source stringlengths 31 168 | text stringlengths 51 3k |
|---|---|
https://en.wikipedia.org/wiki/Raimund%20Seidel | Raimund G. Seidel is a German and Austrian theoretical computer scientist and an expert in computational geometry.
Seidel was born in Graz, Austria, and studied with Hermann Maurer at the Graz University of Technology. He received his M. Sc. in 1981 from University of British Columbia under David G. Kirkpatrick. He received his Ph.D. in 1987 from Cornell University under the supervision of John Gilbert. After teaching at the University of California, Berkeley, he moved in 1994 to Saarland University. In 1997 he and Christoph M. Hoffmann were program chairs for the Symposium on Computational Geometry. In 2014, he took over as Scientific Director of the Leibniz Center for Informatics (LZI) from Reinhard Wilhelm.
Seidel invented backwards analysis of randomized algorithms and used it to analyze a simple linear programming algorithm that runs in linear time for problems of bounded dimension. With his student Cecilia R. Aragon in 1989 he devised the treap data structure, and he is also known for the Kirkpatrick–Seidel algorithm for computing two-dimensional convex hulls.
References
External links
Year of birth missing (living people)
Living people
Austrian computer scientists
German computer scientists
Researchers in geometric algorithms
Cornell University alumni
University of California, Berkeley faculty |
https://en.wikipedia.org/wiki/Desmic%20system | In projective geometry, a desmic system () is a set of three tetrahedra in 3-dimensional projective space, such that any two are desmic (related such that each edge of one cuts a pair of opposite edges of the other). It was introduced by . The three tetrahedra of a desmic system are contained in a pencil of quartic surfaces.
Every line that passes through two vertices of two tetrahedra in the system also passes through a vertex of the third tetrahedron.
The 12 vertices of the desmic system and the 16 lines formed in this way are the points and lines of a Reye configuration.
Example
The three tetrahedra given by the equations
form a desmic system, contained in the pencil of quartics
for a + b + c = 0.
References
.
.
.
External links
Desmic tetrahedra
Projective geometry |
https://en.wikipedia.org/wiki/Absolute%20risk | Absolute risk (or AR) is the probability or chance of an event. It is usually used for the number of events (such as a disease) that occurred in a group, divided by the number of people in that group.
Absolute risk is one of the most understandable ways of communicating health risks to the general public.
See also
Absolute risk reduction
Relative risk
Relative risk reduction
External links
Know Your Chances: Understanding Health Statistics
References
Medical terminology |
https://en.wikipedia.org/wiki/Orbital%20integral | In mathematics, an orbital integral is an integral transform that generalizes the spherical mean operator to homogeneous spaces. Instead of integrating over spheres, one integrates over generalized spheres: for a homogeneous space X = G/H, a generalized sphere centered at a point x0 is an orbit of the isotropy group of x0.
Definition
The model case for orbital integrals is a Riemannian symmetric space G/K, where G is a Lie group and K is a symmetric compact subgroup. Generalized spheres are then actual geodesic spheres and the spherical averaging operator is defined as
where
the dot denotes the action of the group G on the homogeneous space X
g ∈ G is a group element such that x = g·o
y ∈ X is an arbitrary element of the geodesic sphere of radius r centered at x: d(x,y) = r
the integration is taken with respect to the Haar measure on K (since K is compact, it is unimodular and the left and right Haar measures coincide and can be normalized so that the mass of K is 1).
Orbital integrals of suitable functions can also be defined on homogeneous spaces G/K where the subgroup K is no longer assumed to be compact, but instead is assumed to be only unimodular. Lorentzian symmetric spaces are of this kind. The orbital integrals in this case are also obtained by integrating over a K-orbit in G/K with respect to the Haar measure of K. Thus
is the orbital integral centered at x over the orbit through y. As above, g is a group element that represents the coset x.
Integral geometry
A central problem of integral geometry is to reconstruct a function from knowledge of its orbital integrals. The Funk transform and Radon transform are two special cases. When G/K is a Riemannian symmetric space, the problem is trivial, since Mrƒ(x) is the average value of ƒ over the generalized sphere of radius r, and
When K is compact (but not necessarily symmetric), a similar shortcut works. The problem is more interesting when K is non-compact. For example, the Radon transform is the orbital integral that results by taking G to be the Euclidean isometry group and K the isotropy group of a hyperplane.
Orbital integrals are an important technical tool in the theory of automorphic forms, where they enter into the formulation of various trace formulas.
References
Harmonic analysis
Automorphic forms |
https://en.wikipedia.org/wiki/Slash%20distribution | In probability theory, the slash distribution is the probability distribution of a standard normal variate divided by an independent standard uniform variate. In other words, if the random variable Z has a normal distribution with zero mean and unit variance, the random variable U has a uniform distribution on [0,1] and Z and U are statistically independent, then the random variable X = Z / U has a slash distribution. The slash distribution is an example of a ratio distribution. The distribution was named by William H. Rogers and John Tukey in a paper published in 1972.
The probability density function (pdf) is
where is the probability density function of the standard normal distribution. The quotient is undefined at x = 0, but the discontinuity is removable:
The most common use of the slash distribution is in simulation studies. It is a useful distribution in this context because it has heavier tails than a normal distribution, but it is not as pathological as the Cauchy distribution.
References
Continuous distributions
Normal distribution
Compound probability distributions
Probability distributions with non-finite variance |
https://en.wikipedia.org/wiki/Geometric%20progression | In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.
Examples of a geometric sequence are powers rk of a fixed non-zero number r, such as 2k and 3k. The general form of a geometric sequence is
where r ≠ 0 is the common ratio and a ≠ 0 is a scale factor, equal to the sequence's start value.
The sum of a geometric progression's terms is called a geometric series.
Elementary properties
The n-th term of a geometric sequence with initial value a = a1 and common ratio r is given by
and in general
Such a geometric sequence also follows the recursive relation
for every integer
Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio.
The common ratio of a geometric sequence may be negative, resulting in an alternating sequence, with numbers alternating between positive and negative. For instance
1, −3, 9, −27, 81, −243, ...
is a geometric sequence with common ratio −3.
The behaviour of a geometric sequence depends on the value of the common ratio. If the common ratio is:
positive, the terms will all be the same sign as the initial term.
negative, the terms will alternate between positive and negative.
greater than 1, there will be exponential growth towards positive or negative infinity (depending on the sign of the initial term).
1, the progression is a constant sequence.
between −1 and 1 but not zero, there will be exponential decay towards zero (→ 0).
−1, the absolute value of each term in the sequence is constant and terms alternate in sign.
less than −1, for the absolute values there is exponential growth towards (unsigned) infinity, due to the alternating sign.
Geometric sequences (with common ratio not equal to −1, 1 or 0) show exponential growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, … (with common difference 11). This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population.
Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression.
An interesting result of the definition of the geometric progression is that any three consecutive terms a, b and c will satisfy the following equation:
where b is considered to be the geometric mean between a and c.
Geometric series
Product
The product of a geometric progression is the product of all terms. It can be quickly computed b |
https://en.wikipedia.org/wiki/Richard%20Dedekind | Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His best known contribution is the definition of real numbers through the notion of Dedekind cut. He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as Logicism.
Life
Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig. His mother was Caroline Henriette Dedekind (née Emperius), the daughter of a professor at the Collegium. Richard Dedekind had three older siblings. As an adult, he never used the names Julius Wilhelm. He was born in Braunschweig (often called "Brunswick" in English), which is where he lived most of his life and died. His body rests at Braunschweig Main Cemetery.
He first attended the Collegium Carolinum in 1848 before transferring to the University of Göttingen in 1850. There, Dedekind was taught number theory by professor Moritz Stern. Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled Über die Theorie der Eulerschen Integrale ("On the Theory of Eulerian integrals"). This thesis did not display the talent evident by Dedekind's subsequent publications.
At that time, the University of Berlin, not Göttingen, was the main facility for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and Bernhard Riemann were contemporaries; they were both awarded the habilitation in 1854. Dedekind returned to Göttingen to teach as a Privatdozent, giving courses on probability and geometry. He studied for a while with Peter Gustav Lejeune Dirichlet, and they became good friends. Because of lingering weaknesses in his mathematical knowledge, he studied elliptic and abelian functions. Yet he was also the first at Göttingen to lecture concerning Galois theory. About this time, he became one of the first people to understand the importance of the notion of groups for algebra and arithmetic.
In 1858, he began teaching at the Polytechnic school in Zürich (now ETH Zürich). When the Collegium Carolinum was upgraded to a Technische Hochschule (Institute of Technology) in 1862, Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching at the Institute. He retired in 1894, but did occasional teaching and continued to publish. He never married, instead living with his sister Julia.
Dedekind was elected to the Academies of Berlin (1880) and Rome, and to the French Academy of Sciences (1900). He received honorary doctorates from the universities of Oslo, Zurich, and Braunschweig.
Work
While teaching calculus for the first time at the Polytechnic school, Dedekind developed the notion now known as a Dedekind cut (Germa |
https://en.wikipedia.org/wiki/Intersection%20%28set%20theory%29 | In set theory, the intersection of two sets and denoted by is the set containing all elements of that also belong to or equivalently, all elements of that also belong to
Notation and terminology
Intersection is written using the symbol "" between the terms; that is, in infix notation. For example:
The intersection of more than two sets (generalized intersection) can be written as:
which is similar to capital-sigma notation.
For an explanation of the symbols used in this article, refer to the table of mathematical symbols.
Definition
The intersection of two sets and denoted by , is the set of all objects that are members of both the sets and
In symbols:
That is, is an element of the intersection if and only if is both an element of and an element of
For example:
The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.
The number 9 is in the intersection of the set of prime numbers {2, 3, 5, 7, 11, ...} and the set of odd numbers {1, 3, 5, 7, 9, 11, ...}, because 9 is not prime.
Intersecting and disjoint sets
We say that if there exists some that is an element of both and in which case we also say that . Equivalently, intersects if their intersection is an , meaning that there exists some such that
We say that if does not intersect In plain language, they have no elements in common. and are disjoint if their intersection is empty, denoted
For example, the sets and are disjoint, while the set of even numbers intersects the set of multiples of 3 at the multiples of 6.
Algebraic properties
Binary intersection is an associative operation; that is, for any sets and one has
Thus the parentheses may be omitted without ambiguity: either of the above can be written as . Intersection is also commutative. That is, for any and one has
The intersection of any set with the empty set results in the empty set; that is, that for any set ,
Also, the intersection operation is idempotent; that is, any set satisfies that . All these properties follow from analogous facts about logical conjunction.
Intersection distributes over union and union distributes over intersection. That is, for any sets and one has
Inside a universe one may define the complement of to be the set of all elements of not in Furthermore, the intersection of and may be written as the complement of the union of their complements, derived easily from De Morgan's laws:
Arbitrary intersections
The most general notion is the intersection of an arbitrary collection of sets.
If is a nonempty set whose elements are themselves sets, then is an element of the of if and only if for every element of is an element of
In symbols:
The notation for this last concept can vary considerably. Set theorists will sometimes write "", while others will instead write "".
The latter notation can be generalized to "", which refers to the intersection of the collection
Here is a nonempty set, and is a set for every
In the case that th |
https://en.wikipedia.org/wiki/Polar%20hypersurface | In algebraic geometry, given a projective algebraic hypersurface described by the homogeneous equation
and a point
its polar hypersurface is the hypersurface
where are the partial derivatives of .
The intersection of and is the set of points such that the tangent at to meets .
References
Projective geometry |
https://en.wikipedia.org/wiki/Albrecht%20Wellmer | Albrecht Wellmer (9 July 1933 – 13 September 2018) was a German philosopher at the Freie Universität Berlin.
Biography
He studied mathematics and physics at Berlin and Kiel, then philosophy and sociology at Heidelberg and Frankfurt. He was an assistant to Jürgen Habermas at the University of Frankfurt from 1966 to 1970. He has held Professorships at the Universität Konstanz (1974–1990), the New School for Social Research and at the Freie Universität Berlin (1990 until his retirement in 2001). He has held guest Professorships at Haverford, Stony Brook, Collège International de Philosophie, the New School of Social Research and the University of Amsterdam.
Awards
In 2006 he received the Theodor W. Adorno Award, a prestigious award for achievement in philosophy, theatre, music, and film.
In 2011 he received the Anna-Krüger-Preis of the Berlin Institute for Advanced Study.
Works
His works include books and articles about Aesthetics, Music, Critical Theory, Ethics, Modernity, and Postmodernity as well as thinkers such as Adorno, Habermas, Rorty, and Wittgenstein.
Personal life
From 1965 to 1981 Albrecht Wellmer was married to Ilse von Neander. He is the father of composer and artist Anne Wellmer. Albrecht Wellmer's grave is at Alter St. Matthäus Kirchhof in Berlin.
References
External links
List of works by Albrecht Wellmer at the Open Library
Albrecht Wellmer's Homepage
List of articles by Albrecht Wellmer at PhilPapers
Article by Albrecht Wellmer on Richard Rorty: 'Rereading Rorty', Krisis, 2008, 2
Albrecht Wellmer – Adorno and the Difficulties of a Critical Reconstruction of the Historical Present - free MP3 audio recording of Wellmer from November 2009
Frankfurter Allgemeine Zeitung, 03.07.2006
Anna-Krueger-Preis des Wissenschaftskolleg in Berlin, 04.05.2011
Tagesspiegel, 18.9.2018
Frankfurter Allgemeine Zeitung, 18.9.2018
1933 births
2018 deaths
Academic staff of the Free University of Berlin
German political philosophers
German political scientists
German sociologists
20th-century German philosophers
Critical theorists
Philosophers of music
Philosophers of language
Social philosophers
Frankfurt School
German male writers
People from Dachau (district) |
https://en.wikipedia.org/wiki/Khyargas%20Nuur | Khyargas Lake () is a salt lake in Khyargas district, Uvs Province, Western Mongolia.
Some sources are using different Khyargas Lake statistics values:
Water level: 1,035.29 m
Surface area: 1,481.1 km2
Average depth: 50.7 m
Volume: 75.2 km³.
The Khyargas Lake National Park is based on the lake. This protected area was established in 2000 and covers about 3,328 km2. It also includes a freshwater Airag Lake.
References
Lakes of Mongolia
Saline lakes of Asia
Protected areas established in 2000
2000 establishments in Mongolia
Landforms of Uvs Province |
https://en.wikipedia.org/wiki/Peter%20Barlow | Peter Barlow may refer to:
Peter Barlow (mathematician) (1776–1862), English writer on pure and applied mathematics
Peter W. Barlow (1809–1885), English civil engineer and son of the mathematician
Peter Barlow (Coronation Street), a fictional character in the UK television soap opera Coronation Street
Peter Barlow (footballer) (born 1950), former forward in the Football League
Peter Townsend Barlow (1857–1921), American jurist in New York City |
https://en.wikipedia.org/wiki/2009%E2%80%9310%20PFC%20Levski%20Sofia%20season | The 2009–10 season is Levski Sofia's 88th season in the First League. This article shows player statistics and all matches (official and friendly) that the club has played during the 2009–10 season.
First-team squad
Current squad
As of 4 July 2009 (according to latest announcements)
Transfers
Summer transfers
In:
Out:
Winter transfers
In:
Out:
Competitions
Bulgarian Supercup
A Group
Table
Results summary
Results by round
Fixtures and results
Bulgarian Cup
Europe
UEFA Champions League
Second qualifying round
Third qualifying round
Play-off round
UEFA Europa League
Group stage
References
External links
Levski Sofia official website
2009–10 Levski Sofia season
PFC Levski Sofia seasons
Levski Sofia |
https://en.wikipedia.org/wiki/Proofs%20involving%20ordinary%20least%20squares | The purpose of this page is to provide supplementary materials for the ordinary least squares article, reducing the load of the main article with mathematics and improving its accessibility, while at the same time retaining the completeness of exposition.
Derivation of the normal equations
Define the th residual to be
Then the objective can be rewritten
Given that S is convex, it is minimized when its gradient vector is zero (This follows by definition: if the gradient vector is not zero, there is a direction in which we can move to minimize it further – see maxima and minima.) The elements of the gradient vector are the partial derivatives of S with respect to the parameters:
The derivatives are
Substitution of the expressions for the residuals and the derivatives into the gradient equations gives
Thus if minimizes S, we have
Upon rearrangement, we obtain the normal equations:
The normal equations are written in matrix notation as
(where XT is the matrix transpose of X).
The solution of the normal equations yields the vector of the optimal parameter values.
Derivation directly in terms of matrices
The normal equations can be derived directly from a matrix representation of the problem as follows. The objective is to minimize
Here has the dimension 1x1 (the number of columns of ), so it is a scalar and equal to its own transpose, hence
and the quantity to minimize becomes
Differentiating this with respect to and equating to zero to satisfy the first-order conditions gives
which is equivalent to the above-given normal equations. A sufficient condition for satisfaction of the second-order conditions for a minimum is that have full column rank, in which case is positive definite.
Derivation without calculus
When is positive definite, the formula for the minimizing value of can be derived without the use of derivatives. The quantity
can be written as
where depends only on and , and is the inner product defined by
It follows that is equal to
and therefore minimized exactly when
Generalization for complex equations
In general, the coefficients of the matrices and can be complex. By using a Hermitian transpose instead of a simple transpose, it is possible to find a vector which minimizes , just as for the real matrix case. In order to get the normal equations we follow a similar path as in previous derivations:
where stands for Hermitian transpose.
We should now take derivatives of with respect to each of the coefficients , but first we separate real and imaginary parts to deal with the conjugate factors in above expression. For the we have
and the derivatives change into
After rewriting in the summation form and writing explicitly, we can calculate both partial derivatives with result:
which, after adding it together and comparing to zero (minimization condition for ) yields
In matrix form:
Least squares estimator for β
Using matrix notation, the sum of squared residuals is given by
Since this |
https://en.wikipedia.org/wiki/Obesity%20in%20Australia | According to 2007 statistics from the World Health Organization (WHO), Australia has the third-highest prevalence of overweight adults in the English-speaking world. Obesity in Australia is an "epidemic" with "increasing frequency." The Medical Journal of Australia found that obesity in Australia more than doubled in the two decades preceding 2003, and the unprecedented rise in obesity has been compared to the same health crisis in America. The rise in obesity has been attributed to poor eating habits in the country closely related to the availability of fast food since the 1970s, sedentary lifestyles and a decrease in the labour workforce.
Classification of obesity
Weight is measured by using the Body Mass Index scale (BMI). This is determined by dividing weight in kilograms by height in metres, squared. If someone is overweight their BMI will be at 25 or more. If someone is obese their BMI will be at 30 or more.
Prevalence of obesity in the Australian population
, 8% of children and 28% of adults in Australia are obese.
Demographic summary
Queensland
Australian adults
In a study published in 2015 by the US Journal of Economics and Human Biology, obesity is found to have the largest impact on men aged over 75, and women aged between 60 and 74.
In 2005, a study was conducted by the Australian Bureau of Statistics that compared the results of a 2004-05 survey with those conducted in the preceding 15 years. The results showed an increase in the number and proportion of adults who are overweight or obese. Over the four surveys, the number of overweight or obese adults increased from 4.6 million in 1989–90 to 5.4 million in 1995, 6.6 million in 2001 and 7.4 million in 2004–05.
In 2007, the World Health Organization (WHO) found that 67.4% of Australian adults are overweight, ranking 21st in the world, and third out of the major countries in the English-speaking world, behind the United States (ranked 9th) and New Zealand (ranked 17th). A 2005 WHO study found that just over 20% of Australian adults are obese, that number increased to about 29 to 30% being obese in 2017.
In the 2005 National Health Survey, 53.6% of Australians reported being overweight with 18% falling into the "obese" category. Those numbers rose to 65% overweight and 29% obese in 2016. This is nearly double the reported number from 1995, when 30% of adults were overweight and 11% were obese. Such representations would be skewed downward as people tend to overestimate their height and under-report their weight, the two key criteria to determine a BMI reading. In the National Health Survey, obesity reports were fairly common across the board, with no major outliers. Victoria had the lowest incidence of obesity, at 17.0% of the population, with South Australia reporting the highest numbers at 19.6%. By 2014, Canberra recorded an obesity rate of 25% which was placing significant strain on ageing health care infrastructure.
In a study conducted by The Obesity Society, between 2001 |
https://en.wikipedia.org/wiki/Quasiregular%20element | This article addresses the notion of quasiregularity in the context of ring theory, a branch of modern algebra. For other notions of quasiregularity in mathematics, see the disambiguation page quasiregular.
In mathematics, specifically ring theory, the notion of quasiregularity provides a computationally convenient way to work with the Jacobson radical of a ring. In this article, we primarily concern ourselves with the notion of quasiregularity for unital rings. However, one section is devoted to the theory of quasiregularity in non-unital rings, which constitutes an important aspect of noncommutative ring theory.
Definition
Let R be a ring (with unity) and let r be an element of R. Then r is said to be quasiregular, if 1 − r is a unit in R; that is, invertible under multiplication. The notions of right or left quasiregularity correspond to the situations where 1 − r has a right or left inverse, respectively.
An element x of a non-unital ring is said to be right quasiregular if there is y such that . The notion of a left quasiregular element is defined in an analogous manner. The element y is sometimes referred to as a right quasi-inverse of x. If the ring is unital, this definition of quasiregularity coincides with that given above. If one writes , then this binary operation is associative. In fact, the map (where × denotes the multiplication of the ring R) is a monoid isomorphism. Therefore, if an element possesses both a left and right quasi-inverse, they are equal.
Note that some authors use different definitions. They call an element x right quasiregular if there exists y such that , which is equivalent to saying that 1 + x has a right inverse when the ring is unital. If we write , then , so we can easily go from one set-up to the other by changing signs. For example, x is right quasiregular in one set-up if and only if −x is right quasiregular in the other set-up.
Examples
If R is a ring, then the additive identity of R is always quasiregular.
If is right (resp. left) quasiregular, then is right (resp. left) quasiregular.
If R is a rng, every nilpotent element of R is quasiregular. This fact is supported by an elementary computation:
If , then
(or if we follow the second convention).
From this we see easily that the quasi-inverse of x is (or ).
In the second convention, a matrix is quasiregular in a matrix ring if it does not possess −1 as an eigenvalue. More generally, a bounded operator is quasiregular if −1 is not in its spectrum.
In a unital Banach algebra, if , then the geometric series converges. Consequently, every such x is quasiregular.
If R is a ring and S = R[[X1, ..., Xn]] denotes the ring of formal power series in n indeterminants over R, an element of S is quasiregular if and only its constant term is quasiregular as an element of R.
Properties
Every element of the Jacobson radical of a (not necessarily commutative) ring is quasiregular. In fact, the Jacobson radical of a ring can be characterized as |
https://en.wikipedia.org/wiki/Quasiregular | In mathematics, quasiregular may refer to:
Quasiregular element, in the context of ring theory
Quasiregular map in analysis
Quasiregular polyhedron, in the context of geometry
Quasiregular representation, in the context of representation theory |
https://en.wikipedia.org/wiki/Albert%E2%80%93Brauer%E2%80%93Hasse%E2%80%93Noether%20theorem | In algebraic number theory, the Albert–Brauer–Hasse–Noether theorem states that a central simple algebra over an algebraic number field K which splits over every completion Kv is a matrix algebra over K. The theorem is an example of a local-global principle in algebraic number theory and
leads to a complete description of finite-dimensional division algebras over algebraic number fields in terms of their local invariants. It was proved independently by Richard Brauer, Helmut Hasse, and Emmy Noether and by Abraham Adrian Albert.
Statement of the theorem
Let A be a central simple algebra of rank d over an algebraic number field K. Suppose that for any valuation v, A splits over the corresponding local field Kv:
Then A is isomorphic to the matrix algebra Md(K).
Applications
Using the theory of Brauer group, one shows that two central simple algebras A and B over an algebraic number field K are isomorphic over K if and only if their completions Av and Bv are isomorphic over the completion Kv for every v.
Together with the Grunwald–Wang theorem, the Albert–Brauer–Hasse–Noether theorem implies that every central simple algebra over an algebraic number field is cyclic, i.e. can be obtained by an explicit construction from a cyclic field extension L/K .
See also
Class field theory
Hasse norm theorem
References
Revised version —
Albert, Nancy E. (2005), "A3 & His Algebra, iUniverse,
Notes
Class field theory
Theorems in algebraic number theory |
https://en.wikipedia.org/wiki/Uniform%20honeycomb | In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of faces at each vertex. Its dimension can be clarified as -honeycomb for an -dimensional honeycomb.
An -dimensional uniform honeycomb can be constructed on the surface of -spheres, in -dimensional Euclidean space, and -dimensional hyperbolic space. A 2-dimensional uniform honeycomb is more often called a uniform tiling or uniform tessellation.
Nearly all uniform tessellations can be generated by a Wythoff construction, and represented by a Coxeter–Dynkin diagram. The terminology for the convex uniform polytopes used in uniform polyhedron, uniform 4-polytope, uniform 5-polytope, uniform 6-polytope, uniform tiling, and convex uniform honeycomb articles were coined by Norman Johnson.
Wythoffian tessellations can be defined by a vertex figure. For 2-dimensional tilings, they can be given by a vertex configuration listing the sequence of faces around every vertex. For example, represents a regular tessellation, a square tiling, with 4 squares around each vertex. In general an -dimensional uniform tessellation vertex figures are define by an -polytope with edges labeled with integers, representing the number of sides of the polygonal face at each edge radiating from the vertex.
Examples of uniform honeycombs
See also
Uniform tiling
List of uniform tilings
Uniform tilings in hyperbolic plane
Honeycomb (geometry)
Wythoff construction
Convex uniform honeycomb
List of regular polytopes
References
George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49–56.
Norman Johnson Uniform Polytopes, Manuscript (1991)
H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
External links
Tessellations of the Plane
Uniform tilings
Honeycombs (geometry) |
https://en.wikipedia.org/wiki/Myrtle%20Point%20High%20School | Myrtle Point High School is a public high school and junior high in Myrtle Point, Oregon, United States.
Students
According to the National Center for Education Statistics, Myrtle Point High School enrolled 191 students across grades 7–12 in the 2021–2022 school year.
Academics
In 2018, 66% of the school's seniors received their high school diploma on time.
References
High schools in Coos County, Oregon
Public middle schools in Oregon
Education in Coos County, Oregon
Public high schools in Oregon |
https://en.wikipedia.org/wiki/Debridement%20%28dental%29 | In dentistry, debridement refers to the removal by dental cleaning of accumulations of plaque and calculus (tartar) in order to maintain dental health. Debridement may be performed using ultrasonic instruments, which fracture the calculus, thereby facilitating its removal, as well as hand tools, including periodontal scaler and curettes, or through the use of chemicals such as hydrogen peroxide.
Description
Dental debridement is a procedure by which plaque and calculus (tartar) that have accumulated on the teeth is removed. Debridement may be performed in the process of personal or professional teeth cleaning. Professional debridement techniques include the use of ultrasonic instruments (which fracture the calculus, thereby facilitating its removal), as well as the use of hand tools, including periodontal scaler and curettes. Debridement may also be performed using saline solution. .
Procedures
Periodontal Pockets
A periodontal pocket is formed from a disease process; it is defined as the apical extension of the gingiva, resulting in detachment of the periodontal ligament (PDL). The PDL is a ligament that attaches the root of the tooth to the supporting alveolar bone. This ligament allows for occlusal force absorption. Plaque accumulates within the pocket initiating an inflammatory response due to an increased number of spirochetes. There are different types of bacteria that make up dental plaque. In cases of aggressive periodontitis three major species of bacteria have been identified within the periodontal pocket. These bacteria include Porphyromonas gingivalis, Prevotella intermedia, and Actinobacillus actinomycetemcomitans. Healthy gingiva consists of few microorganisms, mostly coccoid cells and straight rods. Diseased gingiva consists of increased numbers of spirochetes and mobile rods. Interactions between plaque and host inflammatory response determine the alterations in pocket depths. Bacterial plaque initiates a nonspecific host inflammatory response with the intention of eliminating necrotic cells and harmful bacteria. During this process cytokines, proteinases, and prostaglandins are produced which can cause damage, or kill healthy tissues such as macrophages, fibroblasts, neutrophiles, and epithelial cells. The exposure to connective tissue and blood capillaries, allows microorganisms to gain an entryway to the circulation. This suppresses host protection mechanisms, leading to further destruction of bone.
Periodontal pockets may occur from either coronal swelling or apical migration. Pockets that occur due to coronal swelling with no clinical attachment loss are considered pseudopockets. There are two types of periodontal pockets that are determined by the type of bone loss present. A suprabony pocket occurs when there is horizontal bone loss, the bottom of the pocket is coronal to the alveolar bone. An infrabony pocket occurs when there is vertical bone loss where the bottom of the pocket is apical to the alveolar bone.
Clin |
https://en.wikipedia.org/wiki/Christoph%20Rudolff | Christoph Rudolff (born 1499 in Jawor, Silesia, died 1545 in Vienna) was the author of the first German textbook on algebra.
From 1517 to 1521, Rudolff was a student of Henricus Grammateus (Schreyber from Erfurt) at the University of Vienna and was the author of a book computing, under the title: (Nimble and beautiful calculation via the artful rules of algebra [which] are so commonly called "coss").
He introduced the radical symbol (√) for the square root. It is believed that this was because it resembled a lowercase "r" (for "radix"), though there is no direct evidence. Cajori only says that a "dot is the embryo of our present symbol for the square root" though it is "possible, perhaps probable" that Rudolff's later symbols are not dots but 'r's.
Furthermore, he used the meaningful definition that x0 = 1.
See also
History of mathematical notation
Notes
References
External links
Die Coss Christoffs Rudolffs
1499 births
1545 deaths
People from Jawor
16th-century German mathematicians
16th-century German writers
16th-century German male writers |
https://en.wikipedia.org/wiki/John%20Akehurst%20%28photographer%29 | John Akehurst is a photographer who specializes in fashion, beauty, and advertising.
Biography
He studied mathematics at the University of Nottingham. After graduation he moved to New York where worked as an assistant to Steven Meisel and Albert Watson. He moved to London, eventually publishing the story "The Egg" in The Face in 1997.
References
External links
American photographers
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Mathematical%20elimination | In statistics, the terms "mathematical elimination" and "mathematically eliminated" mean to be excluded in a decision, based on numerical counts, due to insufficient total numbers, even if all remaining events were 100% in favor. The excluded outcome is considered to be eliminated due to the mathematical probability being zero (0%).
The term is used in elections when a candidate lacks sufficient votes to win, even if that candidate could garner all remaining votes. In sports, the term "mathematically eliminated"
refers to situations where there are not enough future games or competitive events remaining to be played to avoid defeat, even if all future events were won.
History
The term "mathematically eliminated" has been in use for more than 100 years,
although the meaning has varied. In a 1904 article, in the American Journal of Psychology, Volume XV, errors of measurement were described as quantifiable to be "mathematically eliminated" from the analysis of the remaining data.
References
Voting |
https://en.wikipedia.org/wiki/Geometry%20%28Jega%20album%29 | Geometry is the second album by the electronic musician Jega, released in 2000 on the Planet Mu and Matador labels.
Track listing
Reception
Sam Eccleston, writing for Pitchfork, called the album "a fascinating, if not moving, musical experience". Mike Bruno of the Chicago Reader wrote a similarly positive review and compared it with Jega's previous album Spectrum, stating that Geometry for Jega "represents a step toward being taken more seriously as an electronic composer".
References
External links
Geometry at the Planet Mu website
2000 albums
Jega (musician) albums
Planet Mu albums |
https://en.wikipedia.org/wiki/Dense%20set | In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
Formally, is dense in if the smallest closed subset of containing is itself.
The of a topological space is the least cardinality of a dense subset of
Definition
A subset of a topological space is said to be a of if any of the following equivalent conditions are satisfied:
The smallest closed subset of containing is itself.
The closure of in is equal to That is,
The interior of the complement of is empty. That is,
Every point in either belongs to or is a limit point of
For every every neighborhood of intersects that is,
intersects every non-empty open subset of
and if is a basis of open sets for the topology on then this list can be extended to include:
For every every neighborhood of intersects
intersects every non-empty
Density in metric spaces
An alternative definition of dense set in the case of metric spaces is the following. When the topology of is given by a metric, the closure of in is the union of and the set of all limits of sequences of elements in (its limit points),
Then is dense in if
If is a sequence of dense open sets in a complete metric space, then is also dense in This fact is one of the equivalent forms of the Baire category theorem.
Examples
The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The irrational numbers are another dense subset which shows that a topological space may have several disjoint dense subsets (in particular, two dense subsets may be each other's complements), and they need not even be of the same cardinality. Perhaps even more surprisingly, both the rationals and the irrationals have empty interiors, showing that dense sets need not contain any non-empty open set. The intersection of two dense open subsets of a topological space is again dense and open.
The empty set is a dense subset of itself. But every dense subset of a non-empty space must also be non-empty.
By the Weierstrass approximation theorem, any given complex-valued continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. In other words, the polynomial functions are dense in the space of continuous complex-valued functions on the interval equipped with the supremum norm.
Every metric space is dense in its completion.
Properties
Every topological space is a dense subset of itself. For a set equipped with the discrete topol |
https://en.wikipedia.org/wiki/Synge%27s%20theorem | In mathematics, specifically Riemannian geometry, Synge's theorem is a classical result relating the curvature of a Riemannian manifold to its topology. It is named for John Lighton Synge, who proved it in 1936.
Theorem and sketch of proof
Let be a closed Riemannian manifold with positive sectional curvature. The theorem asserts:
If is even-dimensional and orientable, then is simply connected.
If is odd-dimensional, then it is orientable.
In particular, a closed manifold of even dimension can support a positively curved Riemannian metric only if its fundamental group has one or two elements.
The proof of Synge's theorem can be summarized as follows. Given a geodesic with an orthogonal and parallel vector field along the geodesic (i.e. a parallel section of the normal bundle to the geodesic), then Synge's earlier computation of the second variation formula for arclength shows immediately that the geodesic may be deformed so as to shorten its length. The only tool used at this stage is the assumption on sectional curvature.
The construction of a parallel vector field along any path is automatic via parallel transport; the nontriviality in the case of a loop is whether the values at the endpoints coincide. This reduces to a problem of pure linear algebra: let be a finite-dimensional real inner product space with an orthogonal linear map with an eigenvector with eigenvalue one. If the determinant of is positive and the dimension of is even, or alternatively if the determinant of is negative and the dimension of is odd, then there is an eigenvector of with eigenvalue one which is orthogonal to . In context, is the tangent space to at a point of a geodesic loop, is the parallel transport map defined by the loop, and is the tangent vector to the geodesic.
Given any noncontractible loop in a complete Riemannian manifold, there is a representative of its (free) homotopy class which has minimal possible arclength, and it is a geodesic. According to Synge's computation, this implies that there cannot be a parallel and orthogonal vector field along this geodesic. However:
Orientability implies that the parallel transport map along every loop has positive determinant. Even-dimensionality then implies the existence of a parallel vector field, orthogonal to the geodesic.
Non-orientability implies the non-contractible loop can be chosen so that the parallel transport map has negative determinant. Odd-dimensionality then implies the existence of a parallel vector field, orthogonal to the geodesic.
This contradiction establishes the non-existence of noncontractible loops in the first case, and the impossibility of non-orientability in the latter case.
Alan Weinstein later rephrased the proof so as to establish fixed points of isometries, rather than topological properties of the underlying manifold.
References
Sources.
Theorems in Riemannian geometry |
https://en.wikipedia.org/wiki/Alperton%20Community%20School | Alperton Community School is a coeducational secondary school and sixth form with academy status. It has a specialism in maths, computing and arts and it is located in the Alperton area of the London Borough of Brent, England.
The school is divided into two sites: the lower school on Ealing Road near Alperton Underground station, consisting of Years 7, 8 and 9 and the upper school on Stanley Avenue, consisting of Years 10, 11, 12 and 13. It has approximately 2000 students.
In July 2016 the Ofsted report judged the school to be “Good with Outstanding Leadership and Management”.
History
In 1922, Alperton Hall mansion was purchased in order to support the educational needs of the growing industrial town and opened with the overseeing headmaster Mr Edmund Lightley. In 1928 the school adopted the name Wembley County Grammar School and the original mansion was demolished in 1938 to allow for a new appropriate site to be built for a traditional grammar school.
A separate school on Ealing Road named Alperton County Mixed School was developed in 1948 on a new site near Alperton tube station on Kennedy's Farm which required the demolition of the adjacent Joy Cottages sitting alongside the station. The requirement grew from the previous local school named Alperton School, which had existed since 1876 on a site now hosting the Shree Sanatan Hindu temple, becoming inappropriate due to the Education Act of 1944 and the increased demand because of area growth. In 1957 the school was split into Alperton Boys and Alperton Girls both being shaped into secondary moderns with Mr T. Hostler as headmaster for boys and Miss J. Dawson head teacher for girls respectively, although officially the girls site was not completed until 1962.
The three schools, Wembley County Grammar, Alperton Boys, and Alperton Girls were amalgamated as to form Alperton High School in 1967.
Mr Roy Innes was recruited as the headmaster to see through the new comprehensive school merger and development and after a decade retired in 1977.
During the early 1990s through the Local Management of Schools (LMS) initiative the school took control over its own finances and in 1993 was renamed Alperton Community School. From 1991, Mr Pankaj Gulab as Deputy Head saw through the change to a local managed school and in 1992 he became the headmaster.
In 2003 the new headmaster was Miss Margaret Rafee. During her tenure, the school managed to achieve an "Outstanding" Ofsted report in 2011 and had won the British Council's International School Award in 2012. Thereafter it was in the top 5% of schools in the country in terms of student progress over a four-year period.
The school converted to Cooperative Academy status in September 2012, and a year later Rafee left the school to be replaced by a new headmaster, Mr Gerard McKenna. In 2014, its first Ofsted judgment as an academy had dropped to "Requires improvement" before recovering to "Good" in 2016.
The main site at Ealing Road was completely renov |
https://en.wikipedia.org/wiki/Errors-in-variables%20models | In statistics, errors-in-variables models or measurement error models are regression models that account for measurement errors in the independent variables. In contrast, standard regression models assume that those regressors have been measured exactly, or observed without error; as such, those models account only for errors in the dependent variables, or responses.
In the case when some regressors have been measured with errors, estimation based on the standard assumption leads to inconsistent estimates, meaning that the parameter estimates do not tend to the true values even in very large samples. For simple linear regression the effect is an underestimate of the coefficient, known as the attenuation bias. In non-linear models the direction of the bias is likely to be more complicated.
Motivating example
Consider a simple linear regression model of the form
where denotes the true but unobserved regressor. Instead we observe this value with an error:
where the measurement error is assumed to be independent of the true value .
If the ′s are simply regressed on the ′s (see simple linear regression), then the estimator for the slope coefficient is
which converges as the sample size increases without bound:
This is in contrast to the "true" effect of , estimated using the ,:
Variances are non-negative, so that in the limit the estimated is smaller than , an effect which statisticians call attenuation or regression dilution. Thus the ‘naïve’ least squares estimator is an inconsistent estimator for . However, is a consistent estimator of the parameter required for a best linear predictor of given the observed : in some applications this may be what is required, rather than an estimate of the ‘true’ regression coefficient , although that would assume that the variance of the errors in the estimation and prediction is identical. This follows directly from the result quoted immediately above, and the fact that the regression coefficient relating the ′s to the actually observed ′s, in a simple linear regression, is given by
It is this coefficient, rather than , that would be required for constructing a predictor of based on an observed which is subject to noise.
It can be argued that almost all existing data sets contain errors of different nature and magnitude, so that attenuation bias is extremely frequent (although in multivariate regression the direction of bias is ambiguous). Jerry Hausman sees this as an iron law of econometrics: "The magnitude of the estimate is usually smaller than expected."
Specification
Usually measurement error models are described using the latent variables approach. If is the response variable and are observed values of the regressors, then it is assumed there exist some latent variables and which follow the model's “true” functional relationship , and such that the observed quantities are their noisy observations:
where is the model's parameter and are those regressors which are assum |
https://en.wikipedia.org/wiki/International%20Academy%20of%20Mathematical%20Chemistry | The International Academy of Mathematical Chemistry (IAMC) was founded in Dubrovnik, Croatia, in 2005 by Milan Randić. It is an organization for chemistry and mathematics avocation; its predecessors have been around since the 1930s. There are 88 Academy members () from around the world (27 countries), comprising six scientists awarded the Nobel Prize.
Governing body of the IAMC
2005–2007:
President: Alexandru Balaban
Vice-President: Milan Randić
Secretary: Ante Graovac
Treasurer: Dejan Plavšić
2008–2011:
President: Roberto Todeschini
Vice-President: Tomaž Pisanski
Secretary: Ante Graovac
Treasurer: Dražen Vikić-Topić
Member: Ivan Gutman
Member: Nikolai Zefirov
since 2011:
President: Roberto Todeschini
Vice-President: Edward C. Kirby
Vice-President: Sandi Klavžar
Secretary: Ante Graovac
Treasurer: Dražen Vikić-Topić
Member: Ivan Gutman
Member: Nikolai Zefirov
since 2019:
President:
Vice-President: Douglas J. Klein
Vice-President: Xueliang Li
Vice-President: Sandi Klavžar
Vice-President: Tomaž Pisanski
Secretary: Boris Furtula
Treasurer:
Member: Ivan Gutman
IAMC yearly meetings
2005 – Dubrovik, Croatia
2006 – Dubrovik, Croatia
2007 – Dubrovik, Croatia
2008 – Verbania, Italy
2009 – Dubrovik, Croatia
2010 – Dubrovik, Croatia
2011 – Bled, Slovenia
2012 – Verona, Italy
2014 – Split, Croatia
2015 – Kranjska Gora, Slovenia
2016 – Tianjin, China
2017 – Cluj, Romania
2019 – Bled, Slovenia
2023 – Kranjska Gora, Slovenia
See also
Mathematical chemistry
References
Mathematical chemistry
International academies
Scientific organizations established in 2005
2005 establishments in Croatia |
https://en.wikipedia.org/wiki/Coherent%20sheaf%20cohomology | In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.
Much of algebraic geometry and complex analytic geometry is formulated in terms of coherent sheaves and their cohomology.
Coherent sheaves
Coherent sheaves can be seen as a generalization of vector bundles. There is a notion of a coherent analytic sheaf on a complex analytic space, and an analogous notion of a coherent algebraic sheaf on a scheme. In both cases, the given space comes with a sheaf of rings , the sheaf of holomorphic functions or regular functions, and coherent sheaves are defined as a full subcategory of the category of -modules (that is, sheaves of -modules).
Vector bundles such as the tangent bundle play a fundamental role in geometry. More generally, for a closed subvariety of with inclusion , a vector bundle on determines a coherent sheaf on , the direct image sheaf , which is zero outside . In this way, many questions about subvarieties of can be expressed in terms of coherent sheaves on .
Unlike vector bundles, coherent sheaves (in the analytic or algebraic case) form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. On a scheme, the quasi-coherent sheaves are a generalization of coherent sheaves, including the locally free sheaves of infinite rank.
Sheaf cohomology
For a sheaf of abelian groups on a topological space , the sheaf cohomology groups for integers are defined as the right derived functors of the functor of global sections, . As a result, is zero for , and can be identified with . For any short exact sequence of sheaves , there is a long exact sequence of cohomology groups:
If is a sheaf of -modules on a scheme , then the cohomology groups (defined using the underlying topological space of ) are modules over the ring of regular functions. For example, if is a scheme over a field , then the cohomology groups are -vector spaces. The theory becomes powerful when is a coherent or quasi-coherent sheaf, because of the following sequence of results.
Vanishing theorems in the affine case
Complex analysis was revolutionized by Cartan's theorems A and B in 1953. These results say that if is a coherent analytic sheaf on a Stein space , then is spanned by its global sections, and for all . (A complex space is Stein if and only if it is isomorphic to a closed analytic subspace of for some .) These results generalize a large body of older work about the construction of complex analytic functions with |
https://en.wikipedia.org/wiki/Mihalis%20Dafermos | Mihalis Dafermos (Greek: Μιχάλης Δαφέρμος; born October 1976) is a Greek mathematician. He is Professor of Mathematics at Princeton University and holds the Lowndean Chair of Astronomy and Geometry at the University of Cambridge.
He studied mathematics at Harvard University and was awarded a BA in 1997. His PhD thesis titled Stability and Instability of the Cauchy Horizon for the Spherically Symmetric Einstein-Maxwell-Scalar Field Equations was written under the supervision of Demetrios Christodoulou at Princeton University.
He has won the Adams Prize writing on the subject Differential Equations in 2004 and the Whitehead Prize in 2009 for "his work on the rigorous analysis of hyperbolic partial differential equations in general relativity." In 2015 he was elected as a fellow of the American Mathematical Society.
References
External links
Homepage at Cambridge
Living people
20th-century Greek mathematicians
21st-century Greek mathematicians
Cambridge mathematicians
Whitehead Prize winners
Fellows of the American Mathematical Society
Harvard College alumni
1976 births
Princeton University alumni
Lowndean Professors of Astronomy and Geometry |
https://en.wikipedia.org/wiki/Small%20control%20property | For applied mathematics, in nonlinear control theory, a non-linear system of the form is said to satisfy the small control property if for every there exists a so that for all there exists a so that the time derivative of the system's Lyapunov function is negative definite at that point.
In other words, even if the control input is arbitrarily small, a starting configuration close enough to the origin of the system can be found that is asymptotically stabilizable by such an input.
References
Nonlinear control |
https://en.wikipedia.org/wiki/Elise%20Brezis | Elise Scheiner Brezis, professor of economics at Bar-Ilan University, is the director of the Azrieli Center for Economic Policy. She has been the head of the Statistics division at the Research Department in the Bank of Israel, and from 1999 to 2003, she was the president of the Israeli Association for the Study of European Integration. She holds a PhD in economics from Massachusetts Institute of Technology (1989).
Her first works in economic history were on 18th century England and are where she developed previously nonexistent data on the balance of payments of the UK. Her research interests are related to economic growth. Her main works are in two fields: the first is the interaction between sociology and economics emphasizing the stratification of society and focusing on the elites; the second is technology, demography, and economic growth. In the former, she has been on the forefront of research in the field of elites. In the latter, her most cited work dealing with technology and growth on leapfrogging in international competition, a paper on the theory of cycles in national technological leadership that she worked on with Paul Krugman, published in the American Economic Review in 1993. She is also a fellow at the Minerva Center for Growth in Jerusalem and serves on the editorial board of Cliometrica.
Contributions
Elites and Economics
Brezis examines the evolution of recruitment of elites and investigates the nature of the links between recruitment of elites and economic growth. The main change that occurred in the way the Western world recruited its elites is that meritocracy became the basis for their recruitment. Although meritocratic selection should result in the best being chosen, she shows that meritocratic recruitment actually leads to class stratification and auto-recruitment.
Brezis's book examines the relationship between elites, minorities, and economic growth. The novelty of the book lies in its focus on the interaction between social and economic changes during economic growth.
Technology and Economic Growth
1. In Brezis's work with Krugman, they emphasized that endogenous-growth theory suggests that technological change tends to reinforce the position of the leading nations. Yet sometimes this leadership role shifts. They suggest a mechanism that explains this pattern of 'leapfrogging' as a response to occasional major changes in technology.
2. On government policy towards technological development, Brezis argues that the method used for financing projects is too conservative: Peer review mechanism has a clear bias against innovative ideas. In consequence, proposals for new ideas are too often rejected. Brezis proposes to adopt a Focal Randomization mechanism.
3. Brezis and Warren Young have studied together the new views on demographic transition, in which they reassess Malthus and Marx's approach to population. Their paper examines the divergence of views of Marx and Malthus regarding the family and the labor marke |
https://en.wikipedia.org/wiki/Structural%20rigidity | In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges.
Definitions
Rigidity is the property of a structure that it does not bend or flex under an applied force. The opposite of rigidity is flexibility. In structural rigidity theory, structures are formed by collections of objects that are themselves rigid bodies, often assumed to take simple geometric forms such as straight rods (line segments), with pairs of objects connected by flexible hinges. A structure is rigid if it cannot flex; that is, if there is no continuous motion of the structure that preserves the shape of its rigid components and the pattern of their connections at the hinges.
There are two essentially different kinds of rigidity. Finite or macroscopic rigidity means that the structure will not flex, fold, or bend by a positive amount. Infinitesimal rigidity means that the structure will not flex by even an amount that is too small to be detected even in theory. (Technically, that means certain differential equations have no nonzero solutions.) The importance of finite rigidity is obvious, but infinitesimal rigidity is also crucial because infinitesimal flexibility in theory corresponds to real-world minuscule flexing, and consequent deterioration of the structure.
A rigid graph is an embedding of a graph in a Euclidean space which is structurally rigid. That is, a graph is rigid if the structure formed by replacing the edges by rigid rods and the vertices by flexible hinges is rigid. A graph that is not rigid is called flexible. More formally, a graph embedding is flexible if the vertices can be moved continuously, preserving the distances between adjacent vertices, with the result that the distances between some nonadjacent vertices are altered. The latter condition rules out Euclidean congruences such as simple translation and rotation.
It is also possible to consider rigidity problems for graphs in which some edges represent compression elements (able to stretch to a longer length, but not to shrink to a shorter length) while other edges represent tension elements (able to shrink but not stretch). A rigid graph with edges of these types forms a mathematical model of a tensegrity structure.
Mathematics of rigidity
The fundamental problem is how to predict the rigidity of a structure by theoretical analysis, without having to build it. Key results in this area include the following:
In any dimension, the rigidity of rod-and-hinge linkages is described by a matroid. The bases of the two-dimensional rigidity matroid (the minimally rigid graphs in the plane) are the Laman graphs.
Cauchy's theorem states that a three-dimensional convex polyhedron constructed with rigid plates for its faces, connected by hinges along its edges, forms a rigid structure.
Flexible polyhedra, non-convex polyhedra that are not rigid, were constructed by Raoul B |
https://en.wikipedia.org/wiki/Schauder%20estimates | In mathematics, and more precisely, in functional Analysis and PDEs, the Schauder estimates are a collection of results due to concerning the regularity of solutions to linear, uniformly elliptic partial differential equations. The estimates say that when the equation has appropriately smooth terms and appropriately smooth solutions, then the Hölder norm of the solution can be controlled in terms of the Hölder norms for the coefficient and source terms. Since these estimates assume by hypothesis the existence of a solution, they are called a priori estimates.
There is both an interior result, giving a Hölder condition for the solution in interior domains away from the boundary, and a boundary result, giving the Hölder condition for the solution in the entire domain. The former bound depends only on the spatial dimension, the equation, and the distance to the boundary; the latter depends on the smoothness of the boundary as well.
The Schauder estimates are a necessary precondition to using the method of continuity to prove the existence and regularity of solutions to the Dirichlet problem for elliptic PDEs. This result says that when the coefficients of the equation and the nature of the boundary conditions are sufficiently smooth, there is a smooth classical solution to the PDE.
Notation
The Schauder estimates are given in terms of weighted Hölder norms; the notation will follow that given in the text of .
The supremum norm of a continuous function is given by
For a function which is Hölder continuous with exponent , that is to say , the usual Hölder seminorm is given by
The sum of the two is the full Hölder norm of f
For differentiable functions u, it is necessary to consider the higher order norms, involving derivatives. The norm in the space of functions with k continuous derivatives, , is given by
where ranges over all multi-indices of appropriate orders. For functions with kth order derivatives which are Holder continuous with exponent , the appropriate semi-norm is given by
which gives a full norm of
For the interior estimates, the norms are weighted by the distance to the boundary
raised to the same power as the derivative, and the seminorms are weighted by
raised to the appropriate power. The resulting weighted interior norm for a function is given by
It is occasionally necessary to add "extra" powers of the weight, denoted by
Formulation
The formulations in this section are taken from the text of .
Interior estimates
Consider a bounded solution on the domain to the elliptic, second order, partial differential equation
where the source term satisfies . If there exists a constant such that the are strictly elliptic,
for all
and the relevant norms coefficients are all bounded by another constant
Then the weighted norm of u is controlled by the supremum of u and the Holder norm of f:
Boundary estimates
Let be a domain (that is to say, about any point on the boundary of the domain the boundary hypersurface ca |
https://en.wikipedia.org/wiki/Method%20of%20continuity | In the mathematics of Banach spaces, the method of continuity provides sufficient conditions for deducing the invertibility of one bounded linear operator from that of another, related operator.
Formulation
Let B be a Banach space, V a normed vector space, and a norm continuous family of bounded linear operators from B into V. Assume that there exists a positive constant C such that for every and every
Then is surjective if and only if is surjective as well.
Applications
The method of continuity is used in conjunction with a priori estimates to prove the existence of suitably regular solutions to elliptic partial differential equations.
Proof
We assume that is surjective and show that is surjective as well.
Subdividing the interval [0,1] we may assume that . Furthermore, the surjectivity of implies that V is isomorphic to B and thus a Banach space. The hypothesis implies that is a closed subspace.
Assume that is a proper subspace. Riesz's lemma shows that there exists a such that and . Now for some and by the hypothesis. Therefore
which is a contradiction since .
See also
Schauder estimates
Sources
Banach spaces |
https://en.wikipedia.org/wiki/Tetrahedral%20hypothesis | The tetrahedral hypothesis is an obsolete scientific theory attempting to explain the arrangement of the Earth's continents and oceans by referring to the geometry of a tetrahedron. Although it was a historically interesting theory in the late 19th and early 20th century, it was superseded by the concepts of continental drift and modern plate tectonics. The theory was first proposed by William Lowthian Green in 1875.
Theory
This idea, described as ‘"ingenious" by geologist Arthur Holmes, is now of historical interest only, being finally refuted by that same Holmes (see reference 7). It attempted to explain apparent anomalies in the distribution of land and water on the Earth's surface:
More than 75% of the Earth's land area is in the northern hemisphere.
Continents are roughly triangular.
Oceans are roughly triangular.
The north pole is surrounded by water, the south pole by land.
Exactly opposite the Earth from land is almost always water.
The Pacific Ocean occupies about one third of the Earth's surface.
To understand its appeal, consider the "regular solids": the sphere and the 5-member set of Platonic Solids. The solid with the lowest number of sides is the tetrahedron (four equilateral triangles); progressing through the hexahedron or cube, the octahedron, the dodecahedron and the icosahedron (20 sides), the sphere can be considered to have an infinite number of sides. All six regular solids share many symmetries.
Now, for each regular solid, we may relate its surface area and volume by the equation:
where k is a characteristic of each solid, V its volume, and A its area. As we traverse the set in order of increasing number of faces, we find that k increases for each member; it is 0.0227 for a tetrahedron and 0.0940 for a sphere. Thus the tetrahedron is the regular solid with the largest surface area for a given volume, and makes a reasonable endpoint for a shrinking spherical Earth.
History
The theory was first proposed by William Lowthian Green in 1875.
It was still popular in 1917 when summarized as:
"The law of least action … demands that the somewhat rigid crustal portion of the earth keep in contact with the lessening interior with the least possible readjustment of its surface. … a shrinking sphere tends by the law of least action to collapse into a tetrahedron, or a tetra-hedroid, a sphere marked by four equal and equidistant triangular projections; and the earth with its three about equal and equidistant double continental masses triangular southward with three intervening depressed oceans triangular northward, its northern ocean and southern continent, with land everywhere antipodal to water, realizes the tetrahedroid status remarkably.“
This is suggesting that a cooling spherical Earth might have shrunk to form a tetrahedron, with its vertices and edges forming the continents, and four oceans (Pacific Ocean, Atlantic Ocean, Indian Ocean and Arctic Ocean) on its faces.
By 1915 German Alfred Wegener (1880–1930) had prop |
https://en.wikipedia.org/wiki/Herbert%20Arthur%20Frederick%20Turner | Herbert Arthur Frederick Turner (1919–1998) was a British economist, statistician, and academic. His great strength was a thorough understanding of economics and statistics, particularly the operation of labour markets and the limitations of available statistics. This set him apart from most other academic industrial relations specialists. He was an inspiring lecturer and his tutorials and post-graduate supervisions were challenging and provocative as students were prodded and persuaded into thinking.
Personal life
Turner, known as Bert to family, friends, and colleagues, was born in London on 11 December 1919, the eldest son of Frederick Turner and Elizabeth May King; he had three siblings. Turner's fourth marriage, to a French academic, led him to spend much of his later years in France with his family. He died in Veneux- les –Sablons, near Fontainebleau, on 2 December 1998, a few days short of his 79th birthday.
Career
Education
Turner studied at the Henry Thornton School in Clapham before going to the London School of Economics, aged 16, to study with Harold Laski. As a young promising left-wing intellectual, he interacted with the Webbs and, through Leonard Woolf, with the Bloomsbury group. He graduated in June 1939 and spent the war years first in the army then on the Second Sea Lord's staff.
In 1944, Turner joined the research and economic department of the TUC. He served as part of the team that prepared the Interim Report on Post-War Reconstruction, which mapped out the Attlee government's programme. Turner worked under Sir Walter Citrine, which developed his lasting interest in economic policy, trade union activities and management and industrial relations. In 1947, Turner became Assistant Education Secretary for the TUC.
In 1950, Turner was elected to the lectureship in industrial relations at University of Manchester. Senior Lecturer in 1959, he defended his PhD on industrial relations in the cotton industry in 1960, which still is the seminal work on the subject.
Professor
Turner moved to Leeds University in 1961, when he was elected to the Montague Burton Chair of Industrial Relations, then to the Cambridge chair in 1964. He stayed there until his retirement from the Professorship, to be succeeded by William Brown, the son of one of his Leeds colleagues, in 1983. On his election to the Cambridge chair, Bert Turner became a Professorial Fellow of Churchill College, and a Life Fellow on becoming Professor Emeritus in 1984.
From his arrival in Cambridge, Bert Turner's career took two different but complementary directions. The nature of his research together with the world political conjuncture meant that he was very much in demand as an expert and consultant at the time of decolonisation. As a Visiting Professor, he taught at the universities of Lusaka (1969), Harvard and M.I.T. (1971–72), Sydney (1976–77), Hong Kong (1978–79, 1985–88), Bombay and Lucknow (1983) at South China University of Technology (1986), and Zhongshan ( |
https://en.wikipedia.org/wiki/Davey%20Lake%20%28Saskatchewan%29 | Davey Lake is a lake in the Canadian province of Saskatchewan.
See also
List of lakes of Saskatchewan
References
Statistics Canada
Anglersatlas.com
Lakes of Saskatchewan |
https://en.wikipedia.org/wiki/Scott%20Lake%20%28Northwest%20Territories%E2%80%93Saskatchewan%29 | Scott Lake is a lake of northern Saskatchewan and the Northwest Territories of Canada.
See also
List of lakes of Saskatchewan
List of lakes of the Northwest Territories
References
Statistics Canada
Anglersatlas.com
Lakes of Saskatchewan
Lakes of the Northwest Territories |
https://en.wikipedia.org/wiki/Lower%20Foster%20Lake | Lower Foster Lake is a lake in the Canadian province of Saskatchewan.
See also
List of lakes of Saskatchewan
Middle Foster Lake
Upper Foster Lake
References
Statistics Canada
Anglersatlas.com
Lakes of Saskatchewan |
https://en.wikipedia.org/wiki/Upper%20Foster%20Lake | Upper Foster Lake is a lake in the Canadian province of Saskatchewan.
See also
List of lakes of Saskatchewan
Middle Foster Lake
Lower Foster Lake
References
External links
Statistics Canada
Lakes of Saskatchewan |
https://en.wikipedia.org/wiki/2006%20Japan%20national%20football%20team | This page records the details of the Japan national football team in 2006.
Schedule
Players statistics
Top goal scorers for 2006
Manager
The manager was Zico up to 2006 World Cup. He was replaced by Ivica Osim.
Kits
References
External links
Japan Football Association
Japan national football team results
2006 in Japanese football
Japan |
https://en.wikipedia.org/wiki/Non-uniform%20discrete%20Fourier%20transform | In applied mathematics, the nonuniform discrete Fourier transform (NUDFT or NDFT) of a signal is a type of Fourier transform, related to a discrete Fourier transform or discrete-time Fourier transform, but in which the input signal is not sampled at equally spaced points or frequencies (or both). It is a generalization of the shifted DFT. It has important applications in signal processing, magnetic resonance imaging, and the numerical solution of partial differential equations.
As a generalized approach for nonuniform sampling, the NUDFT allows one to obtain frequency domain information of a finite length signal at any frequency. One of the reasons to adopt the NUDFT is that many signals have their energy distributed nonuniformly in the frequency domain. Therefore, a nonuniform sampling scheme could be more convenient and useful in many digital signal processing applications. For example, the NUDFT provides a variable spectral resolution controlled by the user.
Definition
The nonuniform discrete Fourier transform transforms a sequence of complex numbers into another sequence of complex numbers defined by
where are sample points and are frequencies. Note that if and , then equation () reduces to the discrete Fourier transform. There are three types of NUDFTs.
The nonuniform discrete Fourier transform of type I (NUDFT-I) uses uniform sample points but nonuniform (i.e. non-integer) frequencies . This corresponds to evaluating a generalized Fourier series at equispaced points. It is also known as NDFT.
The nonuniform discrete Fourier transform of type II (NUDFT-II) uses uniform (i.e. integer) frequencies but nonuniform sample points . This corresponds to evaluating a Fourier series at nonequispaced points. It is also known as adjoint NDFT.
The nonuniform discrete Fourier transform of type III (NUDFT-III) uses both nonuniform sample points and nonuniform frequencies . This corresponds to evaluating a generalized Fourier series at nonequispaced points. It is also known as NNDFT.
A similar set of NUDFTs can be defined by substituting for in equation ().
Unlike in the uniform case, however, this substitution is unrelated to the inverse Fourier transform.
The inversion of the NUDFT is a separate problem, discussed below.
Multidimensional NUDFT
The multidimensional NUDFT converts a -dimensional array of complex numbers into another -dimensional array of complex numbers defined by
where are sample points, are frequencies, and and are -dimensional vectors of indices from 0 to . The multidimensional NUDFTs of types I, II, and III are defined analogously to the 1D case.
Relationship to Z-transform
The NUDFT-I can be expressed as a Z-transform. The NUDFT-I of a sequence of length is
where is the Z-transform of , and are arbitrarily distinct points in the z-plane. Note that the NUDFT reduces to the DFT when the sampling points are located on the unit circle at equally spaced angles.
Expressing the above as a matrix, we get
wher |
https://en.wikipedia.org/wiki/Aleksandr%20Dovbnya%20%28footballer%2C%20born%201987%29 | Aleksandr Vyacheslavovich Dovbnya (; born 14 April 1987) is a Russian professional football goalkeeper. He plays for FC Shinnik Yaroslavl.
Career statistics
Honours
Torpedo Moscow
Russian Football National League : 2021-22
References
External links
Profile by FNL
1987 births
Footballers from Moscow
Living people
Russian men's footballers
Men's association football goalkeepers
FC Haka players
FC Sibir Novosibirsk players
FC Nizhny Novgorod (2007) players
FC Torpedo Moscow players
FC Luch Vladivostok players
FC SKA-Khabarovsk players
FC Orenburg players
FC Rotor Volgograd players
FC Shinnik Yaroslavl players
Veikkausliiga players
Russian Second League players
Russian First League players
Russian Premier League players
Russian expatriate men's footballers
Expatriate men's footballers in Finland
Russian expatriate sportspeople in Finland |
https://en.wikipedia.org/wiki/Reider%27s%20theorem | In algebraic geometry, Reider's theorem gives conditions for a line bundle on a projective surface to be very ample.
Statement
Let D be a nef divisor on a smooth projective surface X. Denote by KX the canonical divisor of X.
If D2 > 4, then the linear system |KX+D| has no base points unless there exists a nonzero effective divisor E such that
, or
;
If D2 > 8, then the linear system |KX+D| is very ample unless there exists a nonzero effective divisor E satisfying one of the following:
or ;
or ;
;
Applications
Reider's theorem implies the surface case of the Fujita conjecture. Let L be an ample line bundle on a smooth projective surface X. If m > 2, then for D=mL we have
D2 = m2 L2 ≥ m2 > 4;
for any effective divisor E the ampleness of L implies D · E = m(L · E) ≥ m > 2.
Thus by the first part of Reider's theorem |KX+mL| is base-point-free. Similarly, for any m > 3 the linear system |KX+mL| is very ample.
References
Algebraic surfaces
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/SR1%20%28disambiguation%29 | SR1 may refer to:
Science and mathematics
SR1 RNA, a small RNA produced by bacteria
SR1, a candidate phylum of bacteria more commonly called Absconditabacteria
Symmetric rank-one, a mathematical algorithm
Products and technology
HDR-SR1, a Sony camcorder
Peugeot SR1, a hybrid concept car
Radical SR1, a sports car
Vintage Ultralight SR-1 Hornet, an American homebuilt aircraft
VR Class Sr1, a class of Finnish electric locomotives
VSR SR-1 Snoshoo, an American Formula One racing aircraft design
Six Chuter SR1, an American powered parachute design
SR1, a FIA Sportscar Championship classification
SR1, a 4-stroke Yamaha motorcycle
SR.1, a semi-rigid airship built in 1918 for the British navy
SR-1 Vektor, a Russian pistol
Media
Saints Row (2006 video game)
SR1 Europawelle, radio programming by Saarländischer Rundfunk (Saarland Broadcasting)
Other
SR1, a spelling reform proposal
State Route or State Road 1; see List of highways numbered 1 |
https://en.wikipedia.org/wiki/Reference%20point | Reference point or similar may refer to:
Mathematics and science
Reference point (physics), used to define a frame of reference
Reference point, a point within a reference range or reference interval, which is a range of values found in healthy persons
Reference point, a measurement taken during a standard state or reference state, used in chemistry to calculate properties under different conditions
Other uses
Reference Point (horse), a 1980s British racehorse
Reference point, a benchmark utility level in prospect theory
Reference Point, a 1990 Acoustic Alchemy album
See also
Benchmark (disambiguation)
Reference (disambiguation) |
https://en.wikipedia.org/wiki/Cubic%20threefold | In algebraic geometry, a cubic threefold is a hypersurface of degree 3 in 4-dimensional projective space. Cubic threefolds are all unirational, but used intermediate Jacobians to show that non-singular cubic threefolds are not rational. The space of lines on a non-singular cubic 3-fold is a Fano surface.
Examples
Koras–Russell cubic threefold
Klein cubic threefold
Segre cubic
References
Algebraic varieties
3-folds |
https://en.wikipedia.org/wiki/Klein%20cubic%20threefold | In algebraic geometry, the Klein cubic threefold is the non-singular cubic threefold in 4-dimensional projective space given by the equation
studied by .
Its automorphism group is the group PSL2(11) of order 660 . It is unirational but not a rational variety.
showed that it is birational to the moduli space of (1,11)-polarized abelian surfaces.
References
3-folds |
https://en.wikipedia.org/wiki/Quintic%20threefold | In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space . Non-singular quintic threefolds are Calabi–Yau manifolds.
The Hodge diamond of a non-singular quintic 3-fold is
Mathematician Robbert Dijkgraaf said "One number which every algebraic geometer knows is the number 2,875 because obviously, that is the number of lines on a quintic."
Definition
A quintic threefold is a special class of Calabi–Yau manifolds defined by a degree projective variety in . Many examples are constructed as hypersurfaces in , or complete intersections lying in , or as a smooth variety resolving the singularities of another variety. As a set, a Calabi-Yau manifold iswhere is a degree homogeneous polynomial. One of the most studied examples is from the polynomialcalled a Fermat polynomial. Proving that such a polynomial defines a Calabi-Yau requires some more tools, like the Adjunction formula and conditions for smoothness.
Hypersurfaces in P4
Recall that a homogeneous polynomial (where is the Serre-twist of the hyperplane line bundle) defines a projective variety, or projective scheme, , from the algebrawhere is a field, such as . Then, using the adjunction formula to compute its canonical bundle, we havehence in order for the variety to be Calabi-Yau, meaning it has a trivial canonical bundle, its degree must be . It is then a Calabi-Yau manifold if in addition this variety is smooth. This can be checked by looking at the zeros of the polynomialsand making sure the setis empty.
Examples
Fermat Quintic
One of the easiest examples to check of a Calabi-Yau manifold is given by the Fermat quintic threefold, which is defined by the vanishing locus of the polynomialComputing the partial derivatives of gives the four polynomialsSince the only points where they vanish is given by the coordinate axes in , the vanishing locus is empty since is not a point in .
As a Hodge Conjecture testbed
Another application of the quintic threefold is in the study of the infinitesimal generalized Hodge conjecture where this difficult problem can be solved in this case. In fact, all of the lines on this hypersurface can be found explicitly.
Dwork family of quintic three-folds
Another popular class of examples of quintic three-folds, studied in many contexts, is the Dwork family. One popular study of such a family is from Candelas, De La Ossa, Green, and Parkes, when they discovered mirror symmetry. This is given by the family pages 123-125where is a single parameter not equal to a 5-th root of unity. This can be found by computing the partial derivates of and evaluating their zeros. The partial derivates are given byAt a point where the partial derivatives are all zero, this gives the relation . For example, in we getby dividing out the and multiplying each side by . From multiplying these families of equations together we have the relationshowing a solution is either given by an or . But in the first case, the |
https://en.wikipedia.org/wiki/Quadratic%20algebra | In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups. The most important class of graded quadratic algebras is Koszul algebras.
Definition
A graded quadratic algebra A is determined by a vector space of generators V = A1 and a subspace of homogeneous quadratic relations S ⊂ V ⊗ V . Thus
and inherits its grading from the tensor algebra T(V).
If the subspace of relations is instead allowed to also contain inhomogeneous degree 2 elements, i.e. S ⊂ k ⊕ V ⊕ (V ⊗ V), this construction results in a filtered quadratic algebra.
A graded quadratic algebra A as above admits a quadratic dual: the quadratic algebra generated by V* and with quadratic relations forming the orthogonal complement of S in V* ⊗ V*.
Examples
Tensor algebra, symmetric algebra and exterior algebra of a finite-dimensional vector space are graded quadratic (in fact, Koszul) algebras.
Universal enveloping algebra of a finite-dimensional Lie algebra is a filtered quadratic algebra.
References
Algebras |
https://en.wikipedia.org/wiki/Dutch%20East%20Indies%20national%20football%20team%20results | This page details the match results and statistics of the Dutch East Indies national football team.
Key
Key to matches
Att.=Match attendance
(H)=Home ground
(A)=Away ground
(N)=Neutral ground
Key to record by opponent
Pld=Games played
W=Games won
D=Games drawn
L=Games lost
GF=Goals for
GA=Goals against
Results
Dutch East Indies' score is shown first in each case.
Notes
Record by opponent
References
Indonesia national football team results |
https://en.wikipedia.org/wiki/South%20Korea%20national%20football%20team%20records%20and%20statistics | Records and statistics of the South Korea national football team are as follows.
Player records
Other records
Youngest player 17 years and 241 days, Kim Pan-keun, vs. Thailand, 1 November 1983
Youngest goalscorer 18 years and 87 days, Ko Jong-soo, vs. New Zealand, 25 January 1997
Oldest player 39 years and 274 days, Kim Yong-sik, vs. Hong Kong, 15 April 1950
Oldest goalscorer 39 years and 274 days, Kim Yong-sik, vs. Hong Kong, 15 April 1950
Longest career 19 years and 112 days, Lee Dong-gook, from 16 May 1998 to 5 September 2017
Most goals in a calendar year 16, Park Lee-chun (1972) and Hwang Sun-hong (1994)
Most consecutive matches scored in 6, Ha Seok-ju (1993)
Fastest goal from kick-off 16 seconds, Hwang Hee-chan, vs. Qatar, 17 November 2020
Most hat-tricks 3, Cha Bum-kun and Park Sung-hwa
Manager records
Team records
Biggest victory 16–0 vs. Nepal, 29 September 2003 (2004 AFC Asian Cup qualification)
Highest scoring draw 4–4 vs. Malaysia, 11 September 1976 (1976 Korea Cup)
Heaviest defeat 0–12 vs. Sweden, 5 August 1948 (1948 Summer Olympics)
Most consecutive victories 11, from 29 July 1975 (3–1 vs. Malaysia) to 21 December 1975 (3–1 vs. Burma)
Most consecutive matches without defeat 29, from 20 September 1986 (3–0 vs. India) to 26 June 1989 (0–0 vs. Czechoslovakia)
Head-to-head record
See also
South Korea national football team
South Korea national football team results
Korea Football Association
References
External links
Korea Football Association official website
South Korea national football team records and statistics
Football records and statistics in South Korea
Korea |
https://en.wikipedia.org/wiki/Andy%20Roddick%20career%20statistics | This is a list of the main career statistics of retired professional American tennis player, Andy Roddick. Throughout his career, Roddick won thirty-two ATP singles titles including one grand slam singles title and five ATP Masters 1000 singles titles. He was also the runner-up at the Wimbledon Championships in 2004, 2005 and 2009 and the US Open in 2006, losing on all four occasions to Roger Federer. Roddick was also a four-time semifinalist at the Australian Open and a three-time semifinalist at the year-ending ATP World Tour Finals. On November 3, 2003, Roddick became the World No. 1 for the first time in his career.
Career achievements
Roddick reached his first career Grand Slam singles quarterfinal at the 2001 US Open, where he lost to fourth-seeded Australian and eventual champion Lleyton Hewitt in a five-set thriller 7–6, 3–6, 4–6, 6–3, 4–6. A year later, he reached his first masters series singles final at the 2002 Rogers Cup, losing in straight sets to Argentine Guillermo Cañas. The following year, Roddick reached his first grand slam semifinal at the 2003 Australian Open, where he lost to thirty-first seed Rainer Schüttler in four sets, 5–7, 6–2, 3–6, 3–6. In August, Roddick won his first major singles title at the 2003 Rogers Cup, defeating David Nalbandian in the final in straight sets. Three weeks later, Roddick rallied from two sets and a match point down to defeat Nalbandian in five sets to reach his first Grand Slam singles final at the US Open, where he defeated fourth-seeded Spaniard Juan Carlos Ferrero in the final in straight sets, 6–3, 7–6, 6–3, to win his first and only Grand Slam singles title to date. Roddick's strong results throughout the year allowed him to qualify for the year-ending ATP World Tour Finals for the first time in his career. He advanced to the semifinals of the event after victories over Guillermo Coria and Carlos Moyá in the round robin stage but lost in straights to then World No. 2 Roger Federer in the semifinals. Despite the loss, Roddick finished the year as the World No. 1 for the first (and only) time in his career, becoming the first American player to finish a season as World No. 1 since Andre Agassi and the youngest American player to have held the top ranking since computer rankings began in 1973.
In July 2004, Roddick reached his first Wimbledon final but lost in four sets to then World No. 1, Roger Federer. He reached the final of the event again the following year but once again lost to Federer, this time in straight sets. The following year, Roddick reached his fourth grand slam singles final but once again lost to Federer, this time in the final of the US Open. In 2007, Roddick reached the semifinals of the year-ending ATP World Tour Finals for the third and final time in his career, losing in straight sets to Spaniard, David Ferrer.
Roddick enjoyed a resurgent year in 2009, during which he reached the semifinals of the Australian Open for the fourth and final time in his career and th |
https://en.wikipedia.org/wiki/South%20Korea%20national%20football%20team%20results | This article shows the match statistics of the South Korea national football team.
Results by year
1950s
1960s
1970s
1980s
1990s
2000s
2010s
2020s
Largest margins
Biggest victories
Heaviest defeats
See also
South Korea national football team
South Korea national football team records and statistics
External links
KFA Website : Official Fixtures/Results of National Team |
https://en.wikipedia.org/wiki/List%20of%20Ascomycota%20families%20incertae%20sedis | The following fungal families have not been taxonomically classified in any of the classes or orders accepted in the current classification of the Ascomycota with a high degree of probability (incertae sedis):
Alinaceae
Amorphothecaceae
Aphanopsidaceae
Aspidotheliaceae
Batistiaceae
Coniocybaceae
Diporothecaceae
Eoterfeziaceae
Epigloeaceae
Hispidicarpomycetaceae
Koralionastetaceae
Lautosporaceae
Mucomassariaceae
Phyllobatheliaceae
Pleurotremataceae
Pseudeurotiaceae
Saccardiaceae
Seuratiaceae
Strangosporaceae
Thelocarpaceae
Xanthopyreniaceae
References
Families |
https://en.wikipedia.org/wiki/2009%E2%80%9310%20Stevenage%20Borough%20F.C.%20season | The 2009–10 season was Stevenage Borough F.C.'s 16th season in the Conference Premier. This article shows statistics of the club's players in the season, and also lists all matches that the club played during the season. Their fifth-place finish and subsequent play-off semi-final defeat in the 2008–09 season meant it was their sixteenth successive season of playing in the Conference Premier. It also marked the second year in charge for manager Graham Westley during his second spell at the club; having previously managed the Hertfordshire club from 2003 to 2006.
The majority of the squad from the 2008–09 season were retained, with little transfer activity in comparison to previous seasons. Steve Morison, the club's top goalscorer for the past three seasons, moved to Millwall for a fee of £130,000, while both John Martin and Calum Willock were released in late May 2009. Midfielder Gary Mills was the last departure of the close season; rejecting a contract and instead opting to join fellow Conference Premier club Mansfield Town. Five players joined the club during the close season. Charlie Griffin was the first signing of the season, joining Stevenage from Salisbury City on a free transfer. Yemi Odubade, Chris Beardsley, and Joel Byrom signed for the club shortly after; the latter for a transfer fee of £15,000. The last signing of pre-season was Stacy Long; who joined the club on a free transfer from Ebbsfleet United. No players departed the club during the season, with Tim Sills the only addition, signing for an undisclosed fee from Torquay United in January 2010.
Stevenage started the season by recording just one win from their first five games of the season. Following a 2–1 defeat to Oxford United in August 2009, the team went on a 17-game unbeaten run from August to December 2009, moving the club into the top two in the league standings. A 4–1 victory against Cambridge United on New Year's Day meant that Stevenage were in first place for the first time in the season. Two defeats away from home within the space of a week in February meant that Oxford United had an eight–point lead going into March 2010. The team responded by winning eight games consecutively; including a 1–0 victory against Oxford United in late March, subsequently replacing Oxford at the top of the league table. Stevenage earned promotion to the Football League with two games to spare following a 2–0 victory against Kidderminster Harriers at Aggborough. The team won their last six games of the league season without conceding a goal, and recorded 42 points from a possible 45 from their last fifteen league fixtures. Stevenage finished the season having amassed a total of 99 points from 44 games, winning the league by 11 points.
The club also played in the final of the FA Trophy against Barrow – losing 2–1 after extra–time; it was Stevenage's third visit to Wembley Stadium in four seasons. Yemi Odubade finished as the club's top goalscorer for the season after scoring 16 goals, |
https://en.wikipedia.org/wiki/Fermat%20quintic%20threefold | In mathematics, a Fermat quintic threefold is a special quintic threefold, in other words a degree 5, dimension 3 hypersurface in 4-dimensional complex projective space, given by the equation
.
This threefold, so named after Pierre de Fermat, is a Calabi–Yau manifold.
The Hodge diamond of a non-singular quintic 3-fold is
Rational curves
conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. The Fermat quintic threefold is not generic in this sense, and showed that its lines are contained in 50 1-dimensional families of the form
for and . There are 375 lines in more than one family, of the form
for fifth roots of unity and .
References
3-folds
Complex manifolds |
https://en.wikipedia.org/wiki/Variational%20bicomplex | In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical field theory on fiber bundles (covariant classical field theory).
The variational bicomplex is a cochain complex of the differential graded algebra of exterior forms on jet manifolds of sections of a fiber bundle. Lagrangians and Euler–Lagrange operators on a fiber bundle are defined as elements of this bicomplex. Cohomology of the variational bicomplex leads to the global first variational formula and first Noether's theorem.
Extended to Lagrangian theory of even and odd fields on graded manifolds, the variational bicomplex provides strict mathematical formulation of classical field theory in a general case of reducible degenerate Lagrangians and the Lagrangian BRST theory.
See also
Calculus of variations
Lagrangian system
Jet bundle
References
Anderson, I., "Introduction to variational bicomplex", Contemp. Math. 132 (1992) 51.
Barnich, G., Brandt, F., Henneaux, M., "Local BRST cohomology", Phys. Rep. 338 (2000) 439.
Giachetta, G., Mangiarotti, L., Sardanashvily, G., Advanced Classical Field Theory, World Scientific, 2009, .
External links
Dragon, N., BRS symmetry and cohomology,
Sardanashvily, G., Graded infinite-order jet manifolds, Int. G. Geom. Methods Mod. Phys. 4 (2007) 1335;
Calculus of variations
Differential equations
Differential geometry |
https://en.wikipedia.org/wiki/Historical%20Statistics%20of%20the%20United%20States | Historical Statistics of the United States (HSUS) is a compendium of statistics about United States. Published by the United States Census Bureau until 1975, it is now published by Cambridge University Press.
The last free version, the Bicentennial Edition, appeared in two volumes in 1975 and is now available online.
The current commercial version deals with Population, Work and Welfare, Economic Structure and Performance, Economic Sectors and Governance & International Relations, respectively, in five volumes.
The fully searchable and downloadable electronic edition was developed by Data Software Research Company (DSRC) for Cambridge University Press.
References
External links
The last free edition, available from the US Census Bureau
Online Edition
United States Census Bureau
Cambridge University Press books |
https://en.wikipedia.org/wiki/Index%20of%20combinatorics%20articles |
A
Abstract simplicial complex
Addition chain
Scholz conjecture
Algebraic combinatorics
Alternating sign matrix
Almost disjoint sets
Antichain
Arrangement of hyperplanes
Assignment problem
Quadratic assignment problem
Audioactive decay
B
Barcode
Matrix code
QR Code
Universal Product Code
Bell polynomials
Bertrand's ballot theorem
Binary matrix
Binomial theorem
Block design
Balanced incomplete block design(BIBD)
Symmetric balanced incomplete block design (SBIBD)
Partially balanced incomplete block designs (PBIBDs)
Block walking
Boolean satisfiability problem
2-satisfiability
3-satisfiability
Bracelet (combinatorics)
Bruck–Chowla–Ryser theorem
C
Catalan number
Cellular automaton
Collatz conjecture
Combination
Combinatorial design
Combinatorial number system
Combinatorial optimization
Combinatorial search
Constraint satisfaction problem
Conway's Game of Life
Cycles and fixed points
Cyclic order
Cyclic permutation
Cyclotomic identity
D
Data integrity
Alternating bit protocol
Checksum
Cyclic redundancy check
Luhn formula
Error detection
Error-detecting code
Error-detecting system
Message digest
Redundancy check
Summation check
De Bruijn sequence
Deadlock
Delannoy number
Dining philosophers problem
Mutual exclusion
Rendezvous problem
Derangement
Dickson's lemma
Dinitz conjecture
Discrete optimization
Dobinski's formula
E
Eight queens puzzle
Entropy coding
Enumeration
Algebraic enumeration
Combinatorial enumeration
Burnside's lemma
Erdős–Ko–Rado theorem
Euler number
F
Faà di Bruno's formula
Factorial number system
Family of sets
Faulhaber's formula
Fifteen puzzle
Finite geometry
Finite intersection property
G
Game theory
Combinatorial game theory
Combinatorial game theory (history)
Combinatorial game theory (pedagogy)
Star (game theory)
Zero game, fuzzy game
Dots and Boxes
Impartial game
Digital sum
Nim
Nimber
Sprague–Grundy theorem
Partizan game
Solved board games
Col game
Sim (pencil game)
Sprouts (game)
Surreal numbers
Transposition table
Black Path Game
Sylver coinage
Generating function
Golomb coding
Golomb ruler
Graeco-Latin square
Gray code
H
Hadamard matrices
Complex Hadamard matrices
Butson-type Hadamard matrices
Generalized Hadamard matrices
Regular Hadamard matrices
Hall's marriage theorem
Perfect matching
Hamming distance
Hash function
Hash collision
Perfect hash function
Heilbronn triangle problem
Helly family
Hypergeometric function identities
Hypergeometric series
Hypergraph
I
Incidence structure
Induction puzzles
Integer partition
Ferrers graph
K
Kakeya needle problem
Kirkman's schoolgirl problem
Knapsack problem
Kruskal–Katona theorem
L
Lagrange inversion theorem
Lagrange reversion theorem
Lah number
Large number
Latin square
Levenshtein distance
Lexicographical order
Littlewood–Offord problem
Lubell–Yamamoto–Meshalkin inequality (known as the LYM inequality)
Lucas chain
M
MacMahon Mas |
https://en.wikipedia.org/wiki/Sexual%20violence%20in%20South%20Africa | The rate of sexual violence in South Africa is among the highest recorded in the world. Police statistics of reported rapes as a per capita figure has been dropping in recent years, although the reasons for the drop has not been analysed and it is not known how many rapes go unreported. More women are attacked than men, and children have also been targeted, partly owing to a myth that having sex with a virgin will cure a man of HIV/AIDS. Rape victims are at high risk of contracting HIV/AIDS owing to the high prevalence of the disease in South Africa. "Corrective rape" is also perpetrated against LGBT men and women.
The South African Government has established several measures, including legislation and initiatives to prevent and combat the problem. These include the establishment of the Sexual Offences and Community Affairs Unit (SOCA) in 1999, and a network of Thuthuzela Care Centres. These are sexual violence support centres which employ a transdisciplinary approach to dealing with the aftermath of an assault, and are considered by the UN as best practice model.
Sexual violence in South Africa has been widely reported in both local and international media.
Statistics
Official police statistics
South Africa's Police Service (SAPS) releases the country's crime statistics. The crime category "sexual offences" includes a wide range of sexual offences, including rape, sexual assault, incest, bestiality, flashing, and other crimes. SAPS releases statistics on reported rapes every quarter, as well as an annual report (financial year, April thru March each year). The figures in the following table include reported rapes only.
Prevalence
According to the report by the United Nations Office on Drugs and Crime for the period 1998–2000, South Africa was ranked first for rapes per capita. In 1998, one in three of the 4,000 women questioned in Johannesburg had been raped, according to Community Information, Empowerment and Transparency (CIET) Africa. While women's groups in South Africa estimate that a woman is raped every 26 seconds, the South African police estimates that a woman is raped every 36 seconds.
The comprehensive study "Rape in South Africa" in 2000 indicated that 2.1% of women aged 16 years or older across population groups reported that they had been sexually abused at least once between the beginning of 1993 and March 1998, results which seem to starkly conflict the MRC survey results. Similarly, The South African demographic and health survey of 1998 gave results of rape prevalence at 4.0% of all women aged between 15 and 49 years in the sampled households (a survey also performed by the Medical Research Council and Department of Health).
According to the World Population Review, South Africa had the highest rate of rape in the world at 132.4 incidents per 100,000 people in 2010, but had decreased to 72.1 in 2019–2020. It has not been ascertained whether this decline is due to actual reduction in offending, or fewer people reportin |
https://en.wikipedia.org/wiki/Chung%20Kai-lai | Kai Lai Chung (traditional Chinese: 鍾開萊; simplified Chinese: 钟开莱; September 19, 1917 – June 2, 2009) was a Chinese-American mathematician known for his significant contributions to modern probability theory.
Biography
Chung was a native of Hangzhou, the capital city of Zhejiang Province. Chung entered Tsinghua University in 1936, and initially studied physics at its Department of Physics. In 1940, Chung graduated from the Department of Mathematics of the National Southwestern Associated University, where he later worked as a teaching assistant. During this period, he first studied number theory with Lo-Keng Hua and then probability theory with Pao-Lu Hsu.
In 1944, Chung was chosen to be one of the recipients of the 6th Boxer Indemnity Scholarship Program for study in the United States. He arrived at Princeton University in December 1945 and obtained his PhD in 1947. Chung's dissertation was titled “On the maximum partial sum of sequences of independent random variables” and was under the supervision of John Wilder Tukey and Harald Cramér.
In 1950s, Chung taught at the University of Chicago, Columbia University, UC-Berkeley, Cornell University and Syracuse University. He then transferred to Stanford University in 1961, where he made fundamental contributions to the study of Brownian motion and laid the framework for the general mathematical theory of Markov chains. Chung would later be appointed Professor Emeritus of Mathematics of the Department of Mathematics at Stanford.
Chung was regarded as one of the leading probabilists after World War II. He was an Invited Speaker at the ICM in 1958 in Edinburgh and in 1970 in Nice. Some of his most influential contributions have been in the form of his expositions in his textbooks on elementary probability and Markov chains. In addition, Chung also explored other branches of mathematics, such as probabilistic potential theory and gauge theorems for the Schrödinger equation.
Chung's visit to China in 1979 (together with Joseph L. Doob and Jacques Neveu), and his subsequent visits, served as a point of renewed exchange between Chinese probabilists and their Western counterparts. He also served as an external examiner for several universities in the Asian region, including the National University of Singapore.
In 1981, Chung initiated, with Erhan Cinlar and Ronald Getoor, the "Seminars on Stochastic Processes", a popular annual national meeting covering Markov processes, Brownian motion and probability.
Chung also possessed a wide-ranging and intimate knowledge of literature and music, especially opera. He also had an interest in Italian culture and taught himself Italian after he retired. Chung spoke several languages and translated a probability book from Russian to English.
Chung died of natural causes on June 1, 2009, at the age of 91.
Publications
Elementary Probability Theory; by Kai Lai Chung & Farid Aitsahlia; Springer; .
A Course in Probability Theory; by Kai Lai Chung.
Markov Processes |
https://en.wikipedia.org/wiki/Koras%E2%80%93Russell%20cubic%20threefold | In algebraic geometry, the Koras–Russell cubic threefolds are smooth affine complex threefolds diffeomorphic to studied by . They have a hyperbolic action of a one-dimensional torus with a unique fixed point, such that the quotients of the threefold and the tangent space of the fixed point by this action are isomorphic. They were discovered in the process of proving the Linearization Conjecture in dimension 3. A linear action of on the affine space is one of the form , where and . The Linearization Conjecture in dimension says that every algebraic action of on the complex affine space is linear in some algebraic coordinates on . M. Koras and P. Russell made a key step towards the solution in dimension 3, providing a list of threefolds (now called Koras-Russell threefolds) and proving that the Linearization Conjecture for holds if all those threefolds are exotic affine 3-spaces, that is, none of them is isomorphic to . This was later shown by Kaliman and Makar-Limanov using the ML-invariant of an affine variety, which had been invented exactly for this purpose.
Earlier than the above referred paper, Russell noticed that the hypersurface has properties very similar to the affine 3-space like contractibility and was interested in distinguishing them as algebraic varieties. This now follows from the computation that and .
References
Sources
3-folds |
https://en.wikipedia.org/wiki/Quartic%20threefold | In algebraic geometry, a quartic threefold is a degree 4 hypersurface of dimension 3 in 4-dimensional projective space.
showed that all non-singular quartic threefolds are irrational, though some of them are unirational.
Examples
Burkhardt quartic
Igusa quartic
References
3-folds |
https://en.wikipedia.org/wiki/1985%E2%80%9386%20Libyan%20Premier%20League | Following are the statistics of the Libyan Premier League for the 1985–86 season which it was the 19th edition of the competition. The Libyan Premier League () is the highest division of Libyan football championship, organised by Libyan Football Federation. It was founded in 1963 and features mostly professional players.
Overview
16 teams were split into two groups, depending on their geographic location. Top two teams in each group advanced to the semifinals.
Teams
Group A (East)
Afriqi
Akhdar
Al-Ahly Benghazi
Hilal
Nasr
Tahaddy
Suqoor
Group B (West)
Al-Ahl Tripoli
Shabab al Arabi
Dhahra
Madina
Mahalla
Olomby
Wahda
Sweahly
Ittihad Tripoli
Playoff
Semifinal
Al-Ahly (Benghazi) 0-1 ; 0-0 Al-Ittihad (Tripoli)
Al-Ahl (Tripoli) 6-2 ; 2-3 Al-Nasr (Benghazi)
Final
Al-Ittihad (Tripoli) 2-1 Al-Ahly (Tripoli)
References
Libya - List of final tables (RSSSF)
Libyan Premier League seasons
1
Libya |
https://en.wikipedia.org/wiki/1987%E2%80%9388%20Libyan%20Premier%20League | Following are the statistics of the Libyan Premier League for the 1987–88 season which was the 21st edition of the competition.. The Libyan Premier League () is the highest division of Libyan football championship, organised by Libyan Football Federation. It was founded in 1963 and features mostly professional players.
Overview
It was contested by 18 teams, and Al-Ittihad (Tripoli) won the championship.
References
Libya - List of final tables (RSSSF)
Libyan Premier League seasons
1
Libya |
https://en.wikipedia.org/wiki/1988%E2%80%9389%20Libyan%20Premier%20League | Following are the statistics of the Libyan Premier League for the 1988–89 season which was the 22nd edition of the competition. The Libyan Premier League () is the highest division of Libyan football championship, organised by Libyan Football Federation. It was founded in 1963 and features mostly professional players.
Overview
Al-Ittihad (Tripoli) won the championship.
References
Libya - List of final tables (RSSSF)
Libyan Premier League seasons
1
Libya |
https://en.wikipedia.org/wiki/1989%E2%80%9390%20Libyan%20Premier%20League | Statistics of Libyan Premier League for the 1989–90 season which was the 23rd edition of the competition.
Overview
Al-Ittihad (Tripoli) won the championship.
References
Libya - List of final tables (RSSSF)
Libyan Premier League seasons
1
Libya |
https://en.wikipedia.org/wiki/1990%E2%80%9391%20Libyan%20Premier%20League | Statistics of Libyan Premier League for the 1990–91 season which was the 24th edition of the competition.
Overview
Al-Ittihad (Tripoli) won the championship.
References
Libya - List of final tables (RSSSF)
Libyan Premier League seasons
1
Libya |
https://en.wikipedia.org/wiki/1994%E2%80%9395%20Libyan%20Premier%20League | Statistics of Libyan Premier League for the 1994–95 season which was the 28th edition of the competition.
Overview
Al-Ahly (Tripoli) won the championship.
References
Libya - List of final tables (RSSSF)
Libyan Premier League seasons
1
Libya |
https://en.wikipedia.org/wiki/1995%E2%80%9396%20Libyan%20Premier%20League | Following are the statistics of the Libyan Premier League for the 1995–96 season which was the 29th edition of the competition. The Libyan Premier League () is the highest division of Libyan football championship, organised by Libyan Football Federation. It was founded in 1963 and features mostly professional players.
Overview
Al Shat Tripoli won the championship.
References
Libya - List of final tables (RSSSF)
Libyan Premier League seasons
1
Libya |
https://en.wikipedia.org/wiki/1997%E2%80%9398%20Libyan%20Premier%20League | Statistics of Libyan Premier League for the 1997–98 season which was the 31st edition of the competition.
Overview
It was contested by 16 teams, and Al Tahaddy Benghazi won the championship.
League standings
References
Libya - List of final tables (RSSSF)
Libyan Premier League seasons
1
Libya |
https://en.wikipedia.org/wiki/1998%E2%80%9399%20Libyan%20Premier%20League | Following are the statistics of the Libyan Premier League for the 1998–99 season which was the 32nd edition of the competition. The Libyan Premier League () is the highest division of Libyan football championship, organised by Libyan Football Federation. It was founded in 1963 and features mostly professional players.
Overview
It was contested by 16 teams, and Al Mahalah Tripoli won the championship.
Group stage
Group A
Group B
Playoff
Championship Group
Relegation Group
References
Libya - List of final tables (RSSSF)
Libyan Premier League seasons
1
Libya |
https://en.wikipedia.org/wiki/2000%20Libyan%20Premier%20League | Statistics of Libyan Premier League in season 2000 which was the 33rd edition of the competition.
Overview
It was contested by 15 teams, and Al-Ahly (Tripoli) won the championship.
Group stage
Group A
Group B
Final
Al-Ahly (Tripoli) 1-0 Al-Hilal (Benghazi)
References
Libya - List of final tables (RSSSF)
Libyan Premier League seasons
1
Libya
Libya |
https://en.wikipedia.org/wiki/2000%E2%80%9301%20Libyan%20Premier%20League | Following are the statistics of the Libyan Premier League for the 2000–01 season which was the 34th edition of the competition. The Libyan Premier League () is the highest division of Libyan football championship, organised by Libyan Football Federation. It was founded in 1963 and features mostly professional players.
Overview
It was contested by 14 teams, and Al Madina Tripoli won the championship.
Final
Al Madina Tripoli 1-1 Al Tahaddy Benghazi
Al Madina Tripoli won on PK
References
Libya - List of final tables (RSSSF)
Libyan Premier League seasons
1
Libyan Premier League |
https://en.wikipedia.org/wiki/Pregaussian%20class | In probability theory, a pregaussian class or pregaussian set of functions is a set of functions, square integrable with respect to some probability measure, such that there exists a certain Gaussian process, indexed by this set, satisfying the conditions below.
Definition
For a probability space (S, Σ, P), denote by a set of square integrable with respect to P functions , that is
Consider a set . There exists a Gaussian process , indexed by , with mean 0 and covariance
Such a process exists because the given covariance is positive definite. This covariance defines a semi-inner product as well as a pseudometric on given by
Definition A class is called pregaussian if for each the function on is bounded, -uniformly continuous, and prelinear.
Brownian bridge
The process is a generalization of the brownian bridge. Consider with P being the uniform measure. In this case, the process indexed by the indicator functions , for is in fact the standard brownian bridge B(x). This set of the indicator functions is pregaussian, moreover, it is the Donsker class.
References
Stochastic processes
Empirical process
Normal distribution |
https://en.wikipedia.org/wiki/Vinicius%20%28footballer%2C%20born%201989%29 | Vinicius Galvão Leal (born August 12, 1989, Brazil) is a Brazilian footballer currently under contract for Austrian side Union Sparkasse Pettenbach.
Club career
Debreceni VSC
Club Statistics
Club statistics
Updated to games played as of August 4, 2012.
References
External links
MLSZ
1989 births
Living people
Footballers from Porto Alegre
Brazilian men's footballers
Men's association football forwards
Debreceni VSC players
Nyíregyháza Spartacus FC players
Expatriate men's footballers in Hungary
Brazilian expatriate sportspeople in Hungary
Brazilian expatriate men's footballers |
https://en.wikipedia.org/wiki/Valentina%20Harizanov | Valentina Harizanov is a Serbian-American mathematician and professor of mathematics at The George Washington University. Her main research contributions are in computable structure theory (roughly at the intersection of computability theory and model theory), where she introduced the notion of degree spectra of relations on computable structures and obtained the first significant results concerning uncountable, countable, and finite Turing degree spectra. Her recent interests include algorithmic learning theory and spaces of orders on groups.
Education
She obtained her Bachelor of Science in mathematics in 1978 at the University of Belgrade and her Ph.D. in mathematics in 1987 at the University of Wisconsin–Madison under the direction of Terry Millar.
Career
At The George Washington University, Harizanov was an assistant professor of mathematics from 1987 to 1993, an associate professor of mathematics from 1994 to 2002, and a professor of mathematics from 2003 to the present.
She has held two visiting professor positions, one in 1994 at the University of Maryland, College Park and one in 2014 at the Kurt Gödel Research Center at the University of Vienna.
Harizanov has co-directed the Center for Quantum Computing, Information, Logic, and Topology at The George Washington University since 2011.
Research
In 2009, Harizanov received a grant from the National Science Foundation to research how algebraic, topological, and algorithmic properties of mathematical structures relate.
Awards and honors
Harizanov won the Oscar and Shoshana Trachtenberg Prize for Faculty Scholarship from The George Washington University (GWU) in 2016.
This award is presented each year to a tenured GWU faculty member to recognize outstanding research accomplishments. She was named MSRI Eisenbud Professor for Fall 2020.
Publications
Harizanov has over 40 publications in peer-reviewed journals, including
V.S. Harizanov, "Some effects of Ash-Nerode and other decidability conditions on degree spectra " Annals of Pure and Applied Logic 55 (1), pp. 51–65 (1991), cited 21 times according to Web of Science
In addition, she has published the following book-length survey paper and co-edited, co-authored book:
V.S. Harizanov, “Pure computable model theory,” in the volume: Handbook of Recursive Mathematics, vol. 1, Yu.L. Ershov, S.S. Goncharov, A. Nerode, and J.B. Remmel, editors (North-Holland, Amsterdam, 1998), pp. 3–114.
M. Friend, N.B. Goethe, and V.S. Harizanov, Induction, Algorithmic Learning Theory, and Philosophy, Series: Logic, Epistemology, and the Unity of Science, vol. 9, Springer, Dordrecht, 304 pp., 2007.
Degree spectra of relations are introduced and first studied in Harizanov's dissertation: Degree Spectrum of a Recursive Relation on a Recursive Structure(1987).
References
External links
Valentina Harizanov's home page
20th-century American mathematicians
21st-century American mathematicians
Serbian mathematicians
American women mathematicians
George |
https://en.wikipedia.org/wiki/Bahraini%20football%20club%20records%20and%20statistics | Among Bahraini football clubs the one that has won by far the greatest number of trophies is Al-Muharraq Sports Club, which has won both the Bahraini Premier League and the King's Cup on 30 or more occasions.
Successful teams
Football in Bahrain |
https://en.wikipedia.org/wiki/Naihua%20Duan | Naihua Duan (; born 31 October 1949) is a Taiwanese biostatistician specializing in mental health services and policy research at Columbia University. Duan is a professor of biostatistics (in psychiatry) with tenure in the Departments of Psychiatry and Biostatistics at Columbia University Medical Center, and a senior research scientist at NYSPI.
Duan received a B.S. in mathematics from National Taiwan University, an M.A. in mathematical statistics from Columbia University, and a Ph.D. in statistics from Stanford University.
He is an elected fellow of the American Statistical Association and the Institute of Mathematical Statistics. He has also served as a member of the editorial board for Statistica Sinica and Health Services & Outcomes Research Methodology, and associate editor for the Journal of the American Statistical Association. In addition, he served on a number of national and international panels, such as the Institute of Medicine's Committee on Organ Procurement and Transplantation Policy and Committee on Assessing the Medical Risks of Human Oocyte Donation for Stem Cell Research; the National Research Council’s Committee on Carbon Monoxide Episodes in Meteorological and Topological Problems Areas; and the National Institute of Mental Health’s Behavioral Sciences Workgroup.
References
External links
Duan, N. (no date) A Quest for Evidence Beyond Evidence-Based Medicine: Unleashing Clinical Experience through Evidence Farming. (PDF) Manuscript, University of California, Davis.
Department of Psychiatry, Columbia University College of Physicians and Surgeons
Taiwanese statisticians
Living people
Biostatisticians
Columbia Graduate School of Arts and Sciences alumni
National Taiwan University alumni
Columbia University faculty
Stanford University alumni
Fellows of the American Statistical Association
Fellows of the Institute of Mathematical Statistics
1949 births
Academic journal editors
20th-century Taiwanese mathematicians
21st-century Taiwanese mathematicians
Taiwanese expatriates in the United States |
https://en.wikipedia.org/wiki/Kirill%20Shestakov | Kirill Sergeyevich Shestakov (; born 19 June 1985) is a Russian former professional footballer.
His father Sergei Shestakov was also a professional footballer.
Career statistics
Club
References
External links
Profile at playerhistory.com
1985 births
Living people
Russian men's footballers
Russian expatriate men's footballers
Expatriate men's footballers in Kazakhstan
Russian expatriate sportspeople in Kazakhstan
Kazakhstan Premier League players
FC Sodovik Sterlitamak players
FC Kairat players
FC Kaisar players
FC Aktobe players
Men's association football midfielders
FC Nosta Novotroitsk players |
https://en.wikipedia.org/wiki/BMW%20N57 | The BMW N57 is a family of aluminium, turbocharged straight-6 common rail diesel engines. The engines utilize variable geometry turbochargers and Bosch piezo-electric injectors. The engine jointly replaced the M57 straight-6 and M67 V8 engines. In 2015 the N57 started to be replaced with the B57 engine, beginning with the G11 730d.
Summary
N57D30
N57D30Ox has 1800 bar fuel pressure, while N57D30Tx has 2000 bar fuel pressure.
N57D30Ox uses a single turbocharger, while N57D30Tx uses twin-turbochargers, and N57S uses three turbochargers of varying size.
Bore x stroke:
BMW released an M Performance Kit for N57D30O1 in some markets boosting power to and torque to . This kit included the larger intercooler of the N57D30T0
Applications:
N57D30U0
2010 - 2011 BMW 5 Series F10/F11 525d
2010 - 2013 BMW 3 Series E90/E91/E92/E93 325d
N57D30O0
2008 - 2013 BMW 3 Series E90/E91/E92/E93 330d/330xd
2010 - 2011 BMW 5 Series F10/F11 530d
2009 - BMW 5 Series GT F07 530d GT/530d xDrive GT
2008 - 2012 BMW 7 Series F01/F02 730d/730Ld
2010 - 2013 BMW X5 E70 xDrive30d
2010 - 2014 BMW X6 E71 xDrive30d
N57D30O1
2011 - 2016 BMW 5 Series F10/F11 530d
2011 - BMW X3 F25 xDrive30d
2011 - 2016 BMW X5 F15 xDrive35d
2012 - 2019 BMW 3 Series F30/F31 330d
2014 - BMW 4 Series F32/F33/F36 430d
2012 - 2015 BMW 7 Series F01/F02 730d/730Ld
N57D30T0
2010 - 2011 BMW 5 Series F10/F11 535d
2009 - 2017 BMW 5 Series GT F07 535d GT/535d xDrive GT
2009 - 2015 BMW 7 Series F01 740d/740d xDrive
2010 - 2013 BMW X5 E70 xDrive40d
2010 - 2014 BMW X6 E71 xDrive40d
N57D30T1
2011 - 2016 BMW 5 Series F10/F11 535d
2011 - 2018 BMW 6 Series F12/F13 640d
2011 - BMW X3 F25 xDrive35d
2013 - 2019 BMW 3 Series F30/F31 335d
2014 - BMW 4 Series F32/F33/F36 435d
2014 - BMW X4 F26 X4 xDrive35d
2014 - BMW X5 F15 X5 xDrive40d
2015 - BMW X6 F16 X6 xDrive40d
N57S (Tri-Turbo)
2012 - 2017 BMW 5 Series F10/F11 M550d xDrive
2012 - 2015 BMW 7 Series F01 750d xDrive
2012 - 2015 BMW 7 Series F01 750Ld xDrive
2012 - 2013 BMW X5 E70 M50d
2012 - 2014 BMW X6 M50d
2013 - 2018 BMW X5 F15 M50d
2015 - 2018 BMW X6 F16 M50d
Engine fires in police vehicles
In January 2022, BMW released a statement acknowledging the presence of a technical issue with the N57 engine which have contributed to instances of police vehicles in the United Kingdom catching fire, including one case in January 2020 which resulted in the death of a British police officer. This issue led to police forces across the United Kingdom withdrawing, retiring or limiting the speed of vehicles powered by the N57 engine, preventing their use in pursuits. In the press release, BMW stated “This issue is associated with the particular way in which the police operate these high-performance vehicles […] there is no need for action on civilian vehicles”.
It was reported that the issue was caused by high-speed driving after long periods of engine idling.
Safety concerns about this engine in 2016, and the 2022 inquest into the death of PC Nicholas Dumphreys on 26 Januar |
https://en.wikipedia.org/wiki/Crop%20reports | Crop reports are reports compiled by the National Agricultural Statistics Service (NASS) on various commodities that are released throughout the year. Information in the reports includes estimates on planted acreage, yield, and expected production, as well as comparison of production from previous years.
References
United States Department of Agriculture |
https://en.wikipedia.org/wiki/Lawson%20topology | In mathematics and theoretical computer science the Lawson topology, named after Jimmie D. Lawson, is a topology on partially ordered sets used in the study of domain theory. The lower topology on a poset P is generated by the subbasis consisting of all complements of principal filters on P. The Lawson topology on P is the smallest common refinement of the lower topology and the Scott topology on P.
Properties
If P is a complete upper semilattice, the Lawson topology on P is always a complete T1 topology.
See also
Formal ball
References
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott (2003), Continuous Lattices and Domains, Encyclopedia of Mathematics and its Applications, Cambridge University Press.
External links
"How Do Domains Model Topologies?," Paweł Waszkiewicz, Electronic Notes in Theoretical Computer Science 83 (2004)
Domain theory
General topology
Order theory |
https://en.wikipedia.org/wiki/Anatoli%20Tebloyev | Anatoli Grigoryevich Tebloyev (; born July 16, 1974) is a Russian retired professional footballer. His last club was Gabala.
Career statistics
Honours
Neftchi Baku
Azerbaijan Premier League champion: 2004–05
References
1974 births
Living people
Sportspeople from Arkhangelsk
Russian men's footballers
Russian Premier League players
Russian expatriate men's footballers
Expatriate men's footballers in Azerbaijan
FC Anzhi Makhachkala players
FC Spartak Vladikavkaz players
FC Volgar Astrakhan players
FC Oryol players
Gabala SC players
FC Irtysh Omsk players
Men's association football midfielders
Neftçi PFK players
FC Znamya Truda Orekhovo-Zuyevo players
FC Mashuk-KMV Pyatigorsk players |
https://en.wikipedia.org/wiki/Noether%20identities | In mathematics, Noether identities characterize the degeneracy of a Lagrangian system. Given a Lagrangian system and its Lagrangian L, Noether identities can be defined as a differential operator whose kernel contains a range of the Euler–Lagrange operator of L. Any Euler–Lagrange operator obeys Noether identities which therefore are separated into the trivial and non-trivial ones. A Lagrangian L is called degenerate if the Euler–Lagrange operator of L satisfies non-trivial Noether identities. In this case Euler–Lagrange equations are not independent.
Noether identities need not be independent, but satisfy first-stage Noether identities, which are subject to the second-stage Noether identities and so on. Higher-stage Noether identities also are separated into the trivial and non-trivial once. A degenerate Lagrangian is called reducible if there exist non-trivial higher-stage Noether identities. Yang–Mills gauge theory and gauge gravitation theory exemplify irreducible Lagrangian field theories.
Different variants of second Noether’s theorem state the one-to-one correspondence between the non-trivial reducible Noether identities and the non-trivial reducible gauge symmetries. Formulated in a very general setting, second Noether’s theorem associates to the Koszul–Tate complex of reducible Noether identities, parameterized by antifields, the BRST complex of reducible gauge symmetries parameterized by ghosts. This is the case of covariant classical field theory and Lagrangian BRST theory.
See also
Noether's second theorem
Emmy Noether
Lagrangian system
Variational bicomplex
Gauge symmetry (mathematics)
References
Gomis, J., Paris, J., Samuel, S., Antibracket, antifields and gauge theory quantization, Phys. Rep. 259 (1995) 1.
Fulp, R., Lada, T., Stasheff, J. Noether variational theorem II and the BV formalism,
Bashkirov, D., Giachetta, G., Mangiarotti, L., Sardanashvily, G., The KT-BRST complex of a degenerate Lagrangian system, Lett. Math. Phys. 83 (2008) 237; .
Sardanashvily, G., Noether theorems in a general setting, .
Calculus of variations
Differential equations
Theoretical physics
Mathematical identities |
https://en.wikipedia.org/wiki/Spirangle | In geometry, a spirangle is a spiral polygonal chain. Spirangles are similar to spirals in that they expand from a center point as they grow larger, but they are made out of straight line segments, instead of curves. Spirangle vectographs are used in vision therapy to promote stereopsis and help resolve problems with hand–eye coordination.
Two-dimensional spirangles
A two-dimensional spirangle is an open figure consisting of a line bent into angles similar to a corresponding polygon. The spirangle can start at a center point, or a distance from the center, and has some number of turns around the center point.
Three-dimensional spirangles
Three-dimensional spirangles have layers that slant upward, progressively gaining height from the previous segment. This is similar to staircases in large buildings that turn at the top of each flight. The segments also may progressively lose an amount of length and resemble a pyramid.
Uses
Ophthalmology —
Electronics — printed inductors
Architecture — ‘spiral’ staircases
Jewelry — earrings, pendants
Search algorithms — optimal scanning of a region of interest, for example a crime scene or a region of the celestial sphere
See also
Turtle graphics
References
Michael Scheiman & Bruce Wick (2013) Clinical Management of Binocular Vision, pp. 216, 256, 272, Wolters Kluwer, Fourth edition, .
Jaime Aquilera & Roc Berenquer (2007) Design and Test of Integrated Inductors for RF Applications, p. 24, Springer Science & Business Media .
External links
Colorado College field archeology site with a sample of ancient spirangle art
Ophthalmology
Spirals
Applied mathematics |
https://en.wikipedia.org/wiki/Wilkes%20University%20Election%20Statistics%20Project | The Wilkes University Election Statistics Project is a free online resource documenting Pennsylvania political election results dating back to 1796.
Currently, the database documents Pennsylvania's county-level vote totals for President, Governor, United States Senator, and Congressional elections back to 1796. The database also contains directories for members of the Pennsylvania Provincial Assembly and the Pennsylvania General Assembly, dating back to 1682.
According to the database's designer, Wilkes University Professor Harold E. Cox, "No other state has anything like it." The project's impetus began in 1996, when Cox inquired about 19th century election statistics, only to find that the data would cost $1,000.
The project has been cataloged by the Pennsylvania State University Libraries and the Van Pelt Library at the University of Pennsylvania. It has been cited as a source in academic books about the Supreme Court of the United States, Communist politicians in Pennsylvania, and a survey of state-level political parties.
See also
Elections in Pennsylvania
References
External links
Pennsylvania Election Statistics: 1682-2006 - The Wilkes University Election Statistics Project
Pennsylvania Election Statistics: 1682-2006 - Mirror site
Wilkes University
Politics of Pennsylvania
Online databases
Elections in the United States
Internet properties established in 1996
1996 establishments in Pennsylvania |
https://en.wikipedia.org/wiki/Smooth%20functor | In differential topology, a branch of mathematics, a smooth functor is a type of functor defined on finite-dimensional real vector spaces. Intuitively, a smooth functor is smooth in the sense that it sends smoothly parameterized families of vector spaces to smoothly parameterized families of vector spaces. Smooth functors may therefore be uniquely extended to functors defined on vector bundles.
Let Vect be the category of finite-dimensional real vector spaces whose morphisms consist of all linear mappings, and let F be a covariant functor that maps Vect to itself. For vector spaces T, U ∈ Vect, the functor F induces a mapping
where Hom is notation for Hom functor. If this map is smooth as a map of infinitely differentiable manifolds then F is said to be a smooth functor.
Common smooth functors include, for some vector space W:
F(W) = ⊗nW, the nth iterated tensor product;
F(W) = Λn(W), the nth exterior power; and
F(W) = Symn(W), the nth symmetric power.
Smooth functors are significant because any smooth functor can be applied fiberwise to a differentiable vector bundle on a manifold. Smoothness of the functor is the condition required to ensure that the patching data for the bundle are smooth as mappings of manifolds. For instance, because the nth exterior power of a vector space defines a smooth functor, the nth exterior power of a smooth vector bundle is also a smooth vector bundle.
Although there are established methods for proving smoothness of standard constructions on finite-dimensional vector bundles, smooth functors can be generalized to categories of topological vector spaces and vector bundles on infinite-dimensional Fréchet manifolds.
See also
Smooth infinitesimal analysis
Synthetic differential geometry
Notes
References
.
.
.
Functors |
https://en.wikipedia.org/wiki/Prices%20received%20index | The prices received index is an index that measures changes in the prices received for crops and livestock within the United States. The National Agricultural Statistics Service currently publishes the index on a 1990-92 = 100 base. A ratio of the prices received index to the prices paid index on the 1990-92 base that is greater than 100% indicates that farm commodity prices have increased at a faster rate than farm input prices. When the ratio is less than 100%, farm input prices are increasing a more rapid pace than farm commodity prices. The prices received index and the prices paid index are used to calculate the parity ratio.
See also
Prices paid index
References
Agricultural economics
Price indices |
https://en.wikipedia.org/wiki/MacRobert%20E%20function | In mathematics, the E-function was introduced by to extend the generalized hypergeometric series pFq(·) to the case p > q + 1. The underlying objective was to define a very general function that includes as particular cases the majority of the special functions known until then. However, this function had no great impact on the literature as it can always be expressed in terms of the Meijer G-function, while the opposite is not true, so that the G-function is of a still more general nature. It is defined as:
Definition
There are several ways to define the MacRobert E-function; the following definition is in terms of the generalized hypergeometric function:
when p ≤ q and x ≠ 0, or p = q + 1 and |x| > 1:
when p ≥ q + 2, or p = q + 1 and |x| < 1:
The asterisks here remind us to ignore the contribution with index j = h as follows: In the product this amounts to replacing Γ(0) with 1, and in the argument of the hypergeometric function this amounts to shortening the vector length from p to p − 1. Evidently, this definition covers all values of p and q.
Relationship with the Meijer G-function
The MacRobert E-function can always be expressed in terms of the Meijer G-function:
where the parameter values are unrestricted, i.e. this relation holds without exception.
References
(see § 5.2, "Definition of the E-Function", p. 203)
External links
Hypergeometric functions |
https://en.wikipedia.org/wiki/Indexed%20family | In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers, is a collection of real numbers, where a given function selects one real number for each integer (possibly the same) as indexing.
More formally, an indexed family is a mathematical function together with its domain and image (that is, indexed families and mathematical functions are technically identical, just point of views are different). Often the elements of the set are referred to as making up the family. In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set is called the index set of the family, and is the indexed set.
Sequences are one type of families indexed by natural numbers. In general, the index set is not restricted to be countable. For example, one could consider an uncountable family of subsets of the natural numbers indexed by the real numbers.
Formal definition
Let and be sets and a function such that
where is an element of and the image of under the function is denoted by . For example, is denoted by The symbol is used to indicate that is the element of indexed by The function thus establishes a family of elements in indexed by which is denoted by or simply if the index set is assumed to be known. Sometimes angle brackets or braces are used instead of parentheses, although the use of braces risks confusing indexed families with sets.
Functions and indexed families are formally equivalent, since any function with a domain induces a family and conversely. Being an element of a family is equivalent to being in the range of the corresponding function. In practice, however, a family is viewed as a collection, rather than a function.
Any set gives rise to a family where is indexed by itself (meaning that is the identity function). However, families differ from sets in that the same object can appear multiple times with different indices in a family, whereas a set is a collection of distinct objects. A family contains any element exactly once if and only if the corresponding function is injective.
An indexed family defines a set that is, the image of under Since the mapping is not required to be injective, there may exist with such that Thus, , where denotes the cardinality of the set For example, the sequence indexed by the natural numbers has image set In addition, the set does not carry information about any structures on Hence, by using a set instead of the family, some information might be lost. For example, an ordering on the index set of a family induces an ordering on the family, but no ordering on the corresponding image set.
Indexed subfamily
An indexed family is a subfamily of an indexed family if and only if is a subset of and holds for all
Examples
Indexed vectors
For example, consider the following |
https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20Poland | This page details football records in Poland.
Team records
Most top-division championships won
Overall
15, Legia Warsaw.
Consecutive
5, Górnik Zabrze (1962/63 season to 1967/68 season).
Most Polish Cups won
Overall
19, Legia Warsaw.
Consecutive
5, Górnik Zabrze (1967/68 season to 1971/72 season).
Most League Cups won
Overall
2, Dyskobolia Grodzisk Wielkopolski.
Consecutive
2, Dyskobolia Grodzisk Wielkopolski (2006/07 season to 2007/08 season).
Most SuperCups won
Overall
6, Lech Poznań.
Consecutive
2, Amica Wronki (1998 edition to 1999 edition)
Most successful clubs overall
See also
Football in Poland
Poland national football team
Polish Championship in Football
Polish Cup
Records
Poland |
https://en.wikipedia.org/wiki/Crime%20in%20New%20Zealand | Crime in New Zealand encompasses criminal law, crime statistics, the nature and characteristics of crime, sentencing, punishment, and public perceptions of crime. New Zealand criminal law has its origins in English criminal law, which was codified into statute by the New Zealand parliament in 1893. Although New Zealand remains a common law jurisdiction, all criminal offences and their penalties are codified in New Zealand statutes.
Criminal justice system
Criminal Law
Criminal law in New Zealand is based on English criminal law that the New Zealand parliament initially codified in statute in 1893. Although New Zealand remains a common law jurisdiction, all criminal offences and their penalties are codified in New Zealand statutes.
Most criminal offences that would result in imprisonment in New Zealand are set out in the Crimes Act 1961 and its amendments. Criminal offences related to specific situations also appear in other legislation, such as the Misuse of Drugs Act 1975 for drug offences and the Land Transport Act 1998 for traffic offences. Less serious breaches of the law are dealt with under legislation such as the Summary Offences Act 1981, where penalties are more often a fine or other community sanctions rather than imprisonment.
The age of criminal responsibility in New Zealand is 10 years; however, children aged 10 and 11 can only be convicted of murder and manslaughter, while children aged 12 and 13 can only be convicted of a crime with a maximum sentence of 14 years or more imprisonment.
Enforcement
The primary enforcement agency is the New Zealand Police, however more specialised crimes are enforced by other agencies such as the Serious Fraud Office, Ministry for Primary Industries, Immigration New Zealand and the New Zealand Customs Service among others. Local councils and other individuals appointed by the Police Commissioner also have the power to enforce laws and bylaws. The enforcement agency may charge an individual accused of breaking the law by filing a charging document with the registry of a district court.
Adult Diversion Scheme
First offenders charged with minor crimes and accepting full responsibility of their actions are considered for the New Zealand Police Adult Diversion Scheme. Given offenders agree to the conditions of diversion (which usually involves a written agreement tailored to change the offending behaviour), the offender may have the charge withdrawn.
Family Group Conferences
Family Group Conferences (FGC) are a type of statutory forum for youth offenders in which a child or young person, the victim of an alleged offence, family, whānau, hapū, iwi and supporters, and state and community representatives meet to decide how to best respond to the offending behaviour. FGCs may be invoked in a variety of scenarios including when the police has the intention to charge a child or a young person, when a child or young person is appearing before a court and does not deny the charge, following a prosecution |
https://en.wikipedia.org/wiki/Normal%20crossing%20singularity | In algebraic geometry a normal crossing singularity is a singularity similar to a union of coordinate hyperplanes. The term can be confusing because normal crossing singularities are not usually normal schemes (in the sense of the local rings being integrally closed).
Normal crossing divisors
In algebraic geometry, normal crossing divisors are a class of divisors which generalize the smooth divisors. Intuitively they cross only in a transversal way.
Let A be an algebraic variety, and a reduced Cartier divisor, with its irreducible components. Then Z is called a smooth normal crossing divisor if either
(i) A is a curve, or
(ii) all are smooth, and for each component , is a smooth normal crossing divisor.
Equivalently, one says that a reduced divisor has normal crossings if each point étale locally looks like the intersection of coordinate hyperplanes.
Normal crossing singularity
In algebraic geometry a normal crossings singularity is a point in an algebraic variety that is locally isomorphic to a normal crossings divisor.
Simple normal crossing singularity
In algebraic geometry a simple normal crossings singularity is a point in an algebraic variety, the latter having smooth irreducible components, that is locally isomorphic to a normal crossings divisor.
Examples
The normal crossing points in the algebraic variety called the Whitney umbrella are not simple normal crossings singularities.
The origin in the algebraic variety defined by is a simple normal crossings singularity. The variety itself, seen as a subvariety of the two-dimensional affine plane is an example of a normal crossings divisor.
Any variety which is the union of smooth varieties which all have smooth intersections is a variety with normal crossing singularities. For example, let be irreducible polynomials defining smooth hypersurfaces such that the ideal defines a smooth curve. Then is a surface with normal crossing singularities.
References
Robert Lazarsfeld, Positivity in algebraic geometry, Springer-Verlag, Berlin, 1994.
Algebraic geometry
Geometry of divisors |
https://en.wikipedia.org/wiki/Cheng%27s%20eigenvalue%20comparison%20theorem | In Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplace–Beltrami operator is small. This general characterization is not precise, in part because the notion of "size" of the domain must also account for its curvature. The theorem is due to by Shiu-Yuen Cheng. Using geodesic balls, it can be generalized to certain tubular domains .
Theorem
Let M be a Riemannian manifold with dimension n, and let BM(p, r) be a geodesic ball centered at p with radius r less than the injectivity radius of p ∈ M. For each real number k, let N(k) denote the simply connected space form of dimension n and constant sectional curvature k. Cheng's eigenvalue comparison theorem compares the first eigenvalue λ1(BM(p, r)) of the Dirichlet problem in BM(p, r) with the first eigenvalue in BN(k)(r) for suitable values of k. There are two parts to the theorem:
Suppose that KM, the sectional curvature of M, satisfies
Then
The second part is a comparison theorem for the Ricci curvature of M:
Suppose that the Ricci curvature of M satisfies, for every vector field X,
Then, with the same notation as above,
S.Y. Cheng used Barta's theorem to derive the eigenvalue comparison theorem. As a special case, if k = −1 and inj(p) = ∞, Cheng’s inequality becomes λ*(N) ≥ λ*(H n(−1)) which is McKean’s inequality.
See also
Comparison theorem
Eigenvalue comparison theorem
References
Citations
Bibliography
.
.
.
.
.
/
Theorems in Riemannian geometry
Chinese mathematical discoveries |
https://en.wikipedia.org/wiki/Charles%20B.%20Morrey%20Jr. | Charles Bradfield Morrey Jr. (July 23, 1907 – April 29, 1984) was an American mathematician who made fundamental contributions to the calculus of variations and the theory of partial differential equations.
Life
Charles Bradfield Morrey Jr. was born July 23, 1907, in Columbus, Ohio; his father was a professor of bacteriology at Ohio State University, and his mother was president of a school of music in Columbus, therefore it can be said that his one was a family of academicians. Perhaps from his mother's influence, he had a lifelong love for piano, even if mathematics was his main interest since his childhood. He was at first educated in the public schools of Columbus and, before going to the university, he spent a year at Staunton Military Academy in Staunton, Virginia.
In 1933, during his stay at the Department of Mathematics of the University of California, Berkeley as an instructor, he met Frances Eleonor Moss, who had just started studying for her M.A.: they married in 1937 and had three children. With summers off the family enjoyed traveling: they crossed the United States by car at least 20 times, visiting many natural wonders, and looked forward to the AMS meetings, held each year in August. They usually spent abroad their sabbatical leaves, and doing so they visited nearly every European country, witnessing many changes succeeding during the period from the 1950s to the 1980s.
Academic career
Morrey graduated from Ohio State University with a B.A. in 1927 and a M.A. in 1928, and then studied at Harvard University under the supervision of George Birkhoff, obtaining a Ph.D. in 1931 with a thesis entitled Invariant functions of Conservative Surface Transformations. After being awarded his Ph.D, he was a National Research Council Fellow at Princeton, at the Rice Institute and finally at the University of Chicago. He became a professor of mathematics at UC Berkeley in 1933, hired by Griffith Conrad Evans, and was a faculty member until his retirement in 1973. In Berkeley, he was early given several administrative duties, for example being the Chairman of the Department of Mathematics during the period 1949–1954, and being the Acting Chairman, the Vice Chairman and the Director of the Institute of Pure and Applied Mathematics at various times. During the years 1937–1938 and 1954–1955 he was a member of the Institute for Advanced Study: he was also Visiting Assistant Professor at Northwestern University, Visiting Professor at the University of Chicago and Miller Research Professor at Berkeley. During World War II he was employed as a mathematician at the U.S. Ballistic Research Laboratory in Maryland.
Honors
In 1962 he was elected member of the National Academy of Sciences: on May 12, 1965, he was elected fellow member of the American Academy of Arts and Sciences. From 1967 to 1968 he was president of the American Mathematical Society. On the fifth of June 1973 he was awarded the prestigious Berkeley Citation. refers also that other honor |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.