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https://en.wikipedia.org/wiki/Institute%20for%20the%20Promotion%20of%20Teaching%20Science%20and%20Technology | The Institute for the Promotion of Teaching Science and Technology (IPST) is a Thai state agency, founded in 1972. Its responsibilities include the development of national science and mathematics curricula, and sponsorship of science education, as well as the promotion of science in general. It is also Thailand's coordinator for the International Science Olympiad.
See also
Thailand at the International Science Olympiad
References
Public organizations of Thailand
Educational organizations based in Thailand
Science education in Thailand
Government agencies established in 1972
1972 establishments in Thailand |
https://en.wikipedia.org/wiki/Peter%20Bouwknegt | Pier Gerard "Peter" Bouwknegt (born 20 April 1961, Geldrop) is professor of theoretical physics and mathematics at the Australian National University (ANU), and deputy director of their Mathematical Sciences Institute. He is an adjunct professor at University of Adelaide.
Biography
He studied Theoretical Physics and Mathematics at the University of Utrecht, Netherlands, and at the University of Amsterdam under the direction of Prof F.A. Bais, obtaining his PhD in 1988. After that, he became a postdoctoral fellow at MIT, CERN, and the University of Southern California. He moved to Australia in 1995 and worked at the University of Adelaide as an ARC QEII Fellow and subsequently as an ARC Senior Research Fellow. In 2005, he was appointed Professor of Theoretical Physics and Mathematics at the Australian National University.
Awards
In 2001, he received the 2001 Australian Mathematical Society Medal, and from 2009–2011, he served on the Australian Research Council's College of Experts. He was formerly director of the Mathematical Sciences Institute at ANU, where is now deputy director.
Academic work
Bowknegt specializes in the mathematical foundations of String Theory and Conformal Field Theory. According to his web site at ANU, his specific interests are "the investigation of mathematical aspects of physical theories, in particular quantum field theories. Main expertise is the structure of two-dimensional conformal field theory and their applications in diverse areas such as condensed matter physics, integrable models of statistical mechanics and string theory, as well as the mathematical structures underlying string theory and D-branes, using mathematical techniques such as K-theory and gerbes."
References
Sources
P.G. Bouwknegt, 1961 - at the University of Amsterdam Album Academicum website
External links
K-theory (physics)
Tpsrv.anu.edu.au
1961 births
Academic staff of the Australian National University
People associated with CERN
Dutch expatriates in Australia
Dutch mathematicians
Living people
People from Geldrop
University of Amsterdam alumni
Utrecht University alumni |
https://en.wikipedia.org/wiki/Smooth%20scheme | In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology.
Definition
First, let X be an affine scheme of finite type over a field k. Equivalently, X has a closed immersion into affine space An over k for some natural number n. Then X is the closed subscheme defined by some equations g1 = 0, ..., gr = 0, where each gi is in the polynomial ring k[x1,..., xn]. The affine scheme X is smooth of dimension m over k if X has dimension at least m in a neighborhood of each point, and the matrix of derivatives (∂gi/∂xj) has rank at least n−m everywhere on X. (It follows that X has dimension equal to m in a neighborhood of each point.) Smoothness is independent of the choice of immersion of X into affine space.
The condition on the matrix of derivatives is understood to mean that the closed subset of X where all (n−m) × (n − m) minors of the matrix of derivatives are zero is the empty set. Equivalently, the ideal in the polynomial ring generated by all gi and all those minors is the whole polynomial ring.
In geometric terms, the matrix of derivatives (∂gi/∂xj) at a point p in X gives a linear map Fn → Fr, where F is the residue field of p. The kernel of this map is called the Zariski tangent space of X at p. Smoothness of X means that the dimension of the Zariski tangent space is equal to the dimension of X near each point; at a singular point, the Zariski tangent space would be bigger.
More generally, a scheme X over a field k is smooth over k if each point of X has an open neighborhood which is a smooth affine scheme of some dimension over k. In particular, a smooth scheme over k is locally of finite type.
There is a more general notion of a smooth morphism of schemes, which is roughly a morphism with smooth fibers. In particular, a scheme X is smooth over a field k if and only if the morphism X → Spec k is smooth.
Properties
A smooth scheme over a field is regular and hence normal. In particular, a smooth scheme over a field is reduced.
Define a variety over a field k to be an integral separated scheme of finite type over k. Then any smooth separated scheme of finite type over k is a finite disjoint union of smooth varieties over k.
For a smooth variety X over the complex numbers, the space X(C) of complex points of X is a complex manifold, using the classical (Euclidean) topology. Likewise, for a smooth variety X over the real numbers, the space X(R) of real points is a real manifold, possibly empty.
For any scheme X that is locally of finite type over a field k, there is a coherent sheaf Ω1 of differentials on X. The scheme X is smooth over k if and only if Ω1 is a vector bundle of rank equal to the dimension of X near each point. In that case, Ω1 is cal |
https://en.wikipedia.org/wiki/Distortion%20risk%20measure | In financial mathematics and economics, a distortion risk measure is a type of risk measure which is related to the cumulative distribution function of the return of a financial portfolio.
Mathematical definition
The function associated with the distortion function is a distortion risk measure if for any random variable of gains (where is the Lp space) then
where is the cumulative distribution function for and is the dual distortion function .
If almost surely then is given by the Choquet integral, i.e. Equivalently, such that is the probability measure generated by , i.e. for any the sigma-algebra then .
Properties
In addition to the properties of general risk measures, distortion risk measures also have:
Law invariant: If the distribution of and are the same then .
Monotone with respect to first order stochastic dominance.
If is a concave distortion function, then is monotone with respect to second order stochastic dominance.
is a concave distortion function if and only if is a coherent risk measure.
Examples
Value at risk is a distortion risk measure with associated distortion function
Conditional value at risk is a distortion risk measure with associated distortion function
The negative expectation is a distortion risk measure with associated distortion function .
See also
Risk measure
Coherent risk measure
Deviation risk measure
Spectral risk measure
References
Financial risk modeling |
https://en.wikipedia.org/wiki/1943%E2%80%9344%20Galatasaray%20S.K.%20season | The 1943–44 season was Galatasaray SK's 40th in existence and the club's 32nd consecutive season in the Istanbul Football League.
Squad statistics
Squad changes for the 1943–1944 season
In:
Istanbul Football League
Classification
Matches
Kick-off listed in local time (EEST)
Istanbul Futbol Kupası
3rd Round
1/4 final
1/2 final
Friendly matches
References
Atabeyoğlu, Cem. 1453-1991 Türk Spor Tarihi Ansiklopedisi. page(155-159).(1991) An Grafik Basın Sanayi ve Ticaret AŞ
Tekil, Süleyman. Dünden bugüne Galatasaray, (1983), page(88, 123-125, 184). Arset Matbaacılık Kol.Şti.
Futbol vol.2. Galatasaray. Page: 565, 586. Tercüman Spor Ansiklopedisi. (1981)Tercüman Gazetecilik ve Matbaacılık AŞ.
1940 Milli Küme Maçları. Türk Futbol Tarihi vol.1. page(81). (June 1992) Türkiye Futbol Federasyonu Yayınları.
External links
Galatasaray Sports Club Official Website
Turkish Football Federation - Galatasaray A.Ş.
uefa.com - Galatasaray AŞ
Galatasaray S.K. (football) seasons
Turkish football clubs 1943–44 season
1940s in Istanbul |
https://en.wikipedia.org/wiki/1944%E2%80%9345%20Galatasaray%20S.K.%20season | The 1944–45 season was Galatasaray SK's 41st in existence and the club's 33rd consecutive season in the Istanbul Football League.
Squad statistics
Squad changes for the 1944–1945 season
In:
Competitions
Istanbul Football League
Classification
Matches
Kick-off listed in local time (EEST)
Milli Küme
Classification
Matches
Istanbul Futbol Kupası
Matches
References
Atabeyoğlu, Cem. 1453-1991 Türk Spor Tarihi Ansiklopedisi. page(155-159).(1991) An Grafik Basın Sanayi ve Ticaret AŞ
Tekil, Süleyman. Dünden bugüne Galatasaray, (1983), page(88, 123-125, 184). Arset Matbaacılık Kol.Şti.
Futbol vol.2. Galatasaray. Page: 565, 586. Tercüman Spor Ansiklopedisi. (1981)Tercüman Gazetecilik ve Matbaacılık AŞ.
1940 Milli Küme Maçları. Türk Futbol Tarihi vol.1. page(81). (June 1992) Türkiye Futbol Federasyonu Yayınları.
External links
Galatasaray Sports Club Official Website
Turkish Football Federation - Galatasaray A.Ş.
uefa.com - Galatasaray AŞ
Galatasaray S.K. (football) seasons
Turkish football clubs 1944–45 season
1940s in Istanbul |
https://en.wikipedia.org/wiki/Eberlein%20compactum | In mathematics an Eberlein compactum, studied by William Frederick Eberlein, is a compact topological space homeomorphic to a subset of a Banach space with the weak topology.
Every compact metric space, more generally every one-point compactification of a locally compact metric space, is Eberlein compact. The converse is not true.
References
General topology |
https://en.wikipedia.org/wiki/Reflected%20Brownian%20motion | In probability theory, reflected Brownian motion (or regulated Brownian motion, both with the acronym RBM) is a Wiener process in a space with reflecting boundaries. In the physical literature, this process describes diffusion in a confined space and it is often called confined Brownian motion. For example it can describe the motion of hard spheres in water confined between two walls.
RBMs have been shown to describe queueing models experiencing heavy traffic as first proposed by Kingman and proven by Iglehart and Whitt.
Definition
A d–dimensional reflected Brownian motion Z is a stochastic process on uniquely defined by
a d–dimensional drift vector μ
a d×d non-singular covariance matrix Σ and
a d×d reflection matrix R.
where X(t) is an unconstrained Brownian motion and
with Y(t) a d–dimensional vector where
Y is continuous and non–decreasing with Y(0) = 0
Yj only increases at times for which Zj = 0 for j = 1,2,...,d
Z(t) ∈ , t ≥ 0.
The reflection matrix describes boundary behaviour. In the interior of the process behaves like a Wiener process; on the boundary "roughly speaking, Z is pushed in direction Rj whenever the boundary surface is hit, where Rj is the jth column of the matrix R."
Stability conditions
Stability conditions are known for RBMs in 1, 2, and 3 dimensions. "The problem of recurrence classification for SRBMs in four and higher dimensions remains open." In the special case where R is an M-matrix then necessary and sufficient conditions for stability are
R is a non-singular matrix and
R−1μ < 0.
Marginal and stationary distribution
One dimension
The marginal distribution (transient distribution) of a one-dimensional Brownian motion starting at 0 restricted to positive values (a single reflecting barrier at 0) with drift μ and variance σ2 is
for all t ≥ 0, (with Φ the cumulative distribution function of the normal distribution) which yields (for μ < 0) when taking t → ∞ an exponential distribution
For fixed t, the distribution of Z(t) coincides with the distribution of the running maximum M(t) of the Brownian motion,
But be aware that the distributions of the processes as a whole are very different. In particular, M(t) is increasing in t, which is not the case for Z(t).
The heat kernel for reflected Brownian motion at :
For the plane above
Multiple dimensions
The stationary distribution of a reflected Brownian motion in multiple dimensions is tractable analytically when there is a product form stationary distribution, which occurs when the process is stable and
where D = diag(Σ). In this case the probability density function is
where ηk = 2μkγk/Σkk and γ = R−1μ. Closed-form expressions for situations where the product form condition does not hold can be computed numerically as described below in the simulation section.
Simulation
One dimension
In one dimension the simulated process is the absolute value of a Wiener process. The following MATLAB program creates a sample path.
% rbm.m
n = 10^4; h=10^(- |
https://en.wikipedia.org/wiki/Bott%20cannibalistic%20class | In mathematics, the Bott cannibalistic class, introduced by , is an element of the representation ring of a compact Lie group that describes the action of the Adams operation on the Thom class of a complex representation . The term "cannibalistic" for these classes was introduced by .
References
Representation theory
K-theory |
https://en.wikipedia.org/wiki/List%20of%20Philippine%20Basketball%20Association%20career%20rebounding%20leaders | This is a list of the Philippine Basketball Association players in total career rebounds.
Statistics accurate and correct as of December 22, 2022.
See also
List of Philippine Basketball Association players
References
External links
Philippine Basketball Association All-time Most Rebounds Leaders – PBA Online.net
Philippine Basketball Association All-time Most Defensive Rebounds Leaders – PBA Online.net
Philippine Basketball Association All-time Most Offensive Rebounds Leaders – PBA Online.net
Rebounding, Career |
https://en.wikipedia.org/wiki/Subhash%20Suri | Subhash Suri (born July 7, 1960) is an Indian-American computer scientist, a professor at the University of California, Santa Barbara. He is known for his research in computational geometry, computer networks, and algorithmic game theory.
Biography
Suri did his undergraduate studies at the Indian Institute of Technology Roorkee, graduating in 1981. He then worked as a programmer in India before beginning his graduate studies in 1984 at Johns Hopkins University, where he earned a Ph.D. in computer science in 1987 under the supervision of Joseph O'Rourke. He was a member of the technical staff at Bellcore until 1994, when he returned to academia as an associate professor at Washington University in St. Louis. He moved to a full professorship at UCSB in 2000.
He was program committee chair for the 7th Annual International Symposium on Algorithms and Computation in 1996, and program committee co-chair for the 18th ACM Symposium on Computational Geometry in 2002.
Selected publications
.
.
.
.
Awards and honors
Suri was elected as a fellow of the IEEE in 2009, of the Association for Computing Machinery in 2010, and of the American Association for the Advancement of Science in 2011.
References
External links
Home page at UCSB
1960 births
Living people
American computer scientists
Indian computer scientists
20th-century Indian mathematicians
Researchers in geometric algorithms
Johns Hopkins University alumni
Washington University in St. Louis faculty
University of California, Santa Barbara faculty
Fellows of the American Association for the Advancement of Science
Fellows of the Association for Computing Machinery
Fellow Members of the IEEE |
https://en.wikipedia.org/wiki/Posterior%20predictive%20distribution | In Bayesian statistics, the posterior predictive distribution is the distribution of possible unobserved values conditional on the observed values.
Given a set of N i.i.d. observations , a new value will be drawn from a distribution that depends on a parameter , where is the parameter space.
It may seem tempting to plug in a single best estimate for , but this ignores uncertainty about , and because a source of uncertainty is ignored, the predictive distribution will be too narrow. Put another way, predictions of extreme values of will have a lower probability than if the uncertainty in the parameters as given by their posterior distribution is accounted for.
A posterior predictive distribution accounts for uncertainty about . The posterior distribution of possible values depends on :
And the posterior predictive distribution of given is calculated by marginalizing the distribution of given over the posterior distribution of given :
Because it accounts for uncertainty about , the posterior predictive distribution will in general be wider than a predictive distribution which plugs in a single best estimate for .
Prior vs. posterior predictive distribution
The prior predictive distribution, in a Bayesian context, is the distribution of a data point marginalized over its prior distribution . That is, if and , then the prior predictive distribution is the corresponding distribution , where
This is similar to the posterior predictive distribution except that the marginalization (or equivalently, expectation) is taken with respect to the prior distribution instead of the posterior distribution.
Furthermore, if the prior distribution is a conjugate prior, then the posterior predictive distribution will belong to the same family of distributions as the prior predictive distribution. This is easy to see. If the prior distribution is conjugate, then
i.e. the posterior distribution also belongs to but simply with a different parameter instead of the original parameter Then,
Hence, the posterior predictive distribution follows the same distribution H as the prior predictive distribution, but with the posterior values of the hyperparameters substituted for the prior ones.
The prior predictive distribution is in the form of a compound distribution, and in fact is often used to define a compound distribution, because of the lack of any complicating factors such as the dependence on the data and the issue of conjugacy. For example, the Student's t-distribution can be defined as the prior predictive distribution of a normal distribution with known mean μ but unknown variance σx2, with a conjugate prior scaled-inverse-chi-squared distribution placed on σx2, with hyperparameters ν and σ2. The resulting compound distribution is indeed a non-standardized Student's t-distribution, and follows one of the two most common parameterizations of this distribution. Then, the corresponding posterior predictive distribution would again be S |
https://en.wikipedia.org/wiki/William%20S.%20Zwicker | William Seymour Zwicker (born 1949) is an American mathematician and the William D. Williams Professor of Mathematics at Union College in Schenectady, New York.
Zwicker earned a bachelor's degree from Harvard University in 1971, and a Ph.D from Massachusetts Institute of Technology in 1976, under the supervision of Eugene M. Kleinberg. He joined the Union College faculty in 1975, was given his named chair in 2006, and retired in 2021.
Zwicker has done research in set theory and social choice theory. He is credited with inventing the concept of a supergame and the related hypergame paradox. With Alan D. Taylor, he is the author of Simple Games: Desirability Relations, Trading, Pseudoweightings (Princeton University Press, 1999).
References
1949 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Harvard University alumni
Union College (New York) faculty
Set theorists
Massachusetts Institute of Technology School of Science alumni |
https://en.wikipedia.org/wiki/Polygraph%20%28mathematics%29 | In mathematics, and particularly in category theory, a polygraph is a generalisation of a directed graph. It is also known as a computad. They were introduced as "polygraphs" by Albert Burroni and as "computads" by Ross Street.
In the same way that a directed multigraph can freely generate a category, an n-computad is the "most general" structure which can generate a free n-category.
References
Category theory
Directed graphs |
https://en.wikipedia.org/wiki/Gunnar%20Carlsson | Gunnar E. Carlsson (born August 22, 1952 in Stockholm, Sweden) is an American mathematician, working in algebraic topology. He is known for his work on the Segal conjecture, and for his work on applied algebraic topology, especially topological data analysis. He is a Professor Emeritus in the Department of Mathematics at Stanford University. He is the founder and president of the predictive technology company Ayasdi.
Biography
Carlsson was born in Sweden and was educated in the United States. He graduated from Redwood High School (Larkspur, California) in 1969. He received a Ph.D. from Stanford University in 1976, with a dissertation written under the supervision of R. J. Milgram. He was a Dickson Assistant Professor at the University of Chicago (1976-1978) and Professor at the University of California, San Diego (1978–86), Princeton University (1986-1991), and Stanford University (1991–2015) where he held the Anne and Bill Swindells Professorship and was Chair of the Department of Mathematics from 1995 to 1998.
He was an Ordway Visiting Professor at the University of Minnesota (May–June 1991) and held a Sloan Foundation Research Fellowship (1984-1986). He has delivered an invited address at the International Congress of Mathematicians in Berkeley, California, in 1986; a plenary address at the annual meeting of the American Mathematical Society (1984); the Whittaker Colloquium at the University of Edinburgh (2011); the Rademacher Lectures at the University of Pennsylvania (2011); and an invited plenary address at the annual meeting of the Society of Industrial and Applied Mathematics (2012).
He was elected as a member of the 2017 class of Fellows of the American Mathematical Society "for contributions to algebraic topology, particularly equivariant stable homotopy theory, algebraic K-theory, and applied algebraic topology".
In 2008, Carlsson cofounded Ayasdi, a predictive technology based on big data, machine learning and artificial intelligence.
Work
Equivariant methods in homotopy theory
Segal's Burnside conjecture provides a description of the stable cohomotopy theory of the classifying space of a finite group. It is the analogue for cohomotopy of the work of Michael Atiyah and Graeme Segal on the K-theory of these classifying spaces. Building on earlier work by Frank Adams, Jeremy Gunawardena, Haynes Miller, J. Peter May, James McClure, and L. Gaunce Lewis, Carlsson proved this conjecture in 1982. He also adapted the techniques to provide a proof of Sullivan's fixed point conjecture, which was also proved simultaneously and independently by Miller and Jean Lannes.
Algebraic K-theory
Algebraic K-theory is a topological construction that assigns spaces (ultimately spectra) to rings, schemes, and other non-topological input. It has connections with important questions in high-dimensional topology, notably the conjectures of Novikov and Borel. Carlsson has proved, jointly with E. Pedersen and B. Goldfarb Novikov's conjecture for larg |
https://en.wikipedia.org/wiki/Ditkin%20set | In mathematics, a Ditkin set, introduced by , is a closed subset of the circle such that a function f vanishing on the set can be approximated by functions φnf with φ vanishing in a neighborhood of the set.
References
Mathematical analysis |
https://en.wikipedia.org/wiki/2009%E2%80%9310%201.%20FC%20N%C3%BCrnberg%20season | The 2009–10 1. FC Nürnberg season was the 110th season in the club's football history.
Match results
Legend
Bundesliga
Playoff
DFB-Pokal
Player information
Roster and statistics
Transfers
In
Pula Pizda Coaiele
Out
Kits
Sources
1. FC Nürnberg seasons
Nuremberg |
https://en.wikipedia.org/wiki/Endrass%20surface | In algebraic geometry, an Endrass surface is a nodal surface of degree 8 with 168 real nodes, found by . , it remained the record-holder for the most number of real nodes for its degree; however, the best proven upper bound, 174, does not match the lower bound given by this surface.
See also
Barth surface
Sarti surface
Togliatti surface
References
Algebraic surfaces |
https://en.wikipedia.org/wiki/Eta%20invariant | In mathematics, the eta invariant of a self-adjoint elliptic differential operator on a compact manifold is formally the number of positive eigenvalues minus the number of negative eigenvalues. In practice both numbers are often infinite so are defined using zeta function regularization. It was introduced by who used it to extend the Hirzebruch signature theorem to manifolds with boundary. The name comes from the fact that it is a generalization of the Dirichlet eta function.
They also later used the eta invariant of a self-adjoint operator to define the eta invariant of a compact odd-dimensional smooth manifold.
defined the signature defect of the boundary of a manifold as the eta invariant, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at s=0 or 1 of a Shimizu L-function.
Definition
The eta invariant of self-adjoint operator A is given by ηA(0), where η is the analytic continuation of
and the sum is over the nonzero eigenvalues λ of A.
References
Differential operators |
https://en.wikipedia.org/wiki/List%20of%20Philippine%20Basketball%20Association%20career%20games%20played%20leaders | This is a list of Philippine Basketball Association players by total career games played.
Statistics accurate as of December 22, 2022.
See also
List of Philippine Basketball Association players
References
External links
Philippine Basketball Association All-time Leaders in Most Games Played – PBA Online.net
Games Played |
https://en.wikipedia.org/wiki/Hartshorne%20ellipse | In mathematics, a Hartshorne ellipse is an ellipse in the unit ball bounded by the 4-sphere S4 such that the ellipse and the circle given by intersection of its plane with S4 satisfy the Poncelet condition that there is a triangle with vertices on the circle and edges tangent to the ellipse. They were introduced by , who showed that they correspond to k = 2 instantons on S4.
References
Algebraic geometry |
https://en.wikipedia.org/wiki/List%20of%20Philippine%20Basketball%20Association%20career%20free%20throw%20scoring%20leaders | This is a list of Philippine Basketball Association players by total career free throws made.
Statistics accurate as of December 22, 2022.
See also
List of Philippine Basketball Association players
References
External links
Philippine Basketball Association All-time Leaders in Most Free Throws Made – PBA Online.net
Free Throw, Career |
https://en.wikipedia.org/wiki/Lebrun%20manifold | In mathematics, a LeBrun manifold is a connected sum of copies of the complex projective plane, equipped with an explicit self-dual metric. Here, self-dual means that the Weyl tensor is its own Hodge star. The metric
is determined by the choice of a finite collection of points in hyperbolic 3-space. These metrics were discovered by , and named after LeBrun by .
References
Differential geometry |
https://en.wikipedia.org/wiki/Reversed%20compound%20agent%20theorem | In probability theory, the reversed compound agent theorem (RCAT) is a set of sufficient conditions for a stochastic process expressed in any formalism to have a product form stationary distribution (assuming that the process is stationary). The theorem shows that product form solutions in Jackson's theorem, the BCMP theorem and G-networks are based on the same fundamental mechanisms.
The theorem identifies a reversed process using Kelly's lemma, from which the stationary distribution can be computed.
Notes
References
A short introduction to RCAT.
Probability theorems |
https://en.wikipedia.org/wiki/Severi%20variety | In algebraic geometry, a Severi variety, named after Francesco Severi, may be:
a Brauer–Severi variety
A Severi variety, a variety contained in a Hilbert scheme that parametrizes curves in projective space with given degree, arithmetic genus, and number of nodes and no other singularities.
a Scorza variety of dimension n in projective space of dimension 3n/2 + 2 that can be isomorphically projected to a hyperplane. |
https://en.wikipedia.org/wiki/Scorza%20variety | In mathematics, a k-Scorza variety is a smooth projective variety, of maximal dimension among those whose k–1 secant varieties are not the whole of projective space. Scorza varieties were introduced and classified by , who named them after Gaetano Scorza. The special case of 2-Scorza varieties are sometimes called Severi varieties, after Francesco Severi.
Classification
Zak showed that k-Scorza varieties are the projective varieties of the rank 1 matrices of rank k simple Jordan algebras.
Severi varieties
The Severi varieties are the non-singular varieties of dimension n (even) in PN that can be isomorphically projected to a hyperplane and satisfy N=3n/2+2.
Severi showed in 1901 that the only Severi variety with n=2 is the Veronese surface in P5.
The only Severi variety with n=4 is the Segre embedding of P2×P2 into P8, found by Scorza in 1908.
The only Segre variety with n=8 is the 8-dimensional Grassmannian G(1,5) of lines in P5 embedded into P14, found by John Greenlees Semple in 1931.
The only Severi variety with n=16 is a 16-dimensional variety E6/Spin(10)U(1) in P26 found by Robert Lazarsfeld in 1981.
These 4 Severi varieties can be constructed in a uniform way, as orbits of groups acting on the complexifications of the 3 by 3 hermitian matrices over the four real (possibly non-associative) division algebras of dimensions 2k = 1, 2, 4, 8. These representations have complex dimensions 3(2k+1) = 6, 9, 15, and 27, giving varieties of dimension 2k+1 = 2, 4, 8, 16 in projective spaces of dimensions 3(2k)+2 = 5, 8, 14, and 26.
Zak proved that the only Severi varieties are the 4 listed above, of dimensions 2, 4, 8, 16.
References
Algebraic geometry |
https://en.wikipedia.org/wiki/Gaetano%20Scorza | Bernardino Gaetano Scorza (29 September 1876, in Morano Calabro – 6 August 1939, in Rome) was an Italian mathematician working in algebraic geometry, whose work inspired the theory of Scorza varieties.
Publications
References
Italian mathematicians
People from the Province of Cosenza
1876 births
1939 deaths |
https://en.wikipedia.org/wiki/Novikov%E2%80%93Shubin%20invariant | In mathematics, a Novikov–Shubin invariant, introduced by , is an invariant of a compact Riemannian manifold related to the spectrum of the Laplace operator acting on square-integrable differential forms on its universal cover.
The Novikov–Shubin invariant gives a measure of the density of eigenvalues around zero. It can be computed from a triangulation of the manifold, and it is a homotopy invariant. In particular, it does not depend on the chosen Riemannian metric on the manifold.
Notes
References
Differential geometry
Algebraic topology |
https://en.wikipedia.org/wiki/Paul%20Baum%20%28mathematician%29 | Paul Frank Baum (born 1936) is an American mathematician, the Evan Pugh Professor of Mathematics at Pennsylvania State University. He is known for formulating the Baum–Connes conjecture with Alain Connes in the early 1980s.
Baum studied at Harvard University, earning a bachelor's degree summa cum laude in 1958. He went on to Princeton University for his graduate studies, completing his Ph.D. in 1963 under the supervision of John Coleman Moore and Norman Steenrod. He was several times a visiting scholar at the Institute for Advanced Study (1964–65, 1976–77, 2004) After several visiting positions and an assistant professorship at Princeton, he moved to Brown University in 1967, and remained there until 1987 when he moved to Penn State. He became a distinguished professor in 1991 and was given his named chair in 1996.
In 2007, a meeting in honor of his 70th birthday was held in Warsaw by the Polish Academy of Sciences. In 2011, the University of Colorado gave him an honorary doctorate. In 2012 he became a fellow of the American Mathematical Society.
References
1936 births
Brown University faculty
Living people
20th-century American mathematicians
21st-century American mathematicians
Harvard College alumni
Princeton University alumni
Princeton University faculty
Pennsylvania State University faculty
Institute for Advanced Study visiting scholars
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/List%20of%20converts%20to%20Christianity%20from%20Judaism | This is a list of notable converts to Christianity from Judaism.
The Jewish Encyclopedia gives some statistics on conversion of Jews to Protestantism, to Roman Catholicism, and to Orthodox Christianity Some 2,000 European Jews converted to Christianity every year during the 19th century, but in the 1890s the number was running closer to 3,000 per year—1,000 in Austria Hungary (Galizian Poland), 1,000 in Russia (Poland, Belarus, Ukraine, and Lithuania), 500 in Germany (Posen), and the remainder in the English world.
The 19th century saw at least 250,000 Jews convert to Christianity according to existing records of various societies. Data from the Pew Research Center that as of 2013, about 1.6 million adult Americans of Jewish background identify themselves as Christians, most are Protestant. According to same data most of the Americans of Jewish background who identify themselves as some sort of Christian (1.6 million) were raised as Jews or are Jews by ancestry. According to 2012 study 17% of Jews in Russia identify themselves as Christians. According to Heman in Herzog-Hauck, "Real-Encyc." (x. 114), the number of converts during the 19th century exceeded 100,000. Salmon, in his Handbuch der Mission (1893, p. 48) claims 130,000; others claim as many as 250,000. For Russia alone 40,000 are claimed as having been converted from 1836 to 1875 while for England, up to 1875, the estimate is 50,000.
Modern conversions mainly occurred en masse and at critical periods. In England there was a large secession when individuals from the chief Sephardic families, the Bernals, Furtados, Ricardos, Disraelis, Ximenes, Lopez's, Uzziellis, and others, joined the Church (see Picciotto, "Sketches of Anglo-Jewish History"). Germany had three of these periods. The Mendelssohnian era was marked by numerous conversions. In 1811, David Friedlander handed Prussian State Chancellor Hardenberg a list of 32 Jewish families and 18 unmarried Jews who had recently abandoned their ancestral faith (Rabbi Abraham Geiger, "Vor Hundert Jahren," Brunswick, 1899). In the reign of Frederick William III., about 2,200 Jews were baptized (1822–1840), most of these being residents of the larger cities. The 3rd and longest period of secession was the anti-Semitic, beginning with the year 1880. During this time the other German states, besides Austria and France, had an equal share in the number of those who obtained high stations and large revenues as the price for renouncing Judaism. The following is a list of the more prominent modern converts, the rarity of French names in which is probably because conversion was not necessary to a public career in that country.
A
Abd-al-Masih (martyr) (?–died 390 AD) – convert martyred for his faith
Abraham Abramson (1754–1811) – Prussian coiner and medallist. Born into a Jewish family, he later converted to Christianity.
Felix Aderca (1891–1962) – Romanian novelist, playwright, poet, journalist and critic, noted as a representative of rebelliou |
https://en.wikipedia.org/wiki/John%20Greenlees%20Semple | John Greenlees Semple (10 June 1904 in Belfast, Ireland – 23 October 1985 in London, England) was a British mathematician working in algebraic geometry.
Publications
Algebraic Projective Geometry. By J. G. Semple and G. T. Kneebone. Pp. viii, 404. 35s. 1952. (Oxford University Press).
References
20th-century British mathematicians
Scientists from Belfast
20th-century mathematicians from Northern Ireland
1904 births
1985 deaths |
https://en.wikipedia.org/wiki/Budan%27s%20theorem | In mathematics, Budan's theorem is a theorem for bounding the number of real roots of a polynomial in an interval, and computing the parity of this number. It was published in 1807 by François Budan de Boislaurent.
A similar theorem was published independently by Joseph Fourier in 1820. Each of these theorems is a corollary of the other. Fourier's statement appears more often in the literature of 19th century and has been referred to as Fourier's, Budan–Fourier, Fourier–Budan, and even Budan's theorem
Budan's original formulation is used in fast modern algorithms for real-root isolation of polynomials.
Sign variation
Let be a finite sequence of real numbers. A sign variation or sign change in the sequence is a pair of indices such that and either or for all such that .
In other words, a sign variation occurs in the sequence at each place where the signs change, when ignoring zeros.
For studying the real roots of a polynomial, the number of sign variations of several sequences may be used. For Budan's theorem, it is the sequence of the coefficients. For the Fourier's theorem, it is the sequence of values of the successive derivatives at a point. For Sturm's theorem it is the sequence of values at a point of the Sturm sequence.
Descartes' rule of signs
All results described in this article are based on Descartes' rule of signs.
If is a univariate polynomial with real coefficients, let us denote by the number of its positive real roots, counted with their multiplicity, and by the number of sign variations in the sequence of its coefficients. Descartes's rule of signs asserts that
is a nonnegative even integer.
In particular, if , then one has .
Budan's statement
Given a univariate polynomial with real coefficients, let us denote by the number of real roots, counted with their multiplicities, of in a half-open interval (with real numbers). Let us denote also by the number of sign variations in the sequence of the coefficients of the polynomial . In particular, one has with the notation of the preceding section.
Budan's theorem is the following:
is a nonnegative even integer.
As is non negative, this implies
This is a generalization of Descartes' rule of signs, as, if one chooses sufficiently large, it is larger than all real roots of , and all the coefficients of are positive, that is Thus and which makes Descartes' rule of signs a special case of Budan's theorem.
As for Descartes' rule of signs, if one has This means that, if one has a "zero-root test" and a "one-root test".
Examples
1. Given the polynomial and the open interval , one has
Thus, and Budan's theorem asserts that the polynomial has either two or zero real roots in the open interval
2. With the same polynomial one has
Thus, and Budan's theorem asserts that the polynomial has no real root in the open interval This is an example of the use of Budan's theorem as a zero-root test.
Fourier's statement
Fourier's theorem on polyno |
https://en.wikipedia.org/wiki/Fyodor%20Zak | Fyodor L. Zak ( (born December 2, 1949, in Moscow) is a Russian mathematician working on mathematical economics and algebraic geometry who classified the Scorza varieties.
Publications
References
Further reading
Mathematicians from Moscow
1949 births
Living people |
https://en.wikipedia.org/wiki/Zeuthen%E2%80%93Segre%20invariant | In algebraic geometry, the Zeuthen–Segre invariant I is an invariant of a projective surface found in a complex projective space which was introduced by and rediscovered by .
The invariant I is defined to be d – 4g – b if the surface has a pencil of curves, non-singular of genus g except for d curves with 1 ordinary node, and with b base points where the curves are non-singular and transverse.
showed that the Zeuthen–Segre invariant I is χ–4, where χ is the topological Euler–Poincaré characteristic introduced by , which is equal to the Chern number c2 of the surface.
References
Reprinted 2010
Algebraic surfaces |
https://en.wikipedia.org/wiki/List%20of%20census%20divisions%20of%20Quebec | Statistics Canada divides Quebec into 98 census divisions largely coextensive with the regional county municipalities of the province (of Quebec's 87 regional county municipalities, 82 have coextensive borders with Statistics Canada census divisions).
Quebec's census divisions consist of numerous census subdivisions. The types of census subdivisions within a Quebec census division may include:
cities and towns (ville), "ordinary" municipalities (municipalité), parish municipalities (paroisse), townships (canton) and united townships (cantons unis), villages (village)
Cree villages (village cri), northern villages (village nordique, i.e., Inuit), and one Naskapi village (village Naskapi)
Land reserved to Crees (Terres réservées aux Cris), Inuit land (Terre inuite), Naskapi land (Terres réservées aux Naskapis)
Indian reserves and Indian settlements
Unorganized territories
List of census divisions
The following is a list of Quebec's census divisions, as of the 2011 census.
See also
Types of municipalities in Quebec
Administrative divisions of Quebec
List of regional county municipalities in Quebec
Regional county municipality
References
External links
Statistics Canada: Subprovincial geography levels: Quebec
Lists of populated places in Quebec |
https://en.wikipedia.org/wiki/Non-Archimedean%20geometry | In mathematics, non-Archimedean geometry is any of a number of forms of geometry in which the axiom of Archimedes is negated. An example of such a geometry is the Dehn plane. Non-Archimedean geometries may, as the example indicates, have properties significantly different from Euclidean geometry.
There are two senses in which the term may be used, referring to geometries over fields which violate one of the two senses of the Archimedean property (i.e. with respect to order or magnitude).
Geometry over a non-Archimedean ordered field
The first sense of the term is the geometry over a non-Archimedean ordered field, or a subset thereof. The aforementioned Dehn plane takes the self-product of the finite portion of a certain non-Archimedean ordered field based on the field of rational functions. In this geometry, there are significant differences from Euclidean geometry; in particular, there are infinitely many parallels to a straight line through a point—so the parallel postulate fails—but the sum of the angles of a triangle is still a straight angle.
Intuitively, in such a space, the points on a line cannot be described by the real numbers or a subset thereof, and there exist segments of "infinite" or "infinitesimal" length.
Geometry over a non-Archimedean valued field
The second sense of the term is the metric geometry over a non-Archimedean valued field, or ultrametric space. In such a space, even more contradictions to Euclidean geometry result. For example, all triangles are isosceles, and overlapping balls nest. An example of such a space is the p-adic numbers.
Intuitively, in such a space, distances fail to "add up" or "accumulate".
References
Fields of geometry |
https://en.wikipedia.org/wiki/Moment%20closure | In probability theory, moment closure is an approximation method used to estimate moments of a stochastic process.
Introduction
Typically, differential equations describing the i-th moment will depend on the (i + 1)-st moment. To use moment closure, a level is chosen past which all cumulants are set to zero. This leaves a resulting closed system of equations which can be solved for the moments. The approximation is particularly useful in models with a very large state space, such as stochastic population models.
History
The moment closure approximation was first used by Goodman and Whittle who set all third and higher-order cumulants to be zero, approximating the population distribution with a normal distribution.
In 2006, Singh and Hespanha proposed a closure which approximates the population distribution as a log-normal distribution to describe biochemical reactions.
Applications
The approximation has been used successfully to model the spread of the Africanized bee in the Americas, nematode infection in ruminants. and quantum tunneling in ionization experiments.
References
Stochastic processes |
https://en.wikipedia.org/wiki/Peter%20M.%20Gruber | Peter Manfred Gruber (28 August 1941, Klagenfurt – 7 March 2017, Vienna) was an Austrian mathematician working in geometric number theory as well as in convex and discrete geometry.
Biography
Gruber obtained his PhD at the University of Vienna in 1966, under the supervision of Nikolaus Hofreiter. From 1971, he was Professor at the University of Linz, and from 1976, at the TU Wien. He was a member of the Austrian Academy of Sciences, a foreign member of the Russian Academy of Sciences, and a corresponding member of the Bavarian Academy of Sciences and Humanities.
His past doctoral students include Monika Ludwig.
Selected publications
Decorations and awards
1967: Prize of the Austrian Mathematical Society
1978, 1980 and 1982: Chairman of the Austrian Mathematical Society
1991: Full member of the Austrian Academy of Sciences (Corresponding member since 1988)
1996: Medal of the Union of Czech Mathematicians and Physicists
2001: Austrian Cross of Honour for Science and Art, 1st class
2001: Medal of the mathematical and physical faculty of Charles University in Prague
2003: Foreign member of the Russian Academy of Sciences
2008: Grand Silver Medal for Services to the Republic of Austria
2013: Fellow of the American Mathematical Society, for "contributions to the geometry of numbers and to convex and discrete geometry".
Honorary doctorates from the Universities of Siegen, Turin and Salzburg
Member of the Academies of Sciences in Messina and Modena
Corresponding member of the Bavarian Academy of Sciences
Notes
1941 births
2017 deaths
Scientists from Klagenfurt
Austrian mathematicians
Geometers
Number theorists
University of Vienna alumni
Academic staff of Johannes Kepler University Linz
Academic staff of TU Wien
Foreign Members of the Russian Academy of Sciences
Members of the Austrian Academy of Sciences
Recipients of the Austrian Cross of Honour for Science and Art, 1st class
Recipients of the Grand Decoration for Services to the Republic of Austria
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/Monsky%27s%20theorem | In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area. In other words, a square does not have an odd equidissection.
The problem was posed by Fred Richman in the American Mathematical Monthly in 1965, and was proved by Paul Monsky in 1970.
Proof
Monsky's proof combines combinatorial and algebraic techniques, and in outline is as follows:
Take the square to be the unit square with vertices at (0,0), (0,1), (1,0) and (1,1). If there is a dissection into n triangles of equal area then the area of each triangle is 1/n.
Colour each point in the square with one of three colours, depending on the 2-adic valuation of its coordinates.
Show that a straight line can contain points of only two colours.
Use Sperner's lemma to show that every triangulation of the square into triangles meeting edge-to-edge must contain at least one triangle whose vertices have three different colours.
Conclude from the straight-line property that a tricolored triangle must also exist in every dissection of the square into triangles, not necessarily meeting edge-to-edge.
Use Cartesian geometry to show that the 2-adic valuation of the area of a triangle whose vertices have three different colours is greater than 1. So every dissection of the square into triangles must contain at least one triangle whose area has a 2-adic valuation greater than 1.
If n is odd then the 2-adic valuation of 1/n is 1, so it is impossible to dissect the square into triangles all of which have area 1/n.
Optimal dissections
By Monsky's theorem it is necessary to have triangles with different areas to dissect a square into an odd number of triangles. Lower bounds for the area differences that must occur to dissect a square into an odd numbers of triangles and the optimal dissections have been studied.
Generalizations
The theorem can be generalized to higher dimensions: an n-dimensional hypercube can only be divided into simplices of equal volume, if the number of simplices is a multiple of n!.
References
Euclidean plane geometry
Theorems in discrete geometry
Geometric dissection |
https://en.wikipedia.org/wiki/Narasimhan%E2%80%93Seshadri%20theorem | In mathematics, the Narasimhan–Seshadri theorem, proved by , says that a holomorphic vector bundle over a Riemann surface is stable if and only if it comes from an irreducible projective unitary representation of the fundamental group.
The main case to understand is that of topologically trivial bundles, i.e. those of degree zero (and the other cases are a minor
technical extension of this case). This case of the Narasimhan–Seshadri theorem says that a degree zero holomorphic vector bundle over a Riemann surface is stable if and only if it comes from an irreducible unitary representation of the fundamental group of the Riemann surface.
gave another proof using differential geometry, and showed that the stable vector bundles have an essentially unique unitary connection of constant (scalar) curvature. In the degree zero case, Donaldson's version of the theorem says that a degree zero holomorphic vector bundle over a Riemann surface is stable if and only if it admits a flat unitary connection compatible with its holomorphic structure. Then the fundamental group representation appearing in the original statement is just the monodromy representation of this flat unitary connection.
See also
Nonabelian Hodge correspondence
Kobayashi–Hitchin correspondence
Stable vector bundle
References
Riemann surfaces
Theorems in analysis |
https://en.wikipedia.org/wiki/Shimizu%20L-function | In mathematics, the Shimizu L-function, introduced by , is a Dirichlet series associated to a totally real algebraic number field.
defined the signature defect of the boundary of a manifold as the eta invariant, the value as s=0 of their eta function, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at s=0 or 1 of a Shimizu L-function.
Definition
Suppose that K is a totally real algebraic number field, M is a lattice in the field, and V is a subgroup of maximal rank of the group of totally positive units preserving the lattice. The Shimizu L-series is given by
References
Zeta and L-functions |
https://en.wikipedia.org/wiki/Metasymplectic%20space | In mathematics, a metasymplectic space, introduced by and , is a Tits building of type F4 (a specific generalized incidence structure).
The four types of vertices are called points, lines, planes, and symplecta.
References
Incidence geometry |
https://en.wikipedia.org/wiki/Brieskorn%E2%80%93Grothendieck%20resolution | In mathematics, a Brieskorn–Grothendieck resolution is a resolution conjectured by Alexander Grothendieck, that in particular gives a resolution of the universal deformation of a Kleinian singularity. announced the construction of this resolution, and published the details of Brieskorn's construction.
References
Singularity theory |
https://en.wikipedia.org/wiki/Peter%20Slodowy | Peter Slodowy (12 October 1948, in Leverkusen – 19 November 2002, in Bonn) was a German mathematician who worked on singularity theory and algebraic geometry.
He completed his Ph.D. thesis at the University of Regensburg in 1978 under the direction of Theodor Bröcker and Egbert Brieskorn. The Slodowy correspondence is named after him.
Publications
References
20th-century German mathematicians
1948 births
2002 deaths
University of Regensburg alumni |
https://en.wikipedia.org/wiki/Warsaw%20School | Warsaw School may refer to:
Universities
Warsaw School of Economics
Warsaw School of Social Sciences and Humanities
Schools of thought
Warsaw School (mathematics)
Warsaw School (history of ideas)
See also
Lwów-Warsaw School (disambiguation) |
https://en.wikipedia.org/wiki/Iulie%20Aslaksen | Iulie Margrethe Nicolaysen Aslaksen (born 1956) is a Norwegian economist and Senior Researcher at Statistics Norway. She was a member of the Petroleum Price Board from 1990 to 2000. She is an expert on energy and environmental economics, including petroleum economics, climate policy and economics and sustainable development. She is cand.oecon. from the University of Oslo in 1981 and dr.polit. from 1990. She has been a visiting researcher and Fulbright Fellow at Harvard University and the University of California, Berkeley, and Associate Professor of Economics at the University of Oslo. She was a member of the government commissions resulting in the Norwegian Official Report 1988:21 Norsk økonomi i forandring (A Changing Norwegian Economy) and the Norwegian Official Report 1999:11 Analyse av investeringsutviklingen på kontinentalsokkelen (Analysis of Investments on the Norwegian Continental Shelf).
Selected bibliography
Thesis
Chapters in books
(pdf version)
Journal articles
See also
Feminist economics
List of feminist economists
References
External links
Profile: Iulie Aslaksen Statistics Norway
1956 births
Environmental social scientists
Environmental economists
Feminist economists
Harvard University faculty
Living people
Norwegian economists
Norwegian women economists
Norwegian environmentalists
University of Oslo alumni
Academic staff of the University of Oslo |
https://en.wikipedia.org/wiki/Quillen%20determinant%20line%20bundle | In mathematics, the Quillen determinant line bundle is a line bundle over the space of Cauchy–Riemann operators of a vector bundle over a Riemann surface, introduced by . Quillen proved the existence of the Quillen metric on the determinant line bundle, a Hermitian metric defined using the analytic torsion of a family of differential operators.
See also
Quillen metric
References
Riemann surfaces
Vector bundles |
https://en.wikipedia.org/wiki/Weil%20algebra | The term "Weil algebra" is also sometimes used to mean a finite-dimensional real local Artinian ring.
In mathematics, the Weil algebra of a Lie algebra g, introduced by based on unpublished work of André Weil, is a differential graded algebra given by the Koszul algebra Λ(g*)⊗S(g*) of its dual g*.
References
Reprinted in
Lie algebras |
https://en.wikipedia.org/wiki/Chordal%20variety | In algebraic geometry, a chordal variety of a variety is the union of all the chords (lines meeting 2 points), including the limiting cases of tangent lines.
References
Algebraic geometry |
https://en.wikipedia.org/wiki/Multivariate%20Pareto%20distribution | In statistics, a multivariate Pareto distribution is a multivariate extension of a univariate Pareto distribution.
There are several different types of univariate Pareto distributions including Pareto Types I−IV and Feller−Pareto. Multivariate Pareto distributions have been defined for many of these types.
Bivariate Pareto distributions
Bivariate Pareto distribution of the first kind
Mardia (1962) defined a bivariate distribution with cumulative distribution function (CDF) given by
and joint density function
The marginal distributions are Pareto Type 1 with density functions
The means and variances of the marginal distributions are
and for a > 2, X1 and X2 are positively correlated with
Bivariate Pareto distribution of the second kind
Arnold suggests representing the bivariate Pareto Type I complementary CDF by
If the location and scale parameter are allowed to differ, the complementary CDF is
which has Pareto Type II univariate marginal distributions. This distribution is called a multivariate Pareto distribution of type II by Arnold. (This definition is not equivalent to Mardia's bivariate Pareto distribution of the second kind.)
For a > 1, the marginal means are
while for a > 2, the variances, covariance, and correlation are the same as for multivariate Pareto of the first kind.
Multivariate Pareto distributions
Multivariate Pareto distribution of the first kind
Mardia's Multivariate Pareto distribution of the First Kind has the joint probability density function given by
The marginal distributions have the same form as (1), and the one-dimensional marginal distributions have a Pareto Type I distribution. The complementary CDF is
The marginal means and variances are given by
If a > 2 the covariances and correlations are positive with
Multivariate Pareto distribution of the second kind
Arnold suggests representing the multivariate Pareto Type I complementary CDF by
If the location and scale parameter are allowed to differ, the complementary CDF is
which has marginal distributions of the same type (3) and Pareto Type II univariate marginal distributions. This distribution is called a multivariate Pareto distribution of type II by Arnold.
For a > 1, the marginal means are
while for a > 2, the variances, covariances, and correlations are the same as for multivariate Pareto of the first kind.
Multivariate Pareto distribution of the fourth kind
A random vector X has a k-dimensional multivariate Pareto distribution of the Fourth Kind if its joint survival function is
The k1-dimensional marginal distributions (k1<k) are of the same type as (4), and the one-dimensional marginal distributions are Pareto Type IV.
Multivariate Feller–Pareto distribution
A random vector X has a k-dimensional Feller–Pareto distribution if
where
are independent gamma variables. The marginal distributions and conditional distributions are of the same type (5); that is, they are multivariate Feller–Pareto distributions. The one–dimensional |
https://en.wikipedia.org/wiki/Equivariant%20index%20theorem | In differential geometry, the equivariant index theorem, of which there are several variants, computes the (graded) trace of an element of a compact Lie group acting in given setting in terms of the integral over the fixed points of the element. If the element is neutral, then the theorem reduces to the usual index theorem.
The classical formula such as the Atiyah–Bott formula is a special case of the theorem.
Statement
Let be a clifford module bundle. Assume a compact Lie group G acts on both E and M so that is equivariant. Let E be given a connection that is compatible with the action of G. Finally, let D be a Dirac operator on E associated to the given data. In particular, D commutes with G and thus the kernel of D is a finite-dimensional representation of G.
The equivariant index of E is a virtual character given by taking the supertrace:
See also
Equivariant topological K-theory
Kawasaki's Riemann–Roch formula
References
Berline, Nicole; Getzler, E.; Vergne, Michèle (2004), Heat Kernels and Dirac Operators, Berlin, New York: Springer-Verlag
Differential geometry |
https://en.wikipedia.org/wiki/Secondary%20cohomology%20operation | In mathematics, a secondary cohomology operation is a functorial correspondence between cohomology groups. More precisely, it is a natural transformation from the kernel of some primary cohomology operation to the cokernel of another primary operation. They were introduced by in his solution to the Hopf invariant problem. Similarly one can define tertiary cohomology operations from the kernel to the cokernel of secondary operations, and continue like this to define higher cohomology operations, as in .
Michael Atiyah pointed out in the 1960s that many of the classical applications could be proved more easily using generalized cohomology theories, such as in his reproof of the Hopf invariant one theorem. Despite this, secondary cohomology operations still see modern usage, for example, in the obstruction theory of commutative ring spectra.
Examples of secondary and higher cohomology operations include the Massey product, the Toda bracket, and differentials of spectral sequences.
See also
Peterson–Stein formula
References
Algebraic topology |
https://en.wikipedia.org/wiki/Peterson%E2%80%93Stein%20formula | In mathematics, the Peterson–Stein formula, introduced by , describes the Spanier–Whitehead dual of a secondary cohomology operation.
References
Theorems in algebraic topology |
https://en.wikipedia.org/wiki/Berry%E2%80%93Robbins%20problem | In mathematics, the Berry–Robbins problem asks whether there is a continuous map from configurations of n points in R3 to the flag manifold U(n)/Tn that is compatible with the action of the symmetric group on n points. It was posed by and solved positively by .
See also
Atiyah conjecture on configurations
References
Lie groups |
https://en.wikipedia.org/wiki/Mathai%E2%80%93Quillen%20formalism | In mathematics, the Mathai–Quillen formalism is an approach to topological quantum field theory introduced by , based on the Mathai–Quillen form constructed in . In more detail, using the superconnection formalism of Quillen, they obtained a refinement of the Riemann–Roch formula, which links together the Thom classes in K-theory and cohomology, as an equality on the level of differential forms. This has an interpretation in physics as the computation of the classical and quantum (super) partition functions for the fermionic analogue of a harmonic oscillator with source term. In particular, they obtained a nice Gaussian shape representative of the Thom class in cohomology, which has a peak along the zero section.
References
Algebraic topology |
https://en.wikipedia.org/wiki/N%C3%A9ron%E2%80%93Ogg%E2%80%93Shafarevich%20criterion | In mathematics, the Néron–Ogg–Shafarevich criterion states that if A is an elliptic curve or abelian variety over a local field K and ℓ is a prime not dividing the characteristic of the residue field of K then A has good reduction if and only if the ℓ-adic Tate module Tℓ of A is unramified. introduced the criterion for elliptic curves. used the results of to extend it to abelian varieties,
and named the criterion after Ogg, Néron and Igor Shafarevich (commenting that Ogg's result seems to have been known to Shafarevich).
References
Abelian varieties
Elliptic curves
Theorems in algebraic geometry
Arithmetic geometry |
https://en.wikipedia.org/wiki/Bipolar%20theorem | In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set.
In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem.
Preliminaries
Suppose that is a topological vector space (TVS) with a continuous dual space and let for all and
The convex hull of a set denoted by is the smallest convex set containing
The convex balanced hull of a set is the smallest convex balanced set containing
The polar of a subset is defined to be:
while the prepolar of a subset is:
The bipolar of a subset often denoted by is the set
Statement in functional analysis
Let denote the weak topology on (that is, the weakest TVS topology on making all linear functionals in continuous).
The bipolar theorem: The bipolar of a subset is equal to the -closure of the convex balanced hull of
Statement in convex analysis
The bipolar theorem: For any nonempty cone in some linear space the bipolar set is given by:
Special case
A subset is a nonempty closed convex cone if and only if when where denotes the positive dual cone of a set
Or more generally, if is a nonempty convex cone then the bipolar cone is given by
Relation to the Fenchel–Moreau theorem
Let
be the indicator function for a cone
Then the convex conjugate,
is the support function for and
Therefore, if and only if
See also
− A generalization of the bipolar theorem.
References
Bibliography
Convex analysis
Functional analysis
Theorems in analysis
Linear functionals |
https://en.wikipedia.org/wiki/Grammatical%20Man | Grammatical Man: Information, Entropy, Language, and Life is a 1982 book written by Jeremy Campbell, then Washington correspondent for the Evening Standard. The book examines the topics of probability, information theory, cybernetics, genetics, and linguistics.
Information processes are used to frame and examine all of existence, from the Big Bang to DNA to human communication to artificial intelligence.
Part 1: Establishing the Theory of Information
The book's first chapter, The Second Law and the Yellow Peril, introduces the concept of entropy and gives brief outlines of the histories of Information Theory and cybernetics, examining World War II figures such as Claude Shannon and Norbert Wiener.
The Noise of Heat gives an outline of the history of thermodynamics, focusing on Rudolf Clausius's 2nd Law and its relation to order and information.
In The Demon Possessed Campbell examines the concept of entropy and presents entropy as missing information.
Chapter Four, A Nest of Subtleties and Traps, takes its name from a critique of one of the earliest theorems in probability theory, Law of large numbers (Bernoulli, 1713). The chapter outlines the history of probability, touching on characters such as Gerolamo Cardano, Antoine Gombaud, Bernoulli, Richard von Mises, and John Maynard Keynes. Campbell examines information and entropy as a probability distribution of possible messages and says that subjective versus objective interpretations of probability are made largely obsolete by an understanding of the relationship between probability and information.
Not Too Dull, Not Too Exciting addresses the problem of clarifying order from disorder within communication by highlighting the role that redundancy plays in information theory.
In the last chapter of Part 1, The Struggle Against Randomness, Campbell addresses the concepts published by Shannon in 1948—that a message can be sent from one place to another, even under noisy conditions, and be as free from error as the sender cares to make it, as long as it is coded in the proper form.
Part 2: Nature as an Information Process
Campbell uses Arrows in All Directions discusses the potential inverse relation between entropy and novelty, invoking such concepts as Laplace's Superman. Campbell quotes David Layzer: For Laplace's "intelligence," as for the God of Plato, Galileo and Einstein, the past and future coexist on equal terms, like the two rays into which an arbitrarily chosen point divides a straight line. If the theories I have presented are correct, however, not even the ultimate computer --the universe itself-- ever contains enough information to specify completely its own future states. The present moment always contains an element of genuine novelty and the future is never wholly predictable. Because biological processes also generate information and because consciousness enables us to experience those processes directly, the intuitive perception of the world as unfolding in time captures |
https://en.wikipedia.org/wiki/Marot%20ring | In mathematics, a Marot ring, introduced by , is a commutative ring whose regular ideals are generated by regular elements.
References
Ring theory |
https://en.wikipedia.org/wiki/Kronheimer%E2%80%93Mrowka%20basic%20class | In mathematics, the Kronheimer–Mrowka basic classes are elements of the second cohomology H2(X) of a simple smooth 4-manifold X that determine its Donaldson polynomials. They were introduced by .
References
Differential geometry |
https://en.wikipedia.org/wiki/Atiyah%20conjecture%20on%20configurations | In mathematics, the Atiyah conjecture on configurations is a conjecture introduced by stating that a certain n by n matrix depending on n points in R3 is always non-singular.
See also
Berry–Robbins problem
References
Conjectures
Unsolved problems in geometry |
https://en.wikipedia.org/wiki/Gibbons%E2%80%93Hawking%20ansatz | In mathematics, the Gibbons–Hawking ansatz is a method of constructing gravitational instantons introduced by . It gives examples of hyperkähler manifolds in dimension 4 that are invariant under a circle action.
See also
Gibbons–Hawking space
References
1978 introductions
Differential geometry
General relativity
Stephen Hawking |
https://en.wikipedia.org/wiki/Normal-Wishart%20distribution | In probability theory and statistics, the normal-Wishart distribution (or Gaussian-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and precision matrix (the inverse of the covariance matrix).
Definition
Suppose
has a multivariate normal distribution with mean and covariance matrix , where
has a Wishart distribution. Then
has a normal-Wishart distribution, denoted as
Characterization
Probability density function
Properties
Scaling
Marginal distributions
By construction, the marginal distribution over is a Wishart distribution, and the conditional distribution over given is a multivariate normal distribution. The marginal distribution over is a multivariate t-distribution.
Posterior distribution of the parameters
After making observations , the posterior distribution of the parameters is
where
Generating normal-Wishart random variates
Generation of random variates is straightforward:
Sample from a Wishart distribution with parameters and
Sample from a multivariate normal distribution with mean and variance
Related distributions
The normal-inverse Wishart distribution is essentially the same distribution parameterized by variance rather than precision.
The normal-gamma distribution is the one-dimensional equivalent.
The multivariate normal distribution and Wishart distribution are the component distributions out of which this distribution is made.
Notes
References
Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.
Multivariate continuous distributions
Conjugate prior distributions
Normal distribution |
https://en.wikipedia.org/wiki/Patharpratima%20Mahavidyalaya | Patharpratima Mahavidyalaya, established in 2001, is an undergraduate college in Patharpratima, West Bengal, India. It is affiliated with the University of Calcutta.
Departments
Science
Mathematics
Arts and Commerce
Bengali
English
History
Geography
Political Science
Philosophy
Economics
Education
Commerce
See also
List of colleges affiliated to the University of Calcutta
Education in India
Education in West Bengal
References
External links
Patharpratima Mahavidyalaya
University of Calcutta affiliates
Universities and colleges in South 24 Parganas district
Educational institutions established in 2001
2001 establishments in West Bengal |
https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20France | This page details football records and statistics in France.
National team
League
.
Titles
Most top-flight League titles: 11, Paris Saint-Germain
Most consecutive League titles: 7, Lyon
Top-flight appearances
Most appearances: 68 seasons, Marseille
Most consecutive seasons in top-flight: 49 seasons, Paris Saint-Germain (1974–2023)
Wins
Most wins in the top-flight overall: 992, Marseille
Most wins in a top flight season: 30, Paris Saint-Germain (2015–16), Monaco (2016–17)
Most home wins in a top flight season: 19, Saint-Étienne (1974–75)
Most away wins in a top flight season: 13, Monaco (2016–17)
Most consecutive wins: 14, Bordeaux (round 28 2008–09 to round 3 2009–10)
Most consecutive home wins: 28, Saint-Étienne (round 28 1973–74 to round 4 1975–76)
Most consecutive away wins: 9, Marseille (round 24 2008–09 to round 1 2009–10)
Most consecutive wins in a single top flight season: 12, Monaco (round 27 2016–17 to round 38 2016–17)
Draws
Most draws in the top-flight overall: 622, Sochaux
Losses
Most losses in the top-flight overall: 859, Sochaux
Fewest losses in a top flight season: 1, Nantes (1994–95)
Fewest home losses in a top flight season: 0, joint record :
Nantes, 7 seasons
Marseille, Bordeaux, 5 seasons
Saint-Étienne, Monaco, 4 seasons
Sète, Strasbourg, Auxerre, 3 seasons
Nîmes, Bastia, Lens, Paris Saint-Germain, Lyon, 2 seasons
Olympique Lillois, RC Paris, Valenciennes, Angers, Nice, Nancy, Laval, Toulouse, Cannes, 1 season
Fewest away losses in a top flight season: 1, Nantes (1994–95)
Points
Most points in the top-flight overall:
2 points for a win: 2551, Marseille
3 points for a win: 3543, Marseille
Fewest points in a top flight season:
3 points for a win: 17, Lens (1988–89)
Goals
Most goals scored in the top-flight overall: 3611, Marseille
Most goals conceded in the top-flight overall: 3236, FC Sochaux
Highest goal difference in the top-flight overall: +847, Monaco
Most goals scored in a top flight season: 118, Racing CF (1959–60)
Fewest goals conceded in a top flight season: 19, Paris Saint-Germain (2015–16)
Highest goal difference in a top flight season: +83, Paris Saint-Germain (2015–16)
Attendances
Record attendance: Lille v Lyon played at Stade de France (7 March 2009).
Total titles won (1918–present)
Key
Performance by club
(Sorted by overall titles. Use sorting button to change criteria.)
Last updated: 31 July 2022
The figures in bold represent the most times this competition has been won by a French team.
References
Football records and statistics in France
Football in France
French records
France |
https://en.wikipedia.org/wiki/Grothendieck%E2%80%93Ogg%E2%80%93Shafarevich%20formula | In mathematics, the Grothendieck–Ogg–Shafarevich formula describes the Euler characteristic of a complete curve with coefficients in an abelian variety or constructible sheaf, in terms of local data involving the Swan conductor. and proved the formula for abelian varieties with tame ramification over curves, and extended the formula to constructible sheaves over a curve .
Statement
Suppose that F is a constructible sheaf over a genus g smooth projective curve C, of rank n outside a finite set X of points where it has stalk 0. Then
where Sw is the Swan conductor at a point.
References
Elliptic curves
Abelian varieties |
https://en.wikipedia.org/wiki/Normal-inverse-Wishart%20distribution | In probability theory and statistics, the normal-inverse-Wishart distribution (or Gaussian-inverse-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and covariance matrix (the inverse of the precision matrix).
Definition
Suppose
has a multivariate normal distribution with mean and covariance matrix , where
has an inverse Wishart distribution. Then
has a normal-inverse-Wishart distribution, denoted as
Characterization
Probability density function
The full version of the PDF is as follows:
Here is the multivariate gamma function and is the Trace of the given matrix.
Properties
Scaling
Marginal distributions
By construction, the marginal distribution over is an inverse Wishart distribution, and the conditional distribution over given is a multivariate normal distribution. The marginal distribution over is a multivariate t-distribution.
Posterior distribution of the parameters
Suppose the sampling density is a multivariate normal distribution
where is an matrix and (of length ) is row of the matrix .
With the mean and covariance matrix of the sampling distribution is unknown, we can place a Normal-Inverse-Wishart prior on the mean and covariance parameters jointly
The resulting posterior distribution for the mean and covariance matrix will also be a Normal-Inverse-Wishart
where
.
To sample from the joint posterior of , one simply draws samples from , then draw . To draw from the posterior predictive of a new observation, draw , given the already drawn values of and .
Generating normal-inverse-Wishart random variates
Generation of random variates is straightforward:
Sample from an inverse Wishart distribution with parameters and
Sample from a multivariate normal distribution with mean and variance
Related distributions
The normal-Wishart distribution is essentially the same distribution parameterized by precision rather than variance. If then .
The normal-inverse-gamma distribution is the one-dimensional equivalent.
The multivariate normal distribution and inverse Wishart distribution are the component distributions out of which this distribution is made.
Notes
References
Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.
Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution."
Multivariate continuous distributions
Conjugate prior distributions
Normal distribution |
https://en.wikipedia.org/wiki/Guangdong%20Evergrande%20volleyball%20team%20statistics | This is Statistics of China women's volleyball club Guangdong Evergrande
Team Roster
Team member 2009-2010
Head coach: Lang Ping
Team member 2010-2011
Head coach: Lang Ping
Team member 2011-2012
Head coach: Lang Ping
Best Scorer History
External links
2009-2010 season
2010-2011 season
2011-2012 season
Volleyball in China |
https://en.wikipedia.org/wiki/Robert%20J.%20Harrison | Robert J. Harrison (born June 19, 1960) is a distinguished expert in high-performance computing. He is a professor in the Applied Mathematics and Statistics department and founding Director of the Institute for Advanced Computational Science at Stony Brook University with a $20M endowment. Through a joint appointment with Brookhaven National Laboratory, Professor Harrison has also been named Director of the Computational Science Center and New York Center for Computational Sciences at Brookhaven. Dr. Harrison comes to Stony Brook from the University of Tennessee and Oak Ridge National Laboratory, where he was Director of the Joint Institute of Computational Science, Professor of Chemistry and Corporate Fellow. He has a prolific career in high-performance computing with over one hundred publications on the subject, as well as extensive service on national advisory committees.
He has many publications in peer-reviewed journals in the areas of theoretical and computational chemistry, and high-performance computing. His undergraduate (1981) and post-graduate (1984) degrees were obtained at Cambridge University, England. Subsequently, he worked as a postdoctoral research fellow at the Quantum Theory Project, University of Florida, and the Daresbury Laboratory, England, before joining the staff of the theoretical chemistry group at Argonne National Laboratory in 1988. In 1992, he moved to the Environmental Molecular Sciences Laboratory of Pacific Northwest National Laboratory, conducting research in theoretical chemistry and leading the development of NWChem, a computational chemistry code for massively parallel computers. In August 2002, he started the joint faculty appointment with UT/ORNL, and became director of JICS in 2011.
In addition to his DOE Scientific Discovery through Advanced Computing (SciDAC) research into efficient and accurate calculations on large systems, he has been pursuing applications in molecular electronics and chemistry at the nanoscale. In 1999, the NWChem team received an R&D Magazine R&D100 award, in 2002, he received the IEEE Computer Society Sidney Fernbach Award, and in 2011 another R&D Magazine R&D100 award for the development of MADNESS. In 2015-2016, Dr. Harrison co-chaired with Bill Gropp the National Academies of Sciences, Engineering, and Medicine committee on Future Directions for NSF Advanced Computing Infrastructure to Support U.S. Science in 2017-2020.
His interests and expertise are in theoretical and computational chemistry, high-performance computing, electron correlation, electron transport, relativistic quantum chemistry, and response theory.
Bibliography
References
External links
Microsoft Research Paper Search
1960 births
Living people
University of Tennessee faculty
21st-century American chemists
Alumni of the University of Cambridge
People from Birmingham, West Midlands
American computer scientists
Computational chemists |
https://en.wikipedia.org/wiki/Shimura%27s%20reciprocity%20law | In mathematics, Shimura's reciprocity law, introduced by , describes the action of ideles of imaginary quadratic fields on the values of modular functions at singular moduli. It forms a part of the Kronecker Jugendtraum, explicit class field theory for such fields. There are also higher-dimensional generalizations.
References
Theorems in number theory |
https://en.wikipedia.org/wiki/Cartier%20duality | In mathematics,
Cartier duality is an analogue of Pontryagin duality for commutative group schemes. It was introduced by .
Definition using characters
Given any finite flat commutative group scheme G over S, its Cartier dual is the group of characters, defined as the functor that takes any S-scheme T to the abelian group of group scheme homomorphisms from the base change to and any map of S-schemes to the canonical map of character groups. This functor is representable by a finite flat S-group scheme, and Cartier duality forms an additive involutive antiequivalence from the category of finite flat commutative S-group schemes to itself. If G is a constant commutative group scheme, then its Cartier dual is the diagonalizable group D(G), and vice versa. If S is affine, then the duality functor is given by the duality of the Hopf algebras of functions.
Definition using Hopf algebras
A finite commutative group scheme over a field corresponds to a finite dimensional commutative cocommutative Hopf algebra. Cartier duality corresponds to taking the dual of the Hopf algebra, exchanging the multiplication and comultiplication.
More general cases of Cartier duality
The definition of Cartier dual extends usefully to much more general situations where the resulting functor on schemes is no longer represented as a group scheme. Common cases include fppf sheaves of commutative groups over S, and complexes thereof. These more general geometric objects can be useful when one wants to work with categories that have good limit behavior. There are cases of intermediate abstraction, such as commutative algebraic groups over a field, where Cartier duality gives an antiequivalence with commutative affine formal groups, so if G is the additive group , then its Cartier dual is the multiplicative formal group , and if G is a torus, then its Cartier dual is étale and torsion-free. For loop groups of tori, Cartier duality defines the tame symbol in local geometric class field theory. Gérard Laumon introduced a sheaf-theoretic Fourier transform for quasi-coherent modules over 1-motives that specializes to many of these equivalences.
Examples
The Cartier dual of the cyclic group of order n is the n-th roots of unity .
Over a field of characteristic p the group scheme (the kernel of the endomorphism of the additive group induced by taking pth powers) is its own Cartier dual.
References
Algebraic groups |
https://en.wikipedia.org/wiki/Waldspurger%27s%20theorem | In mathematics, Waldspurger's theorem, introduced by , is a result that identifies Fourier coefficients of modular forms of half-integral weight k+1/2 with the value of an L-series at s=k/2.
References
Modular forms
Zeta and L-functions
Theorems in number theory |
https://en.wikipedia.org/wiki/Brumer%20bound | In mathematics, the Brumer bound is a bound for the rank of an elliptic curve, proved by .
See also
Mestre bound
References
Elliptic curves
Theorems in number theory |
https://en.wikipedia.org/wiki/Eichler%20order | In mathematics, an Eichler order, named after Martin Eichler, is an order of a quaternion algebra that is the intersection of two maximal orders.
References
Number theory |
https://en.wikipedia.org/wiki/Isoparametric%20function | In differential geometry, an isoparametric function is a function on a Riemannian manifold whose level surfaces are parallel and of constant mean curvatures. They were introduced by .
See also
Isoparametric manifold
References
Riemannian geometry |
https://en.wikipedia.org/wiki/Mathesis%20%28journal%29 | Mathesis: Recueil Mathématique was a Belgian scientific journal for elementary mathematics, established in 1881 by Paul Mansion and Joseph Jean Baptiste Neuberg.
An earlier Belgian mathematics journal, Nouvelle Correspondance Mathématique, was established in 1874 by Mansion and Neuberg together with Eugène Catalan. In 1880, Nouvelle Correspondance ceased publication, and Mansion and Neuberg together launched its successor, Mathesis, in 1881. Mathesis ceased publication in 1915 because of the war in Europe, but restarted again under the editorship of Neuberg and Adolphe Mineur in 1922
as the official journal of the Belgian Mathematical Society, which itself was founded in 1921. It continued in publication until 1965.
References
Mathematics journals
French-language journals
Publications established in 1881
Publications disestablished in 1965
1881 establishments in Belgium |
https://en.wikipedia.org/wiki/Mojette%20Transform | The Mojette Transform is an application of discrete geometry. More specifically, it is a discrete and exact version of the Radon transform, thus a projection operator.
The IRCCyN laboratory - UMR CNRS 6597 in Nantes, France has been developing it since 1994.
The first characteristic of the Mojette Transform is using only additions and subtractions. The second characteristic is that the Mojette Transform is redundant, spreading the initial geometrical information into several projections.
This transform uses discrete geometry in order to dispatch information onto a discrete geometrical support. This support is then projected by the Mojette operator along discrete directions. When enough projections are available, the initial information can be reconstructed.
The Mojette transform has been already used in numerous applications domains:
Medical tomography
Network packet transfer
Distributed storage on disks or networks
Image fingerprinting and image cryptography schemes
History
After one year of research, the first communication introducing the Mojette Transform was held in May 1995 in the first edition of CORESA National Congress CCITT Rennes. Many others will follow it for 18 years of existence. In 2011, the book The Mojette Transform: Theory and Applications at ISTE-Wiley was well received by the scientific community. All this support has encouraged the IRCCyN research team to continue the research on this topic.
Jeanpierre Guédon, professor and inventor of the transform called it: "Mojette Transform". The word "Mojette" comes from the name of white beans in Vendee, originally written "Moghette" or "Mojhette". In many countries, bean is a basic educational tool representing an exact unit that teaches visually additions and subtractions. Therefore, the choice of the name "Mojette" serves to emphasize the fact that the transform uses only exact unit in additions and subtractions.
The original purpose of the Mojette Transform was to create a discrete tool to divide the Fourier plane into angular and radial sectors. The first attempt of application was the psychovisual encoding of image, reproducing the human vision channel. However, it was never realized.
Mathematics
The "raw" transform Mojette definition is this:
The following figure 1 helps to explain the “raw” transform Mojette.
We start with the function f represented by 16 pixels from p1 to p16. The possible values of the function at the point (k, l) are different according to the applications. This can be a binary value of 0 or 1 that it often used to differentiate the object and the background. This can be a ternary value as in the Mojette game. This can be also a finite set of integers value from 0 to (n-1), or more often we take a set of cardinality equal to a power of 2 or a prime number. But it can be integers and real numbers with an infinite number of possibilities, even though this idea has never been used.
With the index "k" as "kolumn" and “l” as a “line”, we define |
https://en.wikipedia.org/wiki/Padre%20Burgos%20Avenue | {
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Padre Burgos Avenue, also known as Padre Burgos Street, is a 14-lane thoroughfare in Manila, Philippines.
The road was named after Jose Burgos, one of the martyred priests who were executed at the nearby Bagumbayan Field (present-day Rizal Park) in 1872. It is a road in the center of the city providing access to several important |
https://en.wikipedia.org/wiki/Monopole%20moduli%20space | In mathematics, the monopole moduli space is a space parametrizing monopoles (solutions of the Bogomolny equations). studied the moduli space for 2 monopoles in detail and used it to describe the scattering of monopoles.
See also
Hitchin system
References
Differential geometry |
https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20Germany | For association football in Germany, this page details football records in Germany.
National team
Appearances
Most appearances: 150, Lothar Matthäus
Youngest player: Willy Baumgärtner, 17 years, 104 days, 5 April 1908, 3–5 v Switzerland
Oldest player: Lothar Matthäus, 39 years, 91 days, 20 June 2000, 0–3 v Portugal
Oldest debutant: Karl Sesta, 35 years, 83 days, 15 June 1941, 5–1 v Croatia
Goals
First goal: Fritz Becker, 5 April 1908, 3–5 v Switzerland
Most goals: 71, Miroslav Klose
Most goals in a match: 10, Gottfried Fuchs (v Russia, 1912 Summer Olympics)
Youngest goalscorer: Marius Hiller, 17 years, 241 days, 3 April 1910, 3–2 v Switzerland
Men's Honours
Major competitions
FIFA World Cup
Champions (4): 1954, 1974, 1990, 2014
Runners-up (4): 1966, 1982, 1986, 2002
Third place (4): 1934, 1970, 2006, 2010
Fourth place (1): 1958
UEFA European Championship
Champions (3): 1972, 1980, 1996
Runners-up (3): 1976, 1992, 2008
Third place (3): 1988, 2012, 2016
Summer Olympic Games
Gold Medal (1): 1976
Silver Medal (2): 1980, 2016
Bronze Medal (3): 1964, 1972, 1988
Fourth place (1): 1952
FIFA Confederations Cup
Champions (1): 2017
Third place (1): 2005
Women's Honours
Major competitions
FIFA Women's World Cup
Champions (2): 2003, 2007
Runners-up (1): 1995
Fourth place (2): 1991, 2015
UEFA Women's Championship
Champions (8): 1989, 1991, 1995, 1997, 2001, 2005, 2009, 2013
Runners-up (1): 2022
Fourth place (1): 1993
Summer Olympic Games
Gold Medal (1): 2016
Bronze Medal (3): 2000, 2004, 2008
League
Titles
Most championships won: 33, Bayern Munich
Most consecutive championships: 11
Bayern Munich (2013–2023)
Most East German championships: 10, Dynamo Berlin
Most consecutive East German championships: 10, Dynamo Berlin (1979–1988)
Attendances
Record attendance: 100,000, SC Rotation Leipzig v SC Lokomotive Leipzig at Zentralstadion (September 1956)
Individual records
Appearances
Most Bundesliga appearances: 602, Charly Körbel
Goals
Most career league goals: Erwin Helmchen
Most career domestic top-flight goals: 404, Uwe Seeler
Most Bundesliga goals: 365, Gerd Müller
Most Bundesliga goals in one season: 41, Robert Lewandowski (2020–21)
Most successful clubs overall (1902–present)
Key
Performance by club
(Sorted by overall titles. Use sorting button to change criteria.)
Last updated on 12 August 2023, following RB Leipzig winning the 2023 DFL-Supercup.
The figures in bold represent the most times this competition has been won by a German team.
References
Football records and statistics in Germany
Records
Football
Germany |
https://en.wikipedia.org/wiki/Signature%20cocycle | In mathematics, the Meyer signature cocycle, introduced by . is an integer-valued 2-cocyle on a symplectic group that describes the signature of a fiber bundle whose base and fiber are both Riemann surfaces.
References
Manifolds |
https://en.wikipedia.org/wiki/K%C3%A4hler%20quotient | In mathematics, specifically in complex geometry, the Kähler quotient of a Kähler manifold by a Lie group acting on by preserving the Kähler structure and with moment map (with respect to the Kähler form) is the quotient
If acts freely and properly, then is a new Kähler manifold whose Kähler form is given by the symplectic quotient construction.
By the Kempf-Ness theorem, a Kähler quotient by a compact Lie group is closely related to a geometric invariant theory quotient by the complexification of .
See also
Hyperkähler quotient
References
Complex manifolds |
https://en.wikipedia.org/wiki/Hyperk%C3%A4hler%20quotient | In mathematics, the hyperkähler quotient of a hyperkähler manifold acted on by a Lie group G is the quotient of a fiber of a hyperkähler moment map over a G-fixed point by the action of G. It was introduced by Nigel Hitchin, Anders Karlhede, Ulf Lindström, and Martin Roček in 1987. It is a hyperkähler analogue of the Kähler quotient.
References
Differential geometry |
https://en.wikipedia.org/wiki/Transportation%20theory | Transportation theory may refer to:
Transportation theory (mathematics)
Transportation theory (psychology) |
https://en.wikipedia.org/wiki/Torsion%20conjecture | In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field. The torsion conjecture has been completely resolved in the case of elliptic curves.
Elliptic curves
From 1906 to 1911, Beppo Levi published a series of papers investigating the possible finite orders of points on elliptic curves over the rationals. He showed that there are infinitely many elliptic curves over the rationals with the following torsion groups:
Cn with 1 ≤ n ≤ 10, where Cn denotes the cyclic group of order n;
C12;
C2n × C2 with 1 ≤ n ≤ 4, where × denotes the direct sum.
At the 1908 International Mathematical Congress in Rome, Levi conjectured that this is a complete list of torsion groups for elliptic curves over the rationals. The torsion conjecture for elliptic curves over the rationals was independently reformulated by and again by , with the conjecture becoming commonly known as Ogg's conjecture.
drew the connection between the torsion conjecture for elliptic curves over the rationals and the theory of classical modular curves. In the early 1970s, the work of Gérard Ligozat, Daniel Kubert, Barry Mazur, and John Tate showed that several small values of n do not occur as orders of torsion points on elliptic curves over the rationals. proved the full torsion conjecture for elliptic curves over the rationals. His techniques were generalized by and , who obtained uniform boundedness for quadratic fields and number fields of degree at most 8 respectively. Finally, proved the conjecture for elliptic curves over any number field.
An effective bound for the size of the torsion group in terms of the degree of the number field was given by . A complete list of possible torsion groups has also been given for elliptic curves over quadratic number fields. There are substantial partial results for quartic and quintic number fields .
See also
Bombieri–Lang conjecture
Uniform boundedness conjecture for preperiodic points
Uniform boundedness conjecture for rational points
References
Bibliography
Abelian varieties
Conjectures
Diophantine geometry
Theorems in number theory
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Eichler%E2%80%93Shimura%20isomorphism | In mathematics, Eichler cohomology (also called parabolic cohomology or cuspidal cohomology) is a cohomology theory for Fuchsian groups, introduced by , that is a variation of group cohomology analogous to the image of the cohomology with compact support in the ordinary cohomology group. The Eichler–Shimura isomorphism, introduced by Eichler for complex cohomology and by for real cohomology, is an isomorphism between an Eichler cohomology group and a space of cusp forms. There are several variations of the Eichler–Shimura isomorphism, because one can use either real or complex coefficients, and can also use either Eichler cohomology or ordinary group cohomology as in . There is also a variation of the Eichler–Shimura isomorphisms using l-adic cohomology instead of real cohomology, which relates the coefficients of cusp forms to eigenvalues of Frobenius acting on these groups. used this to reduce the Ramanujan conjecture to the Weil conjectures that he later proved.
Eichler cohomology
If G is a Fuchsian group and M is a representation of it then the Eichler cohomology group H(G,M) is defined to be the kernel of the map from H(G,M) to
Πc H(Gc,M), where the product is over the cusps c of a fundamental domain of G, and Gc is the subgroup fixing the cusp c.
References
Modular forms |
https://en.wikipedia.org/wiki/Hypocontinuous%20bilinear%20map | In mathematics, a hypocontinuous is a condition on bilinear maps of topological vector spaces that is weaker than continuity but stronger than separate continuity. Many important bilinear maps that are not continuous are, in fact, hypocontinuous.
Definition
If , and are topological vector spaces then a bilinear map is called hypocontinuous if the following two conditions hold:
for every bounded set the set of linear maps is an equicontinuous subset of , and
for every bounded set the set of linear maps is an equicontinuous subset of .
Sufficient conditions
Theorem: Let X and Y be barreled spaces and let Z be a locally convex space. Then every separately continuous bilinear map of into Z is hypocontinuous.
Examples
If X is a Hausdorff locally convex barreled space over the field , then the bilinear map defined by is hypocontinuous.
See also
References
Bibliography
Topological vector spaces
Bilinear maps |
https://en.wikipedia.org/wiki/Bogomolny%20equations | In mathematics, and especially gauge theory, the Bogomolny equation for magnetic monopoles is the equation
where is the curvature of a connection on a principal -bundle over a 3-manifold , is a section of the corresponding adjoint bundle, is the exterior covariant derivative induced by on the adjoint bundle, and is the Hodge star operator on . These equations are named after E. B. Bogomolny and were studied extensively by Michael Atiyah and Nigel Hitchin.
The equations are a dimensional reduction of the self-dual Yang–Mills equations from four dimensions to three dimensions, and correspond to global minima of the appropriate action. If is closed, there are only trivial (i.e. flat) solutions.
See also
Monopole moduli space
Ginzburg–Landau theory
Seiberg–Witten theory
Bogomol'nyi–Prasad–Sommerfield bound
References
Differential geometry
Magnetic monopoles |
https://en.wikipedia.org/wiki/Kervaire%20semi-characteristic | In mathematics, the Kervaire semi-characteristic, introduced by , is an invariant of closed manifolds M of dimension taking values in , given by
.
showed that the Kervaire semi-characteristic of a differentiable manifold is given by the index of a skew-adjoint elliptic operator.
Assuming M is oriented, the Atiyah vanishing theorem states that if M has two linearly independent vector fields, then .
References
Notes
Differential topology |
https://en.wikipedia.org/wiki/List%20of%20Paris%20Saint-Germain%20F.C.%20records%20and%20statistics | Paris Saint-Germain Football Club holds many records, most notably being the most successful French club in history in terms of official titles won, with 48. They are the record holders of all national competitions, having clinched eleven Ligue 1 championships, fourteen Coupe de France, nine Coupe de la Ligue, and eleven Trophée des Champions. Their trophy cabinet also includes one Ligue 2 title. In international football, PSG have claimed one UEFA Cup Winners' Cup and one UEFA Intertoto Cup. Additionally, they have won 25 unofficial titles.
Their victory in the 1995–96 UEFA Cup Winners' Cup makes PSG the sole French side to have won this trophy, one of only two French clubs to have won a major European competition, and the youngest European team to do so. The Parisians are also the club with the most consecutive seasons in the top-flight (48 seasons in Ligue 1 since 1974–75). Furthermore, PSG are the only side to have won the Ligue 1 title after being at the top of the table from the first until the last round (2022–23), the Coupe de France without conceding a single goal (1992–93 and 2016–17), five Coupe de la Ligue in a row (2014–2018), four back-to-back Coupe de France (2015–2018), and eight consecutive Trophée des Champions (2013–2020).
PSG have won all four national titles in a single season on four occasions. This feat is known as the domestic quadruple. They have completed the domestic double, the league and league cup double, the domestic cup double, the domestic treble and the league three-peat several times as well. Therefore, PSG are the club with the most domestic doubles and league and league cup doubles, one of two sides to have achieved the league three-peat twice, and the only team to have won the domestic cup double, the domestic treble and the domestic quadruple.
Influential officials and players in the club's history include most decorated president Nasser Al-Khelaifi, most decorated manager Laurent Blanc, record appearance maker Jean-Marc Pilorget, all-time top scorer Kylian Mbappé, assist maestro Ángel Di María, clean sheet leader Bernard Lama, most capped and longest-serving captain Thiago Silva, Ballon d'Or winner Lionel Messi, and world-record transfer Neymar.
Honours
.
Official
shared record
Unofficial
shared record
Achievements
shared record
Competitive record
.
Club
Matches
All-time record win: 10–0 (away to Côte Chaude, Coupe de France, 22 January 1994).
All-time record defeat: 0–6 (away to Nantes, Ligue 1, 1 September 1971).
Record home win in Ligue 1: 9–0 (vs. Guingamp, 19 January 2019).
Record away win in Ligue 1: 9–0 (vs. Troyes, 13 March 2016).
Record home defeat in Ligue 1: 0–4 (vs. Nice, 30 April 1988).
Record away defeat in Ligue 1: 0–6 (vs. Nantes, 1 September 1971).
Record home win in UEFA competitions: 7–1.
(vs. Gent, UEFA Intertoto Cup, 1 August 2001).
(vs. Celtic, UEFA Champions League, 22 November 2017).
Record away win in UEFA competitions: 5–0.
(vs. Anderlecht, UEFA Champions |
https://en.wikipedia.org/wiki/Probabilistic%20neural%20network | A probabilistic neural network (PNN) is a feedforward neural network, which is widely used in classification and pattern recognition problems. In the PNN algorithm, the parent probability distribution function (PDF) of each class is approximated by a Parzen window and a non-parametric function. Then, using PDF of each class, the class probability of a new input data is estimated and Bayes’ rule is then employed to allocate the class with highest posterior probability to new input data. By this method, the probability of mis-classification is minimized. This type of artificial neural network (ANN) was derived from the Bayesian network and a statistical algorithm called Kernel Fisher discriminant analysis. It was introduced by D.F. Specht in 1966. In a PNN, the operations are organized into a multilayered feedforward network with four layers:
Input layer
Pattern layer
Summation layer
Output layer
Layers
PNN is often used in classification problems. When an input is present, the first layer computes the distance from the input vector to the training input vectors. This produces a vector where its elements indicate how close the input is to the training input. The second layer sums the contribution for each class of inputs and produces its net output as a vector of probabilities. Finally, a compete transfer function on the output of the second layer picks the maximum of these probabilities, and produces a 1 (positive identification) for that class and a 0 (negative identification) for non-targeted classes.
Input layer
Each neuron in the input layer represents a predictor variable. In categorical variables, N-1 neurons are used when there are N number of categories. It standardizes the range of the values by subtracting the median and dividing by the interquartile range. Then the input neurons feed the values to each of the neurons in the hidden layer.
Pattern layer
This layer contains one neuron for each case in the training data set. It stores the values of the predictor variables for the case along with the target value. A hidden neuron computes the Euclidean distance of the test case from the neuron's center point and then applies the radial basis function kernel using the sigma values.
Summation layer
For PNN there is one pattern neuron for each category of the target variable. The actual target category of each training case is stored with each hidden neuron; the weighted value coming out of a hidden neuron is fed only to the pattern neuron that corresponds to the hidden neuron’s category. The pattern neurons add the values for the class they represent.
Output layer
The output layer compares the weighted votes for each target category accumulated in the pattern layer and uses the largest vote to predict the target category.
Advantages
There are several advantages and disadvantages using PNN instead of multilayer perceptron.
PNNs are much faster than multilayer perceptron networks.
PNNs can be more accurate than multilayer percep |
https://en.wikipedia.org/wiki/Friedrich%20Schur | Friedrich Heinrich Schur (27 January 1856, Maciejewo, Krotoschin, Province of Posen – 18 March 1932, Breslau) was a German mathematician who studied geometry.
Life and work
Schur's family was originally Jewish, but converted to Protestantism. His father owned an estate. He attended high school in Krotoschin and in 1875 studied at University of Wroclaw astronomy and mathematics under Heinrich Schröter and Jacob Rosanes. He then went to the Berlin University, where he studied under Karl Weierstrass, Ernst Eduard Kummer, Leopold Kronecker and Gustav Kirchhoff and received his doctorate in 1879 from Kummer: Geometrische Untersuchungen über Strahlenkomplexe ersten und zweiten Grades. In 1880, he passed the exam and the same year qualified as a teacher at the University of Leipzig. After that, he was an assistant professor and in 1884 became an assistant to Felix Klein in Leipzig. In 1885 he was an associate professor there in 1888 and professor at the University of Tartu. In 1892, he was a professor of descriptive geometry at the RWTH Aachen University and in 1897 was a professor at the University of Karlsruhe, where he was also rector in 1904/1905. In 1909, he became a professor at the University of Strasbourg. After the loss of World War I, he was sacked by the French in 1919 and became a professor in Breslau, where he retired in 1924.
Friedrich Schur studied differential geometry, transformation groups (Lie groups) after Sophus Lie. Many of his results, which he summarized in his book Grundlagen der Geometrie (Foundation of Geometry) of 1909, can also be found in the work of David Hilbert without reference to Schur. He also wrote a textbook of analytical geometry (1898) and the graphical statics (1915).
In 1912, he received the Lobachevsky Prize for his book Grundlagen der Geometrie, a Russian prize. In 1910, he was chairman of the German Mathematical Society. He holds honorary doctorates from the University of Karlsruhe. In 1927, he was selected as a corresponding member of the Bavarian Academy of Sciences.
Among his students was Theodor Molien and Julius Wellstein.
Writings (selection)
Schur: Grundlagen der Geometrie. Teubner, Leipzig 1909.
Schur: Lehrbuch der analytischen Geometrie.
Schur: Zur Theorie der endlichen Transformationsgruppen. Mathematische Annalen, Bd.38, 1891.
Schur: Ueber den Fundamentalsatz der projectiven Geometrie. Mathematische Annalen, Bd.51, 1899.
Schur: Ueber die Grundlagen der Geometrie. Mathematische Annalen, Bd. 55, 1902
See also
Baker–Campbell–Hausdorff formula
Derivative of the exponential map
K3 surface
References
The original article was a Google translation of the corresponding German article.
Biographie von Fritsch, pdf-Datei (86 kB)
Mathematics Genealogy Project
External links
19th-century German Jews
19th-century German mathematicians
1856 births
1932 deaths
20th-century German mathematicians
People from the Province of Posen |
https://en.wikipedia.org/wiki/Signature%20defect | In mathematics, the signature defect of a singularity measures the correction that a singularity contributes to the signature theorem.
introduced the signature defect for the cusp singularities of Hilbert modular surfaces.
defined the signature defect of the boundary of a manifold as the eta invariant, the value as s = 0 of their eta function, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at s = 0 or 1 of a Shimizu L-function.
References
Singularity theory |
https://en.wikipedia.org/wiki/Normal%20degree | In algebraic geometry, the normal degree of a rational curve C on a surface is defined to be –K.C–2 where K is the canonical divisor of the surface.
References
Algebraic curves |
https://en.wikipedia.org/wiki/Projective%20bundle | In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces.
By definition, a scheme X over a Noetherian scheme S is a Pn-bundle if it is locally a projective n-space; i.e., and transition automorphisms are linear. Over a regular scheme S such as a smooth variety, every projective bundle is of the form for some vector bundle (locally free sheaf) E.
The projective bundle of a vector bundle
Every vector bundle over a variety X gives a projective bundle by taking the projective spaces of the fibers, but not all projective bundles arise in this way: there is an obstruction in the cohomology group H2(X,O*). To see why, recall that a projective bundle comes equipped with transition functions on double intersections of a suitable open cover. On triple overlaps, any lift of these transition functions satisfies the cocycle condition up to an invertible function. The collection of these functions forms a 2-cocycle which vanishes in H2(X,O*) only if the projective bundle is the projectivization of a vector bundle. In particular, if X is a compact Riemann surface then H2(X,O*)=0, and so this obstruction vanishes.
The projective bundle of a vector bundle E is the same thing as the Grassmann bundle of 1-planes in E.
The projective bundle P(E) of a vector bundle E is characterized by the universal property that says:
Given a morphism f: T → X, to factorize f through the projection map is to specify a line subbundle of f*E.
For example, taking f to be p, one gets the line subbundle O(-1) of p*E, called the tautological line bundle on P(E). Moreover, this O(-1) is a universal bundle in the sense that when a line bundle L gives a factorization f = p ∘ g, L is the pullback of O(-1) along g. See also Cone#O(1) for a more explicit construction of O(-1).
On P(E), there is a natural exact sequence (called the tautological exact sequence):
where Q is called the tautological quotient-bundle.
Let E ⊂ F be vector bundles (locally free sheaves of finite rank) on X and G = F/E. Let q: P(F) → X be the projection. Then the natural map is a global section of the sheaf hom . Moreover, this natural map vanishes at a point exactly when the point is a line in E; in other words, the zero-locus of this section is P(E).
A particularly useful instance of this construction is when F is the direct sum E ⊕ 1 of E and the trivial line bundle (i.e., the structure sheaf). Then P(E) is a hyperplane in P(E ⊕ 1), called the hyperplane at infinity, and the complement of P(E) can be identified with E. In this way, P(E ⊕ 1) is referred to as the projective completion (or "compactification") of E.
The projective bundle P(E) is stable under twisting E by a line bundle; precisely, given a line bundle L, there is the natural isomorphism:
such that (In fact, one gets g by the universal property applied to the line bundle on the right.)
Examples
Many non-trivial examples of projective bundles can be found using fibrations over such as Lefschetz fibrations |
https://en.wikipedia.org/wiki/Al-Judeida%20%28Jenin%29 | Al-Judeida () is a Palestinian village in the Jenin Governorate in the western area of the West Bank, located south of Jenin. According to the Palestinian Central Bureau of Statistics, the town had a population of 4,738 in the 2007 census and 5,950 by 2017.
History
Al-Judeida is an ancient village, where Byzantine ceramics have been found. Zertal notes that the sherds from the Byzantine era were at the edge of the hilltop upon which al-Judeida stands.
Pottery sherds found in the village mostly date back to the medieval and Ottoman eras. During Crusader rule, in 1168, al-Judeida was an estate called Gidideh.
Ottoman era
Like all of Palestine, al-Judeida was incorporated into the Ottoman Empire in 1517. In the 1596 Ottoman tax registers, al-Judeida was an entirely Muslim village with a population of 10 families, located in the Nahiya Jabal Sami, in the Nablus Sanjak. The inhabitants paid a fixed tax-rate of 33.3% on agricultural products, including wheat, barley, summer crops, olive trees, goats and beehives, in addition to occasional revenues and a press for grape syrup or olive oil; a total of 3,500 akçe.
Most of the buildings in the old core of Judeida date back to the 16th and 17th centuries.
In 1838, Edward Robinson noted the village when he travelled in the region, as bordering the extremely fertile Marj Sanur. He listed it as part of the District of Haritheh, north of Nablus.
In 1870, French traveler Victor Guérin visited al-Judeida, describing it as being amid "gardens of fig trees, pomegranates and olives. It seems to be an ancient site, because of the many rock hewn cisterns and the well-shaped stones contained in the walls of its 35 houses." In 1882, it was described by the PEF's Survey of Western Palestine as "a good-sized village on flat ground, with a few olives".
British Mandate era
In the 1922 British census, Al-Judeida had a population of 361, all Muslims, increasing in the 1931 census to 569 inhabitants, still all Muslims, living in a total of 106 houses.
In the 1945 statistics, the population was 830, all Muslims, with 6,360 dunams of land, according to an official land and population survey. Of the village's lands, 2,211 dunams were used for plantations and irrigable land, 2,850 dunams for cereals, while 20 dunams were built-up (urban) areas.
Jordanian era
Following the 1948 Arab–Israeli War, and the subsequent 1949 Armistice Agreements, Al-Judeida came under Jordanian rule.
The Jordanian census of 1961 found 1,351 inhabitants in Judeida.
Post-1967
Since the Six-Day War in 1967, Al-Judeida has been under Israeli occupation. Under the Oslo Accords, the town was assigned to Area A.
On Saturday 9 January 2016 resident Ali Abu Maryam (23) was shot dead by Israeli soldiers at the Beka'ot roadblock.
Geography
Al-Judeida is situated at the southern edge of the Marj Sanur valley on a small hilltop with an elevation of about 425 meters above sea level. The old core of al-Judeida is in the center of the vi |
https://en.wikipedia.org/wiki/Deuring%E2%80%93Heilbronn%20phenomenon | In mathematics, the Deuring–Heilbronn phenomenon, discovered by and , states that a counterexample to the generalized Riemann hypothesis for one Dirichlet L-function affects the location of the zeros of other Dirichlet L-functions.
See also
Siegel zero
References
Analytic number theory |
https://en.wikipedia.org/wiki/Sphinx%20tiling | In geometry, the sphinx tiling is a tessellation of the plane using the "sphinx", a pentagonal hexiamond formed by gluing six equilateral triangles together. The resultant shape is named for its reminiscence to the Great Sphinx at Giza. A sphinx can be dissected into any square number of copies of itself, some of them mirror images, and repeating this process leads to a non-periodic tiling of the plane. The sphinx is therefore a rep-tile (a self-replicating tessellation). It is one of few known pentagonal rep-tiles and is the only known pentagonal rep-tile whose sub-copies are equal in size.
See also
Mosaic
References
External links
Mathematics Centre Sphinx Album ...
Pentagonal tilings
Aperiodic tilings
Sphinxes |
https://en.wikipedia.org/wiki/Serre%E2%80%93Tate%20theorem | In algebraic geometry, the Serre–Tate theorem says that an abelian scheme and its p-divisible group have the same infinitesimal deformation theory. This was first proved by Jean-Pierre Serre when the reduction of the abelian variety is ordinary, using the Greenberg functor; then John Tate gave a proof in the general case by a different method. Their proofs were not published, but they were summarized in the notes of the Lubin–Serre–Tate seminar (Woods Hole, 1964). Other proofs were published by Messing (1962) and Drinfeld (1976).
References
: see, vol.2, p. 854, comments on Tate's letter from Jan.10, 1964.
Abelian varieties
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Tate%20curve | In mathematics, the Tate curve is a curve defined over the ring of formal power series with integer coefficients. Over the open subscheme where q is invertible, the Tate curve is an elliptic curve. The Tate curve can also be defined for q as an element of a complete field of norm less than 1, in which case the formal power series converge.
The Tate curve was introduced by in a 1959 manuscript originally titled "Rational Points on Elliptic Curves Over Complete Fields"; he did not publish his results until many years later, and his work first appeared in .
Definition
The Tate curve is the projective plane curve over the ring Z of formal power series with integer coefficients given (in an affine open subset of the projective plane) by the equation
where
are power series with integer coefficients.
The Tate curve over a complete field
Suppose that the field k is complete with respect to some absolute value | |, and q is a non-zero element of the field k with |q|<1. Then the series above all converge, and define an elliptic curve over k. If in addition q is non-zero then there is an isomorphism of groups from k*/qZ to this elliptic curve, taking w to (x(w),y(w)) for w not a power of q, where
and taking powers of q to the point at infinity of the elliptic curve. The series x(w) and y(w) are not formal power series in w.
Intuitive example
In the case of the curve over the complete field, , the easiest case to visualize is , where is the discrete subgroup generated by one multiplicative period , where the period . Note that is isomorphic to , where is the complex numbers under addition.
To see why the Tate curve morally corresponds to a torus when the field is C with the usual norm, is already singly periodic; modding out by q's integral powers you are modding out by , which is a torus. In other words, we have an annulus, and we glue inner and outer edges.
But the annulus does not correspond to the circle minus a point: the annulus is the set of complex numbers between two consecutive powers of q; say all complex numbers with magnitude between 1 and q. That gives us two circles, i.e., the inner and outer edges of an annulus.
The image of the torus given here is a bunch of inlaid circles getting narrower and narrower as they approach the origin.
This is slightly different from the usual method beginning with a flat sheet of paper, , and gluing together the sides to make a cylinder , and then gluing together the edges of the cylinder to make a torus, .
This is slightly oversimplified. The Tate curve is really a curve over a formal power series ring rather than a curve over C. Intuitively, it is a family of curves depending on a formal parameter. When that formal parameter is zero it degenerates to a pinched torus, and when it is nonzero it is a torus).
Properties
The j-invariant of the Tate curve is given by a power series in q with leading term q−1. Over a p-adic local field, therefore, j is non-integral and the Tate curve has s |
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