.wikipedia.org/wiki/List%20of%20Houston%20Texans%20records | This article details statistics relating to the Houston Texans American football team.
Houston Texans records
Team records
Wins: 12 (2012)
Home wins: 7 (2016)
Road wins: 6 (2012)
Consecutive wins: 9 (2018)
Consecutive losses: 14 (2013)
Most points For: 416 (2012)
Fewest points Allowed: 278 (2011)
Touchdowns: 46 (2012)
Most points Scored: 57 vs. Titans (Oct 1, 2017; W 57-14)
Biggest comeback: 21 vs. Chargers (Sep 9, 2013; W 31-28)
Biggest lead blown: 24 vs. Chiefs (Jan 12, 2020; L 31-51)
All-time franchise individual records
Most consecutive starts: 171 Jon Weeks
Most games played: 171 Jon Weeks
Most consecutive snaps: 3,884 Chester Pitts
Completions: 1,951 Matt Schaub
Attempts: 3,020 Matt Schaub
Completion percentage: 65.6% Sage Rosenfels
Passing yards: 23,221 Matt Schaub
Passing yards per game: 258.0 Matt Schaub
Passing yards per attempt: 8.0 Ryan Fitzpatrick
Passing touchdowns: 124 Matt Schaub
Passer rating: 90.9 Matt Schaub
Rushing attempts: 1,454 Arian Foster
Rushing yards: 6,472 Arian Foster
Rushing average: 4.7 Ben Tate
Rushing touchdowns: 54 Arian Foster
Most games, 100 or more yards rushing: 32 Arian Foster
Rushing yards per game: 85.2 Arian Foster
Receptions: 1,012 Andre Johnson
Receiving yards: 13,597 Andre Johnson
Receiving yards per reception: 16.4 André Davis
Receiving touchdowns: 64 Andre Johnson
Most games, 100 or more yards receiving: 51 Andre Johnson
Receptions per game: 6.0 Andre Johnson
Receiving yards per game: 80.5 Andre Johnson
Total touchdowns: 68 Arian Foster
Yards from scrimmage: 13,651 Andre Johnson
Total points: 767 Kris Brown
Quarterback sacks: 101 J. J. Watt
Total tackles: 658 Brian Cushing
Solo tackles: 479 DeMeco Ryans
Assists on tackles: 238 Brian Cushing
Pass interceptions: 16 Kareem Jackson & Johnathan Joseph
Pass interception return yards: 410 Johnathan Joseph
Pass interception touchdown returns: 4 Johnathan Joseph
Passes defended: 117 Johnathan Joseph
Forced fumbles: 23 J. J. Watt
Fumble recoveries: 15 J. J. Watt
Fumble return yards: 102 DeMeco Ryans
Forced fumble touchdown returns: 1 (13 players)
Safeties: 1 (7 players)
Kick returns: 117 J.J. Moses
Kick return yards: 2,743 André Davis
Kick return average: 28.5 Jerome Mathis
Kick return touchdowns: 3 Jerome Mathis & André Davis
Punt returns: 179 Jacoby Jones
Punt return yards: 1,820 Jacoby Jones
Punt return average: 15.0 Will Fuller
Punt return touchdowns: 3 Jacoby Jones
Field goals made: 172 Kris Brown
Field goals attempted: 223 Kris Brown
Field goal percentage: 86.8% Neil Rackers
Punts: 437 Chad Stanley
Punting yards: 17,908 Chad Stanley
Punting average: 47.2 Shane Lechler & Donnie Jones
(thru 2020 season)
Single-season individual records
Passing attempts: 583 Matt Schaub (2009)
Passing completions: 396 Matt Schaub (2009)
Completion percentage: 70.2% Deshaun Watson (2020)
Passing yards: 4,823 Deshaun Watson (2020)
Passing yards (rookie): 2,664 Davis Mills (2021)
Passer rating: |
https://en.wikipedia.org/wiki/Restricted%20Lie%20algebra | In mathematics, a restricted Lie algebra is a Lie algebra together with an additional "p operation."
Definition
Let L be a Lie algebra over a field k of characteristic p>0. A p operation on L is a map satisfying
for all ,
for all ,
, for all , where is the coefficient of in the formal expression .
If the characteristic of k is 0, then L is a restricted Lie algebra where the p operation is the identity map.
Examples
For any associative algebra A defined over a field of characteristic p, the bracket operation and p operation make A into a restricted Lie algebra .
Let G be an algebraic group over a field k of characteristic p, and be the Zariski tangent space at the identity element of G. Each element of uniquely defines a left-invariant vector field on G, and the commutator of vector fields defines a Lie algebra structure on just as in the Lie group case. If p>0, the Frobenius map defines a p operation on .
Restricted universal enveloping algebra
The functor has a left adjoint called the restricted universal enveloping algebra. To construct this, let be the universal enveloping algebra of L forgetting the p operation. Letting I be the two-sided ideal generated by elements of the form , we set . It satisfies a form of the PBW theorem.
See also
Restricted Lie algebras are used in Jacobson's Galois correspondence for purely inseparable extensions of fields of exponent 1.
References
.
.
.
Algebraic groups
Lie algebras |
https://en.wikipedia.org/wiki/Toric | Toric may refer to:
Mathematics
relating to a torus
Toric code
Toric hyperkahler manifold
Toric ideal
Toric joint
Toric manifold
Toric orbifold
Toric section
Toric variety
Other uses
Toric lens, a type of optical lens
Torić, a village in Bosnia and Herzegovina
Toric Robinson (born 1986), Jamaican footballer
See also
Thoric, related to, or containing thorium |
https://en.wikipedia.org/wiki/Lino%20Aldani | Lino Aldani (29 March 1926 – 31 January 2009) was an Italian science fiction writer.
Biography
Aldani was born in San Cipriano Po in 1926. He lived in Rome, where he worked as a mathematics teacher until 1968, when he returned to his native San Cipriano Po and devoted his life to writing.
He published science fiction stories starting in the Sixties (his first published short story being "Dove sono i vostri Kumar?", in 1960) and his first novel, Quando le radici, in 1977.
In 1962 he wrote the first Italian critical essay about science fiction, La fantascienza. In 1963 Aldani founded the SF magazine Futuro with Massimo Lo Jacono; the magazine lasted eight issues.
His works have been translated into several languages. He died in Pavia on 31 January 2009.
Bibliography
La Fantascienza (1963), essay;
Aleph 3 (1963), his first novel, first published in 2007;
Quarta Dimensione (1964);
Quando le radici (1977), a novel whose main character, Arno, looks for his past in a future, disturbing Italy;
Eclissi 2000 (1979), a novel about the impossibility of creating a government without resorting to lies and deceit;
Nel segno della luna bianca (1980; also known as Febbre di luna, with Daniela Piegai), a left-wing inspired fantasy novel;
La croce di ghiaccio (1989), novel;
Themoro Korik (2007), a novel about Romani people.
References
Afterword to Eclissi 2000, De Vecchi Editore, 1979.
Simone Brioni and Daniele Comberiati, Italian Science Fiction: The Other in Literature and Film. New York: Palgrave, 2019.
External links
Bibliography of Aldani's works in Catalogo della fantascienza, fantasy e horror edited by E. Vegetti, P. Cottogni, E. Bertoni
Obituary in Locus magazine website
1926 births
2009 deaths
People from the Province of Pavia
Italian science fiction writers
Italian male non-fiction writers
Italian speculative fiction critics
Italian speculative fiction editors
Italian psychological writers
Deaths from lung cancer in Lombardy |
https://en.wikipedia.org/wiki/P%C3%A9ter%20Moln%C3%A1r%20%28footballer%29 | Péter Molnár (born 14 December 1983) is an ethnic Hungarian football player from Slovakia who currently plays for BFC Siófok.
Club statistics
Updated to games played as of 9 December 2017.
References
External links
Profile
1983 births
Living people
Sportspeople from Komárno
Footballers from the Nitra Region
Hungarians in Slovakia
Slovak men's footballers
Hungarian men's footballers
Men's association football goalkeepers
KFC Komárno players
Győri ETO FC players
BFC Siófok players
Paksi FC players
Nemzeti Bajnokság I players
Puskás Akadémia FC II players
Nemzeti Bajnokság III players
Slovak expatriate men's footballers
Expatriate men's footballers in Hungary
Slovak expatriate sportspeople in Hungary |
https://en.wikipedia.org/wiki/McDames%20Creek%20Indian%20Reserve%20No.%202 | McDames Creek 2 is a Statistics Canada census designation for what is properly known as McDames Creek Indian Reserve No. 2, which flanks both sides of the Dease River at its confluence with McDame Creek in northwestern British Columbia, Canada. It was named for the 19th-century gold rush prospector Harry McDame. The reserve is under the administration of the Liard First Nation, a government of the Kaska Dena people and a member government of the Kaska Tribal Council.
References
Reserves/Villages/Settlements of the Liard First Nation, inac.gc.ca
Indian reserves in British Columbia
Cassiar Country
Kaska Dena |
https://en.wikipedia.org/wiki/Math%20Horizons | Math Horizons is a magazine aimed at undergraduates interested in mathematics, published by the Mathematical Association of America. It publishes expository articles about "beautiful mathematics" as well as articles about the culture of mathematics covering mathematical people, institutions, humor, games, cartoons, and book reviews.
The MAA gives the Trevor Evans Awards annually to "authors of exceptional articles that are accessible to undergraduates" that are published in Math Horizons.
Notes
Further reading
External links
Math Horizons at JSTOR
Math Horizons at Taylor & Francis Online
Mathematics journals |
https://en.wikipedia.org/wiki/The%20College%20Mathematics%20Journal | The College Mathematics Journal is an expository magazine aimed at teachers of college mathematics, particularly those teaching the first two years. It is published by Taylor & Francis on behalf of the Mathematical Association of America and is a continuation of Two-Year College Mathematics Journal. It covers all aspects of mathematics. It publishes articles intended to enhance undergraduate instruction and classroom learning, including expository articles, short notes, problems, and "mathematical ephemera" such as fallacious proofs, quotations, cartoons, poetry, and humor.
Paid circulation in 2008 was 9,000 and total circulation was 9,500.
The MAA gives the George Pólya Awards annually "for articles of expository excellence" published in the College Mathematics Journal.
References
External links
The College Mathematics Journal at JSTOR
The College Mathematics Journal at Taylor & Francis Online
Mathematics journals
Magazines established in 1970
1970 establishments in the United States |
https://en.wikipedia.org/wiki/Blue%20River%20Indian%20Reserve%20No.%201 | Blue River 1 is the Statistics Canada census-area designation for what is properly termed the Blue River Indian Reserve No. 1, an Indian reserve in the Cassiar Country of the Northern Interior of British Columbia, Canada. It is located on the left bank of the river of the same name at that river's confluence with the Dease River and is under the administration of the Liard First Nation, a member of the Kaska Tribal Council.
References
Indian reserves in British Columbia
Liard Country
Kaska Dena |
https://en.wikipedia.org/wiki/Generating%20Availability%20Data%20System | The Generating Availability Data System (GADS) is a database produced by the North American Electric Reliability Corporation (NERC). It includes annual summary reports comprising the statistics for power stations in the United States and Canada.
GADS is the main source of power station outage data in North America. This reporting system, initiated by the electric utility industry in 1982, expands and extends the data collection procedures begun by the industry in 1963. NERC GADS is recognized today as a valuable source of reliability, availability, and maintainability (RAM) information.
This information, collected for both total unit and major equipment groups, is used by analysts industry-wide in numerous applications. GADS maintains complete operating histories on more than 5,800 generating units representing 71% of the installed generating capacity of the United States and Canada. GADS is a mandatory industry program for conventional generating units 50 MW and larger starting January 1, 2012 and 20 MW and larger starting January 1, 2013. GADS remains open to all non-required participants in the Regional Entities (shown in Figure I-2 of the NERC GADS DRI) and any other organization (domestic or international) that operate electric generating facilities who is willing to follow the GADS mandatory requirements as presented in the document Final GADSTF Recommendations Report dated July 20, 2011.
GADS data consists of three data types:
Design – equipment descriptions such as manufacturers, number of boiler feedwater pumps, steam turbine MW rating, etc.
Performance – summaries of generation produced, fuels units, start ups, etc.
Event – description of equipment failures such as when the event started/ended, type of outage (forced, maintenance, planned), etc.
One example of such detail is that in its data pertaining to forced outages and unplanned unit failures, it makes the fine distinction between immediate, delayed, and postponed outages.
An important statistic calculated from the raw GADS data is the Equivalent Forced Outage Rate (EFOR), which is the hours of unit failure (unplanned outage hours and equivalent unplanned derated hours) given as a percentage of the total hours of the availability of that unit (unplanned outage, unplanned derated, and service hours).
Recently, in response to the deregulated energy markets, the Equivalent Forced Outage Rate – Demand (EFORd) has taken on greater importance:
The probability that a unit will not meet its demand periods for generating requirements.
Best measure of reliability for all loading types (base, cycling, peaking, etc.)
Best measure of reliability for all unit types (fossil, nuclear, gas turbines, diesels, etc.)
For demand period measures and not for the full 24-hour clock.
Industry Development of GADS
Before any data element was included in GADS, an industry committee to determine its applicability to utility operation and RAM analyses scrutinized it. A series of industry meeti |
https://en.wikipedia.org/wiki/Szil%C3%A1rd%20Devecseri | Szilárd Devecseri (born 13 February 1990) is a Hungarian football player.
Club statistics
Updated to games played as of 29 February 2020.
References
External links
HLSZ
1990 births
Living people
Footballers from Szombathely
Hungarian men's footballers
Men's association football defenders
Szombathelyi Haladás footballers
Mezőkövesdi SE footballers
Zalaegerszegi TE players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Hungary men's international footballers
Hungary men's under-21 international footballers |
https://en.wikipedia.org/wiki/Lindeberg%27s%20condition | In probability theory, Lindeberg's condition is a sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independent random variables. Unlike the classical CLT, which requires that the random variables in question have finite variance and be both independent and identically distributed, Lindeberg's CLT only requires that they have finite variance, satisfy Lindeberg's condition, and be independent. It is named after the Finnish mathematician Jarl Waldemar Lindeberg.
Statement
Let be a probability space, and , be independent random variables defined on that space. Assume the expected values and variances exist and are finite. Also let
If this sequence of independent random variables satisfies Lindeberg's condition:
for all , where 1{…} is the indicator function, then the central limit theorem holds, i.e. the random variables
converge in distribution to a standard normal random variable as
Lindeberg's condition is sufficient, but not in general necessary (i.e. the inverse implication does not hold in general).
However, if the sequence of independent random variables in question satisfies
then Lindeberg's condition is both sufficient and necessary, i.e. it holds if and only if the result of central limit theorem holds.
Remarks
Feller's theorem
Feller's theorem can be used as an alternative method to prove that Lindeberg's condition holds. Letting and for simplicity , the theorem states
if , and converges weakly to a standard normal distribution as then satisfies the Lindeberg's condition.
This theorem can be used to disprove the central limit theorem holds for by using proof by contradiction. This procedure involves proving that Lindeberg's condition fails for .
Interpretation
Because the Lindeberg condition implies as , it guarantees that the contribution of any individual random variable () to the variance is arbitrarily small, for sufficiently large values of .
See also
Lyapunov condition
Central limit theorem
References
Theorems in statistics
Central limit theorem |
https://en.wikipedia.org/wiki/Division%20polynomials | In mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves and to study the fields generated by torsion points. They play a central role in the study of counting points on elliptic curves in Schoof's algorithm.
Definition
The set of division polynomials is a sequence of polynomials in with free variables that is recursively defined by:
The polynomial is called the nth division polynomial.
Properties
In practice, one sets , and then and .
The division polynomials form a generic elliptic divisibility sequence over the ring .
If an elliptic curve is given in the Weierstrass form over some field , i.e. , one can use these values of and consider the division polynomials in the coordinate ring of . The roots of are the -coordinates of the points of , where is the torsion subgroup of . Similarly, the roots of are the -coordinates of the points of .
Given a point on the elliptic curve over some field , we can express the coordinates of the nth multiple of in terms of division polynomials:
where and are defined by:
Using the relation between and , along with the equation of the curve, the functions , , are all in .
Let be prime and let be an elliptic curve over the finite field , i.e., . The -torsion group of over is isomorphic to if , and to or if . Hence the degree of is equal to either , , or 0.
René Schoof observed that working modulo the th division polynomial allows one to work with all -torsion points simultaneously. This is heavily used in Schoof's algorithm for counting points on elliptic curves.
See also
Schoof's algorithm
References
A. Enge: Elliptic Curves and their Applications to Cryptography: An Introduction. Kluwer Academic Publishers, Dordrecht, 1999.
N. Koblitz: A Course in Number Theory and Cryptography, Graduate Texts in Math. No. 114, Springer-Verlag, 1987. Second edition, 1994
Müller : Die Berechnung der Punktanzahl von elliptischen kurvenüber endlichen Primkörpern. Master's Thesis. Universität des Saarlandes, Saarbrücken, 1991.
G. Musiker: Schoof's Algorithm for Counting Points on . Available at http://www-math.mit.edu/~musiker/schoof.pdf
Schoof: Elliptic Curves over Finite Fields and the Computation of Square Roots mod p. Math. Comp., 44(170):483–494, 1985. Available at http://www.mat.uniroma2.it/~schoof/ctpts.pdf
R. Schoof: Counting Points on Elliptic Curves over Finite Fields. J. Theor. Nombres Bordeaux 7:219–254, 1995. Available at http://www.mat.uniroma2.it/~schoof/ctg.pdf
L. C. Washington: Elliptic Curves: Number Theory and Cryptography. Chapman & Hall/CRC, New York, 2003.
J. Silverman: The Arithmetic of Elliptic Curves, Springer-Verlag, GTM 106, 1986.
Polynomials
Algebraic curves |
https://en.wikipedia.org/wiki/David%20Bressoud | David Marius Bressoud (born March 27, 1950, in Bethlehem, Pennsylvania) is an American mathematician who works in number theory, combinatorics, and special functions. As of 2019 he is DeWitt Wallace Professor of Mathematics at Macalester College, Director of the Conference Board of the Mathematical Sciences and a former President of the Mathematical Association of America.
Life and education
Bressoud was born March 27, 1950, in Bethlehem, Pennsylvania.
He became interested in mathematics in the seventh grade, where he had a teacher who encouraged him and gave him challenging problems. He attended Albert Wilansky's National Science Foundation summer program at Lehigh University between his junior and senior years in high school, where he also spent most of his time working on problems.
He graduated from Swarthmore College in 1971. When he started at Swarthmore he had not yet decided on a major, but after his first year he decided to get out of college as quickly as possibly and had no interest in graduate school, and the quickest way out was to major in mathematics.
After graduating Bressoud became a Peace Corps volunteer in Antigua from 1971 to 1973, teaching math and science at Clare Hall School. While in Antigua he realized he missed mathematics, and kept working on it as a hobby. After the Peace Corps he went to graduate school at Temple University, and received his PhD in 1977 under Emil Grosswald.
Career
After receiving his PhD, Bressoud taught at Pennsylvania State University from 1977 to 1994, reaching the rank of full professor in 1986. During this period he held visiting positions at the Institute for Advanced Study (1979–1980), the University of Wisconsin (1980–81 and 1982), the University of Minnesota (1983 and 1998), and the University of Strasbourg (1984–85).
His focus at Penn State was mathematics research, but in the late 1980s he became more interested in teaching and writing textbooks, and he decided to make a move. He said in a 2008 interview, "I needed to be in a place that had a strong focus on teaching and a community of people for whom
teaching was what they were most interested in." He decided on a move to Macalester College in 1994, where he was DeWitt Wallace Professor of Mathematics. Since 2005 he has written a monthly online column for MAA titled "Launchings" that focuses on the CUPM (Committee on the Undergraduate Program in Mathematics) Curriculum Guide.
Bressoud received several of the Mathematical Association of America's awards: the Distinguished Teaching Award for the Allegheny Mountain section in 1994,
the Beckenbach Book Prize in 1999,
and he was a George Pólya Lecturer from 2002 to 2004.
Bressoud was elected president of the Mathematical Association of America in the 2007 elections, and served as President-Elect in 2008 and served as president from 2009 to 2011. In 2012 he became a fellow of the American Mathematical Society. He began phased retirement at Macalester College in 2016 and in 2017 took ove |
https://en.wikipedia.org/wiki/List%20of%20Titan%20launches | This is a list of launches made by the LGM-25 Titan ICBMs, and their derivatives.
Launch statistics
Rockets from the Titan family accumulated 368 launches between 1959 and 2005, 322 of which were successful, yielding a success rate.
Launches
See also
List of Atlas launches
List of Thor and Delta launches
References
Titan
Titan (rocket family) |
https://en.wikipedia.org/wiki/Wuhan%20Sports%20University | Wuhan Sports University (), formerly translated as Wuhan Institute of Physical Education, (WIPE), is a tertiary educational institution in China. According to the statistics released by the Ministry of Education of China, the university is one of the four leading universities in the physical education area in China. The university has many world-famous alumni, such as gymnast Cheng Fei, who won more than 19 gold medals including an Olympic one. And this can be better illustrated during Beijing 2008 Olympic Games, when WIPE alumni alone won 6 gold medals, 2 silver medals and 3 bronze medals.
Wuhan Sports University is located in Hubei Province's capital Wuhan, known as "the thoroughfare leading to nine provinces". It has two campuses which are Majiazhuang Campus () and Canglong Campus.
History
Wuhan Sports University's origins can be traced back to Zhongnan Institute of Physical Education, in Nanchang City, Jiangxi province. The Zhongnan Institute was formed in 1953. Shortly after, in 1955, the institute moved to its present site in Wuhan, Hubei Province, and changed its name to the Wuhan Institute of Physical Education (WIPE) in February 1956. WIPE was formerly administered by the National General Administration of Sports and, now it turned to be under the administration of both the National General Administration of Sports and the Hubei Provincial Government in the year 2001. In the year 1985, WIPE started its first art major; Sport Management. Since then several other art majors emerged, such as sport psychology, sport English and so forth, till the newest Sport Advertising. And this is a quintessence of ongoing reform of Tertiary educational institutions in China, whose aim is to combine USSR-style specific school into comprehensive institution.
Academics
WIPE now has 8 colleges, 2 faculties, 3 departments and 2 schools, 15 educational and teaching units in total. The names of these units are listed as follow:
Physical Education College ()
Sports Training College ()
Chinese Martial Arts College ()
Gymnastics College ()
Sports Economy & Management College ()
Health Science College or College of Health Science ()
Competitive Sports School ()
Adult Education School ()
Sports Information & Technology Faculty ()
Sports Journalism & Foreign Language Faculty ()
the Postgraduate Department ()
Political Theory Department ()
Training Department ()
Since 2006, WIPE launched its 3+1 cooperative education project with University of the West of England. Students enrolled as its "3+1" project participants will study at Wuhan for the first 3 years and then go to UWE for the last year of study. In autumn semester of 2009, the first 15 students participated in the 3+1 project in 2006 were sent to UWE. WIPE was largely aimed for education of undergraduate level, and its graduate school, though famous in a time during the 1980s, was increasingly eclipsed by Beijing Sports University. However, in 2006, WIPE was authorized for the conferment of |
https://en.wikipedia.org/wiki/Romani%20people%20in%20Bosnia%20and%20Herzegovina | The Xoraxane in Bosnia and Herzegovina are the largest of the 17 national minorities in the country, although—due to the stigma attached to the label—this is often not reflected in statistics and censuses.
Demographics
The exact number of Roma persons in Bosnia and Herzegovina is uncertain. Due to the social stigma attached to the label, many members of the community refuse to self-identify as such in official surveys and censuses. Their number is thus consistently underestimated.
The 2013 census recorded 12,583 Bosnian-Herzegovinian residents of self-declared Romani ethnicity (this data is deemed as grossly under-representing the Roma community in Bosnia and Herzegovina).
The July 2012 estimates of the Council of Europe counted a minimum of 40,000 and a maximum of 76,000 Roma in BiH, with an average of 58,000, i.e. the 1.54% of the total population. This would still make Bosnia and Herzegovina the country in the Western Balkans with the lowest percentage of Roma population.
A partial survey by the BiH Ombudsman through Roma associations recorded around 50,000 Roma living in Bosnia and Herzegovina, of which 35,000 in the Federation BiH, 3,000 in Republika Srpska, and 2,000–2,500 in the Brčko District — without counting the Roma population in the Sarajevo Canton.
The Needs Assessment process conducted in 2010 by the state-level BiH Ministry for Human Rights and Refugees (MHRR) directly identified 16,771 Roma persons in BiH. The MHRR estimates that there are at least 25,000 to 30,000 Roma residents in BiH, although they acknowledge that up to 39 percent of Roma did not participate in the registration in some districts. According to the Ministry, around 42 percent of the Romani population in BiH is below 19 years old. 0.44%
According to the 1991 census, there were 8,864 Roma in Bosnia and Herzegovina or 0.2 percent of the population. Yet, the number was probably much higher, as 10,422 Bosnians stated that Romani was their native language.
Kali Sara and other local Roma NGOs put the number of Roma in BiH at between 80,000 and 100,000.
Geographical distribution
Important Roma communities in BiH are living in Brčko, Bijeljina, Sarajevo, Banja Luka, Mostar, Tuzla, Kakanj, Prijedor, Zenica and Teslić.
The largest number of Roma in Bosnia and Herzegovina live in the Tuzla Canton (15,000–17,000), of which a sizeable proportion in the municipality of Tuzla (6,000–6,500), as well as in Živinice (3,500), Lukavac (2,540). The Sarajevo Canton hosts around 7,000 Roma families, mostly in the municipality of Novi Grad, Sarajevo (1,200–1,500 families). The Zenica-Doboj Canton hosts between 7,700 and 8,200 Roma, of which 2,000–2,500 in the Zenica Municipality, 2,160 in Kakanj, 2,800 in Visoko. 2,000–2,500 Roma live in the Central Bosnia Canton, mostly in Donji Vakuf (500–550), Vitez (550) and Travnik (450). In the Una-Sana Canton there are between 2,000–2,200 Roma, of which 700 in the Bihać Municipality.
In the territory of Herzegovina-Neretva Ca |
https://en.wikipedia.org/wiki/Pokemouche%2013 | Pokemouche 13 is the Statistics Canada census area designation for what is properly termed the Pokemouche Indian Reserve No. 13, located 64 km east of Bathurst, New Brunswick, Canada in Gloucester County near the community of Pokemouche. The reserve is under the jurisdiction of the Burnt Church First Nation and is 151.4 ha. in size. Nearby locatlies include Boudreau Road, Cowans Creek, Haut-Sainte-Rose, Landry, and Maltampec.
History
Notable people
See also
List of communities in New Brunswick
List of First Nations in New Brunswick
References
Indian reserves in New Brunswick
Communities in Gloucester County, New Brunswick
Burnt Church First Nation |
https://en.wikipedia.org/wiki/Tabusintac%209 | Tabusintac 9 is the Statistics Canada census area designation for what is properly termed the Tabusintac Indian Reserve No. 9, which is an Indian reserve under the governance of the Burnt Church First Nation of the Mi'kmaq people. It is 3268.7 ha. in size and is adjacent to the town of Tabusintac.
See also
References
Canadian GeoNames Database entry
inac.gc.ca info page
Indian reserves in New Brunswick
Burnt Church First Nation
Geography of Northumberland County, New Brunswick |
https://en.wikipedia.org/wiki/N-group%20%28category%20theory%29 | In mathematics, an n-group, or n-dimensional higher group, is a special kind of n-category that generalises the concept of group to higher-dimensional algebra. Here, may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of 2-groups under the moniker 'gr-category'.
The general definition of -group is a matter of ongoing research. However, it is expected that every topological space will have a homotopy -group at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group , or the entire Postnikov tower for .
Examples
Eilenberg-Maclane spaces
One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group can be turned into an Eilenberg-Maclane space through a simplicial construction, and it behaves functorially. This construction gives an equivalence between groups and 1-groups. Note that some authors write as , and for an abelian group , is written as .
2-groups
The definition and many properties of 2-groups are already known. 2-groups can be described using crossed modules and their classifying spaces. Essentially, these are given by a quadruple where are groups with abelian,a group morphism, and a cohomology class. These groups can be encoded as homotopy -types with and , with the action coming from the action of on higher homotopy groups, and coming from the Postnikov tower since there is a fibrationcoming from a map . Note that this idea can be used to construct other higher groups with group data having trivial middle groups , where the fibration sequence is nowcoming from a map whose homotopy class is an element of .
3-groups
Another interesting and accessible class of examples which requires homotopy theoretic methods, not accessible to strict groupoids, comes from looking at homotopy 3-types of groups. Essential, these are given by a triple of groups with only the first group being non-abelian, and some additional homotopy theoretic data from the Postnikov tower. If we take this 3-group as a homotopy 3-type , the existence of universal covers gives us a homotopy type which fits into a fibration sequencegiving a homotopy type with trivial on which acts on. These can be understood explicitly using the previous model of -groups, shifted up by degree (called delooping). Explicitly, fits into a postnikov tower with associated Serre fibrationgiving where the -bundle comes from a map , giving a cohomology class in . Then, can be reconstructed using a homotopy quotient .
n-groups
The previous construction gives the general idea of how to consider higher groups in general. For an n group with groups with the latter bunch being abelian, we can consider the associated homotopy type and first consider the universal cover . Then, this is a |
https://en.wikipedia.org/wiki/2-group | In mathematics, particularly the branch called category theory, a 2-group is a groupoid with a way to multiply objects, making it resemble a group. They are part of a larger hierarchy of n-groups.
They were introduced by Hoàng Xuân Sính in the late 1960s under the name gr-categories, and they are also known as categorical groups.
Definition
A 2-group is a monoidal category G in which every morphism is invertible and every object has a weak inverse. (Here, a weak inverse of an object x is an object y such that xy and yx are both isomorphic to the unit object.)
Strict 2-groups
Much of the literature focuses on strict 2-groups. A strict 2-group is a strict monoidal category in which every morphism is invertible and every object has a strict inverse (so that xy and yx are actually equal to the unit object).
A strict 2-group is a group object in a category of categories; as such, they could be called groupal categories. Conversely, a strict 2-group is a category object in the category of groups; as such, they are also called categorical groups. They can also be identified with crossed modules, and are most often studied in that form. Thus, 2-groups in general can be seen as a weakening of crossed modules.
Every 2-group is equivalent to a strict 2-group, although this can't be done coherently: it doesn't extend to 2-group homomorphisms.
Properties
Weak inverses can always be assigned coherently: one can define a functor on any 2-group G that assigns a weak inverse to each object and makes that object an adjoint equivalence in the monoidal category G.
Given a bicategory B and an object x of B, there is an automorphism 2-group of x in B, written AutB(x). The objects are the automorphisms of x, with multiplication given by composition, and the morphisms are the invertible 2-morphisms between these. If B is a 2-groupoid (so all objects and morphisms are weakly invertible) and x is its only object, then AutB(x) is the only data left in B. Thus, 2-groups may be identified with one-object 2-groupoids, much as groups may be identified with one-object groupoids and monoidal categories may be identified with one-object bicategories.
If G is a strict 2-group, then the objects of G form a group, called the underlying group of G and written G0. This will not work for arbitrary 2-groups; however, if one identifies isomorphic objects, then the equivalence classes form a group, called the fundamental group of G and written π1(G). (Note that even for a strict 2-group, the fundamental group will only be a quotient group of the underlying group.)
As a monoidal category, any 2-group G has a unit object IG. The automorphism group of IG is an abelian group by the Eckmann–Hilton argument, written Aut(IG) or π2(G).
The fundamental group of G acts on either side of π2(G), and the associator of G (as a monoidal category) defines an element of the cohomology group H3(π1(G),π2(G)). In fact, 2-groups are classified in this way: given a group π1, an abel |
https://en.wikipedia.org/wiki/Hahn%20series | In mathematics, Hahn series (sometimes also known as Hahn–Mal'cev–Neumann series) are a type of formal infinite series. They are a generalization of Puiseux series (themselves a generalization of formal power series) and were first introduced by Hans Hahn in 1907 (and then further generalized by Anatoly Maltsev and Bernhard Neumann to a non-commutative setting). They allow for arbitrary exponents of the indeterminate so long as the set supporting them forms a well-ordered subset of the value group (typically or ). Hahn series were first introduced, as groups, in the course of the proof of the Hahn embedding theorem and then studied by him in relation to Hilbert's second problem.
Formulation
The field of Hahn series (in the indeterminate ) over a field and with value group (an ordered group) is the set of formal expressions of the form
with such that the support of f is well-ordered. The sum and product of
and
are given by
and
(in the latter, the sum over values such that , and is finite because a well-ordered set cannot contain an infinite decreasing sequence).
For example, is a Hahn series (over any field) because the set of rationals
is well-ordered; it is not a Puiseux series because the denominators in the exponents are unbounded. (And if the base field K has characteristic p, then this Hahn series satisfies the equation so it is algebraic over .)
Properties
Properties of the valued field
The valuation of a non-zero Hahn series
is defined as the smallest such that (in other words, the smallest element of the support of ): this makes into a spherically complete valued field with value group and residue field (justifying a posteriori the terminology). In fact, if has characteristic zero, then is up to (non-unique) isomorphism the only spherically complete valued field with residue field and value group .
The valuation defines a topology on . If , then corresponds to an ultrametric absolute value , with respect to which is a complete metric space. However, unlike in the case of formal Laurent series or Puiseux series, the formal sums used in defining the elements of the field do not converge: in the case of for example, the absolute values of the terms tend to 1 (because their valuations tend to 0), so the series is not convergent (such series are sometimes known as "pseudo-convergent").
Algebraic properties
If is algebraically closed (but not necessarily of characteristic zero) and is divisible, then is algebraically closed. Thus, the algebraic closure of is contained in , where is the algebraic closure of (when is of characteristic zero, it is exactly the field of Puiseux series): in fact, it is possible to give a somewhat analogous description of the algebraic closure of in positive characteristic as a subset of .
If is an ordered field then is totally ordered by making the indeterminate infinitesimal (greater than 0 but less than any positive element of ) or, equivalently, by using the l |
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