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https://en.wikipedia.org/wiki/Dependence%20relation
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In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.
Let be a set. A (binary) relation between an element of and a subset of is called a dependence relation, written , if it satisfies the following properties:
if , then ;
if , then there is a finite subset of , such that ;
if is a subset of such that implies , then implies ;
if but for some , then .
Given a dependence relation on , a subset of is said to be independent if for all If , then is said to span if for every is said to be a basis of if is independent and spans
Remark. If is a non-empty set with a dependence relation , then always has a basis with respect to Furthermore, any two bases of have the same cardinality.
Examples
Let be a vector space over a field The relation , defined by if is in the subspace spanned by , is a dependence relation. This is equivalent to the definition of linear dependence.
Let be a field extension of Define by if is algebraic over Then is a dependence relation. This is equivalent to the definition of algebraic dependence.
See also
matroid
Linear algebra
Binary relations
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https://en.wikipedia.org/wiki/Kolmogorov%27s%20inequality
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In probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound.
Statement of the inequality
Let X1, ..., Xn : Ω → R be independent random variables defined on a common probability space (Ω, F, Pr), with expected value E[Xk] = 0 and variance Var[Xk] < +∞ for k = 1, ..., n. Then, for each λ > 0,
where Sk = X1 + ... + Xk.
The convenience of this result is that we can bound the worst case deviation of a random walk at any point of time using its value at the end of time interval.
Proof
The following argument employs discrete martingales.
As argued in the discussion of Doob's martingale inequality, the sequence is a martingale.
Define as follows. Let , and
for all .
Then is also a martingale.
For any martingale with , we have that
Applying this result to the martingale , we have
where the first inequality follows by Chebyshev's inequality.
This inequality was generalized by Hájek and Rényi in 1955.
See also
Chebyshev's inequality
Etemadi's inequality
Landau–Kolmogorov inequality
Markov's inequality
Bernstein inequalities (probability theory)
References
(Theorem 22.4)
Stochastic processes
Probabilistic inequalities
Articles containing proofs
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https://en.wikipedia.org/wiki/Nilpotent%20cone
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In mathematics, the nilpotent cone of a finite-dimensional semisimple Lie algebra is the set of elements that act nilpotently in all representations of In other words,
The nilpotent cone is an irreducible subvariety of (considered as a vector space).
Example
The nilpotent cone of , the Lie algebra of 2×2 matrices with vanishing trace, is the variety of all 2×2 traceless matrices with rank less than or equal to
References
.
.
Lie algebras
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https://en.wikipedia.org/wiki/Verma%20module
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Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.
Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Specifically, although Verma modules themselves are infinite dimensional, quotients of them can be used to construct finite-dimensional representations with highest weight , where is dominant and integral. Their homomorphisms correspond to invariant differential operators over flag manifolds.
Informal construction
We can explain the idea of a Verma module as follows. Let be a semisimple Lie algebra (over , for simplicity). Let be a fixed Cartan subalgebra of and let be the associated root system. Let be a fixed set of positive roots. For each , choose a nonzero element for the corresponding root space and a nonzero element in the root space . We think of the 's as "raising operators" and the 's as "lowering operators."
Now let be an arbitrary linear functional, not necessarily dominant or integral. Our goal is to construct a representation of with highest weight that is generated by a single nonzero vector with weight . The Verma module is one particular such highest-weight module, one that is maximal in the sense that every other highest-weight module with highest weight is a quotient of the Verma module. It will turn out that Verma modules are always infinite dimensional; if is dominant integral, however, one can construct a finite-dimensional quotient module of the Verma module. Thus, Verma modules play an important role in the classification of finite-dimensional representations of . Specifically, they are an important tool in the hard part of the theorem of the highest weight, namely showing that every dominant integral element actually arises as the highest weight of a finite-dimensional irreducible representation of .
We now attempt to understand intuitively what the Verma module with highest weight should look like. Since is to be a highest weight vector with weight , we certainly want
and
.
Then should be spanned by elements obtained by lowering by the action of the 's:
.
We now impose only those relations among vectors of the above form required by the commutation relations among the 's. In particular, the Verma module is always infinite-dimensional. The weights of the Verma module with highest weight will consist of all elements that can be obtained from by subtracting integer combinations of positive roots. The figure shows the weights of a Verma module for .
A simple re-ordering argument shows that there is only one possible way the full Lie algebra can act on this space. Specifically, if is any element of , then by the easy part of the Poincaré–Birkhoff–Witt theorem, we can rewrite
as a linear combination of products of Lie algebra elements with the raising operators acting first, the elements of the Cartan subalgebra, and last the lowering operators . Applyi
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https://en.wikipedia.org/wiki/Lie%27s%20theorem
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In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if is a finite-dimensional representation of a solvable Lie algebra, then there's a flag of invariant subspaces of with , meaning that for each and i.
Put in another way, the theorem says there is a basis for V such that all linear transformations in are represented by upper triangular matrices. This is a generalization of the result of Frobenius that commuting matrices are simultaneously upper triangularizable, as commuting matrices generate an abelian Lie algebra, which is a fortiori solvable.
A consequence of Lie's theorem is that any finite dimensional solvable Lie algebra over a field of characteristic 0 has a nilpotent derived algebra (see #Consequences). Also, to each flag in a finite-dimensional vector space V, there correspond a Borel subalgebra (that consist of linear transformations stabilizing the flag); thus, the theorem says that is contained in some Borel subalgebra of .
Counter-example
For algebraically closed fields of characteristic p>0 Lie's theorem holds provided the dimension of the representation is less than p (see the proof below), but can fail for representations of dimension p. An example is given by the 3-dimensional nilpotent Lie algebra spanned by 1, x, and d/dx acting on the p-dimensional vector space k[x]/(xp), which has no eigenvectors. Taking the semidirect product of this 3-dimensional Lie algebra by the p-dimensional representation (considered as an abelian Lie algebra) gives a solvable Lie algebra whose derived algebra is not nilpotent.
Proof
The proof is by induction on the dimension of and consists of several steps. (Note: the structure of the proof is very similar to that for Engel's theorem.) The basic case is trivial and we assume the dimension of is positive. We also assume V is not zero. For simplicity, we write .
Step 1: Observe that the theorem is equivalent to the statement:<ref>{{harvnb|Serre|loc=Theorem 3{{}}}}</ref>
There exists a vector in V that is an eigenvector for each linear transformation in .
Indeed, the theorem says in particular that a nonzero vector spanning is a common eigenvector for all the linear transformations in . Conversely, if v is a common eigenvector, take to its span and then admits a common eigenvector in the quotient ; repeat the argument.
Step 2: Find an ideal of codimension one in .
Let be the derived algebra. Since is solvable and has positive dimension, and so the quotient is a nonzero abelian Lie algebra, which certainly contains an ideal of codimension one and by the ideal correspondence, it corresponds to an ideal of codimension one in .
Step 3: There exists some linear functional in such that
is nonzero. This follows from the inductive hypothesis (it is easy to check that the eigenvalues determine a linear functional).
Step 4: is a -invariant subspace. (Note this step proves a general fact an
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https://en.wikipedia.org/wiki/Przemys%C5%82aw%20Prusinkiewicz
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Przemysław (Przemek) Prusinkiewicz is a Polish computer scientist who advanced the idea that Fibonacci numbers in nature can be in part understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars. Prusinkiewicz's main work is on the modeling of plant growth through such grammars.
Early life and education
in 1978 Prusinkiewicz received his PhD from Warsaw University of Technology .
Career
As of 2008 he was a professor of Computer Science at the University of Calgary.
Awards
Prusinkiewicz received the 1997 SIGGRAPH Computer Graphics Achievement Award for his work.
Influences
In 2006, Michael Hensel examined the work of Prusinkiewicz and his collaborators - the Calgary team - in an article published in Architectural Design. Hensel argued that the Calgary team's computational plant models or "virtual plants" which culminated in software they developed capable of modeling various plant characteristics, could provide important lessons for architectural design. Architects would learn from "the self-organisation processes underlying the growth of living organisms" and the Calgary team's work uncovered some of that potential. Their computational models allowed for a "quantitative understanding of developmental mechanisms" and had the potential to "lead to a synthetic understanding of the interplay between various aspects of development."
Prusinkiewicz's work was informed by that of the Hungarian biologist Aristid Lindenmayer who developed the theory of L-systems in 1968. Lindenmayer used L-systems to describe the behaviour of plant cells and to model the growth processes, plant development and the branching architecture of plant development.
Publications
References
External links
Biography of Przemysław Prusinkiewicz from the University of Calgary
Laboratory website at the University of Calgary
Warsaw University of Technology alumni
Polish mathematicians
Living people
Computer graphics professionals
Computer graphics researchers
Fibonacci numbers
Polish computer scientists
Year of birth missing (living people)
Academic staff of the University of Calgary
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https://en.wikipedia.org/wiki/Distribution%20%28differential%20geometry%29
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In differential geometry, a discipline within mathematics, a distribution on a manifold is an assignment of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle .
Distributions satisfying a further integrability condition give rise to foliations, i.e. partitions of the manifold into smaller submanifolds. These notions have several applications in many fields of mathematics, e.g. integrable systems, Poisson geometry, non-commutative geometry, sub-Riemannian geometry, differential topology, etc.
Even though they share the same name, distributions presented in this article have nothing to do with distributions in the sense of analysis.
Definition
Let be a smooth manifold; a (smooth) distribution assigns to any point a vector subspace in a smooth way. More precisely, consists in a collection of vector subspaces with the following property. Around any there exist a neighbourhood and a collection of vector fields such that, for any point , span
The set of smooth vector fields is also called a local basis of . Note that the number may be different for different neighbourhoods. The notation is used to denote both the assignment and the subset .
Regular distributions
Given an integer , a smooth distribution on is called regular of rank if all the subspaces have the same dimension. Locally, this amounts to ask that every local basis is given by linearly independent vector fields.
More compactly, a regular distribution is a vector subbundle of rank (this is actually the most commonly used definition). A rank distribution is sometimes called an -plane distribution, and when , one talks about hyperplane distributions.
Special classes of distributions
Unless stated otherwise, by "distribution" we mean a smooth regular distribution (in the sense explained above).
Involutive distributions
Given a distribution , its sections consist of the vector fields which are tangent to , and they form a vector subspace of the space of all vector fields on . A distribution is called involutive if is also a Lie subalgebra: in other words, for any two vector fields , the Lie bracket belongs to .
Locally, this condition means that for every point there exists a local basis of the distribution in a neighbourhood of such that, for all , the Lie bracket is in the span of , i.e. is a linear combination of
Involutive distributions are a fundamental ingredient in the study of integrable systems. A related idea occurs in Hamiltonian mechanics: two functions and on a symplectic manifold are said to be in mutual involution if their Poisson bracket vanishes.
Integrable distributions and foliations
An integral manifold for a rank distribution is a submanifold of dimension such that for every . A distribution is called integrable if through any point there is an integral manifold. The base spaces of the bundle are thus disjoint, maximal, co
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https://en.wikipedia.org/wiki/Harnack%27s%20inequality
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In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by . Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions. , and generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equations. Such results can be used to show the interior regularity of weak solutions.
Perelman's solution of the Poincaré conjecture uses a version of the Harnack inequality, found by , for the Ricci flow.
The statement
Harnack's inequality applies to a non-negative function f defined on a closed ball in Rn with radius R and centre x0. It states that, if f is continuous on the closed ball and harmonic on its interior, then for every point x with |x − x0| = r < R,
In the plane R2 (n = 2) the inequality can be written:
For general domains in the inequality can be stated as follows: If is a bounded domain with , then there is a constant such that
for every twice differentiable, harmonic and nonnegative function . The constant is independent of ; it depends only on the domains and .
Proof of Harnack's inequality in a ball
By Poisson's formula
where ωn − 1 is the area of the unit sphere in Rn and r = |x − x0|.
Since
the kernel in the integrand satisfies
Harnack's inequality follows by substituting this inequality in the above integral and using the fact that the average of a harmonic function over a sphere equals its value at the center of the sphere:
Elliptic partial differential equations
For elliptic partial differential equations, Harnack's inequality states that the supremum of a positive solution in some connected open region is bounded by some constant times the infimum, possibly with an added term containing a functional norm of the data:
The constant depends on the ellipticity of the equation and the connected open region.
Parabolic partial differential equations
There is a version of Harnack's inequality for linear parabolic PDEs such as heat equation.
Let be a smooth (bounded) domain in and consider the linear elliptic operator
with smooth and bounded coefficients and a positive definite matrix . Suppose that is a solution of
in
such that
Let be compactly contained in and choose . Then there exists a constant C > 0 (depending only on K, , , and the coefficients of ) such that, for each ,
See also
Harnack's theorem
Harmonic function
References
Kassmann, Moritz (2007), "Harnack Inequalities: An Introduction" Boundary Value Problems 2007:081415, doi: 10.1155/2007/81415, MR 2291922
L. C. Evans (1998), Partial differential equations. American Mathematical Society, USA. For elliptic PDEs see Theorem 5, p. 334 and for parabolic PDEs see Theorem 10, p. 370.
Harmonic functions
Inequalities
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https://en.wikipedia.org/wiki/Equivariant%20cohomology
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In mathematics, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space with action of a topological group is defined as the ordinary cohomology ring with coefficient ring of the homotopy quotient :
If is the trivial group, this is the ordinary cohomology ring of , whereas if is contractible, it reduces to the cohomology ring of the classifying space (that is, the group cohomology of when G is finite.) If G acts freely on X, then the canonical map is a homotopy equivalence and so one gets:
Definitions
It is also possible to define the equivariant cohomology
of with coefficients in a
-module A; these are abelian groups.
This construction is the analogue of cohomology with local coefficients.
If X is a manifold, G a compact Lie group and is the field of real numbers or the field of complex numbers (the most typical situation), then the above cohomology may be computed using the so-called Cartan model (see equivariant differential forms.)
The construction should not be confused with other cohomology theories,
such as Bredon cohomology or the cohomology of invariant differential forms: if G is a compact Lie group, then, by the averaging argument, any form may be made invariant; thus, cohomology of invariant differential forms does not yield new information.
Koszul duality is known to hold between equivariant cohomology and ordinary cohomology.
Relation with groupoid cohomology
For a Lie groupoid equivariant cohomology of a smooth manifold is a special example of the groupoid cohomology of a Lie groupoid. This is because given a -space for a compact Lie group , there is an associated groupoidwhose equivariant cohomology groups can be computed using the Cartan complex which is the totalization of the de-Rham double complex of the groupoid. The terms in the Cartan complex arewhere is the symmetric algebra of the dual Lie algebra from the Lie group , and corresponds to the -invariant forms. This is a particularly useful tool for computing the cohomology of for a compact Lie group since this can be computed as the cohomology ofwhere the action is trivial on a point. Then,For example,since the -action on the dual Lie algebra is trivial.
Homotopy quotient
The homotopy quotient, also called homotopy orbit space or Borel construction, is a “homotopically correct” version of the orbit space (the quotient of by its -action) in which is first replaced by a larger but homotopy equivalent space so that the action is guaranteed to be free.
To this end, construct the universal bundle EG → BG for G and recall that EG admits a free G-action. Then the product EG × X —which is homotopy equivalent to X since EG is contractible—admits a “diagonal” G-action defined by (e,x).g = (eg,g−1x): mor
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https://en.wikipedia.org/wiki/Mary%20Ellen%20Rudin
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Mary Ellen Rudin (December 7, 1924 – March 18, 2013) was an American mathematician known for her work in set-theoretic topology. In 2013, Elsevier established the Mary Ellen Rudin Young Researcher Award, which is awarded annually to a young researcher, mainly in fields adjacent to general topology.
Early life and education
Mary Ellen (Estill) Rudin was born in Hillsboro, Texas to Joe Jefferson Estill and Irene (Shook) Estill. Her mother Irene was an English teacher before marriage, and her father Joe was a civil engineer. The family moved with her father's work, but spent a great deal of Mary Ellen's childhood around Leakey, Texas. She had one sibling, a younger brother. Both of Rudin's maternal grandmothers had attended Mary Sharp College near their hometown of Winchester, Tennessee. Rudin remarks on this legacy and how much her family valued education in an interview.
She attended the University of Texas, completing her B.A. in 1944 after just three years before moving into the graduate program in mathematics under Robert Lee Moore. Her graduate thesis presented a counterexample to one of "Moore's axioms". She completed her Ph.D. in 1949.
During her time as an undergraduate, she was a member of the Phi Mu Women's Fraternity, and was elected to the Phi Beta Kappa society.
In 1953, she married mathematician Walter Rudin, whom she met while teaching at Duke University. They had four children.
Career
At the beginning of her career, Rudin taught at Duke University and the University of Rochester. She took a position as lecturer at the University of Wisconsin in 1959, and was appointed Professor of Mathematics in 1971. After her retirement in 1991, she continued to serve as a Professor Emerita. She was the first Grace Chisholm Young Professor of Mathematics and also held the Hilidale Professorship,.
She was an Invited Speaker of the International Congress of Mathematicians in 1974 in Vancouver. She served as vice-president of the American Mathematical Society, 1980–1981. In 1984 she was selected to be a Noether Lecturer. She was an honorary member of the Hungarian Academy of Sciences (1995). In 2012 she became a fellow of the American Mathematical Society.
Rudin is best known in topology for her constructions of counterexamples to well-known conjectures. In 1958, she found an unshellable triangulation of the tetrahedron. Most famously, Rudin was the first to construct a Dowker space, which she did in 1971, thus disproving a conjecture of Clifford Hugh Dowker that had stood, and helped drive topological research, for more than twenty years. Her example fueled the search for "small" ZFC Dowker spaces. She also proved the first Morita conjecture and a restricted version of the second. Her last major result was a proof of Nikiel's conjecture. Early proofs that every metric space is paracompact were somewhat involved, but Rudin provided an elementary one.
"Reading the articles of Mary Ellen Rudin, studying them until there is no myster
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https://en.wikipedia.org/wiki/Hartogs%20number
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In mathematics, specifically in axiomatic set theory, a Hartogs number is an ordinal number associated with a set. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X can be well-ordered then the cardinal number of α is a minimal cardinal greater than that of X. If X cannot be well-ordered then there cannot be an injection from X to α. However, the cardinal number of α is still a minimal cardinal not less than or equal to the cardinality of X. (If we restrict to cardinal numbers of well-orderable sets then that of α is the smallest that is not not less than or equal to that of X.) The map taking X to α is sometimes called Hartogs's function. This mapping is used to construct the aleph numbers, which are all the cardinal numbers of infinite well-orderable sets.
The existence of the Hartogs number was proved by Friedrich Hartogs in 1915, using Zermelo–Fraenkel set theory alone (that is, without using the axiom of choice).
Hartogs's theorem
Hartogs's theorem states that for any set X, there exists an ordinal α such that ; that is, such that there is no injection from α to X. As ordinals are well-ordered, this immediately implies the existence of a Hartogs number for any set X. Furthermore, the proof is constructive and yields the Hartogs number of X.
Proof
See .
Let be the class of all ordinal numbers β for which an injective function exists from β into X.
First, we verify that α is a set.
X × X is a set, as can be seen in Axiom of power set.
The power set of X × X is a set, by the axiom of power set.
The class W of all reflexive well-orderings of subsets of X is a definable subclass of the preceding set, so it is a set by the axiom schema of separation.
The class of all order types of well-orderings in W is a set by the axiom schema of replacement, as
(Domain(w), w) (β, ≤)
can be described by a simple formula.
But this last set is exactly α. Now, because a transitive set of ordinals is again an ordinal, α is an ordinal. Furthermore, there is no injection from α into X, because if there were, then we would get the contradiction that α ∈ α. And finally, α is the least such ordinal with no injection into X. This is true because, since α is an ordinal, for any β < α, β ∈ α so there is an injection from β into X.
Historic remark
In 1915, Hartogs could use neither von Neumann-ordinals nor the replacement axiom, and so his result is one of Zermelo set theory and looks rather different from the modern exposition above. Instead, he considered the set of isomorphism classes of well-ordered subsets of X and the relation in which the class of A precedes that of B if A is isomorphic with a proper initial segment of B. Hartogs showed this to be a well-ordering greater than any well-ordered subset of X. (This must have been historically the first genuine construction of an uncountable well-ordering.) However, the main purpose of his contribution was to show th
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https://en.wikipedia.org/wiki/Friedrich%20Hartogs
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Friedrich Moritz "Fritz" Hartogs (20 May 1874 – 18 August 1943) was a German-Jewish mathematician, known for his work on set theory and foundational results on several complex variables.
Life
Hartogs was the son of the merchant Gustav Hartogs and his wife Elise Feist and grew up in Frankfurt am Main.
He studied at the Königliche Technische Hochschule Hannover, at the Technische Hochschule Charlottenburg, at the University of Berlin, and at the Ludwig Maximilian University of Munich, graduating with a doctorate in 1903 (supervised by Alfred Pringsheim). He did his Habilitation in 1905 and was Privatdozent and Professor in Munich (from 1910 to 1927 extraordinary professor and since 1927 ordinary professor).
As a Jew, he suffered greatly under the Nazi regime: he was fired in 1935, was mistreated and briefly interned in KZ Dachau in 1938, and eventually committed suicide in 1943.
Work
Hartogs main work was in several complex variables where he is known for
Hartogs's theorem, Hartogs's lemma (also known as Hartogs's principle or Hartogs's extension theorem) and the concepts of holomorphic hull and domain of holomorphy.
In set theory, he contributed to the theory of wellorders and proved what is also known as Hartogs's theorem: for every set x there is a wellordered set that cannot be injectively embedded in x.
The smallest such set is known as the Hartogs number or Hartogs Aleph of x.
References
.
. Available at the DigiZeitschriften.
. Available at the DigiZeitschriften.
External links
Biography (in German)
1874 births
1943 suicides
1943 deaths
19th-century German mathematicians
20th-century German mathematicians
Scientists from Brussels
Complex analysts
Mathematical analysts
German Jews who died in the Holocaust
Suicides by Jews during the Holocaust
Academic staff of the Ludwig Maximilian University of Munich
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https://en.wikipedia.org/wiki/N-skeleton
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In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of ) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the .
These subspaces increase with . The is a discrete space, and the a topological graph. The skeletons of a space are used in obstruction theory, to construct spectral sequences by means of filtrations, and generally to make inductive arguments. They are particularly important when has infinite dimension, in the sense that the do not become constant as
In geometry
In geometry, a of P (functionally represented as skelk(P)) consists of all elements of dimension up to k.
For example:
skel0(cube) = 8 vertices
skel1(cube) = 8 vertices, 12 edges
skel2(cube) = 8 vertices, 12 edges, 6 square faces
For simplicial sets
The above definition of the skeleton of a simplicial complex is a particular case of the notion of skeleton of a simplicial set. Briefly speaking, a simplicial set can be described by a collection of sets , together with face and degeneracy maps between them satisfying a number of equations. The idea of the n-skeleton is to first discard the sets with and then to complete the collection of the with to the "smallest possible" simplicial set so that the resulting simplicial set contains no non-degenerate simplices in degrees .
More precisely, the restriction functor
has a left adjoint, denoted . (The notations are comparable with the one of image functors for sheaves.) The n-skeleton of some simplicial set is defined as
Coskeleton
Moreover, has a right adjoint . The n-coskeleton is defined as
For example, the 0-skeleton of K is the constant simplicial set defined by . The 0-coskeleton is given by the Cech nerve
(The boundary and degeneracy morphisms are given by various projections and diagonal embeddings, respectively.)
The above constructions work for more general categories (instead of sets) as well, provided that the category has fiber products. The coskeleton is needed to define the concept of hypercovering in homotopical algebra and algebraic geometry.
References
External links
Algebraic topology
General topology
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https://en.wikipedia.org/wiki/Charles%20%C3%89tienne%20Louis%20Camus
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Charles Étienne Louis Camus (25 August 1699 – 2 February 1768), was a French mathematician and mechanician who was born at Crécy-en-Brie, near Meaux.
He studied mathematics, civil and military architecture, and astronomy after leaving Collège de Navarre in Paris. In 1730 he was appointed professor of architecture and, in 1733, associate of the Académie des Sciences. He also became a professor of geometry, secretary to the Academy of Architecture and fellow of the Royal Society of London. In 1727 he presented a memoir to the academy on masting ships, in consequence of which he was named the same year joint mechanician to that body. In 1736 he accompanied Pierre Louis Maupertuis and Alexis Clairaut in the expedition to Lapland for the measurement of a degree of meridian arc. He was the author of a Cours de mathématiques (Paris, 1766), and a number of essays on mathematical and mechanical subjects.
In 1760 he became perpetual secretary of the academy of architecture. He was also employed in a variety of public works, and in 1765 was chosen a fellow of the Royal Society of London. He died in 1768.
Works
Traité des forces mouvantes ("Treatise of moving forces"); 1722.
Opérations faites pour mesurer le degré de méridienne entre Paris et Amiens; 1757.
Cours de mathématique ("Course of mathematics"); 3 parts, 1749–52.
Part 1: Élémens d'arithmétique (1749).
Part 2: Élémens de géométrie, théorique et pratique (1750).
Part 3: Élémens de méchanique statique (1751–52).
External links
References
1699 births
1768 deaths
People from Seine-et-Marne
18th-century French mathematicians
Fellows of the Royal Society
Members of the French Academy of Sciences
Members of the Académie royale d'architecture
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https://en.wikipedia.org/wiki/Normalization%20%28statistics%29
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In statistics and applications of statistics, normalization can have a range of meanings. In the simplest cases, normalization of ratings means adjusting values measured on different scales to a notionally common scale, often prior to averaging. In more complicated cases, normalization may refer to more sophisticated adjustments where the intention is to bring the entire probability distributions of adjusted values into alignment. In the case of normalization of scores in educational assessment, there may be an intention to align distributions to a normal distribution. A different approach to normalization of probability distributions is quantile normalization, where the quantiles of the different measures are brought into alignment.
In another usage in statistics, normalization refers to the creation of shifted and scaled versions of statistics, where the intention is that these normalized values allow the comparison of corresponding normalized values for different datasets in a way that eliminates the effects of certain gross influences, as in an anomaly time series. Some types of normalization involve only a rescaling, to arrive at values relative to some size variable. In terms of levels of measurement, such ratios only make sense for ratio measurements (where ratios of measurements are meaningful), not interval measurements (where only distances are meaningful, but not ratios).
In theoretical statistics, parametric normalization can often lead to pivotal quantities – functions whose sampling distribution does not depend on the parameters – and to ancillary statistics – pivotal quantities that can be computed from observations, without knowing parameters.
Examples
There are different types of normalizations in statistics – nondimensional ratios of errors, residuals, means and standard deviations, which are hence scale invariant – some of which may be summarized as follows. Note that in terms of levels of measurement, these ratios only make sense for ratio measurements (where ratios of measurements are meaningful), not interval measurements (where only distances are meaningful, but not ratios). See also :Category:Statistical ratios.
Note that some other ratios, such as the variance-to-mean ratio , are also done for normalization, but are not nondimensional: the units do not cancel, and thus the ratio has units, and is not scale-invariant.
Other types
Other non-dimensional normalizations that can be used with no assumptions on the distribution include:
Assignment of percentiles. This is common on standardized tests. See also quantile normalization.
Normalization by adding and/or multiplying by constants so values fall between 0 and 1. This is used for probability density functions, with applications in fields such as physical chemistry in assigning probabilities to .
See also
Normal score
Ratio distribution
Standard score
Feature scaling
References
Statistical ratios
Statistical data transformation
Equivalence (mathematics)
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https://en.wikipedia.org/wiki/Hartogs%27s%20theorem%20on%20separate%20holomorphicity
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In mathematics, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if is a function which is analytic in each variable zi, 1 ≤ i ≤ n, while the other variables are held constant, then F is a continuous function.
A corollary is that the function F is then in fact an analytic function in the n-variable sense (i.e. that locally it has a Taylor expansion). Therefore, 'separate analyticity' and 'analyticity' are coincident notions, in the theory of several complex variables.
Starting with the extra hypothesis that the function is continuous (or bounded), the theorem is much easier to prove and in this form is known as Osgood's lemma.
There is no analogue of this theorem for real variables. If we assume that a function
is differentiable (or even analytic) in each variable separately, it is not true that will necessarily be continuous. A counterexample in two dimensions is given by
If in addition we define , this function has well-defined partial derivatives in and at the origin, but it is not continuous at origin. (Indeed, the limits along the lines and are not equal, so there is no way to extend the definition of to include the origin and have the function be continuous there.)
References
Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
External links
Several complex variables
Theorems in complex analysis
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https://en.wikipedia.org/wiki/Minkowski%E2%80%93Hlawka%20theorem
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In mathematics, the Minkowski–Hlawka theorem is a result on the lattice packing of hyperspheres in dimension n > 1. It states that there is a lattice in Euclidean space of dimension n, such that the corresponding best packing of hyperspheres with centres at the lattice points has density Δ satisfying
with ζ the Riemann zeta function. Here as n → ∞, ζ(n) → 1. The proof of this theorem is indirect and does not give an explicit example, however, and there is still no known simple and explicit way to construct lattices with packing densities exceeding this bound for arbitrary n. In principle one can find explicit examples: for example, even just picking a few "random" lattices will work with high probability. The problem is that testing these lattices to see if they are solutions requires finding their shortest vectors, and the number of cases to check grows very fast with the dimension, so this could take a very long time.
This result was stated without proof by and proved by . The result is related to a linear lower bound for the Hermite constant.
Siegel's theorem
proved the following generalization of the Minkowski–Hlawka theorem. If S is a bounded set in Rn with Jordan volume vol(S) then the average number of nonzero lattice vectors in S is vol(S)/D, where the average is taken over all lattices with a fundamental domain of volume D, and similarly the average number of primitive lattice vectors in S is vol(S)/Dζ(n).
The Minkowski–Hlawka theorem follows easily from this, using the fact that if S is a star-shaped centrally symmetric body (such as a ball) containing less than 2 primitive lattice vectors then it contains no nonzero lattice vectors.
See also
Kepler conjecture
References
Geometry of numbers
Theorems in geometry
Hermann Minkowski
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https://en.wikipedia.org/wiki/Clifford%27s%20theorem%20on%20special%20divisors
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In mathematics, Clifford's theorem on special divisors is a result of on algebraic curves, showing the constraints on special linear systems on a curve C.
Statement
A divisor on a Riemann surface C is a formal sum of points P on C with integer coefficients. One considers a divisor as a set of constraints on meromorphic functions in the function field of C, defining as the vector space of functions having poles only at points of D with positive coefficient, at most as bad as the coefficient indicates, and having zeros at points of D with negative coefficient, with at least that multiplicity. The dimension of is finite, and denoted . The linear system of divisors attached to D is the corresponding projective space of dimension .
The other significant invariant of D is its degree d, which is the sum of all its coefficients.
A divisor is called special if ℓ(K − D) > 0, where K is the canonical divisor.
Clifford's theorem states that for an effective special divisor D, one has:
,
and that equality holds only if D is zero or a canonical divisor, or if C is a hyperelliptic curve and D linearly equivalent to an integral multiple of a hyperelliptic divisor.
The Clifford index of C is then defined as the minimum of taken over all special divisors (except canonical and trivial), and Clifford's theorem states this is non-negative. It can be shown that the Clifford index for a generic curve of genus g is equal to the floor function
The Clifford index measures how far the curve is from being hyperelliptic. It may be thought of as a refinement of the gonality: in many cases the Clifford index is equal to the gonality minus 2.
Green's conjecture
A conjecture of Mark Green states that the Clifford index for a curve over the complex numbers that is not hyperelliptic should be determined by the extent to which C as canonical curve has linear syzygies. In detail, one defines the invariant a(C) in terms of the minimal free resolution of the homogeneous coordinate ring of C in its canonical embedding, as the largest index i for which the graded Betti number βi, i + 2 is zero. Green and Robert Lazarsfeld showed that a(C) + 1 is a lower bound for the Clifford index, and Green's conjecture states that equality always holds. There are numerous partial results.
Claire Voisin was awarded the Ruth Lyttle Satter Prize in Mathematics for her solution of the generic case of Green's conjecture in two papers. The case of Green's conjecture for generic curves had attracted a huge amount of effort by algebraic geometers over twenty years before finally being laid to rest by Voisin. The conjecture for arbitrary curves remains open.
Notes
References
External links
Algebraic curves
Theorems in algebraic geometry
Unsolved problems in geometry
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https://en.wikipedia.org/wiki/Finite%20topology
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Finite topology is a mathematical concept which has several different meanings.
Finite topological space
A finite topological space is a topological space, the underlying set of which is finite.
In endomorphism rings and modules
If A and B are abelian groups then the finite topology on the group of homomorphisms Hom(A, B) can be defined using the following base of open neighbourhoods of zero.
This concept finds applications especially in the study of endomorphism rings where we have A = B.
Similarly, if R is a ring and M is a right R-module, then the finite topology on is defined using the following system of neighborhoods of zero:
In vector spaces
In a vector space , the finite open sets are defined as those sets whose intersections with all finite-dimensional subspaces are open. The finite topology on is defined by these open sets and is sometimes denoted .
When V has uncountable dimension, this topology is not locally convex nor does it make V as topological vector space, but when V has countable dimension it coincides with both the finest vector space topology on V and the finest locally convex topology on V.
In manifolds
A manifold M is sometimes said to have finite topology, or finite topological type, if it is homeomorphic to a compact Riemann surface from which a finite number of points have been removed.
Notes
References
General topology
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https://en.wikipedia.org/wiki/Metaplectic%20group
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In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles.
The metaplectic group has a particularly significant infinite-dimensional linear representation, the Weil representation. It was used by André Weil to give a representation-theoretic interpretation of theta functions, and is important in the theory of modular forms of half-integral weight and the theta correspondence.
Definition
The fundamental group of the symplectic Lie group Sp2n(R) is infinite cyclic, so it has a unique connected double cover, which is denoted Mp2n(R) and called the metaplectic group.
The metaplectic group Mp2(R) is not a matrix group: it has no faithful finite-dimensional representations. Therefore, the question of its explicit realization is nontrivial. It has faithful irreducible infinite-dimensional representations, such as the Weil representation described below.
It can be proved that if F is any local field other than C, then the symplectic group Sp2n(F) admits a unique perfect central extension with the kernel Z/2Z, the cyclic group of order 2, which is called the metaplectic group over F.
It serves as an algebraic replacement of the topological notion of a 2-fold cover used when . The approach through the notion of central extension is useful even in the case of real metaplectic group, because it allows a description of the group operation via a certain cocycle.
Explicit construction for n = 1
In the case , the symplectic group coincides with the special linear group SL2(R). This group biholomorphically acts on the complex upper half-plane by fractional-linear transformations,
where
is a real 2-by-2 matrix with the unit determinant and z is in the upper half-plane, and this action can be used to explicitly construct the metaplectic cover of SL2(R).
The elements of the metaplectic group Mp2(R) are the pairs (g, ε), where and ε is a holomorphic function on the upper half-plane such that . The multiplication law is defined by:
where
That this product is well-defined follows from the cocycle relation . The map
is a surjection from Mp2(R) to SL2(R) which does not admit a continuous section. Hence, we have constructed a non-trivial 2-fold cover of the latter group.
Construction of the Weil representation
We first give a rather abstract reason why the Weil representation exists. The Heisenberg group has an irreducible unitary representation on a Hilbert space , that is,
with the center acting as a given nonzero constant. The Stone–von Neumann theorem states that this representation is essentially unique: if is another such representation, there exists an automorphism
such that .
and the conjugating automorphism is projectively unique, i.e., up to a multiplicative modulus 1 constant. So any automorphism of the Heisenberg group, inducing the i
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https://en.wikipedia.org/wiki/Bred%20vector
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In applied mathematics, bred vectors are perturbations related to Lyapunov vectors, that capture fast-growing dynamical instabilities of the solution of a numerical model. They are used, for example, as initial perturbations for ensemble forecasting in numerical weather prediction. They were introduced by Zoltan Toth and Eugenia Kalnay.
Method
Bred vectors are created by adding initially random perturbations to a nonlinear model. The control (unperturbed) and the perturbed models are integrated in time, and periodically the control solution is subtracted from the perturbed solution. This difference is the bred vector. The vector is scaled to be the same size as the initial perturbation and is then added back to the control to create the new perturbed initial condition. After a short transient period, this "breeding" process creates bred vectors dominated by the naturally fastest-growing instabilities of the evolving control solution.
References
Functional analysis
Mathematical physics
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https://en.wikipedia.org/wiki/Plimpton%20322
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Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University. This tablet, believed to have been written about 1800 BC, has a table of four columns and 15 rows of numbers in the cuneiform script of the period.
This table lists two of the three numbers in what are now called Pythagorean triples, i.e., integers , , and satisfying . From a modern perspective, a method for constructing such triples is a significant early achievement, known long before the Greek and Indian mathematicians discovered solutions to this problem. At the same time, one should recall the tablet's author was a scribe, rather than a professional mathematician; it has been suggested that one of his goals may have been to produce examples for school problems.
There has been significant scholarly debate on the nature and purpose of the tablet. For readable popular treatments of this tablet see recipient of the Lester R. Ford Award for expository excellence in mathematics or, more briefly, . is a more detailed and technical discussion of the interpretation of the tablet's numbers, with an extensive bibliography.
Provenance and dating
Plimpton 322 is partly broken, approximately 13 cm wide, 9 cm tall, and 2 cm thick. New York publisher George Arthur Plimpton purchased the tablet from an archaeological dealer, Edgar J. Banks, in about 1922, and bequeathed it with the rest of his collection to Columbia University in the mid-1930s. According to Banks, the tablet came from Senkereh, a site in southern Iraq corresponding to the ancient city of Larsa.
The tablet is believed to have been written about 1800 BC, using the middle chronology, based in part on the style of handwriting used for its cuneiform script: writes that this handwriting "is typical of documents from southern Iraq of 4000–3500 years ago." More specifically, based on formatting similarities with other tablets from Larsa that have explicit dates written on them, Plimpton 322 might well be from the period 1822–1784 BC. Robson points out that Plimpton 322 was written in the same format as other administrative, rather than mathematical, documents of the period.
Content
The main content of Plimpton 322 is a table of numbers, with four columns and fifteen rows, in Babylonian sexagesimal notation. The fourth column is just a row number, in order from 1 to 15. The second and third columns are completely visible in the surviving tablet. However, the edge of the first column has been broken off, and there are two consistent extrapolations for what the missing digits could be; these interpretations differ only in whether or not each number starts with an additional digit equal to 1.
With the differing extrapolations shown in parentheses, damaged portions of the first and fourth columns whose content is surmised shown in italics, and six presumed errors shown in boldface along with the generally proposed c
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https://en.wikipedia.org/wiki/251%20%28number%29
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251 (two hundred [and] fifty-one) is the natural number between 250 and 252. It is also a prime number.
In mathematics
251 is:
a Sophie Germain prime.
the sum of three consecutive primes (79 + 83 + 89) and seven consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47).
a Chen prime.
an Eisenstein prime with no imaginary part.
a de Polignac number, meaning that it is odd and cannot be formed by adding a power of two to a prime number.
the smallest number that can be formed in more than one way by summing three positive cubes:
Every 5 × 5 matrix has exactly 251 square submatrices.
References
Integers
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https://en.wikipedia.org/wiki/Roger%20Heath-Brown
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David Rodney "Roger" Heath-Brown (born 12 October 1952) is a British mathematician working in the field of analytic number theory.
Education
He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervisor was Alan Baker.
Career and research
In 1979 he moved to the University of Oxford, where from 1999 he held a professorship in pure mathematics. He retired in 2016.
Heath-Brown is known for many striking results. He proved that there are infinitely many prime numbers of the form x3 + 2y3.
In collaboration with S. J. Patterson in 1978 he proved the Kummer conjecture on cubic Gauss sums in its equidistribution form.
He has applied Burgess's method on character sums to the ranks of elliptic curves in families.
He proved that every non-singular cubic form over the rational numbers in at least ten variables represents 0.
Heath-Brown also showed that Linnik's constant is less than or equal to 5.5. More recently, Heath-Brown is known for his pioneering work on the so-called determinant method. Using this method he was able to prove a conjecture of Serre in the four variable case in 2002. This particular conjecture of Serre was later dubbed the "dimension growth conjecture" and this was almost completely solved by various works of Browning, Heath-Brown, and Salberger by 2009.
Awards and honours
The London Mathematical Society has awarded Heath-Brown the Junior Berwick Prize (1981), the Senior Berwick Prize (1996), and the Pólya Prize (2009). He was made a Fellow of the Royal Society in 1993, and a corresponding member of the Göttingen Academy of Sciences in 1999.
He was an invited speaker at International Congress of Mathematicians in 1983 in Warsaw and in 2010 in Hyderabad on the topic of "Number Theory."
In 2012 he became a fellow of the American Mathematical Society. In 2022 the Royal Society awarded him the Sylvester Medal "for his many important contributions to the study of prime numbers and solutions to equations in integers".
Other
In September 2007, he co-authored (along with Joseph H. Silverman) the preface to the Oxford University Sixth Edition of An Introduction to the Theory of Numbers by G.H. Hardy and E.M. Wright.
References
Alumni of Trinity College, Cambridge
20th-century British mathematicians
21st-century British mathematicians
Fellows of Magdalen College, Oxford
Fellows of Worcester College, Oxford
Fellows of the Royal Society
Fellows of the American Mathematical Society
Living people
Number theorists
1952 births
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https://en.wikipedia.org/wiki/Hugh%20Lowell%20Montgomery
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Hugh Lowell Montgomery (born August 26, 1944) is an American mathematician, working in the fields of analytic number theory and mathematical analysis. As a Marshall scholar, Montgomery earned his Ph.D. from the University of Cambridge. For many years, Montgomery has been teaching at the University of Michigan.
He is best known for Montgomery's pair correlation conjecture, his development of the large sieve methods and for co-authoring (with Ivan M. Niven and Herbert Zuckerman) one of the standard introductory number theory texts, An Introduction to the Theory of Numbers, now in its fifth edition ().
In 1974 Montgomery was an invited speaker of the International Congress of Mathematicians (ICM) in Vancouver. In 2012 he became a fellow of the American Mathematical Society.
Bibliography
Davenport, Harold. Multiplicative number theory. Third edition. Revised and with a preface by Hugh L. Montgomery. Graduate Texts in Mathematics, 74. Springer-Verlag, New York, 2000. xiv+177 pp. .
Levinson, Norman; Montgomery, Hugh L. "Zeros of the derivatives of the Riemann zeta function". Acta Mathematica 133 (1974), 49–65.
Montgomery, Hugh L. Topics in multiplicative number theory. Lecture Notes in Mathematics, Vol. 227. Springer-Verlag, Berlin-New York, 1971. ix+178 pp.
Montgomery, Hugh L. Ten lectures on the interface between analytic number theory and harmonic analysis. CBMS Regional Conference Series in Mathematics, 84. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. xiv+220 pp. .
Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134.
Montgomery, Hugh L., and Vaughan, Robert C. Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics, 97. Cambridge University Press, Cambridge, 2006. xviii+552 pp. ; 0-521-84903-9.
Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. An introduction to the theory of numbers. Fifth edition. John Wiley & Sons, Inc., New York, 1991. xiv+529 pp.
References
External links
Official Page
An Introduction to the Theory of Numbers, Fifth Edition page
Number theorists
Alumni of the University of Cambridge
20th-century American mathematicians
21st-century American mathematicians
University of Michigan faculty
American Rhodes Scholars
1944 births
Living people
Fellows of the American Mathematical Society
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https://en.wikipedia.org/wiki/Weyl%27s%20theorem
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In mathematics, Weyl's theorem or Weyl's lemma might refer to one of a number of results of Hermann Weyl. These include
the Peter–Weyl theorem
Weyl's theorem on complete reducibility, results originally derived from the unitarian trick on representation theory of semisimple groups and semisimple Lie algebras
Weyl's theorem on eigenvalues
Weyl's criterion for equidistribution (Weyl's criterion)
Weyl's lemma on the hypoellipticity of the Laplace equation
results estimating Weyl sums in the theory of exponential sums
Weyl's inequality
Weyl's criterion for a number to be in the essential spectrum of an operator
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https://en.wikipedia.org/wiki/Matrix%20population%20models
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Matrix population models are a specific type of population model that uses matrix algebra. Population models are used in population ecology to model the dynamics of wildlife or human populations. Matrix algebra, in turn, is simply a form of algebraic shorthand for summarizing a larger number of often repetitious and tedious algebraic computations.
All populations can be modeled
where:
Nt+1 = abundance at time t+1
Nt = abundance at time t
B = number of births within the population between Nt and Nt+1
D = number of deaths within the population between Nt and Nt+1
I = number of individuals immigrating into the population between Nt and Nt+1
E = number of individuals emigrating from the population between Nt and Nt+1
This equation is called a BIDE model (Birth, Immigration, Death, Emigration model).
Although BIDE models are conceptually simple, reliable estimates of the 5 variables contained therein (N, B, D, I and E) are often difficult to obtain. Usually a researcher attempts to estimate current abundance, Nt, often using some form of mark and recapture technique. Estimates of B might be obtained via a ratio of immatures to adults soon after the breeding season, Ri. Number of deaths can be obtained by estimating annual survival probability, usually via mark and recapture methods, then multiplying present abundance and survival rate. Often, immigration and emigration are ignored because they are so difficult to estimate.
For added simplicity it may help to think of time t as the end of the breeding season in year t and to imagine that one is studying a species that has only one discrete breeding season per year.
The BIDE model can then be expressed as:
where:
Nt,a = number of adult females at time t
Nt,i = number of immature females at time t
Sa = annual survival of adult females from time t to time t+1
Si = annual survival of immature females from time t to time t+1
Ri = ratio of surviving young females at the end of the breeding season per breeding female
In matrix notation this model can be expressed as:
Suppose that you are studying a species with a maximum lifespan of 4 years. The following is an age-based Leslie matrix for this species. Each row in the first and third matrices corresponds to animals within a given age range (0–1 years, 1–2 years and 2–3 years). In a Leslie matrix the top row of the middle matrix consists of age-specific fertilities: F1, F2 and F3. Note, that F1 = Si×Ri in the matrix above. Since this species does not live to be 4 years old the matrix does not contain an S3 term.
These models can give rise to interesting cyclical or seemingly chaotic patterns in abundance over time when fertility rates are high.
The terms Fi and Si can be constants or they can be functions of environment, such as habitat or population size. Randomness can also be incorporated into the environmental component.
See also
Population dynamics of fisheries
References
Caswell, H. 2001. Matrix population models: Construction
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https://en.wikipedia.org/wiki/Square-free%20element
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In mathematics, a square-free element is an element r of a unique factorization domain R that is not divisible by a non-trivial square. This means that every s such that is a unit of R.
Alternate characterizations
Square-free elements may be also characterized using their prime decomposition. The unique factorization property means that a non-zero non-unit r can be represented as a product of prime elements
Then r is square-free if and only if the primes pi are pairwise non-associated (i.e. that it doesn't have two of the same prime as factors, which would make it divisible by a square number).
Examples
Common examples of square-free elements include square-free integers and square-free polynomials.
See also
Prime number
References
David Darling (2004) The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes John Wiley & Sons
Baker, R. C. "The square-free divisor problem." The Quarterly Journal of Mathematics 45.3 (1994): 269-277.
Ring theory
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https://en.wikipedia.org/wiki/Signed%20measure
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In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values, i.e., to acquire sign.
Definition
There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. Signed measures are usually only allowed to take finite real values, while some textbooks allow them to take infinite values. To avoid confusion, this article will call these two cases "finite signed measures" and "extended signed measures".
Given a measurable space (that is, a set with a σ-algebra on it), an extended signed measure is a set function
such that and is σ-additive – that is, it satisfies the equality
for any sequence of disjoint sets in
The series on the right must converge absolutely when the value of the left-hand side is finite. One consequence is that an extended signed measure can take or as a value, but not both. The expression is undefined and must be avoided.
A finite signed measure (a.k.a. real measure) is defined in the same way, except that it is only allowed to take real values. That is, it cannot take or
Finite signed measures form a real vector space, while extended signed measures do not because they are not closed under addition. On the other hand, measures are extended signed measures, but are not in general finite signed measures.
Examples
Consider a non-negative measure on the space (X, Σ) and a measurable function f: X → R such that
Then, a finite signed measure is given by
for all A in Σ.
This signed measure takes only finite values. To allow it to take +∞ as a value, one needs to replace the assumption about f being absolutely integrable with the more relaxed condition
where f−(x) = max(−f(x), 0) is the negative part of f.
Properties
What follows are two results which will imply that an extended signed measure is the difference of two non-negative measures, and a finite signed measure is the difference of two finite non-negative measures.
The Hahn decomposition theorem states that given a signed measure μ, there exist two measurable sets P and N such that:
P∪N = X and P∩N = ∅;
μ(E) ≥ 0 for each E in Σ such that E ⊆ P — in other words, P is a positive set;
μ(E) ≤ 0 for each E in Σ such that E ⊆ N — that is, N is a negative set.
Moreover, this decomposition is unique up to adding to/subtracting μ-null sets from P and N.
Consider then two non-negative measures μ+ and μ− defined by
and
for all measurable sets E, that is, E in Σ.
One can check that both μ+ and μ− are non-negative measures, with one taking only finite values, and are called the positive part and negative part of μ, respectively. One has that μ = μ+ − μ−. The measure |μ| = μ+ + μ− is called the variation of μ, and its maximum possible value, ||μ|| = |μ|(X), is called the total variation of μ.
This consequence of the Hahn decomposition theorem is called the Jordan decomposition. The measures μ+, μ− and |μ| are inde
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https://en.wikipedia.org/wiki/Hahn%20decomposition%20theorem
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In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space and any signed measure defined on the -algebra , there exist two -measurable sets, and , of such that:
and .
For every such that , one has , i.e., is a positive set for .
For every such that , one has , i.e., is a negative set for .
Moreover, this decomposition is essentially unique, meaning that for any other pair of -measurable subsets of fulfilling the three conditions above, the symmetric differences and are -null sets in the strong sense that every -measurable subset of them has zero measure. The pair is then called a Hahn decomposition of the signed measure .
Jordan measure decomposition
A consequence of the Hahn decomposition theorem is the , which states that every signed measure defined on has a unique decomposition into a difference of two positive measures, and , at least one of which is finite, such that for every -measurable subset and for every -measurable subset , for any Hahn decomposition of . We call and the positive and negative part of , respectively. The pair is called a Jordan decomposition (or sometimes Hahn–Jordan decomposition) of . The two measures can be defined as
for every and any Hahn decomposition of .
Note that the Jordan decomposition is unique, while the Hahn decomposition is only essentially unique.
The Jordan decomposition has the following corollary: Given a Jordan decomposition of a finite signed measure , one has
for any in . Furthermore, if for a pair of finite non-negative measures on , then
The last expression means that the Jordan decomposition is the minimal decomposition of into a difference of non-negative measures. This is the minimality property of the Jordan decomposition.
Proof of the Jordan decomposition: For an elementary proof of the existence, uniqueness, and minimality of the Jordan measure decomposition see Fischer (2012).
Proof of the Hahn decomposition theorem
Preparation: Assume that does not take the value (otherwise decompose according to ). As mentioned above, a negative set is a set such that for every -measurable subset .
Claim: Suppose that satisfies . Then there is a negative set such that .
Proof of the claim: Define . Inductively assume for that has been constructed. Let
denote the supremum of over all the -measurable subsets of . This supremum might a priori be infinite. As the empty set is a possible candidate for in the definition of , and as , we have . By the definition of , there then exists a -measurable subset satisfying
Set to finish the induction step. Finally, define
As the sets are disjoint subsets of , it follows from the sigma additivity of the signed measure that
This shows that . Assume were not a negative set. This means that there would exist a -measurable subset that satisfies . Then for every , so the series on the right would have to diverge to , implyi
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https://en.wikipedia.org/wiki/Character%20sum
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In mathematics, a character sum is a sum of values of a Dirichlet character χ modulo N, taken over a given range of values of n. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an upper bound for the least quadratic non-residue modulo N. Character sums are often closely linked to exponential sums by the Gauss sums (this is like a finite Mellin transform).
Assume χ is a non-principal Dirichlet character to the modulus N.
Sums over ranges
The sum taken over all residue classes mod N is then zero. This means that the cases of interest will be sums over relatively short ranges, of length R < N say,
A fundamental improvement on the trivial estimate is the Pólya–Vinogradov inequality, established independently by George Pólya and I. M. Vinogradov in 1918, stating in big O notation that
Assuming the generalized Riemann hypothesis, Hugh Montgomery and R. C. Vaughan have shown that there is the further improvement
Summing polynomials
Another significant type of character sum is that formed by
for some function F, generally a polynomial. A classical result is the case of a quadratic, for example,
and χ a Legendre symbol. Here the sum can be evaluated (as −1), a result that is connected to the local zeta-function of a conic section.
More generally, such sums for the Jacobi symbol relate to local zeta-functions of elliptic curves and hyperelliptic curves; this means that by means of André Weil's results, for N = p a prime number, there are non-trivial bounds
The constant implicit in the notation is linear in the genus of the curve in question, and so (Legendre symbol or hyperelliptic case) can be taken as the degree of F. (More general results, for other values of N, can be obtained starting from there.)
Weil's results also led to the Burgess bound, applying to give non-trivial results beyond Pólya–Vinogradov, for R a power of N greater than 1/4.
Assume the modulus N is a prime.
for any integer r ≥ 3.
Notes
References
Further reading
External links
PlanetMath article on the Pólya–Vinogradov inequality
Analytic number theory
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https://en.wikipedia.org/wiki/Kodaira%20embedding%20theorem
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In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomials.
Kunihiko Kodaira's result is that for a compact Kähler manifold M, with a Hodge metric, meaning that the cohomology class in degree 2 defined by the Kähler form ω is an integral cohomology class, there is a complex-analytic embedding of M into complex projective space of some high enough dimension N.
The fact that M embeds as an algebraic variety follows from its compactness by Chow's theorem.
A Kähler manifold with a Hodge metric is occasionally called a Hodge manifold (named after W. V. D. Hodge), so Kodaira's results states that Hodge manifolds are projective.
The converse that projective manifolds are Hodge manifolds is more elementary and was already known.
Kodaira also proved (Kodaira 1963), by recourse to the classification of compact complex surfaces, that every compact Kähler surface is a deformation of a projective Kähler surface. This was later simplified by Buchdahl to remove reliance on the classification (Buchdahl 2008).
Kodaira embedding theorem
Let X be a compact Kähler manifold, and L a holomorphic line bundle on X. Then L is a positive line bundle if and only if there is a holomorphic embedding of X into some projective space such that for some m > 0.
See also
Fujita conjecture
Hodge structure
Moishezon manifold
References
A proof of the embedding theorem without the vanishing theorem (due to Simon Donaldson) appears in the lecture notes here.
Theorems in complex geometry
Theorems in algebraic geometry
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https://en.wikipedia.org/wiki/Product%20measure
|
In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces, except that there can be many natural choices for the product measure.
Let and be two measurable spaces, that is, and are sigma algebras on and respectively, and let and be measures on these spaces. Denote by the sigma algebra on the Cartesian product generated by subsets of the form , where and This sigma algebra is called the tensor-product σ-algebra on the product space.
A product measure
(also denoted by by many authors)
is defined to be a measure on the measurable space satisfying the property
for all
.
(In multiplying measures, some of which are infinite, we define the product to be zero if any factor is zero.)
In fact, when the spaces are -finite, the product measure is uniquely defined, and for every measurable set E,
where and , which are both measurable sets.
The existence of this measure is guaranteed by the Hahn–Kolmogorov theorem. The uniqueness of product measure is guaranteed only in the case that both and are σ-finite.
The Borel measures on the Euclidean space Rn can be obtained as the product of n copies of Borel measures on the real line R.
Even if the two factors of the product space are complete measure spaces, the product space may not be. Consequently, the completion procedure is needed to extend the Borel measure into the Lebesgue measure, or to extend the product of two Lebesgue measures to give the Lebesgue measure on the product space.
The opposite construction to the formation of the product of two measures is disintegration, which in some sense "splits" a given measure into a family of measures that can be integrated to give the original measure.
Examples
Given two measure spaces, there is always a unique maximal product measure μmax on their product, with the property that if μmax(A) is finite for some measurable set A, then μmax(A) = μ(A) for any product measure μ. In particular its value on any measurable set is at least that of any other product measure. This is the measure produced by the Carathéodory extension theorem.
Sometimes there is also a unique minimal product measure μmin, given by μmin(S) = supA⊂S, μmax(A) finite μmax(A), where A and S are assumed to be measurable.
Here is an example where a product has more than one product measure. Take the product X×Y, where X is the unit interval with Lebesgue measure, and Y is the unit interval with counting measure and all sets measurable. Then for the minimal product measure the measure of a set is the sum of the measures of its horizontal sections, while for the maximal product measure a set has measure infinity unless it is contained in the union of a countable number of sets of the form A×B, where either A has Lebesgue measure 0 or B is a single point. (In
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https://en.wikipedia.org/wiki/Kodaira%20vanishing%20theorem
|
In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are automatically zero. The implications for the group with index q = 0 is usually that its dimension — the number of independent global sections — coincides with a holomorphic Euler characteristic that can be computed using the Hirzebruch–Riemann–Roch theorem.
The complex analytic case
The statement of Kunihiko Kodaira's result is that if M is a compact Kähler manifold of complex dimension n, L any holomorphic line bundle on M that is positive, and KM is the canonical line bundle, then
for q > 0. Here stands for the tensor product of line bundles. By means of Serre duality, one also obtains the vanishing of for q < n. There is a generalisation, the Kodaira–Nakano vanishing theorem, in which , where Ωn(L) denotes the sheaf of holomorphic (n,0)-forms on M with values on L, is replaced by Ωr(L), the sheaf of holomorphic (r,0)-forms with values on L. Then the cohomology group Hq(M, Ωr(L)) vanishes whenever q + r > n.
The algebraic case
The Kodaira vanishing theorem can be formulated within the language of algebraic geometry without any reference to transcendental methods such as Kähler metrics. Positivity of the line bundle L translates into the corresponding invertible sheaf being ample (i.e., some tensor power gives a projective embedding). The algebraic Kodaira–Akizuki–Nakano vanishing theorem is the following statement:
If k is a field of characteristic zero, X is a smooth and projective k-scheme of dimension d, and L is an ample invertible sheaf on X, then
where the Ωp denote the sheaves of relative (algebraic) differential forms (see Kähler differential).
showed that this result does not always hold over fields of characteristic p > 0, and in particular fails for Raynaud surfaces. Later give a counterexample for singular varieties with non-log canonical singularities, and also, gave elementary counterexamples inspired by proper homogeneous spaces with non-reduced stabilizers.
Until 1987 the only known proof in characteristic zero was however based on the complex analytic proof and the GAGA comparison theorems. However, in 1987 Pierre Deligne and Luc Illusie gave a purely algebraic proof of the vanishing theorem in . Their proof is based on showing that the Hodge–de Rham spectral sequence for algebraic de Rham cohomology degenerates in degree 1. This is shown by lifting a corresponding more specific result from characteristic p > 0 — the positive-characteristic result does not hold without limitations but can be lifted to provide the full result.
Consequences and applications
Historically, the Kodaira embedding theorem was derived with the help of the vanishing theorem. With application of Serre duality, the vanishing of various sheaf cohomology groups (usually related to the canonical line bundle) of curves and surfaces h
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https://en.wikipedia.org/wiki/Composite%20measure
|
Composite measure in statistics and research design refer to composite measures of variables, i.e. measurements based on multiple data items.
An example of a composite measure is an IQ test, which gives a single score based on a series of responses to various questions.
Three common composite measures include:
indexes - measures that summarize and rank specific observations, usually on the ordinal scale;
scales - advanced indexes whose observations are further transformed (scaled) due to their logical or empirical relationships;
typologies - measures that classify observations in terms of their attributes on multiple variables, usually on a nominal scale.
Indexes versus scales
Indexes are often referred to as scales, but in fact not all indexes are scales.
Whereas indexes are usually created by aggregating scores assigned to individual attributes of various variables, scales are more nuanced and take into account differences in intensity among the attribute of the same variable in question. Indexes and scales should provide an ordinal ranking of cases on a given variable, though scales are usually more efficient at this. While indexes are based on a simple aggregation of indicators of a variable, scales are more advanced, and their calculations may be more complex, using for example scaling procedures such as semantic differential.
Composite measure validation
A good composite measure will ensure that the indicators are independent of one another. It should also successfully predict other indicators of the variable.
References
Measurement
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https://en.wikipedia.org/wiki/Church%20encoding
|
In mathematics, Church encoding is a means of representing data and operators in the lambda calculus. The Church numerals are a representation of the natural numbers using lambda notation. The method is named for Alonzo Church, who first encoded data in the lambda calculus this way.
Terms that are usually considered primitive in other notations (such as integers, booleans, pairs, lists, and tagged unions) are mapped to higher-order functions under Church encoding. The Church–Turing thesis asserts that any computable operator (and its operands) can be represented under Church encoding. In the untyped lambda calculus the only primitive data type is the function.
Use
A straightforward implementation of Church encoding slows some access operations from to , where is the size of the data structure, making Church encoding impractical. Research has shown that this can be addressed by targeted optimizations, but most functional programming languages instead expand their intermediate representations to contain algebraic data types. Nonetheless Church encoding is often used in theoretical arguments, as it is a natural representation for partial evaluation and theorem proving. Operations can be typed using higher-ranked types, and primitive recursion is easily accessible. The assumption that functions are the only primitive data types streamlines many proofs.
Church encoding is complete but only representationally. Additional functions are needed to translate the representation into common data types, for display to people. It is not possible in general to decide if two functions are extensionally equal due to the undecidability of equivalence from Church's theorem. The translation may apply the function in some way to retrieve the value it represents, or look up its value as a literal lambda term. Lambda calculus is usually interpreted as using intensional equality. There are potential problems with the interpretation of results because of the difference between the intensional and extensional definition of equality.
Church numerals
Church numerals are the representations of natural numbers under Church encoding. The higher-order function that represents natural number n is a function that maps any function to its n-fold composition. In simpler terms, the "value" of the numeral is equivalent to the number of times the function encapsulates its argument.
All Church numerals are functions that take two parameters. Church numerals 0, 1, 2, ..., are defined as follows in the lambda calculus.
Starting with 0 not applying the function at all, proceed with 1 applying the function once, 2 applying the function twice, 3 applying the function three times, etc.:
The Church numeral 3 represents the action of applying any given function three times to a value. The supplied function is first applied to a supplied parameter and then successively to its own result. The end result is not the numeral 3 (unless the supplied parameter happens to be 0
|
https://en.wikipedia.org/wiki/%CE%93-convergence
|
In the field of mathematical analysis for the calculus of variations, Γ-convergence (Gamma-convergence) is a notion of convergence for functionals. It was introduced by Ennio de Giorgi.
Definition
Let be a topological space and denote the set of all neighbourhoods of the point . Let further be a sequence of functionals on . The and the are defined as follows:
.
are said to -converge to , if there exist a functional such that .
Definition in first-countable spaces
In first-countable spaces, the above definition can be characterized in terms of sequential -convergence in the following way.
Let be a first-countable space and a sequence of functionals on . Then are said to -converge to the -limit if the following two conditions hold:
Lower bound inequality: For every sequence such that as ,
Upper bound inequality: For every , there is a sequence converging to such that
The first condition means that provides an asymptotic common lower bound for the . The second condition means that this lower bound is optimal.
Relation to Kuratowski convergence
-convergence is connected to the notion of Kuratowski-convergence of sets. Let denote the epigraph of a function and let be a sequence of functionals on . Then
where denotes the Kuratowski limes inferior and the Kuratowski limes superior in the product topology of . In particular, -converges to in if and only if -converges to in . This is the reason why -convergence is sometimes called epi-convergence.
Properties
Minimizers converge to minimizers: If -converge to , and is a minimizer for , then every cluster point of the sequence is a minimizer of .
-limits are always lower semicontinuous.
-convergence is stable under continuous perturbations: If -converges to and is continuous, then will -converge to .
A constant sequence of functionals does not necessarily -converge to , but to the relaxation of , the largest lower semicontinuous functional below .
Applications
An important use for -convergence is in homogenization theory. It can also be used to rigorously justify the passage from discrete to continuum theories for materials, for example, in elasticity theory.
See also
Mosco convergence
Kuratowski convergence
Epi-convergence
References
A. Braides: Γ-convergence for beginners. Oxford University Press, 2002.
G. Dal Maso: An introduction to Γ-convergence. Birkhäuser, Basel 1993.
Calculus of variations
Variational analysis
Convergence (mathematics)
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https://en.wikipedia.org/wiki/263%20%28number%29
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263 is the natural number between 262 and 264. It is also a prime number.
In mathematics
263 is
a balanced prime,
an irregular prime,
a Ramanujan prime, a Chen prime, and
a safe prime.
It is also a strictly non-palindromic number and a happy number.
References
Integers
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https://en.wikipedia.org/wiki/269%20%28number%29
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269 (two hundred [and] sixty-nine) is the natural number between 268 and 270. It is also a prime number.
In mathematics
269 is a twin prime,
and a Ramanujan prime.
It is the largest prime factor of 9! + 1 = 362881,
and the smallest natural number that cannot be represented as the determinant of a 10 × 10 (0,1)-matrix.
References
Integers
|
https://en.wikipedia.org/wiki/AN/APQ-181
|
The AN/APQ-181 is an all-weather, low probability of intercept (LPI) phased array radar system designed by Hughes Aircraft (now Raytheon) for the U.S. Air Force B-2A Spirit bomber aircraft. The system was developed in the mid-1980s and entered service in 1993. The APQ-181 provides a number of precision targeting modes, and also supports terrain-following radar and terrain avoidance. The radar operates in the Ku band (a subset of the J band). The original design uses a TWT-based transmitter with a 2-dimensional passive electronically scanned array (PESA) antenna.
In 1991, the B-2 Industrial Team (including Hughes as a major subcontractor) was awarded the Collier Trophy in recognition of the "design, development, production, and flight testing of the B-2 aircraft, which has contributed significantly to America's enduring leadership in aerospace and the country's future national security."
In 2002, Raytheon was awarded a contract to develop a new, active electronically scanned array (AESA) version of the APQ-181. This upgrade will improve system reliability, and will also eliminate potential conflicts in frequency usage between the B-2 and commercial satellite systems that also use the J band.
In 2008 the Federal Communications Commission accidentally sold the APQ-181 frequency to a commercial user resulting in the need for installing new radar arrays at a cost of over $1 billion. All B-2 aircraft are expected to have the upgraded radar by 2010.
See also
List of radars
Joint Electronics Type Designation System (JETDS)
References
External links
Raytheon product description
spacedaily.com article
Aircraft radars
Raytheon Company products
Radars of the United States Air Force
Military radars of the United States
Military equipment introduced in the 1990s
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https://en.wikipedia.org/wiki/Kloosterman%20sum
|
In mathematics, a Kloosterman sum is a particular kind of exponential sum. They are named for the Dutch mathematician Hendrik Kloosterman, who introduced them in 1926 when he adapted the Hardy–Littlewood circle method to tackle a problem involving positive definite diagonal quadratic forms in four as opposed to five or more variables, which he had dealt with in his dissertation in 1924.
Let be natural numbers. Then
Here x* is the inverse of modulo .
Context
The Kloosterman sums are a finite ring analogue of Bessel functions. They occur (for example) in the Fourier expansion of modular forms.
There are applications to mean values involving the Riemann zeta function, primes in short intervals, primes in arithmetic progressions, the spectral theory of automorphic functions and related topics.
Properties of the Kloosterman sums
If or then the Kloosterman sum reduces to the Ramanujan sum.
depends only on the residue class of and modulo . Furthermore and if .
Let with and coprime. Choose and such that and . Then
This reduces the evaluation of Kloosterman sums to the case where for a prime number and an integer .
The value of is always an algebraic real number. In fact is an element of the subfield which is the compositum of the fields
where ranges over all odd primes such that and
for with .
The Selberg identity:
was stated by Atle Selberg and first proved by Kuznetsov using the spectral theory of modular forms. Nowadays elementary proofs of this identity are known.
For an odd prime, there are no known simple formula for , and the Sato–Tate conjecture suggests that none exist. The lifting formulas below, however, are often as good as an explicit evaluation. If one also has the important transformation:
where denotes the Jacobi symbol.
Let with prime and assume . Then:
where is chosen so that and is defined as follows (note that is odd):
This formula was first found by Hans Salie and there are many simple proofs in the literature.
Estimates
Because Kloosterman sums occur in the Fourier expansion of modular forms, estimates for Kloosterman sums yield estimates for Fourier coefficients of modular forms as well. The most famous estimate is due to André Weil and states:
Here is the number of positive divisors of . Because of the multiplicative properties of Kloosterman sums these estimates may be reduced to the case where is a prime number . A fundamental technique of Weil reduces the estimate
when ab ≠ 0 to his results on local zeta-functions. Geometrically the sum is taken along a 'hyperbola' XY = ab and we consider this as defining an algebraic curve over the finite field with elements. This curve has a ramified Artin–Schreier covering , and Weil showed that the local zeta-function of has a factorization; this is the Artin L-function theory for the case of global fields that are function fields, for which Weil gives a 1938 paper of J. Weissinger as reference (the next year he gave a 1935 paper
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https://en.wikipedia.org/wiki/Kuiper%27s%20theorem
|
In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible bounded endomorphisms of H is such that all maps from any finite complex Y to GL(H) are homotopic to a constant, for the norm topology on operators.
A significant corollary, also referred to as Kuiper's theorem, is that this group is weakly contractible, ie. all its homotopy groups are trivial. This result has important uses in topological K-theory.
General topology of the general linear group
For finite dimensional H, this group would be a complex general linear group and not at all contractible. In fact it is homotopy equivalent to its maximal compact subgroup, the unitary group U of H. The proof that the complex general linear group and unitary group have the same homotopy type is by the Gram-Schmidt process, or through the matrix polar decomposition, and carries over to the infinite-dimensional case of separable Hilbert space, basically because the space of upper triangular matrices is contractible as can be seen quite explicitly. The underlying phenomenon is that passing to infinitely many dimensions causes much of the topological complexity of the unitary groups to vanish; but see the section on Bott's unitary group, where the passage to infinity is more constrained, and the resulting group has non-trivial homotopy groups.
Historical context and topology of spheres
It is a surprising fact that the unit sphere, sometimes denoted S∞, in infinite-dimensional Hilbert space H is a contractible space, while no finite-dimensional spheres are contractible. This result, certainly known decades before Kuiper's, may have the status of mathematical folklore, but it is quite often cited. In fact more is true: S∞ is diffeomorphic to H, which is certainly contractible by its convexity. One consequence is that there are smooth counterexamples to an extension of the Brouwer fixed-point theorem to the unit ball in H. The existence of such counter-examples that are homeomorphisms was shown in 1943 by Shizuo Kakutani, who may have first written down a proof of the contractibility of the unit sphere. But the result was anyway essentially known (in 1935 Andrey Nikolayevich Tychonoff showed that the unit sphere was a retract of the unit ball).
The result on the group of bounded operators was proved by the Dutch mathematician Nicolaas Kuiper, for the case of a separable Hilbert space; the restriction of separability was later lifted. The same result, but for the strong operator topology rather than the norm topology, was published in 1963 by Jacques Dixmier and Adrien Douady. The geometric relationship of the sphere and group of operators is that the unit sphere is a homogeneous space for the unitary group U. The stabiliser of a single vector v of the unit sphere is the unitary group of the orthogonal complement of v; therefore the homotopy long exact sequence predic
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https://en.wikipedia.org/wiki/Tensor%20product%20of%20modules
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In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.
Balanced product
For a ring R, a right R-module M, a left R-module N, and an abelian group G, a map is said to be R-balanced, R-middle-linear or an R-balanced product if for all m, m′ in M, n, n′ in N, and r in R the following hold:
The set of all such balanced products over R from to G is denoted by .
If φ, ψ are balanced products, then each of the operations and −φ defined pointwise is a balanced product. This turns the set into an abelian group.
For M and N fixed, the map is a functor from the category of abelian groups to itself. The morphism part is given by mapping a group homomorphism to the function , which goes from to .
Remarks
Properties (Dl) and (Dr) express biadditivity of φ, which may be regarded as distributivity of φ over addition.
Property (A) resembles some associative property of φ.
Every ring R is an R-bimodule. So the ring multiplication in R is an R-balanced product .
Definition
For a ring R, a right R-module M, a left R-module N, the tensor product over R
is an abelian group together with a balanced product (as defined above)
which is universal in the following sense:
For every abelian group G and every balanced product there is a unique group homomorphism such that
As with all universal properties, the above property defines the tensor product uniquely up to a unique isomorphism: any other abelian group and balanced product with the same properties will be isomorphic to and ⊗. Indeed, the mapping ⊗ is called canonical, or more explicitly: the canonical mapping (or balanced product) of the tensor product.
The definition does not prove the existence of ; see below for a construction.
The tensor product can also be defined as a representing object for the functor ; explicitly, this means there is a natural isomorphism:
This is a succinct way of stating the universal mapping property given above. (If a priori one is given this natural isomorphism, then can be recovered by taking and then mapping the
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https://en.wikipedia.org/wiki/A%E2%99%AF%20%28Axiom%29
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{{DISPLAYTITLE:A♯ (Axiom)}}
A♯ (pronounced: A sharp) is an object-oriented functional programming language distributed as a separable component of Version 2 of the Axiom computer algebra system. A# types and functions are first-class values and can be used freely together with an extensive library of data structures and other mathematical abstractions. A key design guideline for A# was suitability of compiling to portable and efficient machine code. It is distributed as free and open-source software under a BSD-like license.
Development of A# has now changed to the programming language Aldor.
A# has both an optimising compiler, and an intermediate code interpreter. The compiler can emit any of:
Executable stand-alone programs
Libraries, of native operating system format objects, or of portable bytecode
Source code, for languages C, or Lisp
The following C compilers are supported: GNU Compiler Collection (GCC), Xlc, Oracle Developer Studio, Borland, Metaware, and MIPS C.
References
Functional languages
Discontinued programming languages
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https://en.wikipedia.org/wiki/Strong%20prior
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In Bayesian statistics, a strong prior is a preceding assumption, theory, concept or idea upon which, after taking account of new information, a current assumption, theory, concept or idea is founded. The term is used to contrast the case of a weak or uninformative prior probability. A strong prior would be a type of informative prior in which the information contained in the prior distribution dominates the information contained in the data being analysed. The Bayesian analysis combines the information contained in the prior with that extracted from the data to produce the posterior distribution which, in the case of a "strong prior", would be little changed from the prior distribution.
Bayesian statistics
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https://en.wikipedia.org/wiki/Throughput%20%28business%29
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Throughput is rate at which a product is moved through a production process and is consumed by the end-user, usually measured in the form of sales or use statistics. The goal of most organizations is to minimize the investment in inputs as well as operating expenses while increasing throughput of its production systems. Successful organizations which seek to gain market share strive to match throughput to the rate of market demand of its products.
Overview
In the business management theory of constraints, throughput is the rate at which a system achieves its goal. Oftentimes, this is monetary revenue and is in contrast to output, which is inventory that may be sold or stored in a warehouse. In this case, throughput is measured by revenue received (or not) at the point of sale—exactly the right time. Output that becomes part of the inventory in a warehouse may mislead investors or others about the organizations condition by inflating the apparent value of its assets. The theory of constraints and throughput accounting explicitly avoid that trap.
Throughput can be best described as the rate at which a system generates its products or services per unit of time. Businesses often measure their throughput using a mathematical equation known as Little's law, which is related to inventories and process time: time to fully process a single product.
Basic formula
Using Little's Law, one can calculate throughput with the equation:
where:
I is the number of units contained within the system, inventory;
T is the time it takes for all the inventory to go through the process, flow time;
R is the rate at which the process is delivering throughput, flow rate or throughput.
If you solve for R, you will get:
References
Further reading
Goldratt, Eliyahu and Jeff Cox. The Goal. Croton-on-Hudson: North River Press, 2004.
Business terms
Manufacturing
Production economics
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https://en.wikipedia.org/wiki/Rho%20calculus
|
There are two different calculi that use the name rho-calculus:
The first is a formalism intended to combine the higher-order facilities of lambda calculus with the pattern matching of term rewriting.
The second is a reflective higher-order variant of the asynchronous polyadic pi calculus.
References
Lambda calculus
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https://en.wikipedia.org/wiki/Zolotarev%27s%20lemma
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In number theory, Zolotarev's lemma states that the Legendre symbol
for an integer a modulo an odd prime number p, where p does not divide a, can be computed as the sign of a permutation:
where ε denotes the signature of a permutation and πa is the permutation of the nonzero residue classes mod p induced by multiplication by a.
For example, take a = 2 and p = 7. The nonzero squares mod 7 are 1, 2, and 4, so (2|7) = 1 and (6|7) = −1. Multiplication by 2 on the nonzero numbers mod 7 has the cycle decomposition (1,2,4)(3,6,5), so the sign of this permutation is 1, which is (2|7). Multiplication by 6 on the nonzero numbers mod 7 has cycle decomposition (1,6)(2,5)(3,4), whose sign is −1, which is (6|7).
Proof
In general, for any finite group G of order n, it is straightforward to determine the signature of the permutation πg made by left-multiplication by the element g of G. The permutation πg will be even, unless there are an odd number of orbits of even size. Assuming n even, therefore, the condition for πg to be an odd permutation, when g has order k, is that n/k should be odd, or that the subgroup <g> generated by g should have odd index.
We will apply this to the group of nonzero numbers mod p, which is a cyclic group of order p − 1. The jth power of a primitive root modulo p will have index the greatest common divisor
i = (j, p − 1).
The condition for a nonzero number mod p to be a quadratic non-residue is to be an odd power of a primitive root.
The lemma therefore comes down to saying that i is odd when j is odd, which is true a fortiori, and j is odd when i is odd, which is true because p − 1 is even (p is odd).
Another proof
Zolotarev's lemma can be deduced easily from Gauss's lemma and vice versa. The example
,
i.e. the Legendre symbol (a/p) with a = 3 and p = 11, will illustrate how the proof goes. Start with the set {1, 2, . . . , p − 1} arranged as a matrix of two rows such that the sum of the two elements in any column is zero mod p, say:
Apply the permutation :
The columns still have the property that the sum of two elements in one column is zero mod p. Now apply a permutation V which swaps any pairs in which the upper member was originally a lower member:
Finally, apply a permutation W which gets back the original matrix:
We have W−1 = VU. Zolotarev's lemma says (a/p) = 1 if and only if the permutation U is even. Gauss's lemma says (a/p) = 1 iff V is even. But W is even, so the two lemmas are equivalent for the given (but arbitrary) a and p.
Jacobi symbol
This interpretation of the Legendre symbol as the sign of a permutation can be extended to the Jacobi symbol
where a and n are relatively prime integers with odd n > 0: a is invertible mod n, so multiplication by a on Z/nZ is a permutation and a generalization of Zolotarev's lemma is that the Jacobi symbol above is the sign of this permutation.
For example, multiplication by 2 on Z/21Z has cycle decomposition (0)(1,2,4,8,16,11)(3,6,12)(5,10,20,19,17,13)(7,14)(9,18,15),
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https://en.wikipedia.org/wiki/Theta%20correspondence
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In mathematics, the theta correspondence or Howe correspondence is a mathematical relation between representations of two groups of a reductive dual pair. The local theta correspondence relates irreducible admissible representations over a local field, while the global theta correspondence relates irreducible automorphic representations over a global field.
The theta correspondence was introduced by Roger Howe in . Its name arose due to its origin in André Weil's representation theoretical formulation of the theory of theta series in . The Shimura correspondence as constructed by Jean-Loup Waldspurger in and may be viewed as an instance of the theta correspondence.
Statement
Setup
Let be a local or a global field, not of characteristic . Let be a symplectic vector space over , and the symplectic group.
Fix a reductive dual pair in . There is a classification of reductive dual pairs.
Local theta correspondence
is now a local field. Fix a non-trivial additive character of . There exists a Weil representation of the metaplectic group associated to , which we write as .
Given the reductive dual pair in , one obtains a pair of commuting subgroups in by pulling back the projection map from to .
The local theta correspondence is a 1-1 correspondence between certain irreducible admissible representations of and certain irreducible admissible representations of , obtained by restricting the Weil representation of to the subgroup . The correspondence was defined by Roger Howe in . The assertion that this is a 1-1 correspondence is called the Howe duality conjecture.
Key properties of local theta correspondence include its compatibility with Bernstein-Zelevinsky induction and conservation relations concerning the first occurrence indices along Witt towers .
Global theta correspondence
Stephen Rallis showed a version of the global Howe duality conjecture for cuspidal automorphic representations over a global field, assuming the validity of the Howe duality conjecture for all local places.
Howe duality conjecture
Define the set of irreducible admissible representations of , which can be realized as quotients of
. Define and , likewise.
The Howe duality conjecture asserts that is the graph of a bijection between and .
The Howe duality conjecture for archimedean local fields was proved by Roger Howe. For -adic local fields with odd it was proved by Jean-Loup Waldspurger. Alberto Mínguez later gave a proof for dual pairs of general linear groups, that works for arbitrary residue characteristic. For orthogonal-symplectic or unitary dual pairs, it was proved by Wee Teck Gan and Shuichiro Takeda. The final case of quaternionic dual pairs was completed by Wee Teck Gan and Binyong Sun.
See also
Reductive dual pair
Metaplectic group
References
Bibliography
Langlands program
Representation theory
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https://en.wikipedia.org/wiki/Weierstrass%E2%80%93Enneper%20parameterization
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In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry.
Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863.
Let and be functions on either the entire complex plane or the unit disk, where is meromorphic and is analytic, such that wherever has a pole of order , has a zero of order (or equivalently, such that the product is holomorphic), and let be constants. Then the surface with coordinates is minimal, where the are defined using the real part of a complex integral, as follows:
The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type.
For example, Enneper's surface has , .
Parametric surface of complex variables
The Weierstrass-Enneper model defines a minimal surface () on a complex plane (). Let (the complex plane as the space), the Jacobian matrix of the surface can be written as a column of complex entries:
where and are holomorphic functions of .
The Jacobian represents the two orthogonal tangent vectors of the surface:
The surface normal is given by
The Jacobian leads to a number of important properties: , , , . The proofs can be found in Sharma's essay: The Weierstrass representation always gives a minimal surface. The derivatives can be used to construct the first fundamental form matrix:
and the second fundamental form matrix
Finally, a point on the complex plane maps to a point on the minimal surface in by
where for all minimal surfaces throughout this paper except for Costa's minimal surface where .
Embedded minimal surfaces and examples
The classical examples of embedded complete minimal surfaces in with finite topology include the plane, the catenoid, the helicoid, and the Costa's minimal surface. Costa's surface involves Weierstrass's elliptic function :
where is a constant.
Helicatenoid
Choosing the functions and , a one parameter family of minimal surfaces is obtained.
Choosing the parameters of the surface as :
At the extremes, the surface is a catenoid or a helicoid . Otherwise, represents a mixing angle. The resulting surface, with domain chosen to prevent self-intersection, is a catenary rotated around the axis in a helical fashion.
Lines of curvature
One can rewrite each element of second fundamental matrix as a function of and , for example
And consequently the second fundamental form matrix can be simplified as
One of its eigenvectors is which represents the principal direction in the complex domain. Therefore, the two principal directions in the space turn out to be
See also
Associate family
Bryant surface, found by an analogous parameterization in hyperbolic space
References
Differential geometry
Surfaces
Minimal surfaces
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https://en.wikipedia.org/wiki/Absolute%20Galois%20group
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In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group.
(When K is a perfect field, Ksep is the same as an algebraic closure Kalg of K. This holds e.g. for K of characteristic zero, or K a finite field.)
Examples
The absolute Galois group of an algebraically closed field is trivial.
The absolute Galois group of the real numbers is a cyclic group of two elements (complex conjugation and the identity map), since C is the separable closure of R and [C:R] = 2.
The absolute Galois group of a finite field K is isomorphic to the group
(For the notation, see Inverse limit.)
The Frobenius automorphism Fr is a canonical (topological) generator of GK. (Recall that Fr(x) = xq for all x in Kalg, where q is the number of elements in K.)
The absolute Galois group of the field of rational functions with complex coefficients is free (as a profinite group). This result is due to Adrien Douady and has its origins in Riemann's existence theorem.
More generally, let C be an algebraically closed field and x a variable. Then the absolute Galois group of K = C(x) is free of rank equal to the cardinality of C. This result is due to David Harbater and Florian Pop, and was also proved later by Dan Haran and Moshe Jarden using algebraic methods.
Let K be a finite extension of the p-adic numbers Qp. For p ≠ 2, its absolute Galois group is generated by [K:Qp] + 3 elements and has an explicit description by generators and relations. This is a result of Uwe Jannsen and Kay Wingberg. Some results are known in the case p = 2, but the structure for Q2 is not known.
Another case in which the absolute Galois group has been determined is for the largest totally real subfield of the field of algebraic numbers.
Problems
No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the absolute Galois group has a faithful action on the dessins d'enfants of Grothendieck (maps on surfaces), enabling us to "see" the Galois theory of algebraic number fields.
Let K be the maximal abelian extension of the rational numbers. Then Shafarevich's conjecture asserts that the absolute Galois group of K is a free profinite group.
An interesting problem is to settle Ján Mináč and Nguyên Duy Tân's conjecture about vanishing of - Massey products for .
Some general results
Every profinite group occurs as a Galois group of some Galois extension, however not every profinite group occurs as an absolute Galois group. For example, the Artin–Schreier theorem asserts that the only finite absolute Galois groups are either trivial or of order 2, that is only two isomorphism classes.
Every projective profinite group can be realized as an absolute Galois group of
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https://en.wikipedia.org/wiki/United%20Nations%20Statistics%20Division
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The United Nations Statistics Division (UNSD), formerly the United Nations Statistical Office, serves under the United Nations Department of Economic and Social Affairs (DESA) as the central mechanism within the Secretariat of the United Nations to supply the statistical needs and coordinating activities of the global statistical system. The Division is overseen by the United Nations Statistical Commission, established in 1947, as the apex entity of the global statistical system and highest decision making body for coordinating international statistical activities. It brings together the Chief Statisticians from member states from around the world.
The Division compiles and disseminates global statistical information, develops standards and norms for statistical activities, and supports countries' efforts to strengthen their national statistical systems.
The Division regularly publishes data updates, including the Statistical Yearbook and World Statistics Pocketbook, and books and reports on statistics and statistical methods. Many of the Division's databases are also available at its site (See below), as electronic publications and data files in the form of CD-ROMs, diskettes and magnetic tapes, or as printed publications. UNdata, a new internet-based data service for the global user community brings UN Statistical databases within easy reach of users through a single entry point. Users can search and download a variety of statistical resources of the UN system.
Directors
Including acting directors:
Topics
Economy
Industry Statistics
Energy Statistics
Trade Statistics
...
Environment
Environment Statistics
...
Development Indicators
Sustainable Development Goal indicators
...
PET Lab
UNSD leads the Privacy-Enhancing Technologies Lab (PET Lab), which in turn drives TrustworthyAI together with ITU.
See also
Classification of the Functions of Government
International Standard Industrial Classification
UN M49
United Nations geoscheme
List of national and international statistical services
Committee for the Coordination of Statistical Activities
United Nations
United Nations Group of Experts on Geographical Names
References
External links
United Nations Department of Economic and Social Affairs
United Nations Economic and social development
United Nations Statistical Commission
UNdata
UN Comtrade
Organizations established in 1947
Statistical organizations
Statistics Division
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https://en.wikipedia.org/wiki/Watchman%20route%20problem
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The Watchman Problem is an optimization problem in computational geometry where the objective is to compute the shortest route a watchman should take to guard an entire area with obstacles given only a map of the area. The challenge is to make sure the watchman peeks behind every corner and to determine the best order in which corners should be visited in. The problem may be solved in polynomial time when the area to be guarded is a simple polygon. The problem is NP-hard for polygons with holes, but may be approximated in polynomial time by a solution whose length is within a polylogarithmic factor of optimal.
See also
Art gallery problem, which similarly involves viewing all points of a given area, but with multiple stationary watchmen
References
Geometric algorithms
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https://en.wikipedia.org/wiki/Bauer%E2%80%93Fike%20theorem
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In mathematics, the Bauer–Fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex-valued diagonalizable matrix. In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a properly chosen eigenvalue of the exact matrix. Informally speaking, what it says is that the sensitivity of the eigenvalues is estimated by the condition number of the matrix of eigenvectors.
The theorem was proved by Friedrich L. Bauer and C. T. Fike in 1960.
The setup
In what follows we assume that:
is a diagonalizable matrix;
is the non-singular eigenvector matrix such that , where is a diagonal matrix.
If is invertible, its condition number in -norm is denoted by and defined by:
The Bauer–Fike Theorem
Bauer–Fike Theorem. Let be an eigenvalue of . Then there exists such that:
Proof. We can suppose , otherwise take and the result is trivially true since . Since is an eigenvalue of , we have and so
However our assumption, , implies that: and therefore we can write:
This reveals to be an eigenvalue of
Since all -norms are consistent matrix norms we have where is an eigenvalue of . In this instance this gives us:
But is a diagonal matrix, the -norm of which is easily computed:
whence:
An Alternate Formulation
The theorem can also be reformulated to better suit numerical methods. In fact, dealing with real eigensystem problems, one often has an exact matrix , but knows only an approximate eigenvalue-eigenvector couple, and needs to bound the error. The following version comes in help.
Bauer–Fike Theorem (Alternate Formulation). Let be an approximate eigenvalue-eigenvector couple, and . Then there exists such that:
Proof. We can suppose , otherwise take and the result is trivially true since . So exists, so we can write:
since is diagonalizable; taking the -norm of both sides, we obtain:
However
is a diagonal matrix and its -norm is easily computed:
whence:
A Relative Bound
Both formulations of Bauer–Fike theorem yield an absolute bound. The following corollary is useful whenever a relative bound is needed:
Corollary. Suppose is invertible and that is an eigenvalue of . Then there exists such that:
Note. can be formally viewed as the relative variation of , just as is the relative variation of .
Proof. Since is an eigenvalue of and , by multiplying by from left we have:
If we set:
then we have:
which means that is an eigenvalue of , with as an eigenvector. Now, the eigenvalues of are , while it has the same eigenvector matrix as . Applying the Bauer–Fike theorem to with eigenvalue , gives us:
The Case of Normal Matrices
If is normal, is a unitary matrix, therefore:
so that . The Bauer–Fike theorem then becomes:
Or in alternate formulation:
which obviously remains true if is a Hermitian matrix. In this case, however, a much stronger result holds, known as the Weyl's theorem on eigenvalues. In the hermitian case one can als
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https://en.wikipedia.org/wiki/Asymptotic%20distribution
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In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing approximations to the cumulative distribution functions of statistical estimators.
Definition
A sequence of distributions corresponds to a sequence of random variables Zi for i = 1, 2, ..., I . In the simplest case, an asymptotic distribution exists if the probability distribution of Zi converges to a probability distribution (the asymptotic distribution) as i increases: see convergence in distribution. A special case of an asymptotic distribution is when the sequence of random variables is always zero or Zi = 0 as i approaches infinity. Here the asymptotic distribution is a degenerate distribution, corresponding to the value zero.
However, the most usual sense in which the term asymptotic distribution is used arises where the random variables Zi are modified by two sequences of non-random values. Thus if
converges in distribution to a non-degenerate distribution for two sequences {ai} and {bi} then Zi is said to have that distribution as its asymptotic distribution. If the distribution function of the asymptotic distribution is F then, for large n, the following approximations hold
If an asymptotic distribution exists, it is not necessarily true that any one outcome of the sequence of random variables is a convergent sequence of numbers. It is the sequence of probability distributions that converges.
Central limit theorem
Perhaps the most common distribution to arise as an asymptotic distribution is the normal distribution. In particular, the central limit theorem provides an example where the asymptotic distribution is the normal distribution.
Central limit theorem
Suppose is a sequence of i.i.d. random variables with and . Let be the average of . Then as approaches infinity, the random variables converge in distribution to a normal :
The central limit theorem gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.
Local asymptotic normality
Local asymptotic normality is a generalization of the central limit theorem. It is a property of a sequence of statistical models, which allows this sequence to be asymptotically approximated by a normal location model, after a rescaling of the parameter. An important example when the local asymptotic normality holds is in the case of independent and identically distributed sampling from a regular parametric model; this is just the central limit theorem.
Barndorff-Nielson & Cox provide a direct definition of asymptotic normality.
See also
Asymptotic analysis
Asymptotic theory (statistics)
de Moivre–Laplace theorem
Limiting density of discrete
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https://en.wikipedia.org/wiki/Branched%20covering
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In mathematics, a branched covering is a map that is almost a covering map, except on a small set.
In topology
In topology, a map is a branched covering if it is a covering map everywhere except for a nowhere dense set known as the branch set. Examples include the map from a wedge of circles to a single circle, where the map is a homeomorphism on each circle.
In algebraic geometry
In algebraic geometry, the term branched covering is used to describe morphisms from an algebraic variety to another one , the two dimensions being the same, and the typical fibre of being of dimension 0.
In that case, there will be an open set of (for the Zariski topology) that is dense in , such that the restriction of to (from to , that is) is unramified. Depending on the context, we can take this as local homeomorphism for the strong topology, over the complex numbers, or as an étale morphism in general (under some slightly stronger hypotheses, on flatness and separability). Generically, then, such a morphism resembles a covering space in the topological sense. For example, if and are both compact Riemann surfaces, we require only that is holomorphic and not constant, and then there is a finite set of points of , outside of which we do find an honest covering
.
Ramification locus
The set of exceptional points on is called the ramification locus (i.e. this is the complement of the largest possible open set ). In general monodromy occurs according to the fundamental group of acting on the sheets of the covering (this topological picture can be made precise also in the case of a general base field).
Kummer extensions
Branched coverings are easily constructed as Kummer extensions, i.e. as algebraic extension of the function field. The hyperelliptic curves are prototypic examples.
Unramified covering
An unramified covering then is the occurrence of an empty ramification locus.
Examples
Elliptic curve
Morphisms of curves provide many examples of ramified coverings. For example, let be the elliptic curve of equation
The projection of onto the -axis is a ramified cover with ramification locus given by
This is because for these three values of the fiber is the double point while for any other value of , the fiber consists of two distinct points (over an algebraically closed field).
This projection induces an algebraic extension of degree two of the function fields:
Also, if we take the fraction fields of the underlying commutative rings, we get the morphism
Hence this projection is a degree 2 branched covering. This can be homogenized to construct a degree 2 branched covering of the corresponding projective elliptic curve to the projective line.
Plane algebraic curve
The previous example may be generalized to any algebraic plane curve in the following way.
Let be a plane curve defined by the equation , where is a separable and irreducible polynomial in two indeterminates. If is the degree of in , then the fiber consists of distinct
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https://en.wikipedia.org/wiki/Torelli%20theorem
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In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is determined by its Jacobian variety J(C), when the latter is given in the form of a principally polarized abelian variety. In other words, the complex torus J(C), with certain 'markings', is enough to recover C. The same statement holds over any algebraically closed field. From more precise information on the constructed isomorphism of the curves it follows that if the canonically principally polarized Jacobian varieties of curves of genus are k-isomorphic for k any perfect field, so are the curves.
This result has had many important extensions. It can be recast to read that a certain natural morphism, the period mapping, from the moduli space of curves of a fixed genus, to a moduli space of abelian varieties, is injective (on geometric points). Generalizations are in two directions. Firstly, to geometric questions about that morphism, for example the local Torelli theorem. Secondly, to other period mappings. A case that has been investigated deeply is for K3 surfaces (by Viktor S. Kulikov, Ilya Pyatetskii-Shapiro, Igor Shafarevich and Fedor Bogomolov) and hyperkähler manifolds (by Misha Verbitsky, Eyal Markman and Daniel Huybrechts).
Notes
References
Algebraic curves
Abelian varieties
Moduli theory
Theorems in complex geometry
Theorems in algebraic geometry
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https://en.wikipedia.org/wiki/Cue%20validity
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Cue validity is the conditional probability that an object falls in a particular category given a particular feature or cue. The term was popularized by , and especially by Eleanor Rosch in her investigations of the acquisition of so-called basic categories (;).
Definition of cue validity
Formally, the cue validity of a feature with respect to category has been defined in the following ways:
As the conditional probability ; see , , .
As the deviation of the conditional probability from the category base rate, ; see , .
As a function of the linear correlation; see , , , .
Other definitions; see , .
For the definitions based on probability, a high cue validity for a given feature means that the feature or attribute is more diagnostic of the class membership than a feature with low cue validity. Thus, a high-cue validity feature is one which conveys more information about the category or class variable, and may thus be considered as more useful for identifying objects as belonging to that category. Thus, high cue validity expresses high feature informativeness. For the definitions based on linear correlation, the expression of "informativeness" captured by the cue validity measure is not the full expression of the feature's informativeness (as in mutual information, for example), but only that portion of its informativeness that is expressed in a linear relationship. For some purposes, a bilateral measure such as the mutual information or category utility is more appropriate than the cue validity.
Examples
As an example, consider the domain of "numbers" and allow that every number has an attribute (i.e., a cue) named "is_positive_integer", which we call , and which adopts the value 1 if the number is actually a positive integer. Then we can inquire what the validity of this cue is with regard to the following classes: {rational number, irrational number, even integer}:
If we know that a number is a positive integer we know that it is a rational number. Thus, , the cue validity for is_positive_integer as a cue for the category rational number is 1.
If we know that a number is a positive integer then we know that it is not an irrational number. Thus, , the cue validity for is_positive_integer as a cue for the category irrational number is 0.
If we know only that a number is a positive integer, then its chances of being even or odd are 50-50 (there being the same number of even and odd integers). Thus, , the cue validity for is_positive_integer as a cue for the category even integer is 0.5, meaning that the attribute is_positive_integer is entirely uninformative about the number's membership in the class even integer.
In perception, "cue validity" is often short for ecological validity of a perceptual cue, and is defined as a correlation rather than a probability (see above). In this definition, an uninformative perceptual cue has an ecological validity of 0 rather than 0.5.
Use of the cue validity
In much of the work on modeling human
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https://en.wikipedia.org/wiki/Noether%20normalization%20lemma
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In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field k, and any finitely generated commutative k-algebra A, there exist algebraically independent elements y1, y2, ..., yd in A such that A is a finitely generated module over the polynomial ring S = k[y1, y2, ..., yd]. The integer d is equal to the Krull dimension of the ring A; and if A is an integral domain, d is also the transcendence degree of the field of fractions of A over k.
The theorem has a geometric interpretation. Suppose A is the coordinate ring of an affine variety X, and consider S as the coordinate ring of a d-dimensional affine space . Then the inclusion map induces a surjective finite morphism of affine varieties : that is, any affine variety is a branched covering of affine space.
When k is infinite, such a branched covering map can be constructed by taking a general projection from an affine space containing X to a d-dimensional subspace.
More generally, in the language of schemes, the theorem can equivalently be stated as: every affine k-scheme (of finite type) X is finite over an affine n-dimensional space. The theorem can be refined to include a chain of ideals of R (equivalently, closed subsets of X) that are finite over the affine coordinate subspaces of the corresponding dimensions.
The Noether normalization lemma can be used as an important step in proving Hilbert's Nullstellensatz, one of the most fundamental results of classical algebraic geometry. The normalization theorem is also an important tool in establishing the notions of Krull dimension for k-algebras.
Proof
The following proof is due to Nagata, following Mumford's red book. A more geometric proof is given on page 127 of the red book.
The ring A in the lemma is generated as a k-algebra by some elements, . We shall induct on m. If , then the assertion is trivial. Assume now . It is enough to show that there is a subring S of A that is generated by elements, such that A is finite over S. Indeed, by the inductive hypothesis, we can find algebraically independent elements of S such that S is finite over .
Since otherwise there would be nothing to prove, we can also assume that there is a nonzero polynomial f in m variables over k such that
.
Given an integer r which is determined later, set
Then the preceding reads:
.
Now, if is a monomial appearing in the left-hand side of the above equation, with coefficient , the highest term in after expanding the product looks like
Whenever the above exponent agrees with the highest exponent produced by some other monomial, it is possible that the highest term in of will not be of the above form, because it may be affected by cancellation. However, if r is larger than any exponent appearing in f, then each encodes a unique base r number, so this does not occur. Thus is integral over . Since are also integral over that ring, A is integral over S. It follows A is
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https://en.wikipedia.org/wiki/Kennesaw%20Mountain%20High%20School
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Kennesaw Mountain High School is a public high school located in Kennesaw, Cobb County, Georgia, United States. It was founded in 2000 as a magnet school specializing in science and mathematics, and is one of sixteen high schools in the Cobb County School District.
History
Students
Kennesaw Mountain High School was founded in 2000. The high school, built on a site, was intended to have a capacity of 2,000 students, but due to the rapid population growth in Cobb County, the school quickly became overcrowded. Before the school was built, juniors, as opposed to freshman and sophomores, from Harrison High School (the main output) were given the choice whether to stay at their current school or change to Kennesaw Mountain. Approximately 240 juniors decided to change, which was much more than expected.
Construction
Construction of the five buildings at the campus began in August 1999 and was completed in 14 months. The design, by Passantino & Bavier, Inc., used steel bowstring joists to allow for an arched roofline and the clear span required for the gymnasium. Design and construction of the campus is the only Georgia project featured by the Steel Joist Institute. Included in the design, as requested by the school, was a plan so 70% of classrooms would have windows.
Magnet program
The Academy of Mathematics, Science, & Technology at Kennesaw Mountain is one of seven magnet schools at the high school level in the Cobb County School District. For the class of 2023, the Academy of Mathematics, Science, & Technology at Kennesaw Mountain had an acceptance rate of 52%, and the class of 2026, of those who where accepted into the Magnet Program had an average 8th grade PSAT 8/9 NPR of 93% for Math (520), and 92% for Evidence Based Reading and Writing (540).
Athletics
The Mustangs are a member of GHSA and participate in Region 6-AAAAAA. The school offers football, softball, cross country, volleyball, basketball, wrestling, swimming and diving, baseball, golf, marching band, lacrosse, soccer, tennis, and track and field.
Notable alumni
Corey Heim, racecar driver
Omar Jimenez, Journalist
Tyler Stephenson, Baseball Player
References
External links
School website
KMHS Magnet School
Schools in Cobb County, Georgia
Public high schools in Georgia (U.S. state)
Educational institutions established in 2000
Magnet schools in Georgia (U.S. state)
2000 establishments in Georgia (U.S. state)
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https://en.wikipedia.org/wiki/Disjunctive%20sum
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In the mathematics of combinatorial games, the sum or disjunctive sum of two games is a game in which the two games are played in parallel, with each player being allowed to move in just one of the games per turn. The sum game finishes when there are no moves left in either of the two parallel games, at which point (in normal play) the last player to move wins.
This operation may be extended to disjunctive sums of any number of games, again by playing the games in parallel and moving in exactly one of the games per turn. It is the fundamental operation that is used in the Sprague–Grundy theorem for impartial games and which led to the field of combinatorial game theory for partisan games.
Application to common games
Disjunctive sums arise in games that naturally break up into components or regions that do not interact except in that each player in turn must choose just one component to play in. Examples of such games are Go, Nim, Sprouts, Domineering, the Game of the Amazons, and the map-coloring games.
In such games, each component may be analyzed separately for simplifications that do not affect its outcome or the outcome of its disjunctive sum with other games. Once this analysis has been performed, the components can be combined by taking the disjunctive sum of two games at a time, combining them into a single game with the same outcome as the original game.
Mathematics
The sum operation was formalized by . It is a commutative and associative operation: if two games are combined, the outcome is the same regardless of what order they are combined, and if more than two games are combined, the outcome is the same regardless of how they are grouped.
The negation −G of a game G (the game formed by trading the roles of the two players) forms an additive inverse under disjunctive sums: the game G + −G is a zero game (won by whoever goes second) using a simple echoing strategy in which the second player repeatedly copies the first player's move in the other game. For any two games G and H, the game H + G + −G has the same outcome as H itself (although it may have a larger set of available moves).
Based on these properties, the class of combinatorial games may be thought of as having the structure of an abelian group, although with a proper class of elements rather than (as is more standard for groups) a set of elements. For an important subclass of the games called the surreal numbers, there exists a multiplication operator that extends this group to a field.
For impartial misère play games, an analogous theory of sums can be developed, but with fewer of these properties: these games form a commutative monoid with only one nontrivial invertible element, called star (*), of order two.
References
.
Combinatorial game theory
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https://en.wikipedia.org/wiki/List%20of%20ARCA%20drivers
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The following is a list of drivers who are currently competing in a series sanctioned by the Automobile Racing Club of America (ARCA).
ARCA Racing Series drivers
All statistics used in these tables are as of the end of the 2018 Lucas Oil 200 Driven by General Tire. (Race 1/20)
Full-time drivers
Part-time drivers
Cashiers
ARCA Midwest Tour drivers
All statistics used in these tables are as of the end of the 2016 Oktoberfest 200. (Race 10/10)
Full-time drivers
Part-time drivers
ARCA/CRA Super Series drivers
All statistics used in these tables are as of the end of the 2016 Winchester 400. (Race 13/13)
Full-time drivers
Part-time drivers
External links
ARCA Racing Series website
ARCA Racing Series drivers
ARCA Midwest Tour website
ARCA Midwest Tour all-time statistics
CRA Super Series website
Racing-Reference.info
Arca drivers
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https://en.wikipedia.org/wiki/Mathematics%20Subject%20Classification
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The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme that has collaboratively been produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. The MSC is used by many mathematics journals, which ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The current version is MSC2020.
Structure
The MSC is a hierarchical scheme, with three levels of structure. A classification can be two, three or five digits long, depending on how many levels of the classification scheme are used.
The first level is represented by a two-digit number, the second by a letter, and the third by another two-digit number. For example:
53 is the classification for differential geometry
53A is the classification for classical differential geometry
53A45 is the classification for vector and tensor analysis
First level
At the top level, 64 mathematical disciplines are labeled with a unique two-digit number. In addition to the typical areas of mathematical research, there are top-level categories for "History and Biography", "Mathematics Education", and for the overlap with different sciences. Physics (i.e. mathematical physics) is particularly well represented in the classification scheme with a number of different categories including:
Fluid mechanics
Quantum mechanics
Geophysics
Optics and electromagnetic theory
All valid MSC classification codes must have at least the first-level identifier.
Second level
The second-level codes are a single letter from the Latin alphabet. These represent specific areas covered by the first-level discipline. The second-level codes vary from discipline to discipline.
For example, for differential geometry, the top-level code is 53, and the second-level codes are:
A for classical differential geometry
B for local differential geometry
C for global differential geometry
D for symplectic geometry and contact geometry
In addition, the special second-level code "-" is used for specific kinds of materials. These codes are of the form:
53-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
53-01 Instructional exposition (textbooks, tutorial papers, etc.)
53-02 Research exposition (monographs, survey articles)
53-03 Historical (must also be assigned at least one classification number from Section 01)
53-04 Explicit machine computation and programs (not the theory of computation or programming)
53-06 Proceedings, conferences, collections, etc.
The second and third level of these codes are always the same - only the first level changes. For example, it is not valid to use 53- as a classification. Either 53 on its own or, better yet, a more specific code should be used.
Third level
Third-level codes are the most specific, usually corresponding to a specific kind of mathematical object or a well-know
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https://en.wikipedia.org/wiki/Polish%20School%20of%20Mathematics
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The Polish School of Mathematics was the mathematics community that flourished in Poland in the 20th century, particularly during the Interbellum between World Wars I and II.
Overview
The Polish School of Mathematics subsumed:
the Lwów School of Mathematics - mostly focused on functional analysis;
the Warsaw School of Mathematics - mostly focused on set theory, mathematical logic and topology; and
the Kraków School of Mathematics - mostly focused on differential equations, analytic functions, differential geometry.
Nomenclature
Poland's mathematicians provided a name to Polish notation and Polish space.
Background
It has been debated what stimulated the exceptional efflorescence of mathematics in Poland after World War I. Important preparatory work had been done by the Polish "Positivists" following the disastrous January 1863 Uprising. The Positivists extolled science and technology, and popularized slogans of "organic work" and "building from the foundations." In the 20th century, mathematics was a field of endeavor that could be successfully pursued even with the limited resources that Poland commanded in the interbellum period.
Historical Influences
Over the centuries, Polish mathematicians have influenced the course of history. Copernicus used mathematics to buttress his revolutionary heliocentric theory. Four hundred years later, Marian Rejewski — subsequently assisted by fellow mathematician-cryptologists Jerzy Różycki and Henryk Zygalski — in December 1932 first broke the German Enigma machine cipher, thus laying the foundations for British World War II reading of Enigma ciphers ("Ultra"). After the war, Stanisław Ulam showed Edward Teller how to construct a practicable hydrogen bomb.
See also
Lwów-Warsaw School of Logic.
References
Kazimierz Kuratowski (1980) A Half Century of Polish Mathematics: Remembrances and Reflections, Oxford, Pergamon Press, .
Roman Murawski (2014) The Philosophy and Mathematics of Logic in the 1920s and 1930s in Poland, Maria Kantor translator, Birkhäuser
History of mathematics
History of education in Poland
Polish mathematics
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https://en.wikipedia.org/wiki/Taubes%27s%20Gromov%20invariant
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In mathematics, the Gromov invariant of Clifford Taubes counts embedded (possibly disconnected) pseudoholomorphic curves in a symplectic 4-manifold, where the curves are holomorphic with respect to an auxiliary compatible almost complex structure. (Multiple covers of 2-tori with self-intersection 0 are also counted.)
Taubes proved the information contained in this invariant is equivalent to invariants derived from the Seiberg–Witten equations in a series of four long papers. Much of the analytical complexity connected to this invariant comes from properly counting multiply covered pseudoholomorphic curves so that the result is invariant of the choice of almost complex structure. The crux is a topologically defined index for pseudoholomorphic curves which controls embeddedness and bounds the Fredholm index.
Embedded contact homology is an extension due to Michael Hutchings of this work to noncompact four-manifolds of the form , where Y is a compact contact 3-manifold. ECH is a symplectic field theory-like invariant; namely, it is the homology of a chain complex generated by certain combinations of Reeb orbits of a contact form on Y, and whose differential counts certain embedded pseudoholomorphic curves and multiply covered pseudoholomorphic cylinders with "ECH index" 1 in . The ECH index is a version of Taubes's index for the cylindrical case, and again, the curves are pseudoholomorphic with respect to a suitable almost complex structure. The result is a topological invariant of Y, which Taubes proved is isomorphic to monopole Floer homology, a version of Seiberg–Witten homology for Y.
References
Symplectic topology
4-manifolds
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https://en.wikipedia.org/wiki/Null%20%28mathematics%29
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In mathematics, the word null (from meaning "zero", which is from meaning "none") is often associated with the concept of zero or the concept of nothing. It is used in varying context from "having zero members in a set" (e.g., null set) to "having a value of zero" (e.g., null vector).
In a vector space, the null vector is the neutral element of vector addition; depending on the context, a null vector may also be a vector mapped to some null by a function under consideration (such as a quadratic form coming with the vector space, see null vector, a linear mapping given as matrix product or dot product, a seminorm in a Minkowski space, etc.). In set theory, the empty set, that is, the set with zero elements, denoted "{}" or "∅", may also be called null set. In measure theory, a null set is a (possibly nonempty) set with zero measure.
A null space of a mapping is the part of the domain that is mapped into the null element of the image (the inverse image of the null element). For example, in linear algebra, the null space of a linear mapping, also known as kernel, is the set of vectors which map to the null vector under that mapping.
In statistics, a null hypothesis is a proposition that no effect or relationship exists between populations and phenomena. It is the hypothesis which is presumed true—unless statistical evidence indicates otherwise.
See also
0
Null sign
References
Mathematical terminology
0 (number)
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https://en.wikipedia.org/wiki/Moduli%20of%20algebraic%20curves
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In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.
The most basic problem is that of moduli of smooth complete curves of a fixed genus. Over the field of complex numbers these correspond precisely to compact Riemann surfaces of the given genus, for which Bernhard Riemann proved the first results about moduli spaces, in particular their dimensions ("number of parameters on which the complex structure depends").
Moduli stacks of stable curves
The moduli stack classifies families of smooth projective curves, together with their isomorphisms. When , this stack may be compactified by adding new "boundary" points which correspond to stable nodal curves (together with their isomorphisms). A curve is stable if it is complete, connected, has no singularities other than double points, and has only a finite group of automorphisms. The resulting stack is denoted . Both moduli stacks carry universal families of curves.
Both stacks above have dimension ; hence a stable nodal curve can be completely specified by choosing the values of parameters, when . In lower genus, one must account for the presence of smooth families of automorphisms, by subtracting their number. There is exactly one complex curve of genus zero, the Riemann sphere, and its group of isomorphisms is PGL(2). Hence the dimension of is equal to
Likewise, in genus 1, there is a one-dimensional space of curves, but every such curve has a one-dimensional group of automorphisms. Hence, the stack has dimension 0.
Construction and irreducibility
It is a non-trivial theorem, proved by Pierre Deligne and David Mumford, that the moduli stack is irreducible, meaning it cannot be expressed as the union of two proper substacks. They prove this by analyzing the locus of stable curves in the Hilbert scheme of tri-canonically embedded curves (from the embedding of the very ample for every curve) which have Hilbert polynomial . Then, the stack is a construction of the moduli space . Using deformation theory, Deligne and Mumford show this stack is smooth and use the stack of isomorphisms between stable curves , to show that has finite stabilizers, hence it is a Deligne–Mumford stack. Moreover, they find a stratification of as
,
where is the subscheme of smooth stable curves and is an irreducible component of . They analyze the components of (as a GIT quotient). If there existed multiple components of , none of them would be complete. Also, any component of must contain non-singular curves. Consequently, the singular locus is connected, hence it is contained i
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https://en.wikipedia.org/wiki/Biharmonic%20equation
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In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of thin structures that react elastically to external forces.
Notation
It is written as
or
or
where , which is the fourth power of the del operator and the square of the Laplacian operator (or ), is known as the biharmonic operator or the bilaplacian operator. In Cartesian coordinates, it can be written in dimensions as:
Because the formula here contains a summation of indices, many mathematicians prefer the notation over because the former makes clear which of the indices of the four nabla operators are contracted over.
For example, in three dimensional Cartesian coordinates the biharmonic equation has the form
As another example, in n-dimensional Real coordinate space without the origin ,
where
which shows, for n=3 and n=5 only, is a solution to the biharmonic equation.
A solution to the biharmonic equation is called a biharmonic function. Any harmonic function is biharmonic, but the converse is not always true.
In two-dimensional polar coordinates, the biharmonic equation is
which can be solved by separation of variables. The result is the Michell solution.
2-dimensional space
The general solution to the 2-dimensional case is
where , and are harmonic functions and is a harmonic conjugate of .
Just as harmonic functions in 2 variables are closely related to complex analytic functions, so are biharmonic functions in 2 variables. The general form of a biharmonic function in 2 variables can also be written as
where and are analytic functions.
See also
Harmonic function
References
Eric W Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2002. .
S I Hayek, Advanced Mathematical Methods in Science and Engineering, Marcel Dekker, 2000. .
External links
Elliptic partial differential equations
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https://en.wikipedia.org/wiki/Euler%20method
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In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1770).
The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size.
The Euler method often serves as the basis to construct more complex methods, e.g., predictor–corrector method.
Geometrical description
Purpose and why it works
Consider the problem of calculating the shape of an unknown curve which starts at a given point and satisfies a given differential equation. Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at any point on the curve, once the position of that point has been calculated.
The idea is that while the curve is initially unknown, its starting point, which we denote by is known (see Figure 1). Then, from the differential equation, the slope to the curve at can be computed, and so, the tangent line.
Take a small step along that tangent line up to a point Along this small step, the slope does not change too much, so will be close to the curve. If we pretend that is still on the curve, the same reasoning as for the point above can be used. After several steps, a polygonal curve () is computed. In general, this curve does not diverge too far from the original unknown curve, and the error between the two curves can be made small if the step size is small enough and the interval of computation is finite.
First-order process
When given the values for and , and the derivative of is a given function of and denoted as . Begin the process by setting . Next, choose a value for the size of every step along t-axis, and set (or equivalently ). Now, the Euler method is used to find from and :
The value of is an approximation of the solution at time , i.e., . The Euler method is explicit, i.e. the solution is an explicit function of for .
Higher-order process
While the Euler method integrates a first-order ODE, any ODE of order can be represented as a system of first-order ODEs. When given the ODE of order defined as
as well as , , and , we implement the following formula until we reach the approximation of the solution to the ODE at the desired time:
These first-order systems can be handled by Euler's method or, in fact, by any other scheme for first-order systems.
First-order example
Given the initial value problem
we would like to use the Euler method to approximate .
Using step size equal to 1 ()
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https://en.wikipedia.org/wiki/Gilbert%20Ames%20Bliss
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Gilbert Ames Bliss, (9 May 1876 – 8 May 1951), was an American mathematician, known for his work on the calculus of variations.
Life
Bliss grew up in a Chicago family that eventually became affluent; in 1907, his father became president of the company supplying all of Chicago's electricity. The family was not affluent, however, when Bliss entered the University of Chicago in 1893 (its second year of operation). Hence he had to support himself while a student by winning a scholarship, and by playing in a student professional mandolin quartet.
After obtaining the B.Sc. in 1897, he began graduate studies at Chicago in mathematical astronomy (his first publication was in that field), switching in 1898 to mathematics. He discovered his life's work, the calculus of variations, via the lecture notes of Weierstrass's 1879 course, and Bolza's teaching. Bolza went on to supervise Bliss's Ph.D. thesis, The Geodesic Lines on the Anchor Ring, completed in 1900 and published in the Annals of Mathematics in 1902. After two years as an instructor at the University of Minnesota, Bliss spent the 1902–03 academic year at the University of Göttingen, interacting with Felix Klein, David Hilbert, Hermann Minkowski, Ernst Zermelo, Erhard Schmidt, Max Abraham, and Constantin Carathéodory.
Upon returning to the United States, Bliss taught one year each at the University of Chicago and the University of Missouri. In 1904, he published two more papers on the calculus of variations in the Transactions of the American Mathematical Society. Bliss was a Preceptor at Princeton University, 1905–08, joining a strong group of young mathematicians that included Luther P. Eisenhart, Oswald Veblen, and Robert Lee Moore. While at Princeton he was also an associate editor of the Annals of Mathematics.
In 1908, Chicago's Maschke died and Bliss was hired to replace him; Bliss remained at Chicago until his 1941 retirement. While at Chicago, he was an editor of the Transactions of the American Mathematical Society, 1908–16, and chaired the Mathematics Department, 1927–41. That Department was less distinguished under Bliss than it had been under E. H. Moore's previous leadership, and than it would become under Marshall Stone's and Saunders MacLane's direction after World War II. A near-contemporary of Bliss's at Chicago was the algebraist Leonard Dickson.
During World War I, he worked on ballistics, designing new firing tables for artillery, and lectured on navigation. In 1918, he and Oswald Veblen worked together in the Range Firing Section at the Aberdeen Proving Ground, applying the calculus of variations to correct shell trajectories for the effects of wind, changes in air density, the rotation of the Earth, and other perturbations.
Bliss married Helen Hurd in 1912, who died in the 1918 influenza pandemic; their two children survived. Bliss married Olive Hunter in 1920; they had no children.
Bliss was elected to the National Academy of Sciences (United States) in 1916. He was the
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https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch%20theorem%20for%20smooth%20manifolds
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In mathematics, a Riemann–Roch theorem for smooth manifolds is a version of results such as the Hirzebruch–Riemann–Roch theorem or Grothendieck–Riemann–Roch theorem (GRR) without a hypothesis making the smooth manifolds involved carry a complex structure. Results of this kind were obtained by Michael Atiyah and Friedrich Hirzebruch in 1959, reducing the requirements to something like a spin structure.
Formulation
Let X and Y be oriented smooth closed manifolds,
and f: X → Y a continuous map.
Let vf=f*(TY) − TX in the K-group
K(X).
If dim(X) ≡ dim(Y) mod 2, then
where ch is the Chern character, d(vf) an element of
the integral cohomology group H2(Y, Z) satisfying
d(vf) ≡ f* w2(TY)-w2(TX) mod 2,
fK* the Gysin homomorphism for K-theory,
and fH* the Gysin homomorphism for cohomology
.
This theorem was first proven by Atiyah and Hirzebruch.
The theorem is proven by considering several special cases.
If Y is the Thom space of a vector bundle V over X,
then the Gysin maps are just the Thom isomorphism.
Then, using the splitting principle, it suffices to check the theorem via explicit computation for line
bundles.
If f: X → Y is an embedding, then the
Thom space of the normal bundle of X in Y can be viewed as a tubular neighborhood of X
in Y, and excision gives a map
and
.
The Gysin map for K-theory/cohomology is defined to be the composition of the Thom isomorphism with these maps.
Since the theorem holds for the map from X to the Thom space of N,
and since the Chern character commutes with u and v, the theorem is also true for embeddings.
f: X → Y.
Finally, we can factor a general map f: X → Y
into an embedding
and the projection
The theorem is true for the embedding.
The Gysin map for the projection is the Bott-periodicity isomorphism, which commutes with the Chern character,
so the theorem holds in this general case also.
Corollaries
Atiyah and Hirzebruch then specialised and refined in the case X = a point, where the condition becomes the existence of a spin structure on Y. Corollaries are on Pontryagin classes and the J-homomorphism.
Notes
Theorems in differential geometry
Algebraic surfaces
Bernhard Riemann
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https://en.wikipedia.org/wiki/Upper%20and%20lower%20probabilities
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Upper and lower probabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, this method uses two numbers: the upper probability of the event and the lower probability of the event.
Because frequentist statistics disallows metaprobabilities, frequentists have had to propose new solutions. Cedric Smith and Arthur Dempster each developed a theory of upper and lower probabilities. Glenn Shafer developed Dempster's theory further, and it is now known as Dempster–Shafer theory or Choquet (1953).
More precisely, in the work of these authors one considers in a power set, , a mass function satisfying the conditions
In turn, a mass is associated with two non-additive continuous measures called belief and plausibility defined as follows:
In the case where is infinite there can be such that there is no associated mass function. See p. 36 of Halpern (2003). Probability measures are a special case of belief functions in which the mass function assigns positive mass to singletons of the event space only.
A different notion of upper and lower probabilities is obtained by the lower and upper envelopes obtained from a class C of probability distributions by setting
The upper and lower probabilities are also related with probabilistic logic: see Gerla (1994).
Observe also that a necessity measure can be seen as a lower probability and a possibility measure can be seen as an upper probability.
See also
Possibility theory
Fuzzy measure theory
Interval finite element
Probability bounds analysis
References
Exotic probabilities
Probability bounds analysis
Dempster–Shafer theory
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https://en.wikipedia.org/wiki/Helge%20Tverberg
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Helge Arnulf Tverberg (March 6, 1935December 28, 2020) was a Norwegian mathematician. He was a professor in the Mathematics Department at the University of Bergen, his speciality being combinatorics; he retired at the mandatory age of seventy.
He was born in Bergen. He took the cand.real. degree at the University of Bergen in 1958, and the dr.philos. degree in 1968. He was a lecturer from 1958 to 1971 and professor from 1971 to his retirement in 2005. He was a visiting scholar at the University of Reading in 1966 and at the Australian National University, in Canberra, from 1980 to 1981, 1987 to 1988 and in 2004. He was a member of the Norwegian Academy of Science and Letters.
Tverberg, in 1965, proved a result on intersection patterns of partitions of point configurations that has come to be known as Tverberg's partition theorem. It inaugurated a new branch of combinatorial geometry, with many variations and applications. An account by Günter M. Ziegler of Tverberg's work in this direction appeared in the issue of the Notices of the American Mathematical Society for April, 2011.
See also
Geometric separator
References
1935 births
Living people
20th-century Norwegian mathematicians
Combinatorialists
Academic staff of the University of Bergen
University of Bergen alumni
Members of the Norwegian Academy of Science and Letters
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https://en.wikipedia.org/wiki/Zeta%20function%20regularization
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In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly applied to problems in physics, but has its origins in attempts to give precise meanings to ill-conditioned sums appearing in number theory.
Definition
There are several different summation methods called zeta function regularization for defining the sum of a possibly divergent series
One method is to define its zeta regularized sum to be ζA(−1) if this is defined, where the zeta function is defined for large Re(s) by
if this sum converges, and by analytic continuation elsewhere.
In the case when an = n, the zeta function is the ordinary Riemann zeta function. This method was used by Euler to "sum" the series 1 + 2 + 3 + 4 + ... to ζ(−1) = −1/12.
showed that in flat space, in which the eigenvalues of Laplacians are known, the zeta function corresponding to the partition function can be computed explicitly. Consider a scalar field φ contained in a large box of volume V in flat spacetime at the temperature T = β−1. The partition function is defined by a path integral over all fields φ on the Euclidean space obtained by putting τ = it which are zero on the walls of the box and which are periodic in τ with period β. In this situation from the partition function he computes energy, entropy and pressure of the radiation of the field φ. In case of flat spaces the eigenvalues appearing in the physical quantities are generally known, while in case of curved space they are not known: in this case asymptotic methods are needed.
Another method defines the possibly divergent infinite product a1a2.... to be exp(−ζ′A(0)). used this to define the determinant of a positive self-adjoint operator A (the Laplacian of a Riemannian manifold in their application) with eigenvalues a1, a2, ...., and in this case the zeta function is formally the trace of A−s. showed that if A is the Laplacian of a compact Riemannian manifold then the Minakshisundaram–Pleijel zeta function converges and has an analytic continuation as a meromorphic function to all complex numbers, and extended this to elliptic pseudo-differential operators A on compact Riemannian manifolds. So for such operators one can define the determinant using zeta function regularization. See "analytic torsion."
suggested using this idea to evaluate path integrals in curved spacetimes. He studied zeta function regularization in order to calculate the partition functions for thermal graviton and matter's quanta in curved background such as on the horizon of black holes and on de Sitter background using the relation by the inverse Mellin transformation to the trace of the kernel of heat equations.
Example
The first example in which zeta function regularization is available appears in the Casimir effect
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https://en.wikipedia.org/wiki/Carved%20turn
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A carved turn is a skiing and snowboarding term for the technique of turning by shifting the ski or snowboard onto its edges. When edged, the sidecut geometry causes the ski (in the following, snowboard is implicit and not mentioned) to bend into an arc, and the ski naturally follows this arc shape to produce a turning motion. The carve is efficient in allowing the skier to maintain speed because, unlike the older stem Christie and parallel turns, the skis don't create drag by sliding sideways.
Starting a carved turn requires the ski to be rotated onto its edge, which can be accomplished through angulation of the hips and knees applied to both skis, leading them to efficiently carve a naturally parallel turn. Carving turns are generally smoother and longer radius than either stemmed or parallel turns. Carving maintains the skis efficiently turning along the direction of travel as opposed to skidding at an angle across the direction of travel. For a given velocity, carving with shaped skis typically requires less effort than stemming or parallel and offers increased speed and control in even steep descents and highly energetic turns, making it ubiquitous in racing.
Prior to the introduction of "shaped skis" in the 1990s, the technique was not simple to learn. Since then, it has become accessible and carving is commonly taught as a form of parallel skiing alongside the classic parallel "brushed" technique. Modern downhill technique is generally a combination of carving and skidding, varying the ratio between the two when rapid control over the turn or speed is required. Pure carving is a useful technique on "groomers" – slopes of moderate steepness with smooth snow – with skis dedicated to this style. Other situations remain almost pure parallel Christie technique, such as competitive mogul skiing, with edged turn initiation aided by the moguls themselves.
History
Shaped skis, also called parabolic skis, make carved turns possible at low speeds and with short turn radius. Skis had sidecut since they were first carved from wood – typically just 5mm or so on a long ski. But it wasn't until the early 1980s that much deeper cuts were explored. In 1979 Head developed the "Natural Turning Radius" concept and skis with 7.3mm sidecut (~35 m radius). Olin Corp developed a teaching ski with an 8 m radius (31mm sidecut) and the first asymmetric ski, with no up hill cut and, because side cut involves proportionately wide tips, a platform for the boot to allow a very narrow waist. A total of 150 pairs were produced.
In 1990 Volkl released their metal "Explosiv" with a 10-mm sidecut and 28-m turn radius. K2 introduced a 10mm sidecut race ski, whose improved edging and turning ability became a sought after by consumers. Volant released a 12mm cut ski in 1992, followed by Dynastar, and K2.
Elan engineers Jurij Franko and Pavel Skofic experimentally adjusted sidecut and developed a physical model—desired radius, speed, forces and lean one could generate, and
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https://en.wikipedia.org/wiki/Grothendieck%20spectral%20sequence
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In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his Tôhoku paper, is a spectral sequence that computes the derived functors of the composition of two functors , from knowledge of the derived functors of and .
Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.
Statement
If and are two additive and left exact functors between abelian categories such that both and have enough injectives and takes injective objects to -acyclic objects, then for each object of there is a spectral sequence:
where denotes the p-th right-derived functor of , etc., and where the arrow '' means convergence of spectral sequences.
Five term exact sequence
The exact sequence of low degrees reads
Examples
The Leray spectral sequence
If and are topological spaces, let and be the category of sheaves of abelian groups on and , respectively.
For a continuous map there is the (left-exact) direct image functor .
We also have the global section functors
and
Then since and the functors and satisfy the hypotheses (since the direct image functor has an exact left adjoint , pushforwards of injectives are injective and in particular acyclic for the global section functor), the sequence in this case becomes:
for a sheaf of abelian groups on .
Local-to-global Ext spectral sequence
There is a spectral sequence relating the global Ext and the sheaf Ext: let F, G be sheaves of modules over a ringed space ; e.g., a scheme. Then
This is an instance of the Grothendieck spectral sequence: indeed,
, and .
Moreover, sends injective -modules to flasque sheaves, which are -acyclic. Hence, the hypothesis is satisfied.
Derivation
We shall use the following lemma:
Proof: Let be the kernel and the image of . We have
which splits. This implies each is injective. Next we look at
It splits, which implies the first part of the lemma, as well as the exactness of
Similarly we have (using the earlier splitting):
The second part now follows.
We now construct a spectral sequence. Let be an injective resolution of A. Writing for , we have:
Take injective resolutions and of the first and the third nonzero terms. By the horseshoe lemma, their direct sum is an injective resolution of . Hence, we found an injective resolution of the complex:
such that each row satisfies the hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.)
Now, the double complex gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition,
,
which is always zero unless q = 0 since is G-acyclic by hypothesis. Hence, and . On the other hand, by the definition and the lemma,
Since is an injective resolution of (it is a resolution since its cohomology is trivial),
Since and have the same limiting term, the proof is complete.
Notes
Referenc
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https://en.wikipedia.org/wiki/Projective%20object
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In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object.
Definition
An object in a category is projective if for any epimorphism and morphism , there is a morphism such that , i.e. the following diagram commutes:
That is, every morphism factors through every epimorphism .
If C is locally small, i.e., in particular is a set for any object X in C, this definition is equivalent to the condition that the hom functor (also known as corepresentable functor)
preserves epimorphisms.
Projective objects in abelian categories
If the category C is an abelian category such as, for example, the category of abelian groups, then P is projective if and only if
is an exact functor, where Ab is the category of abelian groups.
An abelian category is said to have enough projectives if, for every object of , there is a projective object of and an epimorphism from P to A or, equivalently, a short exact sequence
The purpose of this definition is to ensure that any object A admits a projective resolution, i.e., a (long) exact sequence
where the objects are projective.
Projectivity with respect to restricted classes
discusses the notion of projective (and dually injective) objects relative to a so-called bicategory, which consists of a pair of subcategories of "injections" and "surjections" in the given category C. These subcategories are subject to certain formal properties including the requirement that any surjection is an epimorphism. A projective object (relative to the fixed class of surjections) is then an object P so that Hom(P, −) turns the fixed class of surjections (as opposed to all epimorphisms) into surjections of sets (in the usual sense).
Properties
The coproduct of two projective objects is projective.
The retract of a projective object is projective.
Examples
The statement that all sets are projective is equivalent to the axiom of choice.
The projective objects in the category of abelian groups are the free abelian groups.
Let be a ring with identity. Consider the (abelian) category -Mod of left -modules. The projective objects in -Mod are precisely the projective left R-modules. Consequently, is itself a projective object in -Mod. Dually, the injective objects in -Mod are exactly the injective left R-modules.
The category of left (right) -modules also has enough projectives. This is true since, for every left (right) -module , we can take to be the free (and hence projective) -module generated by a generating set for (for example we can take to be ). Then the canonical projection is the required surjection.
The projective objects in the category of compact Hausdorff spaces are precisely the extremally disconnected spaces. This result is due to , with a simplified proof given by .
In the category of Banach spaces and contract
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https://en.wikipedia.org/wiki/Leray%27s%20theorem
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In algebraic topology and algebraic geometry, Leray's theorem (so named after Jean Leray) relates abstract sheaf cohomology with Čech cohomology.
Let be a sheaf on a topological space and an open cover of If is acyclic on every finite intersection of elements of , then
where is the -th Čech cohomology group of with respect to the open cover
References
Bonavero, Laurent. Cohomology of Line Bundles on Toric Varieties, Vanishing Theorems. Lectures 16-17 from "Summer School 2000: Geometry of Toric Varieties."
Sheaf theory
Theorems in algebraic geometry
Theorems in algebraic topology
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https://en.wikipedia.org/wiki/De%20Rham%E2%80%93Weil%20theorem
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In algebraic topology, the De Rham–Weil theorem allows computation of sheaf cohomology using an acyclic resolution of the sheaf in question.
Let be a sheaf on a topological space and a resolution of by acyclic sheaves. Then
where denotes the -th sheaf cohomology group of with coefficients in
The De Rham–Weil theorem follows from the more general fact that derived functors may be computed using acyclic resolutions instead of simply injective resolutions.
References
Homological algebra
Sheaf theory
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https://en.wikipedia.org/wiki/Gelfand%20pair
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In mathematics, a Gelfand pair is a pair (G,K) consisting of a group G and a subgroup K (called an Euler subgroup of G) that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to the theory of Riemannian symmetric spaces in differential geometry. Broadly speaking, the theory exists to abstract from these theories their content in terms of harmonic analysis and representation theory.
When G is a finite group the simplest definition is, roughly speaking, that the (K,K)-double cosets in G commute. More precisely, the Hecke algebra, the algebra of functions on G that are invariant under translation on either side by K, should be commutative for the convolution on G.
In general, the definition of Gelfand pair is roughly that the restriction to K of any irreducible representation of G contains the trivial representation of K with multiplicity no more than 1. In each case one should specify the class of considered representations and the meaning of contains.
Definitions
In each area, the class of representations and the definition of containment for representations is slightly different. Explicit definitions in several such cases are given here.
Finite group case
When G is a finite group the following are equivalent
(G,K) is a Gelfand pair.
The algebra of (K,K)-double invariant functions on G with multiplication defined by convolution is commutative.
For any irreducible representation π of G, the space πK of K-invariant vectors in π is no-more-than-1-dimensional.
For any irreducible representation π of G, the dimension of HomK(π, C) is less than or equal to 1, where C denotes the trivial representation.
The permutation representation of G on the cosets of K is multiplicity-free, that is, it decomposes into a direct sum of distinct absolutely irreducible representations in characteristic zero.
The centralizer algebra (Schur algebra) of the permutation representation is commutative.
(G/N, K/N) is a Gelfand pair, where N is a normal subgroup of G contained in K.
Compact group case
When G is a compact topological group the following are equivalent:
(G,K) is a Gelfand pair.
The algebra of (K,K)-double invariant compactly supported continuous measures on G with multiplication defined by convolution is commutative.
For any continuous, locally convex, irreducible representation π of G, the space πK of K-invariant vectors in π is no-more-than-1-dimensional.
For any continuous, locally convex, irreducible representation π of G the dimension of HomK(π,C) is less than or equal to 1.
The representation L2(G/K) of G is multiplicity free i.e. it is a direct sum of distinct unitary irreducible representations.
Lie group with compact subgroup
When G is a Lie group and K is a compact subgroup the following are equivalent:
(G,K) is a Gelfand pair.
The algebra of (K,K)-double invariant compactly supported conti
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https://en.wikipedia.org/wiki/Formal%20moduli
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In mathematics, formal moduli are an aspect of the theory of moduli spaces (of algebraic varieties or vector bundles, for example), closely linked to deformation theory and formal geometry. Roughly speaking, deformation theory can provide the Taylor polynomial level of information about deformations, while formal moduli theory can assemble consistent Taylor polynomials to make a formal power series theory. The step to moduli spaces, properly speaking, is an algebraization question, and has been largely put on a firm basis by Artin's approximation theorem.
A formal universal deformation is by definition a formal scheme over a complete local ring, with special fiber the scheme over a field being studied, and with a universal property amongst such set-ups. The local ring in question is then the carrier of the formal moduli.
References
Moduli theory
Algebraic geometry
Geometric algebra
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https://en.wikipedia.org/wiki/Asian%20Pacific%20Mathematics%20Olympiad
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The Asian Pacific Mathematics Olympiad (APMO) starting from 1989 is a regional mathematics competition which involves countries from the Asian Pacific region. The United States also takes part in the APMO. Every year, APMO is held in the afternoon of the second Monday of March for participating countries in the North and South Americas, and in the morning of the second Tuesday of March for participating countries on the Western Pacific and in Asia.
APMO's Aims
the discovering, encouraging and challenging of mathematically gifted school students in all Pacific-Rim countries
the fostering of friendly international relations and cooperation between students and teachers in the Pacific-Rim Region
the creating of an opportunity for the exchange of information on school syllabi and practice throughout the Pacific Region
the encouragement and support of mathematical involvement with Olympiad type activities, not only in the APMO participating countries, but also in other Pacific-Rim countries.
Scoring and Format
The APMO contest consists of one four-hour paper consisting of five questions of varying difficulty and each having a maximum score of 7 points. Contestants should not have formally enrolled at a university (or equivalent post-secondary institution) and they must be younger than 20 years of age on 1 July of the year of the contest.
APMO Member Nations/Regions
Observer Nations
Honduras and South Africa
Results
https://cms.math.ca/Competitions/APMO/
https://www.apmo-official.org/
https://www.apmo-official.org/2017/ResultsByName.html
http://imomath.com/index.php?options=Ap&mod=23&ttn=Asian-Pacific
See also
International Mathematical Olympiad
External links
APMO Official Website
Mathematics competitions
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https://en.wikipedia.org/wiki/Hilbert%20class%20field
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In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime ideals in K.
In this context, the Hilbert class field of K is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of K. That is, every real embedding of K extends to a real embedding of E (rather than to a complex embedding of E).
Examples
If the ring of integers of K is a unique factorization domain, in particular if , then K is its own Hilbert class field.
Let of discriminant . The field has discriminant and so is an everywhere unramified extension of K, and it is abelian. Using the Minkowski bound, one can show that K has class number 2. Hence, its Hilbert class field is . A non-principal ideal of K is (2,(1+)/2), and in L this becomes the principal ideal ((1+)/2).
The field has class number 3. Its Hilbert class field can be formed by adjoining a root of x3 - x - 1, which has discriminant -23.
To see why ramification at the archimedean primes must be taken into account, consider the real quadratic field K obtained by adjoining the square root of 3 to Q. This field has class number 1 and discriminant 12, but the extension K(i)/K of discriminant 9=32 is unramified at all prime ideals in K, so K admits finite abelian extensions of degree greater than 1 in which all finite primes of K are unramified. This doesn't contradict the Hilbert class field of K being K itself: every proper finite abelian extension of K must ramify at some place, and in the extension K(i)/K there is ramification at the archimedean places: the real embeddings of K extend to complex (rather than real) embeddings of K(i).
By the theory of complex multiplication, the Hilbert class field of an imaginary quadratic field is generated by the value of the elliptic modular function at a generator for the ring of integers (as a Z-module).
History
The existence of a (narrow) Hilbert class field for a given number field K was conjectured by and proved by Philipp Furtwängler. The existence of the Hilbert class field is a valuable tool in studying the structure of the ideal class group of a given field.
Additional properties
The Hilbert class field E also satisfies the following:
E is a finite Galois extension of K and [E : K] = hK, where hK is the class number of K.
The ideal class group of K is isomorphic to the Galois group of E over K.
Every ideal of OK extends to a principal ideal of the ring extension OE (principal ideal theorem).
Every prime ideal P of OK decomposes into the product of hK / f prime ideals in OE, where f is the order of [P] in the ideal class group of OK.
In fact, E is the unique field satisfying the first, second, and fourth properties.
Explicit constructions
If K is imaginary quadratic and
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https://en.wikipedia.org/wiki/Class%20number%20formula
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In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function.
General statement of the class number formula
We start with the following data:
is a number field.
, where denotes the number of real embeddings of , and is the number of complex embeddings of .
is the Dedekind zeta function of .
is the class number, the number of elements in the ideal class group of .
is the regulator of .
is the number of roots of unity contained in .
is the discriminant of the extension .
Then:
Theorem (Class Number Formula). converges absolutely for and extends to a meromorphic function defined for all complex with only one simple pole at , with residue
This is the most general "class number formula". In particular cases, for example when is a cyclotomic extension of , there are particular and more refined class number formulas.
Proof
The idea of the proof of the class number formula is most easily seen when K = Q(i). In this case, the ring of integers in K is the Gaussian integers.
An elementary manipulation shows that the residue of the Dedekind zeta function at s = 1 is the average of the coefficients of the Dirichlet series representation of the Dedekind zeta function. The n-th coefficient of the Dirichlet series is essentially the number of representations of n as a sum of two squares of nonnegative integers. So one can compute the residue of the Dedekind zeta function at s = 1 by computing the average number of representations. As in the article on the Gauss circle problem, one can compute this by approximating the number of lattice points inside of a quarter circle centered at the origin, concluding that the residue is one quarter of pi.
The proof when K is an arbitrary imaginary quadratic number field is very similar.
In the general case, by Dirichlet's unit theorem, the group of units in the ring of integers of K is infinite. One can nevertheless reduce the computation of the residue to a lattice point counting problem using the classical theory of real and complex embeddings and approximate the number of lattice points in a region by the volume of the region, to complete the proof.
Dirichlet class number formula
Peter Gustav Lejeune Dirichlet published a proof of the class number formula for quadratic fields in 1839, but it was stated in the language of quadratic forms rather than classes of ideals. It appears that Gauss already knew this formula in 1801.
This exposition follows Davenport.
Let d be a fundamental discriminant, and write h(d) for the number of equivalence classes of quadratic forms with discriminant d. Let be the Kronecker symbol. Then is a Dirichlet character. Write for the Dirichlet L-series based on . For d > 0, let t > 0, u > 0 be the solution to the Pell equation for which u is smallest, and write
(Then is either a fundamental unit of the real quadratic field or the square of a fundamental unit.)
For d < 0, writ
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https://en.wikipedia.org/wiki/Current%20%28mathematics%29
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In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold M. Currents formally behave like Schwartz distributions on a space of differential forms, but in a geometric setting, they can represent integration over a submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out along subsets of M.
Definition
Let denote the space of smooth m-forms with compact support on a smooth manifold A current is a linear functional on which is continuous in the sense of distributions. Thus a linear functional
is an m-dimensional current if it is continuous in the following sense: If a sequence of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when tends to infinity, then tends to 0.
The space of m-dimensional currents on is a real vector space with operations defined by
Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current as the complement of the biggest open set such that
whenever
The linear subspace of consisting of currents with support (in the sense above) that is a compact subset of is denoted
Homological theory
Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m-current, denoted by :
If the boundary ∂M of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has:
This relates the exterior derivative d with the boundary operator ∂ on the homology of M.
In view of this formula we can define a boundary operator on arbitrary currents
via duality with the exterior derivative by
for all compactly supported m-forms
Certain subclasses of currents which are closed under can be used instead of all currents to create a homology theory, which can satisfy the Eilenberg–Steenrod axioms in certain cases. A classical example is the subclass of integral currents on Lipschitz neighborhood retracts.
Topology and norms
The space of currents is naturally endowed with the weak-* topology, which will be further simply called weak convergence. A sequence of currents, converges to a current if
It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm. If is an m-form, then define its comass by
So if is a simple m-form, then its mass norm is the usual L∞-norm of its coefficient. The mass of a current is then defined as
The mass of a current represents the weighted area of the generalized surface. A current such that M(T) < ∞ is representable by integration of a regular Borel measure by a version of the Riesz representation theorem. This is the starting
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https://en.wikipedia.org/wiki/K-homology
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In mathematics, K-homology is a homology theory on the category of locally compact Hausdorff spaces. It classifies the elliptic pseudo-differential operators acting on the vector bundles over a space. In terms of -algebras, it classifies the Fredholm modules over an algebra.
An operator homotopy between two Fredholm modules and is a norm continuous path of Fredholm modules, , Two Fredholm modules are then equivalent if they are related by unitary transformations or operator homotopies. The group is the abelian group of equivalence classes of even Fredholm modules over A. The group is the abelian group of equivalence classes of odd Fredholm modules over A. Addition is given by direct summation of Fredholm modules, and the inverse of is
References
N. Higson and J. Roe, Analytic K-homology. Oxford University Press, 2000.
K-theory
Homology theory
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https://en.wikipedia.org/wiki/Length%20function
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In the mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.
Definition
A length function L : G → R+ on a group G is a function satisfying:
Compare with the axioms for a metric and a filtered algebra.
Word metric
An important example of a length is the word metric: given a presentation of a group by generators and relations, the length of an element is the length of the shortest word expressing it.
Coxeter groups (including the symmetric group) have combinatorial important length functions, using the simple reflections as generators (thus each simple reflection has length 1). See also: length of a Weyl group element.
A longest element of a Coxeter group is both important and unique up to conjugation (up to different choice of simple reflections).
Properties
A group with a length function does not form a filtered group, meaning that the sublevel sets do not form subgroups in general.
However, the group algebra of a group with a length functions forms a filtered algebra: the axiom corresponds to the filtration axiom.
Group theory
Geometric group theory
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https://en.wikipedia.org/wiki/Van%20Aubel%27s%20theorem
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In plane geometry, Van Aubel's theorem describes a relationship between squares constructed on the sides of a quadrilateral. Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at right angles to one another. Another way of saying the same thing is that the center points of the four squares form the vertices of an equidiagonal orthodiagonal quadrilateral. The theorem is named after Belgian mathematician Henricus Hubertus (Henri) Van Aubel (1830–1906), who published it in 1878.
The theorem holds true also for re-entrant quadrilaterals, and when the squares are constructed internally to the given quadrilateral. For complex (self-intersecting) quadrilaterals, the external and internal constructions for the squares are not definable. In this case, the theorem holds true when the constructions are carried out in the more general way:
follow the quadrilateral vertices in a sequential direction and construct each square on the right hand side of each side of the given quadrilateral.
Follow the quadrilateral vertices in the same sequential direction and construct each square on the left hand side of each side of the given quadrilateral.
The segments joining the centers of the squares constructed externally (or internally) to the quadrilateral over two opposite sides have been referred to as Van Aubel segments. The points of intersection of two equal and orthogonal Van Aubel segments (produced when necessary) have been referred to as Van Aubel points: first or outer Van Aubel point for the external construction, second or inner Van Aubel point for the internal one.
The Van Aubel theorem configuration presents some relevant features, among others:
the Van Aubel points are the centers of the two circumscribed squares of the quadrilateral.
The Van Aubel points, the mid-points of the quadrilateral diagonals and the mid-points of the Van Aubel segments are concyclic.
A few extensions of the theorem, considering similar rectangles, similar rhombi and similar parallelograms constructed on the sides of the given quadrilateral, have been published on The Mathematical Gazette.
See also
Petr–Douglas–Neumann theorem
Thébault's theorem
Napoleon's theorem
Napoleon points
Bottema's theorem
References
External links
Van Aubel's Theorem for Quadrilaterals and Van Aubel's Theorem for Triangles by Jay Warendorff, The Wolfram Demonstrations Project.
The Beautiful Geometric Theorem of Van Aubel by Yutaka Nishiyama, International Journal of Pure and Applied Mathematics.
Interactive applet by Tim Brzezinski showing Van Aubel's Theorem made using GeoGebra.
Some generalizations of Van Aubel's theorem to similar quadrilaterals at Dynamic Geometry Sketches, interactive geometry sketches.
QG-2P6: Outer and Inner Van Aubel Points by Chris Van Tienhoven at Encyclopedia of Quadri-Figur
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https://en.wikipedia.org/wiki/Primary%20ideal
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In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n > 0. For example, in the ring of integers Z, (pn) is a primary ideal if p is a prime number.
The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals. This result is known as the Lasker–Noether theorem. Consequently, an irreducible ideal of a Noetherian ring is primary.
Various methods of generalizing primary ideals to noncommutative rings exist, but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity.
Examples and properties
The definition can be rephrased in a more symmetric manner: an ideal is primary if, whenever , we have or or . (Here denotes the radical of .)
An ideal Q of R is primary if and only if every zero divisor in R/Q is nilpotent. (Compare this to the case of prime ideals, where P is prime if and only if every zero divisor in R/P is actually zero.)
Any prime ideal is primary, and moreover an ideal is prime if and only if it is primary and semiprime (also called radical ideal in the commutative case).
Every primary ideal is primal.
If Q is a primary ideal, then the radical of Q is necessarily a prime ideal P, and this ideal is called the associated prime ideal of Q. In this situation, Q is said to be P-primary.
On the other hand, an ideal whose radical is prime is not necessarily primary: for example, if , , and , then is prime and , but we have , , and for all n > 0, so is not primary. The primary decomposition of is ; here is -primary and is -primary.
An ideal whose radical is maximal, however, is primary.
Every ideal with radical is contained in a smallest -primary ideal: all elements such that for some . The smallest -primary ideal containing is called the th symbolic power of .
If P is a maximal prime ideal, then any ideal containing a power of P is P-primary. Not all P-primary ideals need be powers of P, but at least they contain a power of P; for example the ideal (x, y2) is P-primary for the ideal P = (x, y) in the ring k[x, y], but is not a power of P, however it contains P².
If A is a Noetherian ring and P a prime ideal, then the kernel of , the map from A to the localization of A at P, is the intersection of all P-primary ideals.
A finite nonempty product of -primary ideals is -primary but an infinite product of -primary ideals may not be -primary; since for example, in a Noetherian local ring with maximal ideal , (Krull intersection theorem) where each is -primary, for example the infinite product of the maximal (and hence prime and hence primary) ideal of the local ring yields the zero ideal, which in this case is not primary (because the zero di
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https://en.wikipedia.org/wiki/Jasmone
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Jasmone is an organic compound, which is a volatile portion of the oil from jasmine flowers. It is a colorless to pale yellow liquid. Jasmone can exist in two isomeric forms with differing geometry around the pentenyl double bond, cis-jasmone and trans-jasmone. The natural extract contains only the cis form, while synthetic material is often a mixture of both, with the cis form predominating. Both forms have similar odors and chemical properties. Its structure was deduced by Lavoslav Ružička.
Jasmone is produced by some plants by the metabolism of jasmonic acid, via a decarboxylation. It can act as either an attractant or a repellent for various insects. Commercially, jasmone is used primarily in perfumes and cosmetics.
References
Enones
Cyclic ketones
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https://en.wikipedia.org/wiki/Direct%20image%20functor
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In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf F defined on a topological space X and a continuous map f: X → Y, we can define a new sheaf f∗F on Y, called the direct image sheaf or the pushforward sheaf of F along f, such that the global sections of f∗F is given by the global sections of F. This assignment gives rise to a functor f∗ from the category of sheaves on X to the category of sheaves on Y, which is known as the direct image functor. Similar constructions exist in many other algebraic and geometric contexts, including that of quasi-coherent sheaves and étale sheaves on a scheme.
Definition
Let f: X → Y be a continuous map of topological spaces, and let Sh(–) denote the category of sheaves of abelian groups on a topological space. The direct image functor
sends a sheaf F on X to its direct image presheaf f∗F on Y, defined on open subsets U of Y by
This turns out to be a sheaf on Y, and is called the direct image sheaf or pushforward sheaf of F along f.
Since a morphism of sheaves φ: F → G on X gives rise to a morphism of sheaves f∗(φ): f∗(F) → f∗(G) on Y in an obvious way, we indeed have that f∗ is a functor.
Example
If Y is a point, and f: X → Y the unique continuous map, then Sh(Y) is the category Ab of abelian groups, and the direct image functor f∗: Sh(X) → Ab equals the global sections functor.
Variants
If dealing with sheaves of sets instead of sheaves of abelian groups, the same definition applies. Similarly, if f: (X, OX) → (Y, OY) is a morphism of ringed spaces, we obtain a direct image functor f∗: Sh(X,OX) → Sh(Y,OY) from the category of sheaves of OX-modules to the category of sheaves of OY-modules. Moreover, if f is now a morphism of quasi-compact and quasi-separated schemes, then f∗ preserves the property of being quasi-coherent, so we obtain the direct image functor between categories of quasi-coherent sheaves.
A similar definition applies to sheaves on topoi, such as étale sheaves. There, instead of the above preimage f−1(U), one uses the fiber product of U and X over Y.
Properties
Forming sheaf categories and direct image functors itself defines a functor from the category of topological spaces to the category of categories: given continuous maps f: X → Y and g: Y → Z, we have (gf)∗=g∗f∗.
The direct image functor is right adjoint to the inverse image functor, which means that for any continuous and sheaves respectively on X, Y, there is a natural isomorphism:
.
If f is the inclusion of a closed subspace X ⊆ Y then f∗ is exact. Actually, in this case f∗ is an equivalence between the category of sheaves on X and the category of sheaves on Y supported on X. This follows from the fact that the stalk of is if and zero otherwise (here the closedness of X in Y is used).
If f is the morphism of affine schemes determined by a ring ho
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https://en.wikipedia.org/wiki/%C3%89tale%20morphism
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In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.
The word étale is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle.
Definition
Let be a ring homomorphism. This makes an -algebra. Choose a monic polynomial in and a polynomial in such that the derivative of is a unit in . We say that is standard étale if and can be chosen so that is isomorphic as an -algebra to and is the canonical map.
Let be a morphism of schemes. We say that is étale if and only if it has any of the following equivalent properties:
is flat and unramified.
is a smooth morphism and unramified.
is flat, locally of finite presentation, and for every in , the fiber is the disjoint union of points, each of which is the spectrum of a finite separable field extension of the residue field .
is flat, locally of finite presentation, and for every in and every algebraic closure of the residue field , the geometric fiber is the disjoint union of points, each of which is isomorphic to .
is a smooth morphism of relative dimension zero.
is a smooth morphism and a locally quasi-finite morphism.
is locally of finite presentation and is locally a standard étale morphism, that is,
For every in , let . Then there is an open affine neighborhood of and an open affine neighborhood of such that is contained in and such that the ring homomorphism induced by is standard étale.
is locally of finite presentation and is formally étale.
is locally of finite presentation and is formally étale for maps from local rings, that is:
Let be a local ring and be an ideal of such that . Set and , and let be the canonical closed immersion. Let denote the closed point of . Let and be morphisms such that . Then there exists a unique -morphism such that .
Assume that is locally noetherian and f is locally of finite type. For in , let and let be the induced map on completed local rings. Then the following are equivalent:
is étale.
For every in , the induced map on completed local rings is formally étale for the adic topology.
For every in , is a free -module and the fiber is a field which is a finite separable field extension of the residue field . (Here is the maximal ideal of .)
is formally étale for maps of local rings with the following additional properties. The local ring may be assumed Artinian. If is the maximal ideal of , then may be assumed
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https://en.wikipedia.org/wiki/%C3%89tale%20fundamental%20group
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The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
Topological analogue/informal discussion
In algebraic topology, the fundamental group of a pointed topological space is defined as the group of homotopy classes of loops based at . This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the Zariski topology.
In the classification of covering spaces, it is shown that the fundamental group is exactly the group of deck transformations of the universal covering space. This is more promising: finite étale morphisms of algebraic varieties are the appropriate analogue of covering spaces of topological spaces. Unfortunately, an algebraic variety often fails to have a "universal cover" that is finite over , so one must consider the entire category of finite étale coverings of . One can then define the étale fundamental group as an inverse limit of finite automorphism groups.
Formal definition
Let be a connected and locally noetherian scheme, let be a geometric point of and let be the category of pairs such that is a finite étale morphism from a scheme Morphisms in this category are morphisms as schemes over This category has a natural functor to the category of sets, namely the functor
geometrically this is the fiber of over and abstractly it is the Yoneda functor represented by in the category of schemes over . The functor is typically not representable in ; however, it is pro-representable in , in fact by Galois covers of . This means that we have a projective system in , indexed by a directed set where the are Galois covers of , i.e., finite étale schemes over such that . It also means that we have given an isomorphism of functors
.
In particular, we have a marked point of the projective system.
For two such the map induces a group homomorphism
which produces a projective system of automorphism groups from the projective system . We then make the following definition: the étale fundamental group of at is the inverse limit
with the inverse limit topology.
The functor is now a functor from to the category of finite and continuous -sets, and establishes an equivalence of categories between and the category of finite and continuous -sets.
Examples and theorems
The most basic example of is , the étale fundamental group of a field . Essentially by definition, the fundamental group of can be shown to be isomorphic to the absolute Galois group . More precisely, the choice of a geometric point of is equivalent to giving a separably closed extension field , and the étale fundamental group with respect to that base point identifies with the Galois group . This interpretation of the Galois group is known as Grothendieck's Galois theory.
More generally, for any geometrically connected variety over a field (i.e., is such that is connected) there
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https://en.wikipedia.org/wiki/Discriminant%20of%20an%20algebraic%20number%20field
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In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the squared volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified.
The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K. A theorem of Hermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research.
The discriminant of K can be referred to as the absolute discriminant of K to distinguish it from the relative discriminant of an extension K/L of number fields. The latter is an ideal in the ring of integers of L, and like the absolute discriminant it indicates which primes are ramified in K/L. It is a generalization of the absolute discriminant allowing for L to be bigger than Q; in fact, when L = Q, the relative discriminant of K/Q is the principal ideal of Z generated by the absolute discriminant of K.
Definition
Let K be an algebraic number field, and let OK be its ring of integers. Let b1, ..., bn be an integral basis of OK (i.e. a basis as a Z-module), and let {σ1, ..., σn} be the set of embeddings of K into the complex numbers (i.e. injective ring homomorphisms K → C). The discriminant of K is the square of the determinant of the n by n matrix B whose (i,j)-entry is σi(bj). Symbolically,
Equivalently, the trace from K to Q can be used. Specifically, define the trace form to be the matrix whose (i,j)-entry is
TrK/Q(bibj). This matrix equals BTB, so the square of the discriminant of K is the determinant of this matrix.
The discriminant of an order in K with integral basis b1, ..., bn is defined in the same way.
Examples
Quadratic number fields: let d be a square-free integer, then the discriminant of is
An integer that occurs as the discriminant of a quadratic number field is called a fundamental discriminant.
Cyclotomic fields: let n > 2 be an integer, let ζn be a primitive nth root of unity, and let Kn = Q(ζn) be the nth cyclotomic field. The discriminant of Kn is given by
where is Euler's totient function, and the product in the denominator is over primes p dividing n.
Power bases: In the case where the ring of integers has a power integral basis, that is, can be written as OK = Z[α], the discriminant of K is equal to the discriminant of the minimal polynomial of α. To see this, one can choose the integral basis of OK to be b1 = 1, b2 = α, b3 = α2, ..., bn = αn−1. Then, the matrix in the definition is the Vandermonde matrix associated to αi = σi(α), whose determinant squared is
which is exactly the definition of the discriminant of the minimal polynomial.
Let K =
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https://en.wikipedia.org/wiki/Negative%20relationship
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In statistics, there is a negative relationship or inverse relationship between two variables if higher values of one variable tend to be associated with lower values of the other. A negative relationship between two variables usually implies that the correlation between them is negative, or — what is in some contexts equivalent — that the slope in a corresponding graph is negative. A negative correlation between variables is also called anticorrelation or inverse correlation.
Negative correlation can be seen geometrically when two normalized random vectors are viewed as points on a sphere, and the correlation between them is the cosine of the arc of separation of the points on the sphere. When this arc is more than a quarter-circle (θ > π/2), then the cosine is negative. Diametrically opposed points represent a correlation of –1 = cos(π). Any two points not in the same hemisphere have negative correlation.
An example would be a negative cross-sectional relationship between illness and vaccination, if it is observed that where the incidence of one is higher than average, the incidence of the other tends to be lower than average. Similarly, there would be a negative temporal relationship between illness and vaccination if it is observed in one location that times with a higher-than-average incidence of one tend to coincide with a lower-than-average incidence of the other.
A particular inverse relationship is called inverse proportionality, and is given by where k > 0 is a constant. In a Cartesian plane this relationship is displayed as a hyperbola with y decreasing as x increases.
In finance, an inverse correlation between the returns on two different assets enhances the risk-reduction effect of diversifying by holding them both in the same portfolio.
See also
Diminishing returns
References
External links
Michael Palmer Testing for correlation from Oklahoma State University–Stillwater
Independence (probability theory)
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https://en.wikipedia.org/wiki/Generalized%20Jacobian
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In algebraic geometry a generalized Jacobian is a commutative algebraic group associated to a curve with a divisor, generalizing the Jacobian variety of a complete curve. They were introduced by Maxwell Rosenlicht in 1954, and can be used to study ramified coverings of a curve, with abelian Galois group. Generalized Jacobians of a curve are extensions of the Jacobian of the curve by a commutative affine algebraic group, giving nontrivial examples of Chevalley's structure theorem.
Definition
Suppose C is a complete nonsingular curve, m an effective divisor on C, S is the support of m, and P is a fixed base point on C not in S. The generalized Jacobian Jm is a commutative algebraic group with a rational map f from C to Jm such that:
f takes P to the identity of Jm.
f is regular outside S.
f(D) = 0 whenever D is the divisor of a rational function g on C such that g≡1 mod m.
Moreover Jm is the universal group with these properties, in the sense that any rational map from C to a group with the properties above factors uniquely through Jm. The group Jm does not depend on the choice of base point P, though changing P changes that map f by a translation.
Structure of the generalized Jacobian
For m = 0 the generalized Jacobian Jm is just the usual Jacobian J, an abelian variety of dimension g, the genus of C.
For m a nonzero effective divisor the generalized Jacobian is an extension of J by a connected commutative affine algebraic group Lm of dimension deg(m)−1. So we have an exact sequence
0 → Lm → Jm → J → 0
The group Lm is a quotient
0 → Gm → ΠUPi(ni) → Lm → 0
of a product of groups Ri by the multiplicative group Gm of the underlying field. The product runs over the points Pi in the support of m, and the group UPi(ni) is the group of invertible elements of the local ring modulo those that are 1 mod Pini. The group UPi(ni) has dimension ni, the number of times Pi occurs in m. It is the product of the multiplicative group Gm by a unipotent group of dimension ni−1, which in characteristic 0 is isomorphic to a product of ni−1 additive groups.
Complex generalized Jacobians
Over the complex numbers, the algebraic structure of the generalized Jacobian determines an analytic structure of the generalized Jacobian making it a complex Lie group.
The analytic subgroup underlying the generalized Jacobian can be described as follows. (This does not always determine the algebraic structure as two non-isomorphic commutative algebraic groups may be isomorphic as analytic groups.) Suppose that C is a curve with an effective divisor m with support S. There is a natural map from the homology group H1(C − S) to the dual Ω(−m)* of the complex vector space Ω(−m) (1-forms with poles on m) induced by the integral of a 1-form over a 1-cycle. The analytic generalized Jacobian is then the quotient group Ω(−m)*/H1(C − S).
References
Algebraic groups
Algebraic curves
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https://en.wikipedia.org/wiki/Octagonal%20antiprism
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In geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.
Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals.
In the case of a regular 8-sided base, one usually considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. As faces, it has the two n-gonal bases and, connecting those bases, 2n isosceles triangles.
If faces are all regular, it is a semiregular polyhedron.
See also
External links
Octagonal Antiprism -- Interactive Polyhedron Model
Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
VRML model
polyhedronisme A8
Prismatoid polyhedra
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https://en.wikipedia.org/wiki/Octagonal%20prism
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In geometry, the octagonal prism is a prism comprising eight rectangular sides joining two regular octagon caps.
Symmetry
Images
The octagonal prism can also be seen as a tiling on a sphere:
Use
In optics, octagonal prisms are used to generate flicker-free images in movie projectors.
In uniform honeycombs and 4-polytopes
It is an element of three uniform honeycombs:
It is also an element of two four-dimensional uniform 4-polytopes:
Related polyhedra
External links
Interactive model of an Octagonal Prism
Zonohedra
Prismatoid polyhedra
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https://en.wikipedia.org/wiki/Boustrophedon%20transform
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In mathematics, the boustrophedon transform is a procedure which maps one sequence to another. The transformed sequence is computed by an "addition" operation, implemented as if filling a triangular array in a boustrophedon (zigzag- or serpentine-like) manner—as opposed to a "Raster Scan" sawtooth-like manner.
Definition
The boustrophedon transform is a numerical, sequence-generating transformation, which is determined by an "addition" operation.
Generally speaking, given a sequence: , the boustrophedon transform yields another sequence: , where is likely defined equivalent to . The entirety of the transformation itself can be visualized (or imagined) as being constructed by filling-out the triangle as shown in Figure 1.
Boustrophedon Triangle
To fill-out the numerical Isosceles triangle (Figure 1), you start with the input sequence, , and place one value (from the input sequence) per row, using the boustrophedon scan (zigzag- or serpentine-like) approach.
The top vertex of the triangle will be the input value , equivalent to output value , and we number this top row as row 0.
The subsequent rows (going down to the base of the triangle) are numbered consecutively (from 0) as integers—let denote the number of the row currently being filled. These rows are constructed according to the row number () as follows:
For all rows, numbered , there will be exactly values in the row.
If is odd, then put the value on the right-hand end of the row.
Fill-out the interior of this row from right-to-left, where each value (index: ) is the result of "addition" between the value to right (index: ) and the value to the upper right (index: ).
The output value will be on the left-hand end of an odd row (where is odd).
If is even, then put the input value on the left-hand end of the row.
Fill-out the interior of this row from left-to-right, where each value (index: ) is the result of "addition" between the value to its left (index: ) and the value to its upper left (index: ).
The output value will be on the right-hand end of an even row (where is even).
Refer to the arrows in Figure 1 for a visual representation of these "addition" operations.
For a given, finite input-sequence: , of values, there will be exactly rows in the triangle, such that is an integer in the range: (exclusive). In other words, the last row is .
Recurrence relation
A more formal definition uses a recurrence relation. Define the numbers (with k ≥ n ≥ 0) by
.
Then the transformed sequence is defined by (for and greater indices).
Per this definition, note the following definitions for values outside the restrictions (from the relationship above) on pairs:
Special Cases
In the case a0 = 1, an = 0 (n > 0), the resulting triangle is called the Seidel–Entringer–Arnold Triangle and the numbers are called Entringer numbers .
In this case the numbers in the transformed sequence bn are called the Euler up/down numbers. This is sequence A000111 on the On-Line Ency
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https://en.wikipedia.org/wiki/786%20%28number%29
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786 (seven hundred [and] eighty-six) is the natural number following 785 and preceding 787.
In mathematics
786 is:
a sphenic number.
a Harshad number in bases 4, 5, 7, 14 and 16.
the aliquot sum of 510.
part of the 321329-aliquot tree. The complete aliquot sequence starting at 498 is: 498, 510, 786, 798, 1122, 1470, 2634, 2646, 4194, 4932, 7626, 8502, 9978, 9990, 17370, 28026, 35136, 67226, 33616, 37808, 40312, 35288, 37072, 45264, 79728, 146448, 281166, 281178, 363942, 424638, 526338, 722961, 321329, 1, 0
50 can be partitioned into powers of two in 786 different ways .
786 might be the largest n for which the value of the central binomial coefficient is not divisible by an odd prime squared. If there is a larger such number, it would have to be at least 157450 (see ).
Area code
786 is a United States telephone area code in Miami-Dade County. As an overlay area code, it shares the same geographic numbering plan area with other codes for a larger pool of telephone numbers.
In other fields
80786 - 7th generation x86 like Athlon and Intel Pentium 4
The USSD code 786, typically dialed as ##786# or *#786#, opens the RTN dialog on some cell phones. "RTN" is 786 when dialed on an E.161 telephone pad.
In the New General Catalogue, NGC786 is a magnitude 13.5 spiral galaxy in the constellation Aries. Additionally, 786 Bredichina is an asteroid.
In juggling, 786 as fourhanded Siteswap is also known as French threecount.
In Islam, 786 is often used to represent the Arabic phrase Bismillah.
In films
The number is often featured in films, mostly due to its auspiciousness in Islamic culture.
Vijay Verma's (Amitabh Bachchan) coolie number in the 1975 Hindi film Deewaar.
Raja's (Rajnikanth) coolie number in the 1981 Tamil film Thee, a remake of Deewaar.
Iqbal Khan's (Amitabh Bachchan) coolie number in the 1983 Hindi film Coolie.
Bachchan has indicated that he believes the number is auspicious, as he survived a serious injury while wearing this number during the shooting of Coolie.
Chiranjeevi sported this number in the 1988 Telugu film Khaidi No.786.
Veer Pratap Singh's (Shahrukh Khan) prisoner number in the 2004 Hindi film Veer-Zaara.
Sultan's (Ajay Devgan) car in the 2010 Hindi film Once Upon a Time in Mumbaai bears the registration number MRH 786.
In the 2011 Tamil film Mankatha, in the scene where Vinayak Mahadev (Ajith Kumar) shoots Prem (Premgi Amaren), Prem wears has a gold plate on his chest with the number 786 written on it.
Ashish R Mohan's 2012 Hindi film Khiladi 786 features Akshay Kumar in the title role. The same film was released in Pakistan without the number 786.
References
External links
e-786.com Permissible to write 786
United Submitters analysis of 786
Integers
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