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https://en.wikipedia.org/wiki/Olav%20Kallenberg
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Olav Kallenberg (born 1939) is a probability theorist known for his work on exchangeable stochastic processes and for his graduate-level textbooks and monographs. Kallenberg is a professor of mathematics at Auburn University in Alabama in the USA.
From 1991 to 1994, Kallenberg served as the Editor-in-Chief of Probability Theory and Related Fields (a leading journal in probability).
Biography
Olav Kallenberg was educated in Sweden. He has worked as a probabilist in Sweden and in the United States.
Sweden
Kallenberg was born and educated in Sweden, with an undergraduate exam in engineering physics from Royal Institute of Technology (KTH) in Stockholm. Kallenberg entered doctoral studies in mathematical statistics at KTH, but left his studies to work in operations analysis for a consulting firm in Gothenburg. While in Gothenburg, Kallenberg also taught at Chalmers University of Technology, from which he received his Ph.D. in 1972 under the supervision of Harald Bergström.
After earning his doctoral degree, Kallenberg stayed with Chalmers as a lecturer.
Kallenberg was appointed a full professor in Uppsala University.
United States
Later he moved to the United States. Since 1986, he has been Professor of Mathematics and Statistics at Auburn University.
Honours and awards
In 1977, Kallenberg was awarded the Rollo Davidson Prize from Cambridge University, and Kallenberg was only the second recipient of the prize in history.
Kallenberg is a Fellow of the Institute of Mathematical Statistics.
In April 2006 Kallenberg was selected Auburn's 32nd annual Distinguished Graduate Faculty Lecturer at Auburn. Kallenberg delivered the 2003 AACTM Lewis-Parker Lecture at the University of Alabama in Huntsville.
Selected publications
Books
Kallenberg, O., Probabilistic Symmetries and Invariance Principles. Springer -Verlag, New York (2005). 510 pp. .
Kallenberg, O., Foundations of Modern Probability, 2nd ed. Springer Series in Statistics. (2002). 650 pp. ; 3rd ed. Probability Theory and Stochastic Modelling. (2021). 946 pp.
Kallenberg, O., Random Measures, 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin (1986).
Scientific papers
Homogeneity and the strong Markov property. Ann. Probab. 15 (1987), 213–240.
Spreading and predictable sampling in exchangeable sequences and processes. Ann. Probab. 16 (1988), 508–534.
Multiple integration with respect to Poisson and Lévy processes (with J. Szulga). Probab. Th. Rel. Fields (1989), 101–134.
General Wald-type identities for exchangeable sequences and processes. Probab. Th. Rel. Fields 83 (1989), 447–487.
Random time change and an integral representation for marked stopping times. Probab. Th. Rel. Fields 86 (1990), 167–202.
Some dimension-free features of vector-valued martingales (with R. Sztencel). Probab. Th. Rel. Fields 88 (1991), 215–247.
Symmetries on random arrays and set-indexed processes. J. Theor. Probab. 5 (1992), 727–765.
Random arrays and functionals with multivariat
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https://en.wikipedia.org/wiki/Lie%20coalgebra
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In mathematics a Lie coalgebra is the dual structure to a Lie algebra.
In finite dimensions, these are dual objects: the dual vector space to a Lie algebra naturally has the structure of a Lie coalgebra, and conversely.
Definition
Let E be a vector space over a field k equipped with a linear mapping from E to the exterior product of E with itself. It is possible to extend d uniquely to a graded derivation (this means that, for any a, b ∈ E which are homogeneous elements, ) of degree 1 on the exterior algebra of E:
Then the pair (E, d) is said to be a Lie coalgebra if d2 = 0,
i.e., if the graded components of the exterior algebra with derivation
form a cochain complex:
Relation to de Rham complex
Just as the exterior algebra (and tensor algebra) of vector fields on a manifold form a Lie algebra (over the base field K), the de Rham complex of differential forms on a manifold form a Lie coalgebra (over the base field K). Further, there is a pairing between vector fields and differential forms.
However, the situation is subtler: the Lie bracket is not linear over the algebra of smooth functions (the error is the Lie derivative), nor is the exterior derivative: (it is a derivation, not linear over functions): they are not tensors. They are not linear over functions, but they behave in a consistent way, which is not captured simply by the notion of Lie algebra and Lie coalgebra.
Further, in the de Rham complex, the derivation is not only defined for , but is also defined for .
The Lie algebra on the dual
A Lie algebra structure on a vector space is a map which is skew-symmetric, and satisfies the Jacobi identity. Equivalently, a map that satisfies the Jacobi identity.
Dually, a Lie coalgebra structure on a vector space E is a linear map which is antisymmetric (this means that it satisfies , where is the canonical flip ) and satisfies the so-called cocycle condition (also known as the co-Leibniz rule)
.
Due to the antisymmetry condition, the map can be also written as a map .
The dual of the Lie bracket of a Lie algebra yields a map (the cocommutator)
where the isomorphism holds in finite dimension; dually for the dual of Lie comultiplication. In this context, the Jacobi identity corresponds to the cocycle condition.
More explicitly, let E be a Lie coalgebra over a field of characteristic neither 2 nor 3. The dual space E* carries the structure of a bracket defined by
α([x, y]) = dα(x∧y), for all α ∈ E and x,y ∈ E*.
We show that this endows E* with a Lie bracket. It suffices to check the Jacobi identity. For any x, y, z ∈ E* and α ∈ E,
where the latter step follows from the standard identification of the dual of a wedge product with the wedge product of the duals. Finally, this gives
Since d2 = 0, it follows that
, for any α, x, y, and z.
Thus, by the double-duality isomorphism (more precisely, by the double-duality monomorphism, since the vector space needs not be finite-dimensional), the Jacobi identity is satisfied.
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https://en.wikipedia.org/wiki/Max%20Deuring
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Max Deuring (9 December 1907 – 20 December 1984) was a German mathematician. He is known for his work in arithmetic geometry, in particular on elliptic curves in characteristic p. He worked also in analytic number theory.
Deuring graduated from the University of Göttingen in 1930, then began working with Emmy Noether, who noted his mathematical acumen even as an undergraduate. When she was forced to leave Germany in 1933, she urged that the university offer her position to Deuring. In 1935 he published a report entitled Algebren ("Algebras"), which established his notability in the world of mathematics. He went on to serve as Ordinarius at Marburg and Hamburg, then took a position as ordentlicher Lehrstuhl at Göttingen, where he remained until his retirement.
Deuring was a fellow of the Leopoldina. His doctoral students include Max Koecher and Hans-Egon Richert.
Selected works
Algebren, Springer 1935
Sinn und Bedeutung der mathematischen Erkenntnis, Felix Meiner, Hamburg 1949
Klassenkörper der komplexen Multiplikation, Teubner 1958
Lectures on the theory of algebraic functions of one variable, 1973 (from lectures at the Tata Institute, Mumbai)
Sources
Peter Roquette Über die algebraisch-zahlentheoretischen Arbeiten von Max Deuring, Jahresbericht DMV Vol.91, 1989, p. 109
Martin Kneser Max Deuring, Jahresbericht DMV Vol.89, 1987, p. 135
Martin Kneser, Martin Eichler Das wissenschaftliche Werk von Max Deuring, Acta Arithmetica Vol.47, 1986, p. 187
See also
Deuring–Heilbronn phenomenon
Birch and Swinnerton-Dyer conjecture
Supersingular elliptic curve
References
External links
MacTutor biography
Obituary in Acta Arithmetica
Biographical page
1907 births
1984 deaths
20th-century German mathematicians
Scientists from Göttingen
University of Göttingen alumni
Academic staff of the University of Göttingen
Academic staff of the University of Marburg
Academic staff of the University of Hamburg
Algebraic geometers
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https://en.wikipedia.org/wiki/Information%20theory%20and%20measure%20theory
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This article discusses how information theory (a branch of mathematics studying the transmission, processing and storage of information) is related to measure theory (a branch of mathematics related to integration and probability).
Measures in information theory
Many of the concepts in information theory have separate definitions and formulas for continuous and discrete cases. For example, entropy is usually defined for discrete random variables, whereas for continuous random variables the related concept of differential entropy, written , is used (see Cover and Thomas, 2006, chapter 8). Both these concepts are mathematical expectations, but the expectation is defined with an integral for the continuous case, and a sum for the discrete case.
These separate definitions can be more closely related in terms of measure theory. For discrete random variables, probability mass functions can be considered density functions with respect to the counting measure. Thinking of both the integral and the sum as integration on a measure space allows for a unified treatment.
Consider the formula for the differential entropy of a continuous random variable with range and probability density function :
This can usually be interpreted as the following Riemann–Stieltjes integral:
where is the Lebesgue measure.
If instead, is discrete, with range a finite set, is a probability mass function on , and is the counting measure on , we can write:
The integral expression, and the general concept, are identical in the continuous case; the only difference is the measure used. In both cases the probability density function is the Radon–Nikodym derivative of the probability measure with respect to the measure against which the integral is taken.
If is the probability measure induced by , then the integral can also be taken directly with respect to :
If instead of the underlying measure μ we take another probability measure , we are led to the Kullback–Leibler divergence: let and be probability measures over the same space. Then if is absolutely continuous with respect to , written the Radon–Nikodym derivative exists and the Kullback–Leibler divergence can be expressed in its full generality:
where the integral runs over the support of Note that we have dropped the negative sign: the Kullback–Leibler divergence is always non-negative due to Gibbs' inequality.
Entropy as a "measure"
There is an analogy between Shannon's basic "measures" of the information content of random variables and a measure over sets. Namely the joint entropy, conditional entropy, and mutual information can be considered as the measure of a set union, set difference, and set intersection, respectively (Reza pp. 106–108).
If we associate the existence of abstract sets and to arbitrary discrete random variables X and Y, somehow representing the information borne by X and Y, respectively, such that:
whenever X and Y are unconditionally independent, and
whenever X and
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https://en.wikipedia.org/wiki/Music%20and%20mathematics
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Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory.
While music theory has no axiomatic foundation in modern mathematics, the basis of musical sound can be described mathematically (using acoustics) and exhibits "a remarkable array of number properties".
History
Though ancient Chinese, Indians, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound, the Pythagoreans (in particular Philolaus and Archytas) of ancient Greece were the first researchers known to have investigated the expression of musical scales in terms of numerical ratios, particularly the ratios of small integers. Their central doctrine was that "all nature consists of harmony arising out of numbers".
From the time of Plato, harmony was considered a fundamental branch of physics, now known as musical acoustics. Early Indian and Chinese theorists show similar approaches: all sought to show that the mathematical laws of harmonics and rhythms were fundamental not only to our understanding of the world but to human well-being. Confucius, like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection.
Time, rhythm, and meter
Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement of pulse repetition, accent, phrase and duration – music would not be possible. Modern musical use of terms like meter and measure also reflects the historical importance of music, along with astronomy, in the development of counting, arithmetic and the exact measurement of time and periodicity that is fundamental to physics.
The elements of musical form often build strict proportions or hypermetric structures (powers of the numbers 2 and 3).
Musical form
Musical form is the plan by which a short piece of music is extended. The term "plan" is also used in architecture, to which musical form is often compared. Like the architect, the composer must take into account the function for which the work is intended and the means available, practicing economy and making use of repetition and order. The common types of form known as binary and ternary ("twofold" and "threefold") once again demonstrate the importance of small integral values to the intelligibility and appeal of music.
Frequency and harmony
A musical scale is a discrete set of pitches used in making or describing music. The most important scale in the Western tradition is the diatonic scale but many others have been used and proposed in various historical eras and parts of the world. Each pitch corresponds to a particular frequency, expressed in hertz (Hz), sometimes referred to as cycles per second (c.p.s.). A scale has an interval of repetition, normally the octave. The octave of any
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https://en.wikipedia.org/wiki/Latimer%E2%80%93MacDuffee%20theorem
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The Latimer–MacDuffee theorem is a theorem in abstract algebra, a branch of mathematics.
It is named after Claiborne Latimer and Cyrus Colton MacDuffee, who published it in 1933. Significant contributions to its theory were made later by Olga Taussky-Todd.
Let be a monic, irreducible polynomial of degree . The Latimer–MacDuffee theorem gives a one-to-one correspondence between -similarity classes of matrices with characteristic polynomial and the ideal classes in the order
where ideals are considered equivalent if they are equal up to an overall (nonzero) rational scalar multiple. (Note that this order need not be the full ring of integers, so nonzero ideals need not be invertible.) Since an order in a number field has only finitely many ideal classes (even if it is not the maximal order, and we mean here ideals classes for all nonzero ideals, not just the invertible ones), it follows that there are only finitely many conjugacy classes of matrices over the integers with characteristic polynomial .
References
Theorems in abstract algebra
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https://en.wikipedia.org/wiki/Empirical%20measure
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In probability theory, an empirical measure is a random measure arising from a particular realization of a (usually finite) sequence of random variables. The precise definition is found below. Empirical measures are relevant to mathematical statistics.
The motivation for studying empirical measures is that it is often impossible to know the true underlying probability measure . We collect observations and compute relative frequencies. We can estimate , or a related distribution function by means of the empirical measure or empirical distribution function, respectively. These are uniformly good estimates under certain conditions. Theorems in the area of empirical processes provide rates of this convergence.
Definition
Let be a sequence of independent identically distributed random variables with values in the state space S with probability distribution P.
Definition
The empirical measure Pn is defined for measurable subsets of S and given by
where is the indicator function and is the Dirac measure.
Properties
For a fixed measurable set A, nPn(A) is a binomial random variable with mean nP(A) and variance nP(A)(1 − P(A)).
In particular, Pn(A) is an unbiased estimator of P(A).
For a fixed partition of S, random variables form a multinomial distribution with event probabilities
The covariance matrix of this multinomial distribution is .
Definition
is the empirical measure indexed by , a collection of measurable subsets of S.
To generalize this notion further, observe that the empirical measure maps measurable functions to their empirical mean,
In particular, the empirical measure of A is simply the empirical mean of the indicator function, Pn(A) = Pn IA.
For a fixed measurable function , is a random variable with mean and variance .
By the strong law of large numbers, Pn(A) converges to P(A) almost surely for fixed A. Similarly converges to almost surely for a fixed measurable function . The problem of uniform convergence of Pn to P was open until Vapnik and Chervonenkis solved it in 1968.
If the class (or ) is Glivenko–Cantelli with respect to P then Pn converges to P uniformly over (or ). In other words, with probability 1 we have
Empirical distribution function
The empirical distribution function provides an example of empirical measures. For real-valued iid random variables it is given by
In this case, empirical measures are indexed by a class It has been shown that is a uniform Glivenko–Cantelli class, in particular,
with probability 1.
See also
Empirical risk minimization
Poisson random measure
References
Further reading
Measures (measure theory)
Empirical process
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https://en.wikipedia.org/wiki/Biracks%20and%20biquandles
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In mathematics, biquandles and biracks are sets with binary operations that generalize quandles and racks. Biquandles take, in the theory of virtual knots, the place that quandles occupy in the theory of classical knots. Biracks and racks have the same relation, while a biquandle is a birack which satisfies some additional conditions.
Definitions
Biquandles and biracks have two binary operations on a set written and . These satisfy the following three axioms:
1.
2.
3.
These identities appeared in 1992 in reference [FRS] where the object was called a species.
The superscript and subscript notation is useful here because it dispenses with the need for brackets. For example,
if we write for and for then the
three axioms above become
1.
2.
3.
If in addition the two operations are invertible, that is given in the set there are unique in the set such that and then the set together with the two operations define a birack.
For example, if , with the operation , is a rack then it is a birack if we define the other operation to be the identity, .
For a birack the function can be defined by
Then
1. is a bijection
2.
In the second condition, and are defined by and . This condition is sometimes known as the set-theoretic Yang-Baxter equation.
To see that 1. is true note that defined by
is the inverse to
To see that 2. is true let us follow the progress of the triple under . So
On the other hand, . Its progress under is
Any satisfying 1. 2. is said to be a switch (precursor of biquandles and biracks).
Examples of switches are the identity, the twist and where is the operation of a rack.
A switch will define a birack if the operations are invertible. Note that the identity switch does not do this.
Biquandles
A biquandle is a birack which satisfies some additional structure, as described by Nelson and Rische. The axioms of a biquandle are "minimal" in the sense that they are the weakest restrictions that can be placed on the two binary operations while making the biquandle of a virtual knot invariant under Reidemeister moves.
Linear biquandles
Application to virtual links and braids
Birack homology
Further reading
[FJK] Roger Fenn, Mercedes Jordan-Santana, Louis Kauffman Biquandles and Virtual Links, Topology and its Applications, 145 (2004) 157–175
[FRS] Roger Fenn, Colin Rourke, Brian Sanderson An Introduction to Species and the Rack Space in Topics in Knot Theory (1992), Kluwer 33–55
[K] L. H. Kauffman, Virtual Knot Theory, European Journal of Combinatorics 20 (1999), 663–690.
Knot theory
Algebraic structures
Ordered algebraic structures
Non-associative algebra
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https://en.wikipedia.org/wiki/Apeirogon
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In geometry, an apeirogon () or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries.
Definitions
Classical constructive definition
Given a point A0 in a Euclidean space and a translation S, define the point Ai to be the point obtained from i applications of the translation S to A0, so Ai = Si(A0). The set of vertices Ai with i any integer, together with edges connecting adjacent vertices, is a sequence of equal-length segments of a line, and is called the regular apeirogon as defined by H. S. M. Coxeter.
A regular apeirogon can be defined as a partition of the Euclidean line E1 into infinitely many equal-length segments. It generalizes the regular n-gon, which may be defined as a partition of the circle S1 into finitely many equal-length segments.
Modern abstract definition
An abstract polytope is a partially ordered set P (whose elements are called faces) with properties modeling those of the inclusions of faces of convex polytopes. The rank (or dimension) of an abstract polytope is determined by the length of the maximal ordered chains of its faces, and an abstract polytope of rank n is called an abstract n-polytope.
For abstract polytopes of rank 2, this means that: A) the elements of the partially ordered set are sets of vertices with either zero vertex (the empty set), one vertex, two vertices (an edge), or the entire vertex set (a two-dimensional face), ordered by inclusion of sets; B) each vertex belongs to exactly two edges; C) the undirected graph formed by the vertices and edges is connected.
An abstract polytope is called an abstract apeirotope if it has infinitely many elements; an abstract 2-apeirotope is called an abstract apeirogon.
In an abstract polytope, a flag is a collection of one face of each dimension, all incident to each other (that is, comparable in the partial order); an abstract polytope is called regular if it has symmetries (structure-preserving permutations of its elements) that take any flag to any other flag. In the case of a two-dimensional abstract polytope, this is automatically true; the symmetries of the apeirogon form the infinite dihedral group.
Pseudogon
The regular pseudogon is a partition of the hyperbolic line H1 (instead of the Euclidean line) into segments of length 2λ, as an analogue of the regular apeirogon.
Realizations
Definition
A realization of an abstract apeirogon is defined as a mapping from its vertices to a finite-dimensional geometric space (typically a Euclidean space) such that every symmetry of the abstract apeirogon corresponds to an isometry of the images of the mapping. Two realizations are called congruent if the natural bijection between their sets of vertices is induced by an isometry of their ambient Euclidean spaces. The classical definition of an apeirogon as an equal
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https://en.wikipedia.org/wiki/Signed%20distance%20function
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In mathematics and its applications, the signed distance function (or oriented distance function) is the orthogonal distance of a given point x to the boundary of a set Ω in a metric space, with the sign determined by whether or not x is in the interior of Ω. The function has positive values at points x inside Ω, it decreases in value as x approaches the boundary of Ω where the signed distance function is zero, and it takes negative values outside of Ω. However, the alternative convention is also sometimes taken instead (i.e., negative inside Ω and positive outside).
Definition
If Ω is a subset of a metric space X with metric d, then the signed distance function f is defined by
where denotes the boundary of For any
where denotes the infimum.
Properties in Euclidean space
If Ω is a subset of the Euclidean space Rn with piecewise smooth boundary, then the signed distance function is differentiable almost everywhere, and its gradient satisfies the eikonal equation
If the boundary of Ω is Ck for k ≥ 2 (see Differentiability classes) then d is Ck on points sufficiently close to the boundary of Ω. In particular, on the boundary f satisfies
where N is the inward normal vector field. The signed distance function is thus a differentiable extension of the normal vector field. In particular, the Hessian of the signed distance function on the boundary of Ω gives the Weingarten map.
If, further, Γ is a region sufficiently close to the boundary of Ω that f is twice continuously differentiable on it, then there is an explicit formula involving the Weingarten map Wx for the Jacobian of changing variables in terms of the signed distance function and nearest boundary point. Specifically, if T(∂Ω, μ) is the set of points within distance μ of the boundary of Ω (i.e. the tubular neighbourhood of radius μ), and g is an absolutely integrable function on Γ, then
where denotes the determinant and dSu indicates that we are taking the surface integral.
Algorithms
Algorithms for calculating the signed distance function include the efficient fast marching method, fast sweeping method and the more general level-set method.
For voxel rendering, a fast algorithm for calculating the SDF in taxicab geometry uses summed-area tables.
Applications
Signed distance functions are applied, for example, in real-time rendering, for instance the method of SDF ray marching, and computer vision.
SDF has been used to describe object geometry in real-time rendering, usually in a raymarching context, starting in the mid 2000s. By 2007, Valve is using SDFs to render large pixel-size (or high DPI) smooth fonts with GPU acceleration in its games. Valve's method is not perfect as it runs in raster space in order to avoid the computational complexity of solving the problem in the (continuous) vector space. The rendered text often loses sharp corners. In 2014, an improved method was presented by Behdad Esfahbod. Behdad's GLyphy approximates the font's Bézier curves with a
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https://en.wikipedia.org/wiki/Thompson%20subgroup
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In mathematical finite group theory, the Thompson subgroup of a finite p-group P refers to one of several characteristic subgroups of P. originally defined to be the subgroup generated by the abelian subgroups of P of maximal rank. More often the Thompson subgroup is defined to be the subgroup generated by the abelian subgroups of P of maximal order or the subgroup generated by the elementary abelian subgroups of P of maximal rank. In general these three subgroups can be different, though they are all called the Thompson subgroup and denoted by .
See also
Glauberman normal p-complement theorem
ZJ theorem
Puig subgroup, a subgroup analogous to the Thompson subgroup
References
Finite groups
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https://en.wikipedia.org/wiki/Australian%20and%20New%20Zealand%20Standard%20Industrial%20Classification
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Australian and New Zealand Standard Industrial Classification (ANZSIC) was jointly developed by the Australian Bureau of Statistics and Statistics New Zealand in order to make it easier to compare industry statistics between the two countries and with the rest of the world.
The 2006 edition of the ANZSIC replaced the 1993 edition, which was the first version produced. Prior to 1993, Australia and New Zealand had separate industry classifications. It is arranged into 19 broad industry divisions and 96 industry subdivisions There are two more detailed levels called Groups and Classes. ANZSIC codes are four-digit numbers. The Australian Taxation Office (ATO) uses five-digit codes referred to as Business Industry Codes.
In the 2006 edition, Industry Division D has been expanded to include 'Waste Services', and 'hunting' is removed from Industry Division A.
Divisions and subdivisions
A — Agriculture, Forestry and Fishing
01 — Agriculture
011 — Nursery and Floriculture Production
012 — Mushroom and Vegetable Growing
013 — Fruit Tree and Nut Growing
014 — Sheep, Beef Cattle and Grain Farming
015 — Other Crop Growing
016 — Dairy Cattle Farming
017 — Poultry Farming
018 — Deer Farming
019 — Other Livestock Farming
02 — Aquaculture
03 — Forestry and Logging
04 — Fishing, Hunting and Trapping
041 — Fishing
042 — Hunting and Trapping
05 — Agriculture, Forestry and Fishing Support Services
051 — Forestry Support Services
052 — Agriculture and Fishing Support Services
B — Mining
06 — Coal Mining
07 — Oil and Gas Extraction
08 — Metal Ore Mining
09 — Non-Metallic Mineral Mining and Quarrying
091 — Construction Material Mining
099 — Other Non-Metallic Mineral Mining and Quarrying
10 — Exploration and Other Mining Support Services
101 — Exploration
109 — Other Mining Support Services
C — Manufacturing
11 — Food Product Manufacturing
111 — Meat and meat product manufacturing
112 — Seafood processing
113 — Dairy product manufacturing
114 — Fruit and vegetable processing
115 — Oil and fat manufacturing
116 — Grain mill and cereal product manufacturing
117 — Bakery product manufacturing
118 — Sugar and confectionary manufacturing
119 — Other food manufacturing
12 — Beverage and Tobacco Product Manufacturing
121 — Beverage manufacturing
122 — Cigarette and tobacco product manufacturing
13 — Textile, Leather, Clothing and Footwear Manufacturing
131 — Textile manufacturing
132 — Leather tanning, fur dressing and leather product manufacturing
133 — Textile product manufacturing
134 — Knitted product manufacturing
135 — Clothing and footwear manufacturing
14 — Wood Product Manufacturing
141 — Log sawmilling and timber dressing
149 — Other wood product manufacturing
15 — Pulp, Paper and Converted Paper Product Manufacturing
151 — Pulp, paper and paperboard manufacturing
152 — Converted paper product manufacturing
16 — Printing (including the Reproduction of Recorded Media)
161 — Printing and printing support services
162 — Reproduction of recorded
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https://en.wikipedia.org/wiki/Dietrich%20Stoyan
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Dietrich Stoyan (born 1940, Germany) is a German mathematician and statistician who made contributions to queueing theory, stochastic geometry, and spatial statistics.
Education and career
Stoyan studied mathematics at Technical University Dresden; applied research at Deutsches Brennstoffinstitut Freiberg, 1967 PhD, 1975 Habilitation. Since 1976 at TU Bergakademie Freiberg, Rektor of that university in 1991—1997; he became famous by his statistical research of the diffusion of euro coins in Germany and Europe after the introduction of the euro in 2002.
Research
Queueing Theory
Qualitative theory, in particular inequalities, for queueing systems and related stochastic models. The books
D. Stoyan: Comparison Methods for Queues and other Stochastic Models. J. Wiley and Sons, Chichester, 1983 and
A. Mueller and D. Stoyan: Comparison Methods for Stochastic Models and Risks, J. Wiley and Sons, Chichester, 2002
report on the results. The work goes back to 1969 when he discovered the monotonicity of the GI/G/1 waiting times with respect to the convex order.
Stochastic Geometry
Stereological formulae, applications for marked point process, development of stochastic models. Successful joint work with Joseph Mecke led to the first exact proof of the fundamental stereological formulae.
The book Stochastic Geometry and its Applications, by D. Stoyan, W.S. Kendall and J. Mecke reports on the results. The book of 1995 is the key reference for applied stochastic geometry.
Spatial Statistics
Statistical methods for point processes, random sets and many other random geometrical structures such as fibre processes. Results can be found in the 1995 book on stochastic geometry and in the book, Fractals, Random Shapes and Point Fields by D. and H. Stoyan. (J. Wiley and Sons, Chichester, 1994).
A particular strength of Stoyan is second-order methods.
At the moment Dietrich Stoyan is working (together with three colleagues) for a new book on point process statistics. He used packings of hard spheres as models for materials with the aim to solve mechanical problems for random heterogeneous materials.
Stoyan is very active in demonstrating non-mathematicians and non-statisticians the potential of statistical and stochastic geometrical methods. In particular, he co-organized together
with Klaus Mecke conferences where physicists, geometers and statisticians met. See the books
Mecke Klaus R. and Stoyan D. (eds.): Statistical Physics and Spatial Statistics. Lecture Notes in Physics 554, Springer-Verlag, 2000 and
Mecke Klaus R. and Stoyan D. (eds.): Morphology of Condensed Matter. Lecture Notes in Physics 600, Springer-Verlag, 2002.
External links
Homepage
1940 births
Living people
German statisticians
Spatial statisticians
Scientists from Freiberg
Members of the German Academy of Sciences at Berlin
TU Dresden alumni
Academic staff of the Freiberg University of Mining and Technology
Probability theorists
German mathematicians
Members of Academia Europaea
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https://en.wikipedia.org/wiki/Quasi-open%20map
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In topology a branch of mathematics, a quasi-open map or quasi-interior map is a function which has similar properties to continuous maps.
However, continuous maps and quasi-open maps are not related.
Definition
A function between topological spaces and is quasi-open if, for any non-empty open set , the interior of in is non-empty.
Properties
Let be a map between topological spaces.
If is continuous, it need not be quasi-open. Conversely if is quasi-open, it need not be continuous.
If is open, then is quasi-open.
If is a local homeomorphism, then is quasi-open.
The composition of two quasi-open maps is again quasi-open.
See also
Notes
References
Topology
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https://en.wikipedia.org/wiki/Sergei%20Godunov
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Sergei Konstantinovich Godunov (; 17 July 1929 – 15 July 2023) was a Soviet and Russian professor at the Sobolev Institute of Mathematics of the Russian Academy of Sciences in Novosibirsk, Russia.
Biography
Godunov's most influential work is in the area of applied and numerical mathematics, particularly in the development of methodologies used in Computational Fluid Dynamics (CFD) and other computational fields. Godunov's theorem (Godunov 1959) (also known as Godunov's order barrier theorem) : Linear numerical schemes for solving partial differential equations, having the property of not generating new extrema (a monotone scheme), can be at most first-order accurate. Godunov's scheme is a conservative numerical scheme for solving partial differential equations. In this method, the conservative variables are considered as piecewise constant over the mesh cells at each time step and the time evolution is determined by the exact solution of the Riemann (shock tube) problem at the inter-cell boundaries (Hirsch, 1990).
On 1–2 May 1997 a symposium entitled: Godunov-type numerical methods, was held at the University of Michigan to honour Godunov. These methods are widely used to compute continuum processes dominated by wave propagation. On the following day, 3 May, Godunov received an honorary degree from the University of Michigan. Godunov died on 15 July 2023, two days shy of his 94th birthday.
Education
1946–1951 – Department of Mechanics and Mathematics, Moscow State University.
1951 – Diploma (M. S.), Moscow State University.
1954 – Candidate of Physical and Mathematical Sciences (Ph. D.).
1965 – Doctor of Physical and Mathematical Sciences (D. Sc.).
1976 – Corresponding member of the Academy of Sciences of the Soviet Union.
1994 – Member of the Russian Academy of Sciences (Academician).
1997 – Honorary professor of the University of Michigan (Ann-Arbor, USA).
Awards
1954 – Order of the Badge of Honour
1956 – Order of the Red Banner of Labour
1959 – Lenin Prize
1972 – A.N. Krylov Prize of the Academy of Sciences of the Soviet Union
1975 – Order of the Red Banner of Labour
1981 – Order of the Badge of Honour
1993 – M.A. Lavrentyev Prize of the Russian Academy of Sciences
2010 – Order of Honour
2020 - SAE/Ramesh Agarwal Computational Fluid Dynamics Award
2023 – Order of Alexander Nevsky
See also
Riemann solver
Total variation diminishing
Upwind scheme
Notes
References
Godunov, Sergei K. (1954), Ph. D. Dissertation: Difference Methods for Shock Waves, Moscow State University.
Godunov, S. K. (1959), A Difference Scheme for Numerical Solution of Discontinuous Solution of Hydrodynamic Equations, Mat. Sbornik, 47, 271-306, translated US Joint Publ. Res. Service, JPRS 7225 November 29, 1960.
Godunov, Sergei K. and Romenskii, Evgenii I. (2003) Elements of Continuum Mechanics and Conservation Laws, Springer, .
Hirsch, C. (1990), Numerical Computation of Internal and External Flows, vol 2, Wiley.
External links
Godunov's Personal Web
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https://en.wikipedia.org/wiki/Signal-to-noise%20statistic
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In mathematics the signal-to-noise statistic distance between two vectors a and b with mean values and and standard deviation and respectively is:
In the case of Gaussian-distributed data and unbiased class distributions, this statistic can be related to classification accuracy given an ideal linear discrimination, and a decision boundary can be derived.
This distance is frequently used to identify vectors that have significant difference. One usage is in bioinformatics to locate genes that are differential expressed on microarray experiments.
See also
Distance
Uniform norm
Manhattan distance
Signal-to-noise ratio
Signal to noise ratio (imaging)
Notes
Statistical distance
Statistical ratios
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https://en.wikipedia.org/wiki/Rape%20statistics
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Statistics on rape and other sexual assaults are commonly available in industrialized countries, and have become better documented throughout the world. Inconsistent definitions of rape, different rates of reporting, recording, prosecution and conviction for rape can create controversial statistical disparities, and lead to accusations that many rape statistics are unreliable or misleading.
In some jurisdictions, male-female rape is the only form of rape counted in the statistics. Countries may not define forced sex on a spouse as rape. Rape is an under-reported crime. Prevalence of reasons for not reporting rape differ across countries. They may include fear of retaliation, uncertainty about whether a crime was committed or if the offender intended harm, not wanting others to know about the rape, not wanting the offender to get in trouble, fear of prosecution (e.g. due to laws against premarital sex), and doubt in local law enforcement.
A United Nations statistical report compiled from government sources showed that more than 250,000 cases of rape or attempted rape were recorded by police annually. The reported data covered 65 countries. In a survey by United Nations, 23% of Italian women suffered sexual violence in their lifetimes, 3.3% had experienced attempted rape and 2.3% had experienced rape.
Research
Most rape research and reporting to date has been limited to male-female forms of rape. Research on male-male and female-male is beginning to be done. However, almost no research has been done on female-female rape, though women can be charged with rape in a few jurisdictions. A few books, such as Violent Betrayal: Partner Abuse in Lesbian Relationships by Dr. Claire M. Renzetti, No More Secrets: Violence in Lesbian Relationships by Janice Ristock, and Woman-to-Woman Sexual Violence: Does She Call It Rape? by Lori B. Girshick also cover the topic of rape of women by other women.
By country
This table indicates the number of, and per capita cases of recorded rape by country. It does not, and of course cannot, include cases of rape which go unreported or unrecorded. It does not specify whether recorded means reported, brought to trial, or convicted. Each entry is based on that country's definition of rape, which varies widely throughout the world. The list does not include the estimated rape statistics of the countries, per year, such as South Africa having 500,000 rapes per year, China having 31,833 rapes a year, Egypt having more than 20,000 rapes a year, and the United Kingdom at 85,000 rapes a year.
* Changes in definitions and/or counting rules are reported by the Member State to indicate a break in the time series.
As reference and verification of the mentioned numbers for the EU-27 could be used: "Violence against women: an EU-wide survey – Main results" published in 2014 - Table 2.1: Women who have experienced physical and/or sexual violence by current and/or previous partner, or by any other person since the age of 15, by EU Mem
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https://en.wikipedia.org/wiki/Estimation%20lemma
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In mathematics the estimation lemma, also known as the inequality, gives an upper bound for a contour integral. If is a complex-valued, continuous function on the contour and if its absolute value is bounded by a constant for all on , then
where is the arc length of . In particular, we may take the maximum
as upper bound. Intuitively, the lemma is very simple to understand. If a contour is thought of as many smaller contour segments connected together, then there will be a maximum for each segment. Out of all the maximum s for the segments, there will be an overall largest one. Hence, if the overall largest is summed over the entire path then the integral of over the path must be less than or equal to it.
Formally, the inequality can be shown to hold using the definition of contour integral, the absolute value inequality for integrals and the formula for the length of a curve as follows:
The estimation lemma is most commonly used as part of the methods of contour integration with the intent to show that the integral over part of a contour goes to zero as goes to infinity. An example of such a case is shown below.
Example
Problem.
Find an upper bound for
where is the upper half-circle with radius traversed once in the counterclockwise direction.
Solution.
First observe that the length of the path of integration is half the circumference of a circle with radius , hence
Next we seek an upper bound for the integrand when . By the triangle inequality we see that
therefore
because on . Hence
Therefore, we apply the estimation lemma with . The resulting bound is
See also
Jordan's lemma
References
.
.
Theorems in complex analysis
Lemmas in analysis
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https://en.wikipedia.org/wiki/Lockleys%2C%20South%20Australia
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Lockleys is an inner western suburb of Adelaide, in the City of West Torrens.
Australian Bureau of Statistics data from May 2021 revealed that Adelaide's western suburbs had the lowest unemployment rate in South Australia.
History
The area was inhabited by the Kaurna people before the British colonisation of South Australia.
The area was subject to flooding by the River Torrens, which originally ran into an area named "The Reedbeds" in the upper reaches of the Port River. In the 1930s the Torrens Channel, also named Breakout Creek, was cut through the coastal dunes to Gulf St Vincent, to drain the wetlands and eliminate the flooding. A large part of Lockleys is within a bend of the River Torrens.
Hence, prior to subdivision, the area was renowned for its rich soil, market gardens and greenhouses. The name comes from a property (section 145) owned by Charles Brown Fisher, then Edward Meade Bagot and Gabriel Bennett, who built a course there for amateur horse racing. The property was rented by trainers J. Eden Savill and C. Leslie Macdonald for their Lockleys Stables where many good racehorses were prepared.
Hank family
The area was divided for housing. However, the Hank family lived on Torrens Avenue, Lockleys and had established 11 acres of market garden there after world war I. The Hank brothers (Ray, Bill and Bob) all attended the Lockleys Primary School in Brooklyn Park and would all become footballers for the West Torrens Football Club in the SANFL. Bob Hank would go on to become an AFL Hall of Fame inductee, winning the Magarey Medal in both 1946 and 1947 and winning a record 9 league best and fairest awards for his club. A pavilion in the eastern grandstand at Adelaide Oval is named the Bob Hank Pavilion and the grandstand at Thebarton Oval is named the Hank Brothers Stand after these Australian Football legends. Bob Hank also famously clean bowled Sir Donald Bradman in a District Cricket final in March 1947 whilst playing for the West Torrens Cricket Club against Bradman's Kensington Cricket Club.
John Martin's warehouse
The former John Martin's department store had a bulk warehouse on Pierson Street, which was also a storage location for the floats used in the company's annual Christmas Pageant. The warehouse was converted by EDS for a data and call centre, which opened in 1996, and later owned by the Maras Group and operated by Westpac as a mortgage processing centre. In September 2021 a development application was announced for rezoning the call centre and adjacent child care centre, to allow a medium density residential development to be built on the site.
Windsor Theatre
The Windsor Theatre, located at 362 Henley Beach Road, was originally built as a RSL hall in March 1925, with the construction cost of £3,800 covered by community fund-raising, with much of it donated by John Mellor. It was called the Lockleys Memorial Hall. On 10 October in the same year, the hall was used by Lyric Theatres Ltd to screen a film, and soon bec
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https://en.wikipedia.org/wiki/WPO
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WPO may refer to:
Computing and math
Web performance optimization, in website optimization
Well partial order, an ordering relation in mathematics
Whole program optimization, a compiler optimization
Other uses
Weakly Pareto Optimal
North Fork Valley Airport (IATA code), in the List of airports in Colorado, US
Washington Post Company (former NYSE symbol)
World Photography Organisation, for amateur and professional photographers
Wikipediocracy
See also
WPO-3, 1941 plans for the defense of the Philippine Islands in the Battle of Bataan
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https://en.wikipedia.org/wiki/Local%20time%20%28mathematics%29
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In the mathematical theory of stochastic processes, local time is a stochastic process associated with semimartingale processes such as Brownian motion, that characterizes the amount of time a particle has spent at a given level. Local time appears in various stochastic integration formulas, such as Tanaka's formula, if the integrand is not sufficiently smooth. It is also studied in statistical mechanics in the context of random fields.
Formal definition
For a continuous real-valued semimartingale , the local time of at the point is the stochastic process which is informally defined by
where is the Dirac delta function and is the quadratic variation. It is a notion invented by Paul Lévy. The basic idea is that is an (appropriately rescaled and time-parametrized) measure of how much time has spent at up to time . More rigorously, it may be written as the almost sure limit
which may be shown to always exist. Note that in the special case of Brownian motion (or more generally a real-valued diffusion of the form where is a Brownian motion), the term simply reduces to , which explains why it is called the local time of at . For a discrete state-space process , the local time can be expressed more simply as
Tanaka's formula
Tanaka's formula also provides a definition of local time for an arbitrary continuous semimartingale on
A more general form was proven independently by Meyer and Wang; the formula extends Itô's lemma for twice differentiable functions to a more general class of functions. If is absolutely continuous with derivative which is of bounded variation, then
where is the left derivative.
If is a Brownian motion, then for any the field of local times has a modification which is a.s. Hölder continuous in with exponent , uniformly for bounded and . In general, has a modification that is a.s. continuous in and càdlàg in .
Tanaka's formula provides the explicit Doob–Meyer decomposition for the one-dimensional reflecting Brownian motion, .
Ray–Knight theorems
The field of local times associated to a stochastic process on a space is a well studied topic in the area of random fields. Ray–Knight type theorems relate the field Lt to an associated Gaussian process.
In general Ray–Knight type theorems of the first kind consider the field Lt at a hitting time of the underlying process, whilst theorems of the second kind are in terms of a stopping time at which the field of local times first exceeds a given value.
First Ray–Knight theorem
Let (Bt)t ≥ 0 be a one-dimensional Brownian motion started from B0 = a > 0, and (Wt)t≥0 be a standard two-dimensional Brownian motion started from W0 = 0 ∈ R2. Define the stopping time at which B first hits the origin, . Ray and Knight (independently) showed that
where (Lt)t ≥ 0 is the field of local times of (Bt)t ≥ 0, and equality is in distribution on C[0, a]. The process |Wx|2 is known as the squared Bessel process.
Second Ray–Knight theorem
Let (Bt)t ≥ 0 be a standard on
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https://en.wikipedia.org/wiki/N.%20E.%20Cameron
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N. E. Cameron (26 January 1903 – May 1983) was a writer from Guyana who wrote on almost every topic from history and mathematics to politics.
Biography
Early years and education
Norman Eustace Cameron was born in New Amsterdam, Guyana. He attended Queen's College in Georgetown, and in 1921 won the Guyana Scholarship, achieving First-Class Honours at the Oxford and Cambridge Higher Examination, with five distinctions in Latin, French, English, Mathematics and Religious Knowledge, placing him first among candidates from Barbados and Guyana.
At the University of Cambridge, he continued to excel in Mathematics. taking first-class honours in Part 1 of the Mathematical Tripos in 1923, and graduating Senior Optime in 1925.
Academic career
On returning to Guyana he founded his own school, The Guyanese Academy (1926–34), and wrote The Evolution of the Negro, published in two volumes (1929 and 1934), which Kenneth Ramchand has called "A rare and neglected but very useful work".
In 1934 Cameron returned as a Senior Master to his alma mater, Queen's College, to teach mathematics, and to contribute to the development of his school. While teaching at Queen's College, he wrote several dramatic works, published an anthology of Guyanese poetry, wrote numerous essays, memoranda and articles on the culture and politics of Guyanese life, and authored four high-school text books on mathematics (1942), as well as the history of Queen's College (1951). He was appointed Deputy Principal in 1958. In 1968 he became Professor Emeritus, occupying the Chair of Mathematics at the University of Guyana.
Honours
He was awarded the M.B.E. in 1962, Guyana's Golden Arrow of Achievement (A.A.) in 1972, and the Sir Alfred Victor Crane Gold Medal for his contribution to education in Guyana in 1976.
Posthumous accolades
Cameron died in May 1983. His life was documented in Joycelynne Loncke's The Man and his Works. In 2009, to honour his memory and his school, the Queen's College of Guyana Alumni Association (Toronto) re-published his A History of the Queen's College of British Guiana, adding a biographical sketch by Loncke, and extending it beyond 1951 with the Reminiscences (1945–1980) of C. I. Trotz, an outstanding alumnus, master and principal of the school, and other pieces from the six Queen's College alumni associations around the world.
References
Cameron, N.E.
1903 births
1983 deaths
People from New Amsterdam, Guyana
Alumni of Queen's College, Guyana
Alumni of the University of Cambridge
Academic staff of the University of Guyana
Members of the Order of the British Empire
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https://en.wikipedia.org/wiki/Stark%20conjectures
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In number theory, the Stark conjectures, introduced by and later expanded by , give conjectural information about the coefficient of the leading term in the Taylor expansion of an Artin L-function associated with a Galois extension K/k of algebraic number fields. The conjectures generalize the analytic class number formula expressing the leading coefficient of the Taylor series for the Dedekind zeta function of a number field as the product of a regulator related to S-units of the field and a rational number.
When K/k is an abelian extension and the order of vanishing of the L-function at s = 0 is one, Stark gave a refinement of his conjecture, predicting the existence of certain S-units, called Stark units, which generate abelian extensions of number fields.
Formulation
General case
The Stark conjectures, in the most general form, predict that the leading coefficient of an Artin L-function is the product of a type of regulator, the Stark regulator, with an algebraic number.
Abelian rank-one case
When the extension is abelian and the order of vanishing of an L-function at s = 0 is one, Stark's refined conjecture predicts the existence of Stark units, whose roots generate Kummer extensions of K that are abelian over the base field k (and not just abelian over K, as Kummer theory implies). As such, this refinement of his conjecture has theoretical implications for solving Hilbert's twelfth problem.
Computation
Stark units in the abelian rank-one case have been computed in specific examples, allowing verification of the veracity of his refined conjecture. These also provide an important computational tool for generating abelian extensions of number fields, forming the basis for some standard algorithms for computing abelian extensions of number fields.
The first rank-zero cases are used in recent versions of the PARI/GP computer algebra system to compute Hilbert class fields of totally real number fields, and the conjectures provide one solution to Hilbert's twelfth problem, which challenged mathematicians to show how class fields may be constructed over any number field by the methods of complex analysis.
Progress
Stark's principal conjecture has been proven in a few special cases, such as when the character defining the L-function takes on only rational values. Except when the base field is the field of rational numbers or an imaginary quadratic field, which were covered in the work of Stark, the abelian Stark conjectures is still unproved for number fields. More progress has been made in function fields of an algebraic variety.
related Stark's conjectures to the noncommutative geometry of Alain Connes. This provides a conceptual framework for studying the conjectures, although at the moment it is unclear whether Manin's techniques will yield the actual proof.
Variations
In 1980, Benedict Gross formulated the Gross–Stark conjecture, a p-adic analogue of the Stark conjectures relating derivatives of Deligne–Ribet p-adic L-functions (
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https://en.wikipedia.org/wiki/Jos%C3%A9%20Enrique%20Moyal
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José Enrique Moyal (; 1 October 1910 – 22 May 1998) was an Australian mathematician and mathematical physicist who contributed to aeronautical engineering, electrical engineering and statistics, among other fields.
Career
Moyal helped establish the phase space formulation of quantum mechanics in 1949 by bringing together the ideas of Hermann Weyl, John von Neumann, Eugene Wigner, and Hip Groenewold.
This formulation is statistical in nature and makes logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two formulations. Phase space quantization, also known as Moyal quantization, largely avoids the use of operators for quantum mechanical observables prevalent in the canonical formulation. Quantum-mechanical evolution in phase space is specified by a Moyal bracket.
Moyal grew up in Tel Aviv, and attended the Herzliya Hebrew Gymnasium. He studied in Paris in the 1930s, at the École Supérieure d'Electricité, Institut de Statistique, and, finally, at the Institut Henri Poincaré. His work was carried out in wartime England in the 1940s, while employed at the de Havilland Aircraft company.
Moyal was a professor of mathematics at the former School of Mathematics and Physics of Macquarie University, where he was a colleague of John Clive Ward. Previously, he had worked at the Argonne National Laboratory in Illinois.
He published pioneering work on stochastic processes.
Personal life
Moyal was married to Susanna Pollack (1912-2000), with whom he had two children, Orah Young (born in Tel Aviv) and David Moyal (born in Belfast). They divorced in 1956. He was married to Ann Moyal from 1962 until his death.
Works
J.E. Moyal, "Stochastic Processes and Statistical Physics" Journal of the Royal Statistical Society B'', 11, (1949), 150–210.
See also
Moyal bracket
Wigner–Weyl transform
Wigner quasiprobability distribution
References
External links
Maverick Mathematician: The Life and Science of J.E. Moyal
Obituary by Alan McIntosh and photographs
Moyal Medal awarded annually by Macquarie University for research contributions to mathematics, physics or statistics
1910 births
1998 deaths
20th-century Australian mathematicians
Australian physicists
Australian statisticians
Israeli emigrants to Australia
Herzliya Hebrew Gymnasium alumni
Jewish physicists
Academic staff of Macquarie University
Mathematical physicists
Scientists from Jerusalem
Quantum physicists
Mandatory Palestine expatriates in France
Mandatory Palestine expatriates in the United Kingdom
University of Paris alumni
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https://en.wikipedia.org/wiki/Rami%20Grossberg
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Rami Grossberg () is a full professor of mathematics at Carnegie Mellon University and works in model theory.
Work
Grossberg's work in the past few years has revolved around the classification theory of non-elementary classes. In particular, he has provided, in joint work with Monica VanDieren, a proof of an upward "Morley's Categoricity Theorem" (a version of Shelah's categoricity conjecture) for Abstract Elementary Classes with the amalgamation property, that are tame. In another work with VanDieren, they also initiated the study of tame Abstract Elementary Classes. Tameness is both a crucial technical property in categoricity transfer proofs and an independent notion of interest in the area – it has been studied by Baldwin, Hyttinen, Lessmann, Kesälä, Kolesnikov, Kueker among others.
Other results include a best approximation to the main gap conjecture for AECs (with Olivier Lessmann), identifying AECs with JEP, AP, no maximal models and tameness as the uncountable analog to Fraïssé's constructions (with VanDieren), a stability spectrum theorem and the existence of Morley sequences for those classes (also with VanDieren).
In addition to this work on the Categoricity Conjecture, more recently, with Boney and Vasey, new understanding of frames in AECs and forking (in the abstract elementary class setting) has been obtained.
Some of Grossberg's work may be understood as part of the big project on Saharon Shelah's outstanding categoricity conjectures:
Conjecture 1. (Categoricity for ). Let be a sentence. If is categorical in a cardinal then is categorical in all cardinals . See Infinitary logic and Beth number.
Conjecture 2. (Categoricity for AECs) See and . Let K be an AEC. There exists a cardinal μ(K) such that categoricity in a cardinal greater than μ(K) implies categoricity in all cardinals greater than μ(K). Furthermore, μ(K) is the Hanf number of K.
Other examples of his results in pure model theory include: generalizing the Keisler–Shelah omitting types theorem for to successors of singular cardinals; with Shelah, introducing the notion of unsuper-stability for infinitary logics, and proving a nonstructure theorem, which is used to resolve a problem of Fuchs and Salce in the theory of modules; with Hart, proving a structure theorem for , which resolves Morley's conjecture for excellent classes; and the notion of relative saturation and its connection to Shelah's conjecture for .
Examples of his results in applications to algebra include the finding that under the weak continuum hypothesis there is no universal object in the class of uncountable locally finite groups (answering a question of Macintyre and Shelah); with Shelah, showing that there is a jump in cardinality of the abelian group Extp(G, Z) at the first singular strong limit cardinal.
Personal life
In 1986, Grossberg attained his doctorate from the University of Jerusalem. He later married his former doctoral student and frequent collaborator, Monica VanDieren.
Ref
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https://en.wikipedia.org/wiki/Pseudorandom%20graph
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In graph theory, a graph is said to be a pseudorandom graph if it obeys certain properties that random graphs obey with high probability. There is no concrete definition of graph pseudorandomness, but there are many reasonable characterizations of pseudorandomness one can consider.
Pseudorandom properties were first formally considered by Andrew Thomason in 1987. He defined a condition called "jumbledness": a graph is said to be -jumbled for real and with if
for every subset of the vertex set , where is the number of edges among (equivalently, the number of edges in the subgraph induced by the vertex set ). It can be shown that the Erdős–Rényi random graph is almost surely -jumbled. However, graphs with less uniformly distributed edges, for example a graph on vertices consisting of an -vertex complete graph and completely independent vertices, are not -jumbled for any small , making jumbledness a reasonable quantifier for "random-like" properties of a graph's edge distribution.
Connection to local conditions
Thomason showed that the "jumbled" condition is implied by a simpler-to-check condition, only depending on the codegree of two vertices and not every subset of the vertex set of the graph. Letting be the number of common neighbors of two vertices and , Thomason showed that, given a graph on vertices with minimum degree , if for every and , then is -jumbled. This result shows how to check the jumbledness condition algorithmically in polynomial time in the number of vertices, and can be used to show pseudorandomness of specific graphs.
Chung–Graham–Wilson theorem
In the spirit of the conditions considered by Thomason and their alternately global and local nature, several weaker conditions were considered by Chung, Graham, and Wilson in 1989: a graph on vertices with edge density and some can satisfy each of these conditions if
Discrepancy: for any subsets of the vertex set , the number of edges between and is within of .
Discrepancy on individual sets: for any subset of , the number of edges among is within of .
Subgraph counting: for every graph , the number of labeled copies of among the subgraphs of is within of .
4-cycle counting: the number of labeled -cycles among the subgraphs of is within of .
Codegree: letting be the number of common neighbors of two vertices and ,
Eigenvalue bounding: If are the eigenvalues of the adjacency matrix of , then is within of and .
These conditions may all be stated in terms of a sequence of graphs where is on vertices with edges. For example, the 4-cycle counting condition becomes that the number of copies of any graph in is as , and the discrepancy condition becomes that , using little-o notation.
A pivotal result about graph pseudorandomness is the Chung–Graham–Wilson theorem, which states that many of the above conditions are equivalent, up to polynomial changes in . A sequence of graphs which satisfies those conditions is called quasi-rando
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https://en.wikipedia.org/wiki/Theodore%20Slaman
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Theodore Allen Slaman (born April 17, 1954) is a professor of mathematics at the University of California, Berkeley who works in recursion theory.
Slaman and W. Hugh Woodin formulated the Bi-interpretability Conjecture for the Turing degrees, which conjectures that the partial order of the Turing degrees is logically equivalent to second-order arithmetic. They showed that the Bi-interpretability Conjecture is equivalent to there being no nontrivial automorphism of the Turing degrees. They also exhibited limits on the possible automorphisms of the Turing degrees by showing that any automorphism will be arithmetically definable.
References
External links
home page.
Living people
American logicians
20th-century American mathematicians
21st-century American mathematicians
University of California, Berkeley faculty
Harvard University alumni
1954 births
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https://en.wikipedia.org/wiki/Leo%20Harrington
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Leo Anthony Harrington (born May 17, 1946) is a professor of mathematics at the University of California, Berkeley who works in
recursion theory, model theory, and set theory.
Having retired from being a Mathematician, Professor Leo Harrington is now a Philosopher.
His notable results include proving the Paris–Harrington theorem along with Jeff Paris,
showing that if the axiom of determinacy holds for all analytic sets then x# exists for all reals x,
and proving with Saharon Shelah that the first-order theory of the partially ordered set of recursively enumerable Turing degrees is undecidable.
References
External links
Home page.
Living people
American logicians
20th-century American mathematicians
21st-century American mathematicians
Massachusetts Institute of Technology alumni
University of California, Berkeley College of Letters and Science faculty
Model theorists
Set theorists
1946 births
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https://en.wikipedia.org/wiki/Robin%20Hartshorne
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Robin Cope Hartshorne ( ; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry.
Career
Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University (under the name Robert C. Hartshorne). He received a Ph.D. in mathematics from Princeton University in 1963 after completing a doctoral dissertation titled Connectedness of the Hilbert scheme under the supervision of John Coleman Moore and Oscar Zariski. He then became a Junior Fellow at Harvard University, where he taught for several years. In 1972, he was appointed to the faculty at the University of California, Berkeley, where he is a Professor Emeritus as of 2020.
Hartshorne is the author of the text Algebraic Geometry.
Awards
In 1979, Hartshorne was awarded the Leroy P. Steele Prize for "his expository research article Equivalence relations on algebraic cycles and subvarieties of small codimension, Proceedings of Symposia in Pure Mathematics, volume 29, American Mathematical Society, 1975, pp. 129-164; and his book Algebraic geometry, Springer-Verlag, Berlin and New York, 1977." In 2012, Hartshorne became a fellow of the American Mathematical Society.
Personal life
Hartshorne attended high school at Phillips Exeter Academy, graduating in 1955. Hartshorne is married to Edie Churchill and has two sons and an adopted daughter. He is a mountain climber and amateur flute and shakuhachi player.
Selected publications
Foundations of Projective Geometry, New York: W. A. Benjamin, 1967;
Ample Subvarieties of Algebraic Varieties, New York: Springer-Verlag. 1970;
Algebraic Geometry, New York: Springer-Verlag, 1977; corrected 6th printing, 1993. GTM 52,
Families of Curves in P3 and Zeuthen's Problem. Vol. 617. American Mathematical Society, 1997.
Geometry: Euclid and Beyond, New York: Springer-Verlag, 2000; corrected 2nd printing, 2002; corrected 4th printing, 2005.
Local Cohomology: A Seminar Given by A. Grothendieck, Harvard University. Fall, 1961. Vol. 41. Springer, 2006. (lecture notes by R. Hartshorne)
Deformation Theory, Springer-Verlag, GTM 257, 2010,
See also
Hartshorne ellipse
References
External links
Home page at the University of California at Berkeley
Hartshorne's Paintings
1938 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Algebraic geometers
Princeton University alumni
Harvard University Department of Mathematics faculty
Harvard University faculty
University of California, Berkeley College of Letters and Science faculty
Fellows of the American Mathematical Society
American flautists
Harvard University alumni
Phillips Exeter Academy alumni
Putnam Fellows
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https://en.wikipedia.org/wiki/James%20Sethian
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James Albert Sethian is a professor of mathematics at the University of California, Berkeley and the head of the Mathematics Group at the United States Department of Energy's Lawrence Berkeley National Laboratory.
Sethian was born in Washington, D.C., on May 10, 1954. He received a B.A. (1976) from Princeton and a M.A. (1978) and Ph.D (1982) from Berkeley under the direction of Alexandre Chorin. Beginning in 1983, he was a National Science Foundation postdoctoral fellow, lastly at the Courant Institute under Peter Lax. In 1985, he returned to Berkeley to join the mathematics faculty, where he is currently a full professor. Sethian was elected member of the National Academy of Engineering in 2008 as well as the National Academy of Sciences in 2013. Sethian has acted as Interim Director Research at Thinking Machines Corporation and held visiting positions at the National Center for Atmospheric Research and the National Institute of Standards and Technology.
Work
Sethian has worked on numerical algorithms for tracking moving interfaces for over three decades, starting with his seminal 1982 work on curve and surface propagation in combustion, and his 1985 work on entropy conditions, curvature, stability of numerical algorithms. This work led to development of the level-set method in 1988, which was developed jointly with Stanley Osher.
These are numerical algorithms for tracking moving interfaces in complex situations, and have proved instrumental in a wide collection of applications, including semiconductor processing, fluid mechanics, medical imaging, computer graphics, and materials science.
Jointly with D. Adalsteinsson, Sethian then introduced the idea of adaptivity to level set methods, in which computational labor is focused on the evolving front: their Adaptive Narrow Band level set method and its variants are what makes level set methods efficient and practical, and are the most common form of these techniques in practice today.
Together with Alexander Vladimirsky, Sethian developed a class of Dijkstra-like ordered upwind methods for solving static Hamilton–Jacobi equations. In the case of an Eikonal equation, the first method to do so was developed by Jon N. Tsitsiklis using a control-theoretic approach: followed shortly by Sethian's work on high-order finite-difference Dijkstra-like Fast Marching Methods. Ravikanth Malladi and Sethian pioneered the application of these techniques to image segmentation, Ron Kimmel and Sethian introduced them to robotic navigation and extended them to curved domains, and Mihai Popovici and Sethian were the first to use them as fast wave solvers in geophysical seismic imaging.
Together with Sergey Fomel, Sethian invented Escape Arrival Methods for computing multiple arrivals in wave propagation and geophysical imaging.
Algorithms based on the work of Sethian and his colleagues are now commonly used throughout science and engineering. Examples include informing engineers how to design more prec
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https://en.wikipedia.org/wiki/Institute%20For%20Figuring
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The Institute For Figuring (IFF) is an organization based in Los Angeles, California that promotes the public understanding of the poetic and aesthetic dimensions of science, mathematics and the technical arts. Founded by Margaret Wertheim and Christine Wertheim, the institute hosts public lectures and exhibitions, publishes books and maintains a website.
Published works
Robert Kaplan The Figure That Stands Behind Figures: Mosaics of the Mind (2004)
Margaret Wertheim A Field Guide to Hyperbolic Space (2005)
Margaret Wertheim A Field Guide to the Business Card Menger Sponge (2006)
Margaret Wertheim, Christine Wertheim Crochet Coral Reef: A Project (2015)
See also
Mathematics and fiber arts
European Society for Mathematics and the Arts
References
External links
Educational institutions established in 2003
Culture of Los Angeles
Science and culture
Mathematical institutes
2003 establishments in California
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https://en.wikipedia.org/wiki/Fidelity%20of%20quantum%20states
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In quantum mechanics, notably in quantum information theory, fidelity is a measure of the "closeness" of two quantum states. It expresses the probability that one state will pass a test to identify as the other. The fidelity is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space.
Definition
The fidelity between two quantum states and , expressed as density matrices, is commonly defined as:
The square roots in this expression are well-defined because both and are for positive semidefinite matrices, and the square root of a positive semidefinite matrix is defined via the spectral theorem. The Euclidean inner product from the classical definition is replaced by the Hilbert–Schmidt inner product.
As will be discussed in the following sections, this expression can be simplified in various cases of interest. In particular, for pure states, and , it equals:This tells us that the fidelity between pure states has a straightforward interpretation in terms of probability of finding the state when measuring in a basis containing .
Some authors use an alternative definition and call this quantity fidelity. The definition of however is more common. To avoid confusion, could be called "square root fidelity". In any case it is advisable to clarify the adopted definition whenever the fidelity is employed.
Motivation from classical counterpart
Given two random variables with values (categorical random variables) and probabilities and , the fidelity of and is defined to be the quantity
.
The fidelity deals with the marginal distribution of the random variables. It says nothing about the joint distribution of those variables. In other words, the fidelity is the square of the inner product of and viewed as vectors in Euclidean space. Notice that if and only if . In general, . The measure is known as the Bhattacharyya coefficient.
Given a classical measure of the distinguishability of two probability distributions, one can motivate a measure of distinguishability of two quantum states as follows: if an experimenter is attempting to determine whether a quantum state is either of two possibilities or , the most general possible measurement they can make on the state is a POVM, which is described by a set of Hermitian positive semidefinite operators . When measuring a state with this POVM, -th outcome is found with probability , and likewise with probability for . The ability to distinguish between and is then equivalent to their ability to distinguish between the classical probability distributions and . A natural question is then to ask what is the POVM the makes the two distributions as distinguishable as possible, which in this context means to minimize the Bhattacharyya coefficient over the possible choices of POVM. Formally, we are thus led to define the fidelity between quantum states as:
It was shown by Fuchs and Caves that the minimization in this expression can be com
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https://en.wikipedia.org/wiki/318%20%28number%29
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318 is the natural number following 317 and preceding 319.
In mathematics
318 is:
a sphenic number
a nontotient
the number of posets with 6 unlabeled elements
the sum of 12 consecutive primes, 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47.
In religion
In Genesis 14, Abraham takes 318 men to rescue his brother Lot.
References
Integers
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https://en.wikipedia.org/wiki/UCT%20Mathematics%20Competition
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The UCT Mathematics Competition is an annual mathematics competition for schools in the Western Cape province of South Africa, held at the University of Cape Town.
Around 7000 participants from Grade 8 to Grade 12 take part, writing a multiple-choice paper. Individual and pair entries are accepted, but all write the same paper for their grade.
The current holder of the School Trophy is Rondebosch Boys High School, with Diocesan College achieving second place in the 2022 competition. These two schools have held the top positions in the competition for a number of years.
The competition was established in 1977 by Mona Leeuwenberg and Shirley Fitton, who were teachers at Diocesan College and Westerford High School, and since 1987 has been run by Professor John Webb of the University of Cape Town.
Awards
Mona Leeuwenburg Trophy
The Mona Leeuwenburg Trophy is awarded to the school with the best overall performance in the competition.
UCT Trophy
The UCT Trophy is awarded to the school with the best performance that has not participated in the competition more than twice before.
Diane Tucker Trophy
The Diane Tucker Trophy is awarded to the girl with the best performance in the competition. This trophy was first made in year 2000.
Moolla Trophy
The Moolla Trophy was donated to the competition by the Moolla family. Saadiq, Haroon and Ashraf Moolla represented Rondebosch Boys' High School and achieved Gold Awards from 2003 to 2011. The trophy is awarded to a school from a disadvantaged community that shows a notable performance in the competition.
Lesley Reeler Trophy
The Lesley Reeler Trophy is awarded for the best individual performance over five years (grades 8 to 12).
References
External links
UCT Mathematics Competition
University of Cape Town
Mathematics competitions
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https://en.wikipedia.org/wiki/Analysis%20Situs%20%28paper%29
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"Analysis Situs" is a seminal mathematics paper that Henri Poincaré published in 1895. Poincaré published five supplements to the paper between 1899 and 1904.
These papers provided the first systematic treatment of topology and revolutionized the subject by using algebraic structures to distinguish between non-homeomorphic topological spaces, founding the field of algebraic topology. Poincaré's papers introduced the concepts of the fundamental group and simplicial homology, provided an early formulation of the Poincaré duality theorem, introduced the Euler–Poincaré characteristic for chain complexes, and raised several important conjectures, including the celebrated Poincaré conjecture, which was later proven as a theorem. The 1895 paper coined the mathematical term "homeomorphism".
Footnotes
References
Mathematics papers
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https://en.wikipedia.org/wiki/Resolvable%20space
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In topology, a topological space is said to be resolvable if it is expressible as the union of two disjoint dense subsets. For instance, the real numbers form a resolvable topological space because the rationals and irrationals are disjoint dense subsets. A topological space that is not resolvable is termed irresolvable.
Properties
The product of two resolvable spaces is resolvable
Every locally compact topological space without isolated points is resolvable
Every submaximal space is irresolvable
See also
Glossary of topology
References
Properties of topological spaces
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https://en.wikipedia.org/wiki/Toronto%20space
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In mathematics, in the realm of point-set topology, a Toronto space is a topological space that is homeomorphic to every proper subspace of the same cardinality.
There are five homeomorphism classes of countable Toronto spaces, namely: the discrete topology, the indiscrete topology, the cofinite topology and the upper and lower topologies on the natural numbers. The only countable Hausdorff Toronto space is the discrete space.
The Toronto space problem asks for an uncountable Toronto Hausdorff space that is not discrete.
References
Properties of topological spaces
Homeomorphisms
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https://en.wikipedia.org/wiki/Feebly%20compact%20space
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In mathematics, a topological space is feebly compact if every locally finite cover by nonempty open sets is finite. The concept was introduced by S. Mardeĉić and P. Papić in 1955.
Some facts:
Every compact space is feebly compact.
Every feebly compact paracompact space is compact.
Every feebly compact space is pseudocompact but the converse is not necessarily true.
For a completely regular Hausdorff space the properties of being feebly compact and pseudocompact are equivalent.
Any maximal feebly compact space is submaximal.
References
Compactness (mathematics)
Properties of topological spaces
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https://en.wikipedia.org/wiki/Gamma%20%28disambiguation%29
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Gamma is the third letter of the Greek alphabet.
Gamma may also refer to:
Science and mathematics
General
Gamma wave, a type of brain wave
Latin gamma (), used as an IPA symbol for voiced velar fricative and in the alphabets of African languages
Tropical Storm Gamma (2005), a 2005 Tropical Storm, that made landfall in Honduras
Tropical Storm Gamma (2020), a 2020 Tropical Storm, that made landfall on the Yucatán Peninsula
GAMMA, an extensive air shower array in Armenia
SARS-CoV-2 Gamma variant, one of the variants of SARS-CoV-2, the virus that causes COVID-19
Medicine
Gamma-glutamyltransferase (GGT), is a transferase present in the cell membranes of many tissues
Lower case, γ
Gamma correction, a property of images and video displays
Euler's constant, a mathematical constant
Gamma test (statistics), sometimes called Goodman and Kruskal's gamma, a non-parametric statistical test for strength of association.
Gamma ray, also gamma radiation, an electromagnetic ray
Photon, seen as an elementary particle in physics
Propagation constant of an electromagnetic wave
The third-brightest star of a constellation, in Bayer designation
Adiabatic index or heat capacity ratio, the ratio of the heat capacity at constant pressure to that at constant volume
In engineering, used to denote
Shear strain
Surface tension
Body effect on threshold voltage in field-effect transistor technology
gamma-Hydroxybutyric acid, a narcotic (GHB)
Lindane or gamma-hexachlorocyclohexane, an insecticide
Lorentz factor in relativity theory and astronomy
Gamma (eclipse) denotes how central (how close to the middle of the body) an eclipse is.
Voiced velar fricative in phonetics
Non-SI unit of magnetic flux density, 1nT
Upper case, Γ
Gamma distribution, a probability distribution function
Gamma function, a mathematical function
P-adic gamma function, a mathematical function
Christoffel symbols in general relativity
Circulation (fluid dynamics)
Reflection coefficient in electrical engineering
Gamma (finance), a second order derivative of an option pricing formula
Center of the Brillouin zone
Feferman–Schütte ordinal Γ0
The representation of a molecule's symmetry elements in inorganic chemistry
Decay width of a particle in high-energy physics
Typing environment within a type system
Technology
Gamma software, a stage in the software release life cycle
Gamma correction, in video, graphics, color spaces, and photographic imaging
Elias gamma coding, in computer science, encoding, compression
GM Gamma platform, a subcompact automobile platform by General Motors
Bristol Siddeley Gamma, a family of British rocket engines
Hyundai Gamma engine, a family of 1.6 L gasoline inline-4 engines
Suzuki RG250 Gamma, a two-stroke motorcycle
Companies
Gamma (store), a Dutch hardware-store chain
Gamma Group, developers of FinFisher and associated suites of malware
Organizations
Groupe des Architectes Modernes Marocains, an architecture collective in Morocco
Popular culture
Gamma (agency), a French phot
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https://en.wikipedia.org/wiki/%C3%89tale%20topology
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In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale topology was originally introduced by Alexander Grothendieck to define étale cohomology, and this is still the étale topology's most well-known use.
Definitions
For any scheme X, let Ét(X) be the category of all étale morphisms from a scheme to X. This is the analog of the category of open subsets of X (that is, the category whose objects are varieties and whose morphisms are open immersions). Its objects can be informally thought of as étale open subsets of X. The intersection of two objects corresponds to their fiber product over X. Ét(X) is a large category, meaning that its objects do not form a set.
An étale presheaf on X is a contravariant functor from Ét(X) to the category of sets. A presheaf F is called an étale sheaf if it satisfies the analog of the usual gluing condition for sheaves on topological spaces. That is, F is an étale sheaf if and only if the following condition is true. Suppose that is an object of Ét(X) and that is a jointly surjective family of étale morphisms over X. For each i, choose a section xi of F over Ui. The projection map , which is loosely speaking the inclusion of the intersection of Ui and Uj in Ui, induces a restriction map . If for all i and j the restrictions of xi and xj to are equal, then there must exist a unique section x of F over U which restricts to xi for all i.
Suppose that X is a Noetherian scheme. An abelian étale sheaf F on X is called finite locally constant if it is a representable functor which can be represented by an étale cover of X. It is called constructible if X can be covered by a finite family of subschemes on each of which the restriction of F is finite locally constant. It is called torsion if F(U) is a torsion group for all étale covers U of X. Finite locally constant sheaves are constructible, and constructible sheaves are torsion. Every torsion sheaf is a filtered inductive limit of constructible sheaves.
Grothendieck originally introduced the machinery of Grothendieck topologies and topoi to define the étale topology. In this language, the definition of the étale topology is succinct but abstract: It is the topology generated by the pretopology whose covering families are jointly surjective families of étale morphisms. The small étale site of X is the category O(Xét) whose objects are schemes U with a fixed étale morphism U → X. The morphisms are morphisms of schemes compatible with the fixed maps to X. The big étale site of X is the category Ét/X, that is, the category of schemes with a fixed map to X, considered with the étale topology.
The étale topology can be defined using slightly less data. First, notice that the étale topology is finer than the Zariski topology. Consequently, to define an étale cover of
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https://en.wikipedia.org/wiki/Weierstrass%20product%20inequality
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In mathematics, the Weierstrass product inequality states that for any real numbers 0 ≤ x1, ..., xn ≤ 1 we have
where
The inequality is named after the German mathematician Karl Weierstrass.
Proof
The inequality with the subtractions can be proven easily via mathematical induction. The one with the additions is proven identically. We can choose as the base case and see that for this value of we get
which is indeed true. Assuming now that the inequality holds for all natural numbers up to , for we have:
which concludes the proof.
References
Inequalities
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https://en.wikipedia.org/wiki/Cicho%C5%84%27s%20diagram
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In set theory,
Cichoń's diagram or Cichon's diagram is a table of 10 infinite cardinal numbers related to the set theory of the reals displaying the provable relations between these
cardinal characteristics of the continuum. All these cardinals are greater than or equal to , the smallest uncountable cardinal, and they are bounded above by , the cardinality of the continuum. Four cardinals describe properties of the ideal of sets of measure zero; four more describe the corresponding properties of the ideal of meager sets (first category sets).
Definitions
Let I be an ideal of a fixed infinite set X, containing all finite subsets of X. We define the following "cardinal coefficients" of I:
The "additivity" of I is the smallest number of sets from I whose union is not in I any more. As any ideal is closed under finite unions, this number is always at least ; if I is a σ-ideal, then add(I) ≥ .
The "covering number" of I is the smallest number of sets from I whose union is all of X. As X itself is not in I, we must have add(I) ≤ cov(I).
The "uniformity number" of I (sometimes also written ) is the size of the smallest set not in I. By our assumption on I, add(I) ≤ non(I).
The "cofinality" of I is the cofinality of the partial order (I, ⊆). It is easy to see that we must have non(I) ≤ cof(I) and cov(I) ≤ cof(I).
Furthermore, the "bounding number" or "unboundedness number" and the "dominating number" are defined as follows:
where "" means: "there are infinitely many natural numbers n such that …", and "" means "for all except finitely many natural numbers n we have …".
Diagram
Let be the σ-ideal of those subsets of the real line that are meager (or "of the first category") in the euclidean topology, and let
be the σ-ideal of those subsets of the real line that are of Lebesgue measure zero. Then the following inequalities hold:
Where an arrow from to is to mean that . In addition, the following relations hold:
It turns out that the inequalities described by the diagram, together with the relations mentioned above, are all the relations between these cardinals that are provable in ZFC, in the following limited sense. Let A be any assignment of the cardinals and to the 10 cardinals in Cichoń's diagram. Then if A is consistent with the diagram's relations, and if A also satisfies the two additional relations, then A can be realized in some model of ZFC.
For larger continuum sizes, the situation is less clear. It is consistent with ZFC that all of the Cichoń's diagram cardinals are simultaneously different apart from and (which are equal to other entries), but () it remains open whether all combinations of the cardinal orderings consistent with the diagram are consistent.
Some inequalities in the diagram (such as "add ≤ cov") follow immediately from the definitions. The inequalities and
are classical theorems
and follow from the fact that the real line can be partitioned into a meager set and a set of measure zero
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https://en.wikipedia.org/wiki/Kerk-Avezaath
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Kerk-Avezaath is a village in the Dutch province of Gelderland. It is a part of the municipality of Buren, and lies about 3 km west of Tiel.
A small part of the village (not counted in the statistics above) is part of the municipality of Tiel, and consists of about 60 houses.
History
It was first mentioned in 850 as Auansati, and means "church at the house of Avo (person)". The village developed along two parallel roads on a stream, and a stretched esdorp developed. The tower of the Dutch Reformed Church dates from 1640 and has an 11th century base. The church dates from 1861 and has 14th century elements. In 1840, it was home to 449 people.
Gallery
References
Populated places in Gelderland
Buren
Tiel
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https://en.wikipedia.org/wiki/Kenneth%20Kunen
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Herbert Kenneth Kunen (August 2, 1943August 14, 2020) was a professor of mathematics at the University of Wisconsin–Madison who worked in set theory and its applications to various areas of mathematics, such as set-theoretic topology and measure theory. He also worked on non-associative algebraic systems, such as loops, and used computer software, such as the Otter theorem prover, to derive theorems in these areas.
Personal life
Kunen was born in New York City in 1943 and died in 2020. He lived in Madison, Wisconsin, with his wife Anne, with whom he had two sons, Isaac and Adam.
Education
Kunen completed his undergraduate degree at the California Institute of Technology and received his Ph.D. in 1968 from Stanford University, where he was supervised by Dana Scott.
Career and research
Kunen showed that if there exists a nontrivial elementary embedding j : L → L of the constructible universe, then 0# exists.
He proved the consistency of a normal, -saturated ideal on from the consistency of the existence of a huge cardinal. He introduced the method of iterated ultrapowers, with which he proved that if is a measurable cardinal with or is a strongly compact cardinal then there is an inner model of set theory with many measurable cardinals. He proved Kunen's inconsistency theorem showing the impossibility of a nontrivial elementary embedding , which had been suggested as a large cardinal assumption (a Reinhardt cardinal).
Away from the area of large cardinals, Kunen is known for intricate forcing and combinatorial constructions. He proved that it is consistent that Martin's axiom first fails at a singular cardinal and constructed
under the continuum hypothesis a compact L-space supporting a nonseparable measure. He also showed that has no increasing chain of length in the standard Cohen model
where the continuum is . The concept of a Jech–Kunen tree is named after him and Thomas Jech.
Bibliography
The journal Topology and its Applications has dedicated a special issue to "Ken" Kunen, containing a biography by Arnold W. Miller, and surveys about Kunen's research in various fields by Mary Ellen Rudin, Akihiro Kanamori, István Juhász, Jan van Mill, Dikran Dikranjan, and Michael Kinyon.
Selected publications
Set Theory. College Publications, 2011. .
The Foundations of Mathematics. College Publications, 2009. .
Set Theory: An Introduction to Independence Proofs. North-Holland, 1980. .
(co-edited with Jerry E. Vaughan). Handbook of Set-Theoretic Topology. North-Holland, 1984. .
References
External links
Kunen's home page
1943 births
2020 deaths
20th-century American mathematicians
21st-century American mathematicians
Set theorists
Stanford University alumni
Topologists
Educators from New York City
Writers from New York City
University of Wisconsin–Madison faculty
American logicians
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https://en.wikipedia.org/wiki/Paley%20graph
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In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic tools to be applied to the number theory of quadratic residues, and have interesting properties that make them useful in graph theory more generally.
Paley graphs are named after Raymond Paley. They are closely related to the Paley construction for constructing Hadamard matrices from quadratic residues .
They were introduced as graphs independently by and . Sachs was interested in them for their self-complementarity properties, while Erdős and Rényi studied their symmetries.
Paley digraphs are directed analogs of Paley graphs that yield antisymmetric conference matrices. They were introduced by (independently of Sachs, Erdős, and Rényi) as a way of constructing tournaments with a property previously known to be held only by random tournaments: in a Paley digraph, every small subset of vertices is dominated by some other vertex.
Definition
Let q be a prime power such that q = 1 (mod 4). That is, q should either be an arbitrary power of a Pythagorean prime (a prime congruent to 1 mod 4) or an even power of an odd non-Pythagorean prime. This choice of q implies that in the unique finite field Fq of order q, the element −1 has a square root.
Now let V = Fq and let
.
If a pair {a,b} is included in E, it is included under either ordering of its two elements. For, a − b = −(b − a), and −1 is a square, from which it follows that a − b is a square if and only if b − a is a square.
By definition G = (V, E) is the Paley graph of order q.
Example
For q = 13, the field Fq is just integer arithmetic modulo 13. The numbers with square roots mod 13 are:
±1 (square roots ±1 for +1, ±5 for −1)
±3 (square roots ±4 for +3, ±6 for −3)
±4 (square roots ±2 for +4, ±3 for −4).
Thus, in the Paley graph, we form a vertex for each of the integers in the range [0,12], and connect each such integer x to six neighbors: x ± 1 (mod 13), x ± 3 (mod 13), and x ± 4 (mod 13).
Properties
The Paley graphs are self-complementary: the complement of any Paley graph is isomorphic to it. One isomorphism is via the mapping that takes a vertex x to xk (mod q), where k is any nonresidue mod q .
Paley graphs are strongly regular graphs, with parameters
This in fact follows from the fact that the graph is arc-transitive and self-complementary. In addition, Paley graphs form an infinite family of conference graphs.
The eigenvalues of Paley graphs are (with multiplicity 1) and (both with multiplicity ). They can be calculated using the quadratic Gauss sum or by using the theory of strongly regular graphs.
If q is prime, the isoperimetric number i(G) of the Paley graph is known to satisfy the following bounds:
When q is prime,
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https://en.wikipedia.org/wiki/Gillams%2C%20Newfoundland%20and%20Labrador
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Gillams is a town located north west of the city of Corner Brook in the Canadian province of Newfoundland and Labrador.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Gillams had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
See also
List of cities and towns in Newfoundland and Labrador
References
Populated coastal places in Canada
Towns in Newfoundland and Labrador
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https://en.wikipedia.org/wiki/Full-employment%20theorem
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In computer science and mathematics, a full employment theorem is a term used, often humorously, to refer to a theorem which states that no algorithm can optimally perform a particular task done by some class of professionals. The name arises because such a theorem ensures that there is endless scope to keep discovering new techniques to improve the way at least some specific task is done.
For example, the full employment theorem for compiler writers states that there is no such thing as a provably perfect size-optimizing compiler, as such a proof for the compiler would have to detect non-terminating computations and reduce them to a one-instruction infinite loop. Thus, the existence of a provably perfect size-optimizing compiler would imply a solution to the halting problem, which cannot exist. This also implies that there may always be a better compiler since the proof that one has the best compiler cannot exist. Therefore, compiler writers will always be able to speculate that they have something to improve. A similar example in practical computer science is the idea of no free lunch in search and optimization, which states that no efficient general-purpose solver can exist, and hence there will always be some particular problem whose best known solution might be improved.
Similarly, Gödel's incompleteness theorems have been called full employment theorems for mathematicians. Tasks such as virus writing and detection, and spam filtering and filter-breaking are also subject to Rice's theorem.
References
Solomonoff, Ray, "A Preliminary Report on a General Theory of Inductive Inference", Report V-131, Zator Co., Cambridge, Ma. Feb 4, 1960.
p. 401, Modern Compiler Implementation in ML, Andrew W. Appel, Cambridge University Press, 1998. .
p. 27, Retargetable Compiler Technology for Embedded Systems: Tools and Applications, Rainer Leupers and Peter Marwedel, Springer-Verlag, 2001. .
Notes from a course in Modern Programming Languages at the University of Pennsylvania See p. 8.
Mathematical theorems
Theoretical computer science
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https://en.wikipedia.org/wiki/Bhaskaracharya%20Pratishthana
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Bhaskaracharya Pratishthana is a research and education institute for mathematics in Pune, India, founded by noted Indian-American mathematician professor Shreeram Abhyankar.
The institute is named after the great ancient Indian Mathematician Bhaskaracharya (Born in 1114 A.D.). Bhaskaracharya Pratishthana is a Pune, India, based institute founded in 1976. It has researchers working in many areas of mathematics, particularly in algebra and number theory.
Since 1992, the Pratishthan has also been a recognized center for conducting Regional Mathematics Olympiad (RMO) under the National Board for Higher Mathematics (NBHM) for Maharashtra and Goa Region. This has enabled the Pratishthan to train lots of students from std. V to XII for this examination. Many students who received training at BP have won medals in the International Mathematical Olympiad.
Pratishthana publishes the mathematics periodical Bona Mathematica and has published texts in higher and olympiad mathematics. Besides this the Pratishthan holds annual / biennial conferences/Workshops in some research areas in higher mathematics attended by Indian/Foreign scholars and Professors. The Pratishthan has organized a number of workshops for research students and college teachers under the aegis of NBHM/NCM. The National Board for Higher Mathematics has greatly helped Pratishthan to enrich its library and the Department of Atomic Energy and the Mathematics Department of S. P. Pune University have rendered active co-operation in holding conferences/workshops.
It also conducts the BMTSC exam which is a school-level mathematics competition for students studying in the 5th and the 6th grade. The objectives of the competition are listed below:
1. Identify good students of mathematics at an early age.
2. A pre Olympiad type competition.
3. To enhance Mathematical ability and logical thinking.
4. Nurture programs for successful students to improve their ability.
Projects
Recently, in Pratishthana, two projects supported by MHRD have been started. One is on e-learning and other on the use of free open source software in Mathematics education (FOOSME). In the e-learning project video broadcasting of online Maths lectures is being done. The topics are Ring Theory and Complex Analysis. These lectures are at Undergraduate levels. The FOSSME project is all about exploring FOSS for Maths Education. There were 3 national level workshops held on FOSS for Maths education, Scilab, Advanced and LaTeX Advanced.
References
External links
Home Page of Bhaskaracharya Pratishthana
Home Page of FOSSME project
Research institutes in Pune
Schools of mathematics
Indian mathematics
1976 establishments in Maharashtra
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https://en.wikipedia.org/wiki/SQ-universal%20group
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In mathematics, in the realm of group theory, a countable group is said to be SQ-universal if every countable group can be embedded in one of its quotient groups. SQ-universality can be thought of as a measure of largeness or complexity of a group.
History
Many classic results of combinatorial group theory, going back to 1949, are now interpreted as saying that a particular group or class of groups is (are) SQ-universal. However the first explicit use of the term seems to be in an address given by Peter Neumann to The London Algebra Colloquium entitled "SQ-universal groups" on 23 May 1968.
Examples of SQ-universal groups
In 1949 Graham Higman, Bernhard Neumann and Hanna Neumann proved that every countable group can be embedded in a two-generator group. Using the contemporary language of SQ-universality, this result says that F2, the free group (non-abelian) on two generators, is SQ-universal. This is the first known example of an SQ-universal group. Many more examples are now known:
Adding two generators and one arbitrary relator to a nontrivial torsion-free group, always results in an SQ-universal group.
Any non-elementary group that is hyperbolic with respect to a collection of proper subgroups is SQ-universal.
Many HNN extensions, free products and free products with amalgamation.
The four-generator Coxeter group with presentation:
Charles F. Miller III's example of a finitely presented SQ-universal group all of whose non-trivial quotients have unsolvable word problem.
In addition much stronger versions of the Higmann-Neumann-Neumann theorem are now known. Ould Houcine has proved:
For every countable group G there exists a 2-generator SQ-universal group H such that G can be embedded in every non-trivial quotient of H.
Some elementary properties of SQ-universal groups
A free group on countably many generators h1, h2, ..., hn, ... , say, must be embeddable in a quotient of an SQ-universal group G. If are chosen such that for all n, then they must freely generate a free subgroup of G. Hence:
Every SQ-universal group has as a subgroup, a free group on countably many generators.
Since every countable group can be embedded in a countable simple group, it is often sufficient to consider embeddings of simple groups. This observation allows us to easily prove some elementary results about SQ-universal groups, for instance:
If G is an SQ-universal group and N is a normal subgroup of G (i.e. ) then either N is SQ-universal or the quotient group G/N is SQ-universal.
To prove this suppose N is not SQ-universal, then there is a countable group K that cannot be embedded into a quotient group of N. Let H be any countable group, then the direct product H × K is also countable and hence can be embedded in a countable simple group S. Now, by hypothesis, G is SQ-universal so S can be embedded in a quotient group, G/M, say, of G. The second isomorphism theorem tells us:
Now and S is a simple subgroup of G/M so either:
or:
.
The latter cannot be
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https://en.wikipedia.org/wiki/Family%20of%20curves
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In geometry, a family of curves is a set of curves, each of which is given by a function or parametrization in which one or more of the parameters is variable. In general, the parameter(s) influence the shape of the curve in a way that is more complicated than a simple linear transformation. Sets of curves given by an implicit relation may also represent families of curves.
Families of curves appear frequently in solutions of differential equations; when an additive constant of integration is introduced, it will usually be manipulated algebraically until it no longer represents a simple linear transformation.
Families of curves may also arise in other areas. For example, all non-degenerate conic sections can be represented using a single polar equation with one parameter, the eccentricity of the curve:
as the value of changes, the appearance of the curve varies in a relatively complicated way.
Applications
Families of curves may arise in various topics in geometry, including the envelope of a set of curves and the caustic of a given curve.
Generalizations
In algebraic geometry, an algebraic generalization is given by the notion of a linear system of divisors.
External links
Algebraic geometry
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https://en.wikipedia.org/wiki/Junkyard%20tornado
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The junkyard tornado, sometimes known as Hoyle's fallacy, is an argument against abiogenesis, using a calculation of its probability based on false assumptions, as comparable to "a tornado sweeping through a junk-yard might assemble a Boeing 747 from the materials therein" and to compare the chance of obtaining even a single functioning protein by chance combination of amino acids to a solar system full of blind men solving Rubik's Cubes simultaneously. It was used originally by English astronomer Fred Hoyle (1915–2001) in his book The Intelligent Universe, where he tried to apply statistics to evolution and the origin of life. Similar reasoning were advanced in Darwin's time, and indeed as long ago as Cicero in classical antiquity. While Hoyle himself was an atheist, the argument has since become a mainstay in the rejection of evolution by religious groups.
Hoyle's fallacy contradicts many well-established and widely tested principles in the field of evolutionary biology. As the fallacy argues, the odds of the sudden construction of higher lifeforms are indeed improbable. However, what the junkyard tornado postulation fails to take into account is the vast amount of support that evolution proceeds in many smaller stages, each driven by natural selection rather than by random chance, over a long period of time. The Boeing 747 was not designed in a single unlikely burst of creativity, just as modern lifeforms were not constructed in one single unlikely event, as the junkyard tornado scenario suggests.
The theory of evolution has been studied and tested extensively by numerous researchers and scientists and is the most scientifically accurate explanation for the origins of complex life.
Hoyle's statement
According to Fred Hoyle's analysis, the probability of obtaining all of life's approximate 2000 enzymes in a random trial is about one-in-1040,000:
His junkyard analogy:
This echoes his stance, reported elsewhere:
Hoyle used this to argue in favor of panspermia, that the origin of life on Earth was from preexisting life in space.
History and reception
The junkyard tornado derives from arguments most popular in the 1920s, prior to the modern evolutionary synthesis, which are rejected by evolutionary biologists. A preliminary step is to establish that the phase space containing some biological entity (such as humans, working cells, or the eye) is enormous, something not contentious. The argument is then to infer from the huge size of the phase space that the probability that the entity could appear by chance is exceedingly low, ignoring the key process involved, natural selection.
Sometimes, arguments invoking the junkyard tornado analogy also invoke the universal probability bound, which claims that highly improbable events do not occur. It is refuted by the fact that if all possible outcomes of a natural process are highly improbable when taken individually, then one of the highly improbable outcomes is certain. The true law being refere
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https://en.wikipedia.org/wiki/Mean%20absolute%20difference
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The mean absolute difference (univariate) is a measure of statistical dispersion equal to the average absolute difference of two independent values drawn from a probability distribution. A related statistic is the relative mean absolute difference, which is the mean absolute difference divided by the arithmetic mean, and equal to twice the Gini coefficient.
The mean absolute difference is also known as the absolute mean difference (not to be confused with the absolute value of the mean signed difference) and the Gini mean difference (GMD). The mean absolute difference is sometimes denoted by Δ or as MD.
Definition
The mean absolute difference is defined as the "average" or "mean", formally the expected value, of the absolute difference of two random variables X and Y independently and identically distributed with the same (unknown) distribution henceforth called Q.
Calculation
Specifically, in the discrete case,
For a random sample of size n of a population distributed uniformly according to Q, by the law of total expectation the (empirical) mean absolute difference of the sequence of sample values yi, i = 1 to n can be calculated as the arithmetic mean of the absolute value of all possible differences:
if Q has a discrete probability function f(y), where yi, i = 1 to n, are the values with nonzero probabilities:
In the continuous case,
if Q has a probability density function f(x):
An alternative form of the equation is given by:
if Q has a cumulative distribution function F(x) with quantile function Q(F), then, since f(x)=dF(x)/dx and Q(F(x))=x, it follows that:
Relative mean absolute difference
When the probability distribution has a finite and nonzero arithmetic mean AM, the relative mean absolute difference, sometimes denoted by Δ or RMD, is defined by
The relative mean absolute difference quantifies the mean absolute difference in comparison to the size of the mean and is a dimensionless quantity. The relative mean absolute difference is equal to twice the Gini coefficient which is defined in terms of the Lorenz curve. This relationship gives complementary perspectives to both the relative mean absolute difference and the Gini coefficient, including alternative ways of calculating their values.
Properties
The mean absolute difference is invariant to translations and negation, and varies proportionally to positive scaling. That is to say, if X is a random variable and c is a constant:
MD(X + c) = MD(X),
MD(−X) = MD(X), and
MD(c X) = |c| MD(X).
The relative mean absolute difference is invariant to positive scaling, commutes with negation, and varies under translation in proportion to the ratio of the original and translated arithmetic means. That is to say, if X is a random variable and c is a constant:
RMD(X + c) = RMD(X) · mean(X)/(mean(X) + c) = RMD(X) / (1 + c / mean(X)) for c ≠ −mean(X),
RMD(−X) = −RMD(X), and
RMD(c X) = RMD(X) for c > 0.
If a random variable has a positive mean, then its relative mean absolute
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https://en.wikipedia.org/wiki/Ehresmann%20connection
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In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless, linear connections may be viewed as a special case. Another important special case of Ehresmann connections are principal connections on principal bundles, which are required to be equivariant in the principal Lie group action.
Introduction
A covariant derivative in differential geometry is a linear differential operator which takes the directional derivative of a section of a vector bundle in a covariant manner. It also allows one to formulate a notion of a parallel section of a bundle in the direction of a vector: a section s is parallel along a vector X if . So a covariant derivative provides at least two things: a differential operator, and a notion of what it means to be parallel in each direction. An Ehresmann connection drops the differential operator completely and defines a connection axiomatically in terms of the sections parallel in each direction . Specifically, an Ehresmann connection singles out a vector subspace of each tangent space to the total space of the fiber bundle, called the horizontal space. A section s is then horizontal (i.e., parallel) in the direction X if lies in a horizontal space. Here we are regarding s as a function from the base M to the fiber bundle E, so that is then the pushforward of tangent vectors. The horizontal spaces together form a vector subbundle of .
This has the immediate benefit of being definable on a much broader class of structures than mere vector bundles. In particular, it is well-defined on a general fiber bundle. Furthermore, many of the features of the covariant derivative still remain: parallel transport, curvature, and holonomy.
The missing ingredient of the connection, apart from linearity, is covariance. With the classical covariant derivatives, covariance is an a posteriori feature of the derivative. In their construction one specifies the transformation law of the Christoffel symbols – which is not covariant – and then general covariance of the derivative follows as a result. For an Ehresmann connection, it is possible to impose a generalized covariance principle from the beginning by introducing a Lie group acting on the fibers of the fiber bundle. The appropriate condition is to require that the horizontal spaces be, in a certain sense, equivariant with respect to the group action.
The finishing touch for an Ehresmann connection is that it can be represented as a differential form, in much the same way as the case of a connection form. If the group acts on the fibers and the connection is equivariant, then the form will also be equivariant. Furthermore, the connection form allows for a definition of curvature as a curv
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https://en.wikipedia.org/wiki/Van%20Hiele%20model
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In mathematics education, the Van Hiele model is a theory that describes how students learn geometry. The theory originated in 1957 in the doctoral dissertations of Dina van Hiele-Geldof and Pierre van Hiele (wife and husband) at Utrecht University, in the Netherlands. The Soviets did research on the theory in the 1960s and integrated their findings into their curricula. American researchers did several large studies on the van Hiele theory in the late 1970s and early 1980s, concluding that students' low van Hiele levels made it difficult to succeed in proof-oriented geometry courses and advising better preparation at earlier grade levels. Pierre van Hiele published Structure and Insight in 1986, further describing his theory. The model has greatly influenced geometry curricula throughout the world through emphasis on analyzing properties and classification of shapes at early grade levels. In the United States, the theory has influenced the geometry strand of the Standards published by the National Council of Teachers of Mathematics and the Common Core Standards.
Van Hiele levels
The student learns by rote to operate with [mathematical] relations that he does not understand, and of which he has not seen the origin…. Therefore the system of relations is an independent construction having no rapport with other experiences of the child. This means that the student knows only what has been taught to him and what has been deduced from it. He has not learned to establish connections between the system and the sensory world. He will not know how to apply what he has learned in a new situation. - Pierre van Hiele, 1959
The best known part of the van Hiele model are the five levels which the van Hieles postulated to describe how children learn to reason in geometry. Students cannot be expected to prove geometric theorems until they have built up an extensive understanding of the systems of relationships between geometric ideas. These systems cannot be learned by rote, but must be developed through familiarity by experiencing numerous examples and counterexamples, the various properties of geometric figures, the relationships between the properties, and how these properties are ordered. The five levels postulated by the van Hieles describe how students advance through this understanding.
The five van Hiele levels are sometimes misunderstood to be descriptions of how students understand shape classification, but the levels actually describe the way that students reason about shapes and other geometric ideas. Pierre van Hiele noticed that his students tended to "plateau" at certain points in their understanding of geometry and he identified these plateau points as levels. In general, these levels are a product of experience and instruction rather than age. This is in contrast to Piaget's theory of cognitive development, which is age-dependent. A child must have enough experiences (classroom or otherwise) with these geometric ideas to move to a higher leve
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https://en.wikipedia.org/wiki/Arnold%27s%20cat%20map
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In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name.
Thinking of the torus as the quotient space , Arnold's cat map is the transformation given by the formula
Equivalently, in matrix notation, this is
That is, with a unit equal to the width of the square image, the image is sheared one unit up, then two units to the right, and all that lies outside that unit square is shifted back by the unit until it is within the square.
Properties
Γ is invertible because the matrix has determinant 1 and therefore its inverse has integer entries,
Γ is area preserving,
Γ has a unique hyperbolic fixed point (the vertices of the square). The linear transformation which defines the map is hyperbolic: its eigenvalues are irrational numbers, one greater and the other smaller than 1 (in absolute value), so they are associated respectively to an expanding and a contracting eigenspace which are also the stable and unstable manifolds. The eigenspaces are orthogonal because the matrix is symmetric. Since the eigenvectors have rationally independent components both the eigenspaces densely cover the torus. Arnold's cat map is a particularly well-known example of a hyperbolic toral automorphism, which is an automorphism of a torus given by a square unimodular matrix having no eigenvalues of absolute value 1.
The set of the points with a periodic orbit is dense on the torus. Actually a point is periodic if and only if its coordinates are rational.
Γ is topologically transitive (i.e. there is a point whose orbit is dense).
The number of points with period is exactly (where and are the eigenvalues of the matrix). For example, the first few terms of this series are 1, 5, 16, 45, 121, 320, 841, 2205 .... (The same equation holds for any unimodular hyperbolic toral automorphism if the eigenvalues are replaced.)
Γ is ergodic and mixing,
Γ is an Anosov diffeomorphism and in particular it is structurally stable.
The mapping torus of Γ is a solvmanifold, and as with other Anosov diffeomorphisms, this manifold has solv geometry.
The discrete cat map
It is possible to define a discrete analogue of the cat map. One of this map's features is that image being apparently randomized by the transformation but returning to its original state after a number of steps. As can be seen in the adjacent picture, the original image of the cat is sheared and then wrapped around in the first iteration of the transformation. After some iterations, the resulting image appears rather random or disordered, yet after further iterations the image appears to have further order—ghost-like images of the cat, multiple smaller copies arranged in a repeating structure and even upside-down copies of the original image—and ultimately returns to the original image.
The discrete cat map describes the phase space flow corresponding to the discrete dynamics of a
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https://en.wikipedia.org/wiki/Stewart%27s%20theorem
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In geometry, Stewart's theorem yields a relation between the lengths of the sides and the length of a cevian in a triangle. Its name is in honour of the Scottish mathematician Matthew Stewart, who published the theorem in 1746.
Statement
Let , , be the lengths of the sides of a triangle. Let be the length of a cevian to the side of length . If the cevian divides the side of length into two segments of length and , with adjacent to and adjacent to , then Stewart's theorem states that
A common mnemonic used by students to memorize this equation (after rearranging the terms) is:
The theorem may be written more symmetrically using signed lengths of segments. That is, take the length to be positive or negative according to whether is to the left or right of in some fixed orientation of the line. In this formulation, the theorem states that if are collinear points, and is any point, then
In the special case that the cevian is the median (that is, it divides the opposite side into two segments of equal length), the result is known as Apollonius' theorem.
Proof
The theorem can be proved as an application of the law of cosines.
Let be the angle between and and the angle between and . Then is the supplement of , and so . Applying the law of cosines in the two small triangles using angles and produces
Multiplying the first equation by and the third equation by and adding them eliminates . One obtains
which is the required equation.
Alternatively, the theorem can be proved by drawing a perpendicular from the vertex of the triangle to the base and using the Pythagorean theorem to write the distances , , in terms of the altitude. The left and right hand sides of the equation then reduce algebraically to the same expression.
History
According to , Stewart published the result in 1746 when he was a candidate to replace Colin Maclaurin as Professor of Mathematics at the University of Edinburgh. state that the result was probably known to Archimedes around 300 B.C.E. They go on to say (mistakenly) that the first known proof was provided by R. Simson in 1751. state that the result is used by Simson in 1748 and by Simpson in 1752, and its first appearance in Europe given by Lazare Carnot in 1803.
See also
Mass point geometry
Notes
References
Further reading
I.S Amarasinghe, Solutions to the Problem 43.3: Stewart's Theorem (A New Proof for the Stewart's Theorem using Ptolemy's Theorem), Mathematical Spectrum, Vol 43(03), pp. 138 – 139, 2011.
External links
Euclidean plane geometry
Theorems about triangles
Articles containing proofs
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https://en.wikipedia.org/wiki/Brauer%E2%80%93Nesbitt%20theorem
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In mathematics, the Brauer–Nesbitt theorem can refer to several different theorems proved by Richard Brauer and Cecil J. Nesbitt in the representation theory of finite groups.
In modular representation theory,
the Brauer–Nesbitt theorem on blocks of defect zero states that a character whose order is divisible by the highest power of a prime p dividing the order of a finite group remains irreducible when reduced mod p and vanishes on all elements whose order is divisible by p. Moreover, it belongs to a block of defect zero. A block of defect zero contains only one ordinary character and only one modular character.
Another version states that if k is a field of characteristic zero, A is a k-algebra, V, W are semisimple A-modules which are finite dimensional over k, and TrV = TrW as elements of Homk(A,k), then V and W are isomorphic as A-modules.
Let be a group and be some field. If are two finite-dimensional semisimple representations such that the characteristic polynomials of and coincide for all , then and are isomorphic representations. If or , then the condition on the characteristic polynomials can be changed to the condition that Tr=Tr for all .
As a consequence, let be a semisimple (continuous) -adic representations of the absolute Galois group of some field , unramified outside some finite set of primes . Then the representation is uniquely determined by the values of the traces of for (also using the Chebotarev density theorem).
References
Curtis, Reiner, Representation theory of finite groups and associative algebras, Wiley 1962.
Brauer, R.; Nesbitt, C. On the modular characters of groups. Ann. of Math. (2) 42, (1941). 556-590.
Representation theory of finite groups
Theorems about algebras
Theorems in group theory
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https://en.wikipedia.org/wiki/Slender%20group
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In mathematics, a slender group is a torsion-free abelian group that is "small" in a sense that is made precise in the definition below.
Definition
Let ZN denote the Baer–Specker group, that is, the group of all integer sequences, with termwise addition. For each natural number n, let en be the sequence with n-th term equal to 1 and all other terms 0.
A torsion-free abelian group G is said to be slender if every homomorphism from ZN into G maps all but finitely many of the en to the identity element.
Examples
Every free abelian group is slender.
The additive group of rational numbers Q is not slender: any mapping of the en into Q extends to a homomorphism from the free subgroup generated by the en, and as Q is injective this homomorphism extends over the whole of ZN. Therefore, a slender group must be reduced.
Every countable reduced torsion-free abelian group is slender, so every proper subgroup of Q is slender.
Properties
A torsion-free abelian group is slender if and only if it is reduced and contains no copy of the Baer–Specker group and no copy of the p-adic integers for any p.
Direct sums of slender groups are also slender.
Subgroups of slender groups are slender.
Every homomorphism from ZN into a slender group factors through Zn for some natural number n.
References
.
Properties of groups
Abelian group theory
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https://en.wikipedia.org/wiki/Cotorsion%20group
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In abelian group theory, an abelian group is said to be cotorsion if every extension of it by a torsion-free group splits. If the group is , this says that for all torsion-free groups . It suffices to check the condition for the group of rational numbers.
More generally, a module M over a ring R is said to be a cotorsion module if Ext1(F,M)=0 for all flat modules F. This is equivalent to the definition for abelian groups (considered as modules over the ring Z of integers) because over Z flat modules are the same as torsion-free modules.
Some properties of cotorsion groups:
Any quotient of a cotorsion group is cotorsion.
A direct product of groups is cotorsion if and only if each factor is.
Every divisible group or injective group is cotorsion.
The Baer Fomin Theorem states that a torsion group is cotorsion if and only if it is a direct sum of a divisible group and a bounded group, that is, a group of bounded exponent.
A torsion-free abelian group is cotorsion if and only if it is algebraically compact.
Ulm subgroups of cotorsion groups are cotorsion and Ulm factors of cotorsion groups are algebraically compact.
External links
Abelian group theory
Properties of groups
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https://en.wikipedia.org/wiki/Thin%20group%20%28combinatorial%20group%20theory%29
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In mathematics, in the realm of group theory, a group is said to be thin if there is a finite upper bound on the girth of the Cayley graph induced by any finite generating set. The group is called fat if it is not thin.
Given any generating set of the group, we can consider a graph whose vertices are elements of the group with two vertices adjacent if their ratio is in the generating set. The graph is connected and vertex transitive. Paths in the graph correspond to words in the generators.
If the graph has a cycle of a given length, it has a cycle of the same length containing the identity element. Thus, the girth of the graph corresponds to the minimum length of a nontrivial word that reduces to the identity. A nontrivial word is a word that, if viewed as a word in the free group, does not reduce to the identity.
If the graph has no cycles, its girth is set to be infinity.
The girth depends on the choice of generating set. A thin group is a group where the girth has an upper bound for all finite generating sets.
Some facts about thin and fat groups and about girths:
Every finite group is thin.
Every free group is fat.
The girth of a cyclic group equals its order.
The girth of a noncyclic abelian group is at most 4, because any two elements commute and the commutation relation gives a nontrivial word.
The girth of the dihedral group is 2.
Every nilpotent group, and more generally, every solvable group, is thin.
External links
A preliminary paper on girth of groups
Properties of groups
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https://en.wikipedia.org/wiki/Pure%20subgroup
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In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and related areas.
Definition
A subgroup of a (typically abelian) group is said to be pure if whenever an element of has an root in , it necessarily has an root in . Formally: , the existence of an in G such that the existence of a in S such that .
Origins
Pure subgroups are also called isolated subgroups or serving subgroups and were first investigated in Prüfer's 1923 paper which described conditions for the decomposition of primary abelian groups as direct sums of cyclic groups using pure subgroups. The work of Prüfer was complemented by Kulikoff where many results were proved again using pure subgroups systematically. In particular, a proof was given that pure subgroups of finite exponent are direct summands. A more complete discussion of pure subgroups, their relation to infinite abelian group theory, and a survey of their literature is given in Irving Kaplansky's little red book.
Examples
Every direct summand of a group is a pure subgroup.
Every pure subgroup of a pure subgroup is pure.
A divisible subgroup of an Abelian group is pure.
If the quotient group is torsion-free, the subgroup is pure.
The torsion subgroup of an Abelian group is pure.
The directed union of pure subgroups is a pure subgroup.
Since in a finitely generated Abelian group the torsion subgroup is a direct summand, one might ask if the torsion subgroup is always a direct summand of an Abelian group. It turns out that it is not always a summand, but it is a pure subgroup. Under certain mild conditions, pure subgroups are direct summands. So, one can still recover the desired result under those conditions, as in Kulikoff's paper. Pure subgroups can be used as an intermediate property between a result on direct summands with finiteness conditions and a full result on direct summands with less restrictive finiteness conditions. Another example of this use is Prüfer's paper, where the fact that "finite torsion Abelian groups are direct sums of cyclic groups" is extended to the result that "all torsion Abelian groups of finite exponent are direct sums of cyclic groups" via an intermediate consideration of pure subgroups.
Generalizations
Pure subgroups were generalized in several ways in the theory of abelian groups and modules. Pure submodules were defined in a variety of ways, but eventually settled on the modern definition in terms of tensor products or systems of equations; earlier definitions were usually more direct generalizations such as the single equation used above for n'th roots. Pure injective and pure projective modules follow closely from the ideas of Prüfer's 1923 paper. While pure projective modules have not found as many applications as pure injectives, they are more closely related to the original work: A module is pure projective if it is
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https://en.wikipedia.org/wiki/Algebraically%20compact%20group
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In mathematics, in the realm of abelian group theory, a group is said to be algebraically compact if it is a direct summand of every abelian group containing it as a pure subgroup.
Equivalent characterizations of algebraic compactness:
The reduced part of the group is Hausdorff and complete in the adic topology.
The group is pure injective, that is, injective with respect to exact sequences where the embedding is as a pure subgroup.
Relations with other properties:
A torsion-free group is cotorsion if and only if it is algebraically compact.
Every injective group is algebraically compact.
Ulm factors of cotorsion groups are algebraically compact.
External links
On endomorphism rings of Abelian groups
Abelian group theory
Properties of groups
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https://en.wikipedia.org/wiki/Critical%20group
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In mathematics, in the realm of group theory, a group is said to be critical if it is not in the variety generated by all its proper subquotients, which includes all its subgroups and all its quotients.
Any finite monolithic A-group is critical. This result is due to Kovacs and Newman.
The variety generated by a finite group has a finite number of nonisomorphic critical groups.
External links
Definition of critical group
Properties of groups
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https://en.wikipedia.org/wiki/Stochastic%20volatility
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In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. The name derives from the models' treatment of the underlying security's volatility as a random process, governed by state variables such as the price level of the underlying security, the tendency of volatility to revert to some long-run mean value, and the variance of the volatility process itself, among others.
Stochastic volatility models are one approach to resolve a shortcoming of the Black–Scholes model. In particular, models based on Black-Scholes assume that the underlying volatility is constant over the life of the derivative, and unaffected by the changes in the price level of the underlying security. However, these models cannot explain long-observed features of the implied volatility surface such as volatility smile and skew, which indicate that implied volatility does tend to vary with respect to strike price and expiry. By assuming that the volatility of the underlying price is a stochastic process rather than a constant, it becomes possible to model derivatives more accurately.
A middle ground between the bare Black-Scholes model and stochastic volatility models is covered by local volatility models. In these models the underlying volatility does not feature any new randomness but it isn't a constant either. In local volatility models the volatility is a non-trivial function of the underlying asset, without any extra randomness. According to this definition, models like constant elasticity of variance would be local volatility models, although they are sometimes classified as stochastic volatility models. The classification can be a little ambiguous in some cases.
The early history of stochastic volatility has multiple roots (i.e. stochastic process, option pricing and econometrics), it is reviewed in Chapter 1 of Neil Shephard (2005) "Stochastic Volatility," Oxford University Press.
Basic model
Starting from a constant volatility approach, assume that the derivative's underlying asset price follows a standard model for geometric Brownian motion:
where is the constant drift (i.e. expected return) of the security price , is the constant volatility, and is a standard Wiener process with zero mean and unit rate of variance. The explicit solution of this stochastic differential equation is
The maximum likelihood estimator to estimate the constant volatility for given stock prices at different times is
its expected value is
This basic model with constant volatility is the starting point for non-stochastic volatility models such as Black–Scholes model and Cox–Ross–Rubinstein model.
For a stochastic volatility model, replace the constant volatility with a function that models the variance of . This variance function is also modeled as Brownian motion, and the form of depends on th
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https://en.wikipedia.org/wiki/Partial%20geometry
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An incidence structure consists of points , lines , and flags where a point is said to be incident with a line if . It is a (finite) partial geometry if there are integers such that:
For any pair of distinct points and , there is at most one line incident with both of them.
Each line is incident with points.
Each point is incident with lines.
If a point and a line are not incident, there are exactly pairs , such that is incident with and is incident with .
A partial geometry with these parameters is denoted by .
Properties
The number of points is given by and the number of lines by .
The point graph (also known as the collinearity graph) of a is a strongly regular graph: .
Partial geometries are dual structures: the dual of a is simply a .
Special case
The generalized quadrangles are exactly those partial geometries with .
The Steiner systems are precisely those partial geometries with .
Generalisations
A partial linear space of order is called a semipartial geometry if there are integers such that:
If a point and a line are not incident, there are either or exactly pairs , such that is incident with and is incident with .
Every pair of non-collinear points have exactly common neighbours.
A semipartial geometry is a partial geometry if and only if .
It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters
.
A nice example of such a geometry is obtained by taking the affine points of and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters .
See also
Strongly regular graph
Maximal arc
References
Incidence geometry
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https://en.wikipedia.org/wiki/Epigraph%20%28mathematics%29
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In mathematics, the epigraph or supergraph of a function valued in the extended real numbers is the set, denoted by of all points in the Cartesian product lying on or above its graph. The strict epigraph is the set of points in lying strictly above its graph.
Importantly, although both the graph and epigraph of consists of points in the epigraph consists of points in the subset which is not necessarily true of the graph of
If the function takes as a value then will be a subset of its epigraph
For example, if then the point will belong to but not to
These two sets are nevertheless closely related because the graph can always be reconstructed from the epigraph, and vice versa.
The study of continuous real-valued functions in real analysis has traditionally been closely associated with the study of their graphs, which are sets that provide geometric information (and intuition) about these functions. Epigraphs serve this same purpose in the fields of convex analysis and variational analysis, in which the primary focus is on convex functions valued in instead of continuous functions valued in a vector space (such as or ). This is because in general, for such functions, geometric intuition is more readily obtained from a function's epigraph than from its graph. Similarly to how graphs are used in real analysis, the epigraph can often be used to give geometrical interpretations of a convex function's properties, to help formulate or prove hypotheses, or to aid in constructing counterexamples.
Definition
The definition of the epigraph was inspired by that of the graph of a function, where the of is defined to be the set
The or of a function valued in the extended real numbers is the set
In the union over that appears above on the right hand side of the last line, the set may be interpreted as being a "vertical ray" consisting of and all points in "directly above" it.
Similarly, the set of points on or below the graph of a function is its .
The is the epigraph with the graph removed:
Relationships with other sets
Despite the fact that might take one (or both) of as a value (in which case its graph would be a subset of ), the epigraph of is nevertheless defined to be a subset of rather than of This is intentional because when is a vector space then so is but is a vector space (since the extended real number line is not a vector space). More generally, if is only a non-empty subset of some vector space then is never even a of vector space. The epigraph being a subset of a vector space allows for tools related to real analysis and functional analysis (and other fields) to be more readily applied.
The domain (rather than the codomain) of the function is not particularly important for this definition; it can be any linear space or even an arbitrary set instead of .
The strict epigraph and the graph are always disjoint.
The epigraph of a function is related to its graph and strict epigraph
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https://en.wikipedia.org/wiki/Arc%20%28projective%20geometry%29
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An (simple) arc in finite projective geometry is a set of points which satisfies, in an intuitive way, a feature of curved figures in continuous geometries. Loosely speaking, they are sets of points that are far from "line-like" in a plane or far from "plane-like" in a three-dimensional space. In this finite setting it is typical to include the number of points in the set in the name, so these simple arcs are called -arcs. An important generalization of the -arc concept, also referred to as arcs in the literature, are the ()-arcs.
-arcs in a projective plane
In a finite projective plane (not necessarily Desarguesian) a set of points such that no three points of are collinear (on a line) is called a {{math|k - arc}}. If the plane has order then , however the maximum value of can only be achieved if is even. In a plane of order , a -arc is called an oval and, if is even, a -arc is called a hyperoval.
Every conic in the Desarguesian projective plane PG(2,), i.e., the set of zeros of an irreducible homogeneous quadratic equation, is an oval. A celebrated result of Beniamino Segre states that when is odd, every -arc in PG(2,) is a conic (Segre's theorem). This is one of the pioneering results in finite geometry.
If is even and is a -arc in , then it can be shown via combinatorial arguments that there must exist a unique point in (called the nucleus of ) such that the union of and this point is a ( + 2)-arc. Thus, every oval can be uniquely extended to a hyperoval in a finite projective plane of even order.
A -arc which can not be extended to a larger arc is called a complete arc. In the Desarguesian projective planes, PG(2,), no -arc is complete, so they may all be extended to ovals.
-arcs in a projective space
In the finite projective space PG() with , a set of points such that no points lie in a common hyperplane is called a (spatial) -arc. This definition generalizes the definition of a -arc in a plane (where ).
()-arcs in a projective plane
A ()-arc () in a finite projective plane (not necessarily Desarguesian) is a set, of points of such that each line intersects in at most points, and there is at least one line that does intersect in points. A ()-arc is a -arc and may be referred to as simply an arc if the size is not a concern.
The number of points of a ()-arc in a projective plane of order is at most . When equality occurs, one calls a maximal arc.
Hyperovals are maximal arcs. Complete arcs need not be maximal arcs.
See also
Normal rational curve
Notes
References
External links
Projective geometry
Incidence geometry
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https://en.wikipedia.org/wiki/CA-group
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In mathematics, in the realm of group theory, a group is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian subgroup. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in the Feit–Thompson theorem and the classification of finite simple groups. Several important infinite groups are CA-groups, such as free groups, Tarski monsters, and some Burnside groups, and the locally finite CA-groups have been classified explicitly. CA-groups are also called commutative-transitive groups (or CT-groups for short) because commutativity is a transitive relation amongst the non-identity elements of a group if and only if the group is a CA-group.
History
Locally finite CA-groups were classified by several mathematicians from 1925 to 1998. First, finite CA-groups were shown to be simple or solvable in . Then in the Brauer–Suzuki–Wall theorem , finite CA-groups of even order were shown to be Frobenius groups, abelian groups, or two dimensional projective special linear groups over a finite field of even order, PSL(2, 2f) for f ≥ 2. Finally, finite CA-groups of odd order were shown to be Frobenius groups or abelian groups in , and so in particular, are never non-abelian simple.
CA-groups were important in the context of the classification of finite simple groups. Michio Suzuki showed that every finite, simple, non-abelian, CA-group is of even order. This result was first extended to the Feit–Hall–Thompson theorem showing that finite, simple, non-abelian, CN-groups had even order, and then to the Feit–Thompson theorem which states that every finite, simple, non-abelian group is of even order. A textbook exposition of the classification of finite CA-groups is given as example 1 and 2 in . A more detailed description of the Frobenius groups appearing is included in , where it is shown that a finite, solvable CA-group is a semidirect product of an abelian group and a fixed-point-free automorphism, and that conversely every such semidirect product is a finite, solvable CA-group. Wu also extended the classification of Suzuki et al. to locally finite groups.
Examples
Every abelian group is a CA-group, and a group with a non-trivial center is a CA-group if and only if it is abelian. The finite CA-groups are classified: the solvable ones are semidirect products of abelian groups by cyclic groups such that every non-trivial element acts fixed-point-freely and include groups such as the dihedral groups of order 4k+2, and the alternating group on 4 points of order 12, while the nonsolvable ones are all simple and are the 2-dimensional projective special linear groups PSL(2, 2n) for n ≥ 2. Infinite CA-groups include free groups, PSL(2, R), and Burnside groups of large prime exponent, . Some more recent results in the infinite case are included in , including a classification of locally finite CA-groups. Wu also observes that Tarski monsters a
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https://en.wikipedia.org/wiki/CN-group
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In mathematics, in the area of algebra known as group theory, a more than fifty-year effort was made to answer a conjecture of : are all groups of odd order solvable? Progress was made by showing that CA-groups, groups in which the centralizer of a non-identity element is abelian, of odd order are solvable . Further progress was made showing that CN-groups, groups in which the centralizer of a non-identity element is nilpotent, of odd order are solvable . The complete solution was given in , but further work on CN-groups was done in , giving more detailed information about the structure of these groups. For instance, a non-solvable CN-group G is such that its largest solvable normal subgroup O∞(G) is a 2-group, and the quotient is a group of even order.
Examples
Solvable CN groups include
Nilpotent groups
Frobenius groups whose Frobenius complement is nilpotent
3-step groups, such as the symmetric group S4
Non-solvable CN groups include:
The Suzuki simple groups
The groups PSL2(F2n) for n>1
The group PSL2(Fp) for p>3 a Fermat prime or Mersenne prime.
The group PSL2(F9)
The group PSL3(F4)
References
Finite groups
Group theory
Properties of groups
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https://en.wikipedia.org/wiki/Special%20abelian%20subgroup
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In mathematical group theory, a subgroup of a group is termed a special abelian subgroup or SA-subgroup if the centralizer of any nonidentity element in the subgroup is precisely the subgroup. Equivalently, an SA subgroup is a centrally closed abelian subgroup.
Any SA subgroup is a maximal abelian subgroup, that is, it is not properly contained in another abelian subgroup.
For a CA group, the SA subgroups are precisely the maximal abelian subgroups.
SA subgroups are known for certain characters associated with them termed exceptional characters.
References
Subgroup properties
Finite groups
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https://en.wikipedia.org/wiki/Centrally%20closed%20subgroup
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In mathematics, in the realm of group theory, a subgroup of a group is said to be centrally closed if the centralizer of any nonidentity element of the subgroup lies inside the subgroup.
Some facts about centrally closed subgroups:
Every malnormal subgroup is centrally closed.
Every Frobenius kernel is centrally closed.
SA subgroups are precisely the centrally closed Abelian subgroups.
The trivial subgroup and the whole group are centrally closed.
Subgroup properties
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Nevada
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This is a list of law enforcement agencies in the U.S. state of Nevada.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 76 law enforcement agencies employing 6,643 sworn police officers, about 254 for each 100,000 residents.
Law enforcement in Nevada
The State of Nevada Peace Officers' Standards and Training Commission is responsible to:
provide for and encourage the training and education of persons whose primary duty is law enforcement to ensure the safety of the residents of and visitors to this state.
Shall adopt regulations establishing minimum standards for the certification and decertification, recruitment, selection and training of peace officers.
Within Nevada, Peace Officers are grouped into one of three classes, Category I, Category II, or Category III:
Category I peace officers include traditional law enforcement officers such as Police Officers, Deputy Sheriffs, Deputy Marshals, Parole & Probation Officers, and State Troopers of the Nevada Highway Patrol. The Category I peace officer training is a minimum of 480 hours.
Category II peace officers are specialists and include officers such as Taxicab Authority Officers, Gaming Control Agents, and Constables. The Category II training is a minimum of 330 hours.
Category III peace officers are those officers assigned solely to State Corrections & City/County Detention. The Category III training is a minimum of 6 weeks.
State agencies
Nevada Department of Agricultural
Agriculture Enforcement Division
Nevada Department of Conservation and Natural Resources
Division of State Parks
Nevada Department of Corrections
Nevada Department of Public Safety
Capitol Police
Nevada Highway Patrol
Investigations Division
Nevada Department of Wildlife
Law Enforcement Division
Nevada Gaming Control Board
Nevada Taxicab Authority
University of Las Vegas Police Department
University of Nevada Reno Department
County agencies
Churchill County Sheriff's Office
Clark County Park Police
Clark County Marshal's Office (Nevada)
Clark County Sheriffs Office (formerly Constables)
Clark County Department of Juvenile Justice Services
Douglas County Sheriff's Department
Elko County Sheriff's Office
Esmeralda County Sheriff's Office
Eureka County Sheriff's Office
Humboldt County Sheriff's Office
Mineral County Sheriff's Office
Lander County Sheriff's Office
Lincoln County Sheriff's Office
Lyon County Sheriff's Office
Nye County Sheriff's Office
Pershing County Sheriff's Office
Storey County Sheriff's Office
White Pine County Sheriff's Office
Washoe County Sheriff's Office
Joint jurisdiction/city-county agencies
Carson City Sheriff’s Office
Las Vegas Metropolitan Police Department
City agencies
Boulder City Police Department
Carlin Police Department
Elko Police Department
Ely Police Department
Fallon Police Department
Henderson Police Department
Henderson Township Constable's Office
City of Las Veg
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https://en.wikipedia.org/wiki/World%20Rugby%20Coach%20of%20the%20Year
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The World Rugby Coach of the Year is awarded by World Rugby in the autumn each year. From 2004 to 2007, the award was called the IRB International Coach of the Year.
List of winners
Statistics
Notes
References
External links
Coach
IRB Award
Coaching awards
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https://en.wikipedia.org/wiki/Heronian%20mean
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In mathematics, the Heronian mean H of two non-negative real numbers A and B is given by the formula
It is named after Hero of Alexandria.
Properties
Just like all means, the Heronian mean is symmetric (it does not depend on the order in which its two arguments are given) and idempotent (the mean of any number with itself is the same number).
The Heronian mean of the numbers A and B is a weighted mean of their arithmetic and geometric means:
Therefore, it lies between these two means, and between the two given numbers.
Application in solid geometry
The Heronian mean may be used in finding the volume of a frustum of a pyramid or cone. The volume is equal to the product of the height of the frustum and the Heronian mean of the areas of the opposing parallel faces.
A version of this formula, for square frusta, appears in the Moscow Mathematical Papyrus from Ancient Egyptian mathematics, whose content dates to roughly 1850 BC.
References
Means
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https://en.wikipedia.org/wiki/Anders%20Martin-L%C3%B6f
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Anders Martin-Löf (born 16 March 1940) is a Swedish physicist and mathematician. He has been a professor at the Department of Mathematics of Stockholm University.
Martin-Löf did his undergraduate studies at the KTH Royal Institute of Technology in Stockholm and got his exam in engineering physics in 1963. He continued with graduate studies in optimization at KTH and at MIT in the United States from 1967–1968, later on followed by a position as Research Associate at the Rockefeller University in New York City 1970–1971, working with probability theory and applications to statistical mechanics. he received his Ph.D. degree in 1973.
During the following 10 years he continued working with similar issues as "docent" in Uppsala and Stockholm. In the 1980s he changed to insurance mathematics with the Folksam company including development of theories for controlling movements of insurances. From 1987 he has been working with theoretical and applied aspects of his assignment as professor with the Stockholm University.
Martin-Löf has two children from his first marriage and a daughter from his second. Anders is the brother of Per Martin-Löf, who was responsible for a pioneering definition of randomness, as well as a foundation for constructive mathematics based on intuitionistic type theory. Per is also a professor at Stockholm University, with joint appointments in the departments of mathematics and philosophy. They share an interest in statistics, and in statistical mechanics, though Per has been more interested in the foundations of statistics, while Anders has been more interested in financial mathematics. Their elder brother Johan is also an engineering physicist, but more inclined to space technology.
Martin-Löf was a fellow student at KTH with Olav Kallenberg.
References
External links
Official site
Swedish statisticians
KTH Royal Institute of Technology alumni
Academic staff of Stockholm University
20th-century Swedish mathematicians
21st-century Swedish mathematicians
Members of the Royal Swedish Academy of Engineering Sciences
1940 births
Living people
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https://en.wikipedia.org/wiki/Pasch%27s%20axiom
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In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them. Its essential role was discovered by Moritz Pasch in 1882.
Statement
The axiom states that,
The fact that segments AC and BC are not both intersected by the line is proved in Supplement I,1, which was written by P. Bernays.
A more modern version of this axiom is as follows:
(In case the third side is parallel to our line, we count an "intersection at infinity" as external.) A more informal version of the axiom is often seen:
History
Pasch published this axiom in 1882, and showed that Euclid's axioms were incomplete. The axiom was part of Pasch's approach to introducing the concept of order into plane geometry.
Equivalences
In other treatments of elementary geometry, using different sets of axioms, Pasch's axiom can be proved as a theorem; it is a consequence of the plane separation axiom when that is taken as one of the axioms. Hilbert uses Pasch's axiom in his axiomatic treatment of Euclidean geometry. Given the remaining axioms in Hilbert's system, it can be shown that Pasch's axiom is logically equivalent to the plane separation axiom.
Hilbert's use of Pasch's axiom
David Hilbert uses Pasch's axiom in his book Foundations of Geometry which provides an axiomatic basis for Euclidean geometry. Depending upon the edition, it is numbered either II.4 or II.5. His statement is given above.
In Hilbert's treatment, this axiom appears in the section concerning axioms of order and is referred to as a plane axiom of order. Since he does not phrase the axiom in terms of the sides of a triangle (considered as lines rather than line segments) there is no need to talk about internal and external intersections of the line with the sides of the triangle ABC.
Caveats
Pasch's axiom is distinct from Pasch's theorem which is a statement about the order of four points on a line. However, in literature there are many instances where Pasch's axiom is referred to as Pasch's theorem. A notable instance of this is .
Pasch's axiom should not be confused with the Veblen-Young axiom for projective geometry, which may be stated as:
There is no mention of internal and external intersections in the statement of the Veblen-Young axiom which is only concerned with the incidence property of the lines meeting. In projective geometry the concept of betweeness (required to define internal and external) is not valid and all lines meet (so the issue of parallel lines does not arise).
Notes
References
External links
Euclidean plane geometry
Foundations of geometry
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https://en.wikipedia.org/wiki/2011%20Canadian%20census
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The 2011 Canadian census was a detailed enumeration of the Canadian population on May 10, 2011. Statistics Canada, an agency of the Canadian government, conducts a nationwide census every five years. In 2011, it consisted of a mandatory short form census questionnaire and an inaugural National Household Survey (NHS), a voluntary survey which replaced the mandatory long form census questionnaire; this substitution was the focus of much controversy. Completion of the (short form) census is mandatory for all Canadians, and those who do not complete it may face penalties ranging from fines to prison sentences.
The Statistics Act mandates a Senate and/or House of Commons (joint) committee review of the opt-in clause (for the release of one's census records after 92 years) by 2014.
The 2011 census was the fifteenth decennial census and, like other censuses, was required by section 8 of the Constitution Act, 1867. As with other decennial censuses, the data was used to adjust federal electoral district boundaries.
As of August 24, 2011, Canada's overall collection response rate was 98.1%, up over a percentage point from 96.5% in the 2006 census. Ontario and Prince Edward Island each held the highest response rate at 98.3%, while Nunavut held the lowest response rate at 92.7%.
In an article in the New York Times in August 2015, journalist Stephen Marche argued that by ending the mandatory long-form census in 2011, the federal government "stripped Canada of its capacity to gather information about itself" in the "age of information." Nearly 500 organizations in Canada, including the Canadian Medical Association, the Canadian Chamber of Commerce, the Canadian Federation of Students, and the Canadian Catholic Council of Bishops protested the decision to replace the long form census in 2011 with a shorter version.
Questionnaire revision
Short form
The original schedule of the short-form questions for the 2011 Census of Population was published in the Canada Gazette, Part I on August 21, 2010. The 2011 census consisted of the same eight questions that appeared on the 2006 census short-form questionnaire, with the addition of two questions on language. The federal Minister of Industry Tony Clement's announcement that questions about language would appear on the mandatory short-form census came in response to a lawsuit brought by the Federation of Francophone and Acadian Communities, which claimed that the voluntary status of the long-form census would impact language-related government services.
In addition to possible questions on activity limitation, various organizations called for the following changes to the 2011 census:
Adding "Aboriginal identifier" to the short form (already found on the long form).
Relationship of same-sex married couples.
Place of work and transportation-related questions.
Food security questions.
National Household Survey
The National Household Survey (NHS) began within four weeks of the May 2011 census and included app
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https://en.wikipedia.org/wiki/Artin%E2%80%93Hasse%20exponential
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In mathematics, the Artin–Hasse exponential, introduced by , is the power series given by
Motivation
One motivation for considering this series to be analogous to the exponential function comes from infinite products. In the ring of formal power series Q[[x]] we have the identity
where μ(n) is the Möbius function. This identity can be verified by showing the logarithmic derivative of the two sides are equal and that both sides have the same constant term. In a similar way, one can verify a product expansion for the Artin–Hasse exponential:
So passing from a product over all n to a product over only n prime to p, which is a typical operation in p-adic analysis, leads from ex to Ep(x).
Properties
The coefficients of Ep(x) are rational. We can use either formula for Ep(x) to prove that, unlike ex, all of its coefficients are p-integral; in other words, the denominators of the coefficients of Ep(x) are not divisible by p. A first proof uses the definition of Ep(x) and Dwork's lemma, which says that a power series f(x) = 1 + ... with rational coefficients has p-integral coefficients if and only if f(xp)/f(x)p ≡ 1 mod pZp[[x]]. When f(x) = Ep(x), we have f(xp)/f(x)p = e−px, whose constant term is 1 and all higher coefficients are in pZp.
A second proof comes from the infinite product for Ep(x): each exponent -μ(n)/n for n not divisible by p is a p-integral, and when a rational number a is p-integral all coefficients in the binomial expansion of (1 - xn)a are p-integral by p-adic continuity of the binomial coefficient polynomials t(t-1)...(t-k+1)/k! in t together with their obvious integrality when t is a nonnegative integer (a is a p-adic limit of nonnegative integers) . Thus each factor in the product of Ep(x) has p-integral coefficients, so Ep(x) itself has p-integral coefficients.
The (p-integral) series expansion has radius of convergence 1.
Combinatorial interpretation
The Artin–Hasse exponential is the generating function for the probability a uniformly randomly selected element of Sn (the symmetric group with n elements) has p-power order (the number of which is denoted by tp,n):
This gives a third proof that the coefficients of Ep(x) are p-integral, using the theorem of Frobenius that in a finite group of order divisible by d the number of elements of order dividing d is also divisible by d. Apply this theorem to the nth symmetric group with d equal to the highest power of p dividing n!.
More generally, for any topologically finitely generated profinite group G there is an identity
where H runs over open subgroups of G with finite index (there are finitely many of each index since G is topologically finitely generated) and aG,n is the number of continuous homomorphisms from G to Sn. Two special cases are worth noting. (1) If G is the p-adic integers, it has exactly one open subgroup of each p-power index and a continuous homomorphism from G to Sn is essentially the same thing as choosing an element of p-power order in Sn, so w
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https://en.wikipedia.org/wiki/Jean%20Cavaill%C3%A8s
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Jean Cavaillès (; ; 15 May 1903 – 4 April 1944) was a French philosopher and logician who specialized in philosophy of mathematics and philosophy of science. He took part in the French Resistance within the Libération movement and was arrested by the Gestapo on 17 February 1944 and shot on 4 April 1944.
Early life and education
Cavaillès was born in Saint-Maixent, Deux-Sèvres. After passing his first baccalauréat in 1919 and baccalauréats in mathematics and philosophy the following year, he studied at the Lycée Louis-le-Grand, including two years of classes préparatoires, before entering the École Normale Supérieure in 1923, reading philosophy. In 1927 he passed the agrégation competitive exam. He began graduate studies in Philosophy in 1928 under the supervision of Léon Brunschvicg. Cavaillès won a Rockefeller Foundation scholarship in 1929–1930. In 1931 he travelled extensively in Germany; in Göttingen he conceived, jointly with Emmy Noether, the project of publishing the Cantor-Dedekind correspondence. He was a teaching assistant at the École Normale Supérieure between 1929 and 1935, then teacher at the lycée d'Amiens (now lycée Louis-Thuillier) in 1936. In 1937, he successfully defended his doctoral theses at the University of Paris and became a Doctor of Letters in Philosophy. He was then appointed maître de conférences in Logic and in General Philosophy at the University of Strasbourg.
World War II
After the outbreak of World War II, he was mobilized in 1939 as an infantry lieutenant with the 43rd Regiment and was later attached to the Staff of the 4th Colonial Division. He was honoured for bravery twice and was captured on 11 June 1940. At the end of July 1940 he escaped from Belgium and fled to Clermont-Ferrand, where the university of Strasbourg was re-organized.
At the end of December 1940, he met Emmanuel d'Astier de la Vigerie with whom he created a small group of resistance fighters, known as "the Last Column". To reach a broader audience, they created a newspaper which was to become Libération. It served as the mouthpiece of both Libération-Sud and Libération-Nord. Cavaillès took an active part in editing the paper. The first edition appeared in July 1941.
In 1941, he was appointed professor at the Sorbonne and left Clermont-Ferrand for Paris, where he helped form the Libération-Nord resistance group, becoming part of its management committee.
In April 1942, at the instigation of Christian Pineau, the central Office of Information and Action (BCRA) of London charged him with the task of forming an intelligence network in the Northern Zone, known as "Cohors". He was ordered by Christian Pineau to pass into the Southern Zone, and Cavaillès headed the network and formed similar groups in Belgium and the north of France.
In September 1942 he was arrested with Pineau in Narbonne by the French police. After a failed escape attempt to London, he was interned in Montpellier at the Saint-Paul d' Eyjeaux prison camp from where he escape
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https://en.wikipedia.org/wiki/Harmonic%20polynomial
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In mathematics, in abstract algebra, a multivariate polynomial over a field such that the Laplacian of is zero is termed a harmonic polynomial.
The harmonic polynomials form a vector subspace of the vector space of polynomials over the field. In fact, they form a graded subspace. For the real field, the harmonic polynomials are important in mathematical physics.
The Laplacian is the sum of second partials with respect to all the variables, and is an invariant differential operator under the action of the orthogonal group via the group of rotations.
The standard separation of variables theorem states that every multivariate polynomial over a field can be decomposed as a finite sum of products of a radial polynomial and a harmonic polynomial. This is equivalent to the statement that the polynomial ring is a free module over the ring of radial polynomials.
See also
Harmonic function
Spherical harmonics
Zonal spherical harmonics
Multilinear polynomial
References
Lie Group Representations of Polynomial Rings by Bertram Kostant published in the American Journal of Mathematics Vol 85 No 3 (July 1963)
Abstract algebra
Polynomials
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https://en.wikipedia.org/wiki/Radical%20polynomial
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In mathematics, in the realm of abstract algebra, a radical polynomial is a multivariate polynomial over a field that can be expressed as a polynomial in the sum of squares of the variables. That is, if
is a polynomial ring, the ring of radical polynomials is the subring generated by the polynomial
Radical polynomials are characterized as precisely those polynomials that are invariant under the action of the orthogonal group.
The ring of radical polynomials is a graded subalgebra of the ring of all polynomials.
The standard separation of variables theorem asserts that every polynomial can be expressed as a finite sum of terms, each term being a product of a radical polynomial and a harmonic polynomial. This is equivalent to the statement that the ring of all polynomials is a free module over the ring of radical polynomials.
References
Abstract algebra
Polynomials
Invariant theory
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https://en.wikipedia.org/wiki/Luzin%20N%20property
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In mathematics, a function f on the interval [a, b] has the Luzin N property, named after Nikolai Luzin (also called Luzin property or N property) if for all such that , there holds: , where stands for the Lebesgue measure.
Note that the image of such a set N is not necessarily measurable, but since the Lebesgue measure is complete, it follows that if the Lebesgue outer measure of that set is zero, then it is measurable and its Lebesgue measure is zero as well.
Properties
Any differentiable function has the Luzin N property. This extends to functions that are differentiable on a cocountable set, as the image of a countable set is countable and thus a null set, but not to functions differentiable on a conull set:
The Cantor function does not have the Luzin N property, as the Lebesgue measure of the Cantor set is zero, but its image is the complete [0,1] interval.
A function f on the interval [a,b] is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin N property.
References
External links
Luzin-N-property in the Encyclopedia of Mathematics
Real analysis
Measure theory
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https://en.wikipedia.org/wiki/Mathsoft
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MathSoft was founded in 1984 by Allen Razdow and David Blohm to provide mathematical programs to students, teachers, and professionals. The company is best known for its Mathcad software, an application for solving and visualizing mathematical problems. The company also created the StudyWorks series of math and science education packages aimed at interactively teaching those subjects to middle school and high school students.
Mathsoft Engineering and Education, Inc., the company that sells these products was one division of the former MathSoft. In 2001, MathSoft sold its Engineering & Education Products Division to the division managers as a privately held company. One of the former MathSoft divisions, FreeScholarships.com, was shut down, and MathSoft renamed itself Insightful. Insightful develops custom math programs for businesses and also sells a data analysis package called StatServer. Mathsoft Engineering & Education continued to develop and sell its flagship Mathcad product as well as the Mathcad Calculation Server, an enterprise server for sharing technical calculations.
Mathsoft Engineering & Education, Inc. promoted the term calculation management which included designing, documenting, and managing technical calculations and intellectual property.
In April 2006, Mathsoft was acquired by Parametric Technology Corporation for $62 million. At the time it had an annual income of $20 million with 130 employees in seven countries, including the UK, Germany and Japan.
References
All of Mathcad
Software companies based in Massachusetts
Defunct software companies of the United States
American companies established in 1984
1984 establishments in Massachusetts
2006 mergers and acquisitions
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https://en.wikipedia.org/wiki/Linear%20connection
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In the mathematical field of differential geometry, the term linear connection can refer to either of the following overlapping concepts:
a connection on a vector bundle, often viewed as a differential operator (a Koszul connection or covariant derivative);
a principal connection on the frame bundle of a manifold or the induced connection on any associated bundle — such a connection is equivalently given by a Cartan connection for the affine group of affine space, and is often called an affine connection.
The two meanings overlap, for example, in the notion of a linear connection on the tangent bundle of a manifold.
In older literature, the term linear connection is occasionally used for an Ehresmann connection or Cartan connection on an arbitrary fiber bundle, to emphasise that these connections are "linear in the horizontal direction" (i.e., the horizontal bundle is a vector subbundle of the tangent bundle of the fiber bundle), even if they are not "linear in the vertical (fiber) direction". However, connections which are not linear in this sense have received little attention outside the study of spray structures and Finsler geometry.
References
Connection (mathematics)
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https://en.wikipedia.org/wiki/National%20Numeracy%20Strategy
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The National Numeracy Strategy arose out of the National Numeracy Project in 1996, led by a Numeracy Task Force in England. The strategy included an outline of expected teaching in mathematics for all pupils from Reception to Year 6.
In 2003, the strategy, including the framework for teaching, was absorbed into the broader Primary National Strategy. The framework for teaching was updated in 2006.
See also
National Curriculum (England, Wales and Northern Ireland)
Key Stage
Chunking (division)
Grid method multiplication
Number bond
Further reading
Department for Education and Employment (1998), The implementation of the National Numeracy Strategy: The final report of the Numeracy Task Force, London: DfEE
Department for Education and Employment (1999), The National Numeracy Strategy: framework for teaching mathematics from reception to Year 6, London: DfEE.
QCA (1999), Standards in mathematics: exemplification of key learning objectives from reception to year 6
Rob Eastaway, Why parents can't do maths today, BBC News, 10 September 2010
Ian Thompson (2000), Is the National Numeracy Strategy evidence based?, Mathematics Teaching, 171, 23–27
Dylan V. Jones (2002), National numeracy initiatives in England and Wales: a comparative study of policy, The Curriculum Journal, 13 (1), 5–23.
Chris Kyriacou and Maria Goulding (2004), A systematic review of the impact of the Daily Mathematics Lesson in enhancing pupil confidence and competence in early mathematics, Evidence for Policy and Practice Information and Co-ordinating Centre (EPPI), Institute of Education, London.
External links
Government Primary Frameworks website (Archived, via the National Archives)
Education in England
Mathematics education in the United Kingdom
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https://en.wikipedia.org/wiki/International%20Financial%20Statistics
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The IMF International Financial Statistics (IFS) is a compilation of financial data collected from various sources, covering the economies of 194 countries and areas worldwide, which is published monthly by the International Monetary Fund (IMF).
Methodology and scope
The IFS is the IMF’s principal statistical publication, covering numerous topics of international and domestic finance. It includes, for most countries, data on exchange rates, balance of payments, international liquidity, money and banking, interest rates, prices, etc. Most annual data begins in 1948, quarterly and monthly data dates back to 1957, and most balance of payments data begins in 1970.
The IMF compiles the data from various sources including government departments, national accounts, central banks, the United Nations (UN), Eurostat, the International Labour Organization (ILO), and private financial institutions.
Accessing the data
The Economic and Social Data Service (ESDS) International provides the macro-economic datasets free of charge for members of UK higher and further education institutions. In order to access the data, users have to be registered, which can be done here. Alternatively, the data is available to explore and download free of charge on the IMF data portal. In addition, the IMF offers an API based on the SDMX standard for automated downloads.
References
External links
International Financial Statistics online
About ESDS International
SDMX - Statistical Data and Metadata eXchange
See also
Classification of the Functions of Government
ESDS International
International Monetary Fund
Online databases
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https://en.wikipedia.org/wiki/Indecomposable%20distribution
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In probability theory, an indecomposable distribution is a probability distribution that cannot be represented as the distribution of the sum of two or more non-constant independent random variables: Z ≠ X + Y. If it can be so expressed, it is decomposable: Z = X + Y. If, further, it can be expressed as the distribution of the sum of two or more independent identically distributed random variables, then it is divisible: Z = X1 + X2.
Examples
Indecomposable
The simplest examples are Bernoulli-distributeds: if
then the probability distribution of X is indecomposable.
Proof: Given non-constant distributions U and V, so that U assumes at least two values a, b and V assumes two values c, d, with a < b and c < d, then U + V assumes at least three distinct values: a + c, a + d, b + d (b + c may be equal to a + d, for example if one uses 0, 1 and 0, 1). Thus the sum of non-constant distributions assumes at least three values, so the Bernoulli distribution is not the sum of non-constant distributions.
Suppose a + b + c = 1, a, b, c ≥ 0, and
This probability distribution is decomposable (as the distribution of the sum of two Bernoulli-distributed random variables) if
and otherwise indecomposable. To see, this, suppose U and V are independent random variables and U + V has this probability distribution. Then we must have
for some p, q ∈ [0, 1], by similar reasoning to the Bernoulli case (otherwise the sum U + V will assume more than three values). It follows that
This system of two quadratic equations in two variables p and q has a solution (p, q) ∈ [0, 1]2 if and only if
Thus, for example, the discrete uniform distribution on the set {0, 1, 2} is indecomposable, but the binomial distribution for two trials each having probabilities 1/2, thus giving respective probabilities a, b, c as 1/4, 1/2, 1/4, is decomposable.
An absolutely continuous indecomposable distribution. It can be shown that the distribution whose density function is
is indecomposable.
Decomposable
All infinitely divisible distributions are a fortiori decomposable; in particular, this includes the stable distributions, such as the normal distribution.
The uniform distribution on the interval [0, 1] is decomposable, since it is the sum of the Bernoulli variable that assumes 0 or 1/2 with equal probabilities and the uniform distribution on [0, 1/2]. Iterating this yields the infinite decomposition:
where the independent random variables Xn are each equal to 0 or 1 with equal probabilities – this is a Bernoulli trial of each digit of the binary expansion.
A sum of indecomposable random variables is decomposable into the original summands. But it may turn out to be infinitely divisible. Suppose a random variable Y has a geometric distribution
on {0, 1, 2, ...}.
For any positive integer k, there is a sequence of negative-binomially distributed random variables Yj, j = 1, ..., k, such that Y1 + ... + Yk has this geometric distribution. Therefore, this distribution is
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https://en.wikipedia.org/wiki/Lo%C3%A8ve%20Prize
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The Line and Michel Loève International Prize in Probability (Loève Prize) was created in 1992 in honor of Michel Loève by his widow Line. The prize, awarded every two years, is intended to recognize outstanding contributions by researchers in mathematical probability who are under 45 years old. With a prize value of around $30,000, this is one of the most generous awards in any specific mathematical subdiscipline.
Winners
2023 – Jian Ding
2021 – Ivan Corwin
2019 – Allan Sly
2017 – Hugo Duminil-Copin
2015 – Alexei Borodin
2013 – Sourav Chatterjee
2011 – Scott Sheffield
2009 – Alice Guionnet
2007 – Richard Kenyon
2005 – Wendelin Werner
2003 – Oded Schramm
2001 – Yuval Peres
1999 – Alain-Sol Sznitman
1997 – Jean-François Le Gall
1995 – Michel Talagrand
1993 – David Aldous
See also
List of mathematics awards
External links
References
Mathematics awards
Awards established in 1992
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https://en.wikipedia.org/wiki/MCQ
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MCQ may refer to
McQ, a 1974 famous crime drama
McQ Inc, an American defense company based in Pennsylvania
Mathematical Citation Quotient, a measure of the impact of a mathematics journal
Multiple choice question
Malvern College Qingdao
IATA code for Miskolc Airport
McQ, a clothing line from Alexander McQueen (brand)
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https://en.wikipedia.org/wiki/Lebesgue%20differentiation%20theorem
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In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for Henri Lebesgue.
Statement
For a Lebesgue integrable real or complex-valued function f on Rn, the indefinite integral is a set function which maps a measurable set A to the Lebesgue integral of , where denotes the characteristic function of the set A. It is usually written
with λ the n–dimensional Lebesgue measure.
The derivative of this integral at x is defined to be
where |B| denotes the volume (i.e., the Lebesgue measure) of a ball B centered at x, and B → x means that the diameter of B tends to 0.
The Lebesgue differentiation theorem states that this derivative exists and is equal to f(x) at almost every point x ∈ Rn. In fact a slightly stronger statement is true. Note that:
The stronger assertion is that the right hand side tends to zero for almost every point x. The points x for which this is true are called the Lebesgue points of f.
A more general version also holds. One may replace the balls B by a family of sets U of bounded eccentricity. This means that there exists some fixed c > 0 such that each set U from the family is contained in a ball B with . It is also assumed that every point x ∈ Rn is contained in arbitrarily small sets from . When these sets shrink to x, the same result holds: for almost every point x,
The family of cubes is an example of such a family , as is the family (m) of rectangles in R2 such that the ratio of sides stays between m−1 and m, for some fixed m ≥ 1. If an arbitrary norm is given on Rn, the family of balls for the metric associated to the norm is another example.
The one-dimensional case was proved earlier by . If f is integrable on the real line, the function
is almost everywhere differentiable, with Were defined by a Riemann integral this would be essentially the fundamental theorem of calculus, but Lebesgue proved that it remains true when using the Lebesgue integral.
Proof
The theorem in its stronger form—that almost every point is a Lebesgue point of a locally integrable function f—can be proved as a consequence of the weak–L1 estimates for the Hardy–Littlewood maximal function. The proof below follows the standard treatment that can be found in , , and .
Since the statement is local in character, f can be assumed to be zero outside some ball of finite radius and hence integrable. It is then sufficient to prove that the set
has measure 0 for all α > 0.
Let ε > 0 be given. Using the density of continuous functions of compact support in L1(Rn), one can find such a function g satisfying
It is then helpful to rewrite the main difference as
The first term can be bounded by the value at x of the maximal function for f − g, denoted here by :
The second term disappears in the limit since g is a continuous function, and the thi
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https://en.wikipedia.org/wiki/Zsigmondy%27s%20theorem
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In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if are coprime integers, then for any integer , there is a prime number p (called a primitive prime divisor) that divides and does not divide for any positive integer , with the following exceptions:
, ; then which has no prime divisors
, a power of two; then any odd prime factors of must be contained in , which is also even
, , ; then
This generalizes Bang's theorem, which states that if and is not equal to 6, then has a prime divisor not dividing any with .
Similarly, has at least one primitive prime divisor with the exception .
Zsigmondy's theorem is often useful, especially in group theory, where it is used to prove that various groups have distinct orders except when they are known to be the same.
History
The theorem was discovered by Zsigmondy working in Vienna from 1894 until 1925.
Generalizations
Let be a sequence of nonzero integers.
The Zsigmondy set associated to the sequence is the set
i.e., the set of indices such that every prime dividing also divides some for some . Thus Zsigmondy's theorem implies that , and Carmichael's theorem says that the Zsigmondy set of the Fibonacci sequence is , and that of the Pell sequence is . In 2001 Bilu, Hanrot, and Voutier
proved that in general, if is a Lucas sequence or a Lehmer sequence, then (see , there are only 13 such s, namely 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 18, 30).
Lucas and Lehmer sequences are examples of divisibility sequences.
It is also known that if is an elliptic divisibility sequence, then its Zsigmondy
set is finite. However, the result is ineffective in the sense that the proof does not give an explicit upper bound for the largest element in ,
although it is possible to give an effective upper bound for the number of elements in .
See also
Carmichael's theorem
References
External links
Theorems in number theory
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https://en.wikipedia.org/wiki/2006%20World%20Series%20of%20Poker%20results
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This list of 2006 World Series of Poker (WSOP) results includes statistics, final table results and payouts.
The total money paid out in the 2006 events was $156,409,974.
Results
Event 1: $500 No Limit Hold 'em--Casino Employees
This event kicked off the 2006 WSOP. It was a $500 buy-in no limit Texas hold 'em tournament reserved for casino employees that work in Nevada.
Number of buy-ins: 1,232
Total Prize Pool: $554,400
Number of Payouts: 101
Winning Hand: 3♣
Event 2: $1,500 No Limit Hold 'em
This event was a $1,500 buy-in no-limit Texas hold 'em tournament. It was the first public tournament of the 2006 WSOP.
Number of buy-ins: 2,776
Total Prize Pool: $3,789,240
Number of Payouts: 277
Winning Hand: Q♠
Event 3: $1,500 Pot Limit Hold 'em
This event was a $1,500 buy-in pot limit Texas hold 'em tournament. It was a three-day event with a first prize of $345,984.
Number of buy-ins: 1,102
Total Prize Pool: $1,504,230
Number of Payouts: 99
Winning Hand:
Event 4: $1,500 Limit Hold 'em
This event was a $1,500 buy-in limit Texas hold 'em tournament. It was a three-day event with a first prize of $335,289.
Number of buy-ins: 1,068
Total Prize Pool: $1,457,820
Number of Payouts: 100
Winning Hand: 6♠
Event 5: $2,500 No Limit Hold 'em short-handed
This event was a $2,500 buy-in no limit Texas hold 'em tournament, with a maximum of six players per table instead of the normal nine. It was a three-day event with a first prize of $475,712.
Number of buy-ins: 824
Total Prize Pool: $1,895,200
Number of Payouts: 90
Winning Hand:
Event 6: $2,000 No Limit Hold 'em
This event was a $2,000 buy-in no limit Texas hold 'em tournament. It was a three-day event with a first prize of $803,274.
Number of buy-ins: 1,919
Total Prize Pool: $3,492,580
Number of Payouts: 155
Winning Hand: Q♣ T♣
Event 7: $3,000 Limit Hold 'em
This event was a $3,000 buy-in limit Texas hold 'em tournament. It was a three-day event with a first prize of $343,618.
Number of buy-ins: 415
Total Prize Pool: $1,145,400
Number of Payouts: 46
Winning Hand:
Event 8: $2,000 Omaha Hi-low Split
This event was a $2,000 buy-in Limit Omaha High-low split. It was a three-day event with a first prize of $341,426.
Number of buy-ins: 670
Total Prize Pool: $1,219,400
Number of Payouts: 65
Winning Hand: 7♣
This was the first of four top-three finishes for Madsen at this year's WSOP.
Event 9: $5,000 No Limit Hold'em
Number of buy-ins: 622
Total Prize Pool: $2,293,400
Number of Payouts: 63
Winning Hand:
Event 10: $1,500 7 Card Stud
Number of buy-ins: 478
Total Prize Pool: $652,470
Number of Payouts: 40
Winning Hand: K♠ 4♣ 6♠ 4♠
Event 11: $1,500 Limit Hold'em
Number of buy-ins: 701
Total prize pool: $956,865
Number of Payouts: 72
Winning Hand: 3♠
Event 12: $5,000 Omaha Hi-low Split
This event was a $5,000 buy-in Limit Omaha High-low split. It was a two-day event with a first prize of $398,560.
Number of buy-ins: 265
Total Prize Pool: $1,245,
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https://en.wikipedia.org/wiki/Don%20Rees
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Dr. Donald Rees is the former warden of Hugh Stewart Hall in the University of Nottingham for 29 years (1975–2004). Dr. Rees was a highly respected academic, being a professor of mathematics, and a leading member of the University community. He was the last warden to inhabit the Warden's House at Hugh Stewart in its entirety. The Hall library is now named after Dr. Rees in recognition of his service to the Hall, the University and the City.
He arrived in Nottingham with a London Ph.D. in Mathematics and was one of the first tutors recruited at the opening of Lincoln Hall in 1962. As with many of his Welsh countrymen, his twin passions were rugby football and music, being an accomplished pianist in the latter field. According to old Lincolnites of that era he was a lively and popular tutor, and the rugby team especially flourished with his encouragement.
In 1972 he was appointed as Deputy Warden of Cripps Hall and then three years later he made the short move to Hugh Stewart. Certainly he fought hard to maintain the best traditions of Nottingham's Halls, for example keeping a schedule of one Formal Dinner each week, open to all students, when almost all other Halls had reduced to two per term. Additionally Hugh Stewart continually staged concerts of a high standard, often hosting the Sinfonia String Quartet. HM the Queen visited the hall in 1981.
During his time at Hugh Stewart, the University progressively reduced the power and influence of all Wardens, especially with regard to finance, staffing and maintenance of the fabric. Nevertheless, in the history of the University, perhaps only Dr Harry Lucas, Warden of first Wortley and then Cripps (who was said to be one of Vice-Chancellor Hallward's most trusted advisers in the 1950s and 60s) worked harder to the benefit of the entire Hall system.
References
Year of birth missing (living people)
Living people
Academics of the University of Nottingham
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https://en.wikipedia.org/wiki/Lakes%20of%20Wada
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In mathematics, the are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary. In other words, for any point selected on the boundary of one of the lakes, the other two lakes' boundaries also contain that point.
More than two sets with the same boundary are said to have the Wada property; examples include Wada basins in dynamical systems. This property is rare in real-world systems.
The lakes of Wada were introduced by , who credited the discovery to Takeo Wada. His construction is similar to the construction by of an indecomposable continuum, and in fact it is possible for the common boundary of the three sets to be an indecomposable continuum.
Construction of the lakes of Wada
The Lakes of Wada are formed by starting with a closed unit square of dry land, and then digging 3 lakes according to the following rule:
On day n = 1, 2, 3,... extend lake n mod 3 (= 0, 1, 2) so that it is open and connected and passes within a distance 1/n of all remaining dry land. This should be done so that the remaining dry land remains homeomorphic to a closed unit square.
After an infinite number of days, the three lakes are still disjoint connected open sets, and the remaining dry land is the boundary of each of the 3 lakes.
For example, the first five days might be (see the image on the right):
Dig a blue lake of width 1/3 passing within /3 of all dry land.
Dig a red lake of width 1/32 passing within /32 of all dry land.
Dig a green lake of width 1/33 passing within /33 of all dry land.
Extend the blue lake by a channel of width 1/34 passing within /34 of all dry land. (The small channel connects the thin blue lake to the thick one, near the middle of the image.)
Extend the red lake by a channel of width 1/35 passing within /35 of all dry land. (The tiny channel connects the thin red lake to the thick one, near the top left of the image.)
A variation of this construction can produce a countable infinite number of connected lakes with the same boundary: instead of extending the lakes in the order 1, 2, 0, 1, 2, 0, 1, 2, 0, ...., extend them in the order 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, ... and so on.
Wada basins
Wada basins are certain special basins of attraction studied in the mathematics of non-linear systems. A basin having the property that every neighborhood of every point on the boundary of that basin intersects at least three basins is called a Wada basin, or said to have the Wada property. Unlike the Lakes of Wada, Wada basins are often disconnected.
An example of Wada basins is given by the Newton fractal describing the basins of attraction of the Newton–Raphson method for finding the roots of a cubic polynomial with distinct roots, such as see the picture.
Wada basins in chaos theory
In chaos theory, Wada basins arise very frequently. Usually, the Wada property can be seen in the basin of attraction of dissipative dynamical systems.
Bu
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https://en.wikipedia.org/wiki/Projective%20connection
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In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold.
The structure of a projective connection is modeled on the geometry of projective space, rather than the affine space corresponding to an affine connection. Much like affine connections, projective connections also define geodesics. However, these geodesics are not affinely parametrized. Rather they are projectively parametrized, meaning that their preferred class of parameterizations is acted upon by the group of fractional linear transformations.
Like an affine connection, projective connections have associated torsion and curvature.
Projective space as the model geometry
The first step in defining any Cartan connection is to consider the flat case: in which the connection corresponds to the Maurer-Cartan form on a homogeneous space.
In the projective setting, the underlying manifold of the homogeneous space is the projective space RPn which we shall represent by homogeneous coordinates . The symmetry group of is G = PSL(n+1,R). Let H be the isotropy group of the point . Thus, M = G/H presents as a homogeneous space.
Let be the Lie algebra of G, and that of H. Note that . As matrices relative to the homogeneous basis, consists of trace-free matrices:
.
And consists of all these matrices with . Relative to the matrix representation above, the Maurer-Cartan form of G is a system of 1-forms satisfying the structural equations (written using the Einstein summation convention):
Projective structures on manifolds
A projective structure is a linear geometry on a manifold in which two nearby points are connected by a line (i.e., an unparametrized geodesic) in a unique manner. Furthermore, an infinitesimal neighborhood of each point is equipped with a class of projective frames. According to Cartan (1924),
Une variété (ou espace) à connexion projective est une variété numérique qui, au voisinage immédiat de chaque point, présente tous les caractères d'un espace projectif et douée de plus d'une loi permettant de raccorder en un seul espace projectif les deux petits morceaux qui entourent deux points infiniment voisins. ...
Analytiquement, on choisira, d'une manière d'ailleurs arbitraire, dans l'espace projectif attaché à chaque point a de la variété, un repére définissant un système de coordonnées projectives. ... Le raccord entre les espaces projectifs attachés à deux points infiniment voisins a et a' se traduira analytiquement par une transformation homographique. ...
This is analogous to Cartan's notion of an affine connection, in which nearby points are thus connected and have an affine frame of reference which is transported from one to the other (Cartan, 1923):
La variété sera dite à "connexion affine" lorsqu'on aura défini, d'une manière d'ailleurs arbitraire, une loi permettant de repérer l'un par rapport à l'autre les espaces affines attachés à deux points infiniment voisins quelconques m et m' de la variété; c
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https://en.wikipedia.org/wiki/Rabi%20frequency
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The Rabi frequency is the frequency at which the probability amplitudes of two atomic energy levels fluctuate in an oscillating electromagnetic field. It is proportional to the Transition Dipole Moment of the two levels and to the amplitude (not intensity) of the Electromagnetic field. Population transfer between the levels of such a 2-level system illuminated with light exactly resonant with the difference in energy between the two levels will occur at the Rabi frequency; when the incident light is detuned from this energy difference (detuned from resonance) then the population transfer occurs at the generalized Rabi frequency. The Rabi frequency is a semiclassical concept since it treats the atom as an object with quantized energy levels and the electromagnetic field as a continuous wave.
In the context of a nuclear magnetic resonance experiment, the Rabi frequency is the nutation frequency of a sample's net nuclear magnetization vector about a radio-frequency field. (Note that this is distinct from the Larmor frequency, which characterizes the precession of a transverse nuclear magnetization about a static magnetic field.)
Derivation
Consider two energy eigenstates of a quantum system with Hamiltonian (for example, this could be the Hamiltonian of a particle in a potential, like the Hydrogen atom or the Alkali atoms):
We want to consider the time dependent Hamiltonian
where is the potential of the electromagnetic field. Treating the potential as a perturbation, we can expect the eigenstates of the perturbed Hamiltonian to be some mixture of the eigenstates of the original Hamiltonian with time dependent coefficients:
Plugging this into the time dependent Schrödinger equation
taking the inner product with each of and , and using the orthogonality condition of eigenstates , we arrive at two equations in the coefficients and :
where . The two terms in parentheses are dipole matrix elements dotted into the polarization vector of the electromagnetic field. In considering the spherically symmetric spatial eigenfunctions of the Hydrogen atom potential, the diagonal matrix elements go to zero, leaving us with
or
Here , where is the Rabi Frequency.
Intuition
In the numerator we have the transition dipole moment for the transition, whose squared amplitude represents the strength of the interaction between the electromagnetic field and the atom, and is the vector electric field amplitude, which includes the polarization. The numerator has dimensions of energy, so dividing by gives an angular frequency.
By analogy with a classical dipole, it is clear that an atom with a large dipole moment will be more susceptible to perturbation by an electric field. The dot product includes a factor of , where is the angle between the polarization of the light and the transition dipole moment. When they are parallel the interaction is strongest, when they are perpendicular there is no interaction at all.
If we rewrite the differential equat
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https://en.wikipedia.org/wiki/Residually%20finite%20group
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In the mathematical field of group theory, a group G is residually finite or finitely approximable if for every element g that is not the identity in G there is a homomorphism h from G to a finite group, such that
There are a number of equivalent definitions:
A group is residually finite if for each non-identity element in the group, there is a normal subgroup of finite index not containing that element.
A group is residually finite if and only if the intersection of all its subgroups of finite index is trivial.
A group is residually finite if and only if the intersection of all its normal subgroups of finite index is trivial.
A group is residually finite if and only if it can be embedded inside the direct product of a family of finite groups.
Examples
Examples of groups that are residually finite are finite groups, free groups, finitely generated nilpotent groups, polycyclic-by-finite groups, finitely generated linear groups, and fundamental groups of compact 3-manifolds.
Subgroups of residually finite groups are residually finite, and direct products of residually finite groups are residually finite. Any inverse limit of residually finite groups is residually finite. In particular, all profinite groups are residually finite.
Examples of non-residually finite groups can be constructed using the fact that all finitely generated residually finite groups are Hopfian groups. For example the Baumslag–Solitar group B(2,3) is not Hopfian, and therefore not residually finite.
Profinite topology
Every group G may be made into a topological group by taking as a basis of open neighbourhoods of the identity, the collection of all normal subgroups of finite index in G. The resulting topology is called the profinite topology on G. A group is residually finite if, and only if, its profinite topology is Hausdorff.
A group whose cyclic subgroups are closed in the profinite topology is said to be .
Groups each of whose finitely generated subgroups are closed in the profinite topology are called subgroup separable (also LERF, for locally extended residually finite).
A group in which every conjugacy class is closed in the profinite topology is called conjugacy separable.
Varieties of residually finite groups
One question is: what are the properties of a variety all of whose groups are residually finite? Two results about these are:
Any variety comprising only residually finite groups is generated by an A-group.
For any variety comprising only residually finite groups, it contains a finite group such that all members are embedded in a direct product of that finite group.
See also
Residual property (mathematics)
References
External links
Article with proof of some of the above statements
Infinite group theory
Properties of groups
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https://en.wikipedia.org/wiki/Parafree%20group
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In mathematics, in the realm of group theory, a group is said to be parafree if its quotients by the terms of its lower central series are the same as those of a free group and if it is residually nilpotent (the intersection of the terms of its lower central series is trivial).
Parafree groups share many properties with free groups, making it difficult to distinguish between these two types. Gilbert Baumslag was led to the study of parafree groups in attempts to resolve the conjecture that a group of cohomological dimension one is free. One of his fundamental results is that there exist parafree groups that are not free. With Urs Stammbach, he proved there exists a non-free parafree group with every countable subgroup being free.
References
Baumslag, Gilbert, Groups with the same lower central sequence as a relatively free group. I. The groups. Trans. Amer. Math. Soc. 129 1967 308--321.
Baumslag, Gilbert; Stammbach, Urs, A non-free parafree group all of whose countable subgroups are free. Math. Z. 148 (1976), no. 1, 63--65.
External links
Parafree one-relator groups
Properties of groups
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https://en.wikipedia.org/wiki/Complemented%20group
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In mathematics, in the realm of group theory, the term complemented group is used in two distinct, but similar ways.
In , a complemented group is one in which every subgroup has a group-theoretic complement. Such groups are called completely factorizable groups in the Russian literature, following and .
The following are equivalent for any finite group G:
G is complemented
G is a subgroup of a direct product of groups of square-free order (a special type of Z-group)
G is a supersolvable group with elementary abelian Sylow subgroups (a special type of A-group), .
Later, in , a group is said to be complemented if the lattice of subgroups is a complemented lattice, that is, if for every subgroup H there is a subgroup K such that H ∩ K = 1 and ⟨H, K ⟩ is the whole group. Hall's definition required in addition that H and K permute, that is, that HK = { hk : h in H, k in K } form a subgroup. Such groups are also called K-groups in the Italian and lattice theoretic literature, such as . The Frattini subgroup of a K-group is trivial; if a group has a core-free maximal subgroup that is a K-group, then it itself is a K-group; hence subgroups of K-groups need not be K-groups, but quotient groups and direct products of K-groups are K-groups, . In it is shown that every finite simple group is a complemented group. Note that in the classification of finite simple groups, K-group is more used to mean a group whose proper subgroups only have composition factors amongst the known finite simple groups.
An example of a group that is not complemented (in either sense) is the cyclic group of order p2, where p is a prime number. This group only has one nontrivial subgroup H, the cyclic group of order p, so there can be no other subgroup L to be the complement of H.
References
Properties of groups
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https://en.wikipedia.org/wiki/Quasiconvex%20function
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In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave.
All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. Univariate unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with star-convex sublevel sets can be unimodal without being quasiconvex.
Definition and properties
A function defined on a convex subset of a real vector space is quasiconvex if for all and we have
In words, if is such that it is always true that a point directly between two other points does not give a higher value of the function than both of the other points do, then is quasiconvex. Note that the points and , and the point directly between them, can be points on a line or more generally points in n-dimensional space.
An alternative way (see introduction) of defining a quasi-convex function is to require that each sublevel set
is a convex set.
If furthermore
for all and , then is strictly quasiconvex. That is, strict quasiconvexity requires that a point directly between two other points must give a lower value of the function than one of the other points does.
A quasiconcave function is a function whose negative is quasiconvex, and a strictly quasiconcave function is a function whose negative is strictly quasiconvex. Equivalently a function is quasiconcave if
and strictly quasiconcave if
A (strictly) quasiconvex function has (strictly) convex lower contour sets, while a (strictly) quasiconcave function has (strictly) convex upper contour sets.
A function that is both quasiconvex and quasiconcave is quasilinear.
A particular case of quasi-concavity, if , is unimodality, in which there is a locally maximal value.
Applications
Quasiconvex functions have applications in mathematical analysis, in mathematical optimization, and in game theory and economics.
Mathematical optimization
In nonlinear optimization, quasiconvex programming studies iterative methods that converge to a minimum (if one exists) for quasiconvex functions. Quasiconvex programming is a generalization of convex programming. Quasiconvex programming is used in the solution of "surrogate" dual problems, whose biduals provide quasiconvex closures of the primal problem, which therefore provide tighter bounds than do the convex closures provided by Lagrangian dual problems. In theory, quasiconvex programming and convex programming problems can be solved in reasonable amount of time, where the number o
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