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https://en.wikipedia.org/wiki/Rotations%20and%20reflections%20in%20two%20dimensions
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In Euclidean geometry, two-dimensional rotations and reflections are two kinds of Euclidean plane isometries which are related to one another.
Process
A rotation in the plane can be formed by composing a pair of reflections. First reflect a point to its image on the other side of line . Then reflect to its image on the other side of line . If lines and make an angle with one another, then points and will make an angle around point , the intersection of and . I.e., angle will measure .
A pair of rotations about the same point will be equivalent to another rotation about point . On the other hand, the composition of a reflection and a rotation, or of a rotation and a reflection (composition is not commutative), will be equivalent to a reflection.
Mathematical expression
The statements above can be expressed more mathematically. Let a rotation about the origin by an angle be denoted as . Let a reflection about a line through the origin which makes an angle with the -axis be denoted as . Let these rotations and reflections operate on all points on the plane, and let these points be represented by position vectors. Then a rotation can be represented as a matrix,
and likewise for a reflection,
With these definitions of coordinate rotation and reflection, the following four identities hold:
Proof
These equations can be proved through straightforward matrix multiplication and application of trigonometric identities, specifically the sum and difference identities.
The set of all reflections in lines through the origin and rotations about the origin, together with the operation of composition of reflections and rotations, forms a group. The group has an identity: . Every rotation has an inverse . Every reflection is its own inverse. Composition has closure and is associative, since matrix multiplication is associative.
Notice that both and have been represented with orthogonal matrices. These matrices all have a determinant whose absolute value is unity. Rotation matrices have a determinant of +1, and reflection matrices have a determinant of −1.
The set of all orthogonal two-dimensional matrices together with matrix multiplication form the orthogonal group: .
The following table gives examples of rotation and reflection matrix :
Rotation of axes
See also
2D_computer_graphics#Rotation
Cartan–Dieudonné theorem
Clockwise
Dihedral group
Euclidean plane isometry
Euclidean symmetries
Instant centre of rotation
Orthogonal group
Rotation group SO(3) – 3 dimensions
References
Sources
Euclidean symmetries
Euclidean plane geometry
Rotation
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https://en.wikipedia.org/wiki/Class%20number%20problem
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In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields (for negative integers d) having class number n. It is named after Carl Friedrich Gauss. It can also be stated in terms of discriminants. There are related questions for real quadratic fields and for the behavior as .
The difficulty is in effective computation of bounds: for a given discriminant, it is easy to compute the class number, and there are several ineffective lower bounds on class number (meaning that they involve a constant that is not computed), but effective bounds (and explicit proofs of completeness of lists) are harder.
Gauss's original conjectures
The problems are posed in Gauss's Disquisitiones Arithmeticae of 1801 (Section V, Articles 303 and 304).
Gauss discusses imaginary quadratic fields in Article 303, stating the first two conjectures, and discusses real quadratic fields in Article 304, stating the third conjecture.
Gauss conjecture (class number tends to infinity)
Gauss class number problem (low class number lists) For given low class number (such as 1, 2, and 3), Gauss gives lists of imaginary quadratic fields with the given class number and believes them to be complete.
Infinitely many real quadratic fields with class number one Gauss conjectures that there are infinitely many real quadratic fields with class number one.
The original Gauss class number problem for imaginary quadratic fields is significantly different and easier than the modern statement: he restricted to even discriminants, and allowed non-fundamental discriminants.
Status
Gauss conjecture solved, Heilbronn, 1934.
Low class number lists class number 1: solved, Baker (1966), Stark (1967), Heegner (1952).
Class number 2: solved, Baker (1971), Stark (1971)
Class number 3: solved, Oesterlé (1985)
Class numbers h up to 100: solved, Watkins 2004
Infinitely many real quadratic fields with class number one Open.
Lists of discriminants of class number 1
For imaginary quadratic number fields, the (fundamental) discriminants of class number 1 are:
The non-fundamental discriminants of class number 1 are:
Thus, the even discriminants of class number 1, fundamental and non-fundamental (Gauss's original question) are:
Modern developments
In 1934, Hans Heilbronn proved the Gauss conjecture. Equivalently, for any given class number, there are only finitely many imaginary quadratic number fields with that class number.
Also in 1934, Heilbronn and Edward Linfoot showed that there were at most 10 imaginary quadratic number fields with class number 1 (the 9 known ones, and at most one further).
The result was ineffective (see effective results in number theory): it did not give bounds on the size of the remaining field.
In later developments, the case n = 1 was first discussed by Kurt Heegner, using modular forms and modular equations to show that no further such field could exist. Th
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https://en.wikipedia.org/wiki/Hartley%20transform
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In mathematics, the Hartley transform (HT) is an integral transform closely related to the Fourier transform (FT), but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform by Ralph V. L. Hartley in 1942, and is one of many known Fourier-related transforms. Compared to the Fourier transform, the Hartley transform has the advantages of transforming real functions to real functions (as opposed to requiring complex numbers) and of being its own inverse.
The discrete version of the transform, the discrete Hartley transform (DHT), was introduced by Ronald N. Bracewell in 1983.
The two-dimensional Hartley transform can be computed by an analog optical process similar to an optical Fourier transform (OFT), with the proposed advantage that only its amplitude and sign need to be determined rather than its complex phase. However, optical Hartley transforms do not seem to have seen widespread use.
Definition
The Hartley transform of a function is defined by:
where can in applications be an angular frequency and
is the cosine-and-sine (cas) or Hartley kernel. In engineering terms, this transform takes a signal (function) from the time-domain to the Hartley spectral domain (frequency domain).
Inverse transform
The Hartley transform has the convenient property of being its own inverse (an involution):
Conventions
The above is in accord with Hartley's original definition, but (as with the Fourier transform) various minor details are matters of convention and can be changed without altering the essential properties:
Instead of using the same transform for forward and inverse, one can remove the from the forward transform and use for the inverse—or, indeed, any pair of normalizations whose product is (Such asymmetrical normalizations are sometimes found in both purely mathematical and engineering contexts.)
One can also use instead of (i.e., frequency instead of angular frequency), in which case the coefficient is omitted entirely.
One can use instead of as the kernel.
Relation to Fourier transform
This transform differs from the classic Fourier transform
in the choice of the kernel. In the Fourier transform, we have the exponential kernel,
where is the imaginary unit.
The two transforms are closely related, however, and the Fourier transform (assuming it uses the same normalization convention) can be computed from the Hartley transform via:
That is, the real and imaginary parts of the Fourier transform are simply given by the even and odd parts of the Hartley transform, respectively.
Conversely, for real-valued functions the Hartley transform is given from the Fourier transform's real and imaginary parts:
where and denote the real and imaginary parts.
Properties
The Hartley transform is a real linear operator, and is symmetric (and Hermitian). From the symmetric and self-inverse properties, it follows that the transform is a unitary operator (indeed, orthogo
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https://en.wikipedia.org/wiki/Relatively%20compact%20subspace
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In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). And in an arbitrary topological space every subset of a relatively compact set is relatively compact.
Every compact subset of a Hausdorff space is relatively compact. In a non-Hausdorff space, such as the particular point topology on an infinite set, the closure of a compact subset is not necessarily compact; said differently, a compact subset of a non-Hausdorff space is not necessarily relatively compact.
Every compact subset of a (possibly non-Hausdorff) topological vector space is complete and relatively compact.
In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in has a subsequence convergent in .
Some major theorems characterize relatively compact subsets, in particular in function spaces. An example is the Arzelà–Ascoli theorem. Other cases of interest relate to uniform integrability, and the concept of normal family in complex analysis. Mahler's compactness theorem in the geometry of numbers characterizes relatively compact subsets in certain non-compact homogeneous spaces (specifically spaces of lattices).
Counterexample
As a counterexample take any neighbourhood of the particular point of an infinite particular point space. The neighbourhood itself may be compact but is not relatively compact because its closure is the whole non-compact space.
Almost periodic functions
The definition of an almost periodic function at a conceptual level has to do with the translates of being a relatively compact set. This needs to be made precise in terms of the topology used, in a particular theory.
See also
Compactly embedded
Totally bounded space
References
page 12 of V. Khatskevich, D.Shoikhet, Differentiable Operators and Nonlinear Equations, Birkhäuser Verlag AG, Basel, 1993, 270 pp. at google books
Properties of topological spaces
Compactness (mathematics)
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https://en.wikipedia.org/wiki/Sato%E2%80%93Tate%20conjecture
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In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep obtained from an elliptic curve E over the rational numbers by reduction modulo almost all prime numbers p. Mikio Sato and John Tate independently posed the conjecture around 1960.
If Np denotes the number of points on the elliptic curve Ep defined over the finite field with p elements, the conjecture gives an answer to the distribution of the second-order term for Np. By Hasse's theorem on elliptic curves,
as , and the point of the conjecture is to predict how the O-term varies.
The original conjecture and its generalization to all totally real fields was proved by Laurent Clozel, Michael Harris, Nicholas Shepherd-Barron, and Richard Taylor under mild assumptions in 2008, and completed by Thomas Barnet-Lamb, David Geraghty, Harris, and Taylor in 2011. Several generalizations to other algebraic varieties and fields are open.
Statement
Let E be an elliptic curve defined over the rational numbers without complex multiplication. For a prime number p, define θp as the solution to the equation
Then, for every two real numbers and for which
Details
By Hasse's theorem on elliptic curves, the ratio
is between -1 and 1. Thus it can be expressed as cos θ for an angle θ; in geometric terms there are two eigenvalues accounting for the remainder and with the denominator as given they are complex conjugate and of absolute value 1. The Sato–Tate conjecture, when E doesn't have complex multiplication, states that the probability measure of θ is proportional to
This is due to Mikio Sato and John Tate (independently, and around 1960, published somewhat later).
Proof
In 2008, Clozel, Harris, Shepherd-Barron, and Taylor published a proof of the Sato–Tate conjecture for elliptic curves over totally real fields satisfying a certain condition: of having multiplicative reduction at some prime, in a series of three joint papers.
Further results are conditional on improved forms of the Arthur–Selberg trace formula. Harris has a conditional proof of a result for the product of two elliptic curves (not isogenous) following from such a hypothetical trace formula. In 2011, Barnet-Lamb, Geraghty, Harris, and Taylor proved a generalized version of the Sato–Tate conjecture for an arbitrary non-CM holomorphic modular form of weight greater than or equal to two, by improving the potential modularity results of previous papers. The prior issues involved with the trace formula were solved by Michael Harris, and Sug Woo Shin.
In 2015, Richard Taylor was awarded the Breakthrough Prize in Mathematics "for numerous breakthrough results in (...) the Sato–Tate conjecture."
Generalisations
There are generalisations, involving the distribution of Frobenius elements in Galois groups involved in the Galois representations on étale cohomology. In particular there is a conjectural theory for curves of genus n > 1.
Under the random matrix model developed by Nick Katz and Pet
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https://en.wikipedia.org/wiki/Multinomial%20theorem
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In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials.
Theorem
For any positive integer and any non-negative integer , the multinomial formula describes how a sum with terms expands when raised to an arbitrary power :
where
is a multinomial coefficient. The sum is taken over all combinations of nonnegative integer indices through such that the sum of all is . That is, for each term in the expansion, the exponents of the must add up to . Also, as with the binomial theorem, quantities of the form that appear are taken to equal 1 (even when equals zero).
In the case , this statement reduces to that of the binomial theorem.
Example
The third power of the trinomial is given by
This can be computed by hand using the distributive property of multiplication over addition, but it can also be done (perhaps more easily) with the multinomial theorem. It is possible to "read off" the multinomial coefficients from the terms by using the multinomial coefficient formula. For example:
has the coefficient
has the coefficient
Alternate expression
The statement of the theorem can be written concisely using multiindices:
where
and
Proof
This proof of the multinomial theorem uses the binomial theorem and induction on .
First, for , both sides equal since there is only one term in the sum. For the induction step, suppose the multinomial theorem holds for . Then
by the induction hypothesis. Applying the binomial theorem to the last factor,
which completes the induction. The last step follows because
as can easily be seen by writing the three coefficients using factorials as follows:
Multinomial coefficients
The numbers
appearing in the theorem are the multinomial coefficients. They can be expressed in numerous ways, including as a product of binomial coefficients or of factorials:
Sum of all multinomial coefficients
The substitution of for all into the multinomial theorem
gives immediately that
Number of multinomial coefficients
The number of terms in a multinomial sum, , is equal to the number of monomials of degree on the variables :
The count can be performed easily using the method of stars and bars.
Valuation of multinomial coefficients
The largest power of a prime that divides a multinomial coefficient may be computed using a generalization of Kummer's theorem.
Asymptotics
By Stirling's approximation, or equivalently the log-gamma function's asymptotic expansion, so for example,
Interpretations
Ways to put objects into bins
The multinomial coefficients have a direct combinatorial interpretation, as the number of ways of depositing distinct objects into distinct bins, with objects in the first bin, objects in the second bin, and so on.
Number of ways to select according to a distribution
In statistical mechanics and combinatorics, if one has a number distri
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https://en.wikipedia.org/wiki/Algebraic%20K-theory
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Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.
K-theory was discovered in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only K0, the zeroth K-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic K-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes classical number-theoretic topics like quadratic reciprocity and embeddings of number fields into the real numbers and complex numbers, as well as more modern concerns like the construction of higher regulators and special values of L-functions.
The lower K-groups were discovered first, in the sense that adequate descriptions of these groups in terms of other algebraic structures were found. For example, if F is a field, then is isomorphic to the integers Z and is closely related to the notion of vector space dimension. For a commutative ring R, the group is related to the Picard group of R, and when R is the ring of integers in a number field, this generalizes the classical construction of the class group. The group K1(R) is closely related to the group of units , and if R is a field, it is exactly the group of units. For a number field F, the group K2(F) is related to class field theory, the Hilbert symbol, and the solvability of quadratic equations over completions. In contrast, finding the correct definition of the higher K-groups of rings was a difficult achievement of Daniel Quillen, and many of the basic facts about the higher K-groups of algebraic varieties were not known until the work of Robert Thomason.
History
The history of K-theory was detailed by Charles Weibel.
The Grothendieck group K0
In the 19th century, Bernhard Riemann and his student Gustav Roch proved what is now known as the Riemann–Roch theorem. If X is a Riemann surface, then the sets of meromorphic functions and meromorphic differential forms on X form vector spaces. A line bundle on X determines subspaces of these vector spaces, and if X is projective, then these subspaces are finite dimensional. The Riemann–Roch theorem states that the difference in dimensions between these subspaces is equal to the degree of the line bundle (a measure of twistedness) plus one minus the genus of X. In the mid-20th century, the Riemann–Roch theorem was generalized by Friedrich Hirzebruch to all algebraic varieties. In Hirzebruch's
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https://en.wikipedia.org/wiki/Stirling%20%28disambiguation%29
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Stirling is a city and former ancient burgh in Scotland.
Stirling may also refer to:
Mathematics
Stirling's approximation, a formula to approximate large factorials
Stirling number
Stirling permutation
Physics and Engineering
Stirling cycle, a thermodynamic cycle for Stirling devices.
Stirling engine, a type of heat engine. See also Applications of the Stirling engine.
Stirling radioisotope generator, a type of radioisotope generator based on a Stirling engine.
Advanced Stirling radioisotope generator, a power system developed at NÄSA's Glenn Research Center.
Places
Scotland
Stirling (council area)
Stirling (Scottish Parliament constituency)
Stirling (UK Parliament constituency)
Stirling (Parliament of Scotland constituency), which ceased to exist in 1707
Stirling Sill, an outcropping or sill that underlies a large part of central Scotland
Stirling Village, Aberdeenshire
Stirlingshire, Scotland, a historic county and registration county.
Australia
Mount Stirling, Victoria
Stirling, South Australia, a town east of Adelaide
Stirling, Australian Capital Territory
Stirling, Victoria, an abandoned township near Tambo Crossing
Stirling Park, part of Stirling Linear Park, South Australia
Western Australia
City of Stirling, Perth
Stirling, Western Australia, a Perth suburb within the City of Stirling
Division of Stirling, electoral district in the Australian House of Representatives
Stirling County, Western Australia
Electoral district of Stirling, an electoral district in the Western Australian Legislative Assembly
Stirling Range, a bighill range in southern Western Australia
Stirling Estate, Western Australia, a locality in the Shire of Capel
Canada
Stirling, Alberta, a village
Stirling, a village in the township of Stirling-Rawdon, Ontario
Stirlingville, Alberta, Canada, an unincorporated community.
New Zealand
Stirling, New Zealand, a settlement
Solomon Islands
Stirling Island
United States
Stirling, New Jersey, an unincorporated community
Stirling (Massaponax, Virginia), a historic plantation
Stirling City, California, U.S, an unincorporated community
Buildings
Stirling Castle, Stirling, Scotland
Stirling (Reading, Pennsylvania), a historic mansion
Schools
University of Stirling, Stirling, Scotland
Stirling High School, Stirling, Scotland
Stirling High School, East London, England
Stirling Theological College, Carlton, Victoria, Australia
Stirling School, Stirling, Alberta, Canada, a public school
Transportation
Stirling Highway, connecting Perth and Freemantle, Western Australia
Stirling railway station (Scotland) in Stirling
Stirling railway station, Perth in Perth, Western Australia
MCV Stirling, a bus body
GNR Stirling 4-2-2, a locomotive class
Military
Short Stirling, a Second World War British bomber aircraft
Battle of Stirling (1648), in the Scottish Civil War
HMAS Stirling, a Royal Australian Navy base
People
Stirling (surname), a surname (including a list of people wi
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https://en.wikipedia.org/wiki/Informant%20%28statistics%29
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In statistics, the informant (or score) is the gradient of the log-likelihood function with respect to the parameter vector. Evaluated at a particular point of the parameter vector, the score indicates the steepness of the log-likelihood function and thereby the sensitivity to infinitesimal changes to the parameter values. If the log-likelihood function is continuous over the parameter space, the score will vanish at a local maximum or minimum; this fact is used in maximum likelihood estimation to find the parameter values that maximize the likelihood function.
Since the score is a function of the observations that are subject to sampling error, it lends itself to a test statistic known as score test in which the parameter is held at a particular value. Further, the ratio of two likelihood functions evaluated at two distinct parameter values can be understood as a definite integral of the score function.
Definition
The score is the gradient (the vector of partial derivatives) of , the natural logarithm of the likelihood function, with respect to an m-dimensional parameter vector .
This differentiation yields a row vector, and indicates the sensitivity of the likelihood (its derivative normalized by its value).
In older literature, "linear score" may refer to the score with respect to infinitesimal translation of a given density. This convention arises from a time when the primary parameter of interest was the mean or median of a distribution. In this case, the likelihood of an observation is given by a density of the form . The "linear score" is then defined as
Properties
Mean
While the score is a function of , it also depends on the observations at which the likelihood function is evaluated, and in view of the random character of sampling one may take its expected value over the sample space. Under certain regularity conditions on the density functions of the random variables, the expected value of the score, evaluated at the true parameter value , is zero. To see this, rewrite the likelihood function as a probability density function , and denote the sample space . Then:
The assumed regularity conditions allow the interchange of derivative and integral (see Leibniz integral rule), hence the above expression may be rewritten as
It is worth restating the above result in words: the expected value of the score, at true parameter value is zero. Thus, if one were to repeatedly sample from some distribution, and repeatedly calculate the score, then the mean value of the scores would tend to zero asymptotically.
Variance
The variance of the score, , can be derived from the above expression for the expected value.
Hence the variance of the score is equal to the negative expected value of the Hessian matrix of the log-likelihood.
The latter is known as the Fisher information and is written . Note that the Fisher information is not a function of any particular observation, as the random variable has been averaged out. This concept of in
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https://en.wikipedia.org/wiki/Fisher%20information
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In mathematical statistics, the Fisher information (sometimes simply called information) is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information.
The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized by the statistician Sir Ronald Fisher (following some initial results by Francis Ysidro Edgeworth). The Fisher information matrix is used to calculate the covariance matrices associated with maximum-likelihood estimates. It can also be used in the formulation of test statistics, such as the Wald test.
In Bayesian statistics, the Fisher information plays a role in the derivation of non-informative prior distributions according to Jeffreys' rule. It also appears as the large-sample covariance of the posterior distribution, provided that the prior is sufficiently smooth (a result known as Bernstein–von Mises theorem, which was anticipated by Laplace for exponential families). The same result is used when approximating the posterior with Laplace's approximation, where the Fisher information appears as the covariance of the fitted Gaussian.
Statistical systems of a scientific nature (physical, biological, etc.) whose likelihood functions obey shift invariance have been shown to obey maximum Fisher information. The level of the maximum depends upon the nature of the system constraints.
Definition
The Fisher information is a way of measuring the amount of information that an observable random variable carries about an unknown parameter upon which the probability of depends. Let be the probability density function (or probability mass function) for conditioned on the value of . It describes the probability that we observe a given outcome of , given a known value of . If is sharply peaked with respect to changes in , it is easy to indicate the "correct" value of from the data, or equivalently, that the data provides a lot of information about the parameter . If is flat and spread-out, then it would take many samples of to estimate the actual "true" value of that would be obtained using the entire population being sampled. This suggests studying some kind of variance with respect to .
Formally, the partial derivative with respect to of the natural logarithm of the likelihood function is called the score. Under certain regularity conditions, if is the true parameter (i.e. is actually distributed as ), it can be shown that the expected value (the first moment) of the score, evaluated at the true parameter value , is 0:
The Fisher information is defined to be the variance of the score:
Note that . A random variable carrying high Fisher information implies that the absolute value of the score is often high. The Fisher information is not a function of a particular observation, as the random variable X
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https://en.wikipedia.org/wiki/Brill%E2%80%93Noether%20theory
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In algebraic geometry, Brill–Noether theory, introduced by , is the study of special divisors, certain divisors on a curve that determine more compatible functions than would be predicted. In classical language, special divisors move on the curve in a "larger than expected" linear system of divisors.
Throughout, we consider a projective smooth curve over the complex numbers (or over some other algebraically closed field).
The condition to be a special divisor can be formulated in sheaf cohomology terms, as the non-vanishing of the cohomology of the sheaf of sections of the invertible sheaf or line bundle associated to . This means that, by the Riemann–Roch theorem, the cohomology or space of holomorphic sections is larger than expected.
Alternatively, by Serre duality, the condition is that there exist holomorphic differentials with divisor on the curve.
Main theorems of Brill–Noether theory
For a given genus , the moduli space for curves of genus should contain a dense subset parameterizing those curves with the minimum in the way of special divisors. One goal of the theory is to 'count constants', for those curves: to predict the dimension of the space of special divisors (up to linear equivalence) of a given degree , as a function of , that must be present on a curve of that genus.
The basic statement can be formulated in terms of the Picard variety of a smooth curve , and the subset of corresponding to divisor classes of divisors , with given values of and of in the notation of the Riemann–Roch theorem. There is a lower bound for the dimension of this subscheme in :
called the Brill–Noether number. The formula can be memorized via the mnemonic (using our desired and Riemann-Roch)
For smooth curves and for , the basic results about the space of linear systems on of degree and dimension are as follows.
George Kempf proved that if then is not empty, and every component has dimension at least .
William Fulton and Robert Lazarsfeld proved that if then is connected.
showed that if is generic then is reduced and all components have dimension exactly (so in particular is empty if ).
David Gieseker proved that if is generic then is smooth. By the connectedness result this implies it is irreducible if .
Other more recent results not necessarily in terms of space of linear systems are:
Eric Larson (2017) proved that if , , and , the restriction maps are of maximal rank, also known as the maximal rank conjecture.
Eric Larson and Isabel Vogt (2022) proved that if then there is a curve interpolating through general points in if and only if except in 4 exceptional cases:
References
Notes
Algebraic curves
Algebraic surfaces
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https://en.wikipedia.org/wiki/Poincar%C3%A9%20lemma
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In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed p-form on an open ball in Rn is exact for p with . The lemma was introduced by Henri Poincaré in 1886.
Especially in calculus, the Poincaré lemma also says that every closed 1-form on a simply connected open subset in is exact.
In the language of cohomology, the Poincaré lemma says that the k-th de Rham cohomology group of a contractible open subset of a manifold M (e.g., ) vanishes for . In particular, it implies that the de Rham complex yields a resolution of the constant sheaf on M. The singular cohomology of a contractible space vanishes in positive degree, but the Poincaré lemma does not follow from this, since the fact that the singular cohomology of a manifold can be computed as the de Rham cohomology of it, that is, the de Rham theorem, relies on the Poincaré lemma. It does, however, mean that it is enough to prove the Poincaré lemma for open balls; the version for contractible manifolds then follows from the topological consideration.
The Poincaré lemma is also a special case of the homotopy invariance of de Rham cohomology; in fact, it is common to establish the lemma by showing the homotopy invariance or at least a version of it.
Proofs
Direct proof
We shall prove the lemma for an open subset that is star-shaped or a cone over ; i.e., if is in , then is in for . This case in particular covers the open ball case, since an open ball can be assumed to centered at the origin without loss of generality.
The trick is to consider differential forms on (we use for the coordinate on ). First define the operator (called the fiber integration) for k-forms on by
where , and similarly for and . Now, for , since , using the differentiation under the integral sign, we have:
where denote the restrictions of to the hyperplanes and they are zero since is zero there. If , then a similar computation gives
.
Thus, the above formula holds for any -form on . Finally, let and then set . Then, with the notation , we get: for any -form on ,
the formula known as the homotopy formula. The operator is called the homotopy operator (also called a chain homotopy). Now, if is closed, . On the other hand, and . Hence,
which proves the Poincaré lemma.
The same proof in fact shows the Poincaré lemma for any contractible open subset U of a manifold. Indeed, given such a U, we have the homotopy with the identity and a point. Approximating such , we can assume is in fact smooth. The fiber integration is also defined for . Hence, the same argument goes through.
Proof using Lie derivatives
Cartan's magic formula for Lie derivatives can be used to give a short proof of the Poincaré lemma. The formula states that the Lie derivative along a vector field is given as:
where denotes the interior product; i.e., .
Let be a smooth family of smooth maps
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https://en.wikipedia.org/wiki/Mary%20Somerville
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Mary Somerville (; , formerly Greig; 26 December 1780 – 29 November 1872) was a Scottish scientist, writer, and polymath. She studied mathematics and astronomy, and in 1835 she and Caroline Herschel were elected as the first female Honorary Members of the Royal Astronomical Society.
When John Stuart Mill organized a massive petition to Parliament to give women the right to vote, he made sure that the first signature on the petition would be Somerville's.
When she died in 1872, The Morning Post declared in her obituary that "Whatever difficulty we might experience in the middle of the nineteenth century in choosing a king of science, there could be no question whatever as to the queen of science". One of the earliest uses of the word scientist was in a review by William Whewell of Somerville's second book On the Connexion of the Physical Sciences. However, the word was not used to describe Somerville herself; she was known and celebrated as a mathematician or a philosopher.
Somerville College, a college of the University of Oxford, is named after her, reflecting the virtues of liberalism and academic success which the college wished to embody. She is featured on the front of the Royal Bank of Scotland polymer £10 note launched in 2017 along with a quotation from her work On the Connection of the Physical Sciences.
Early life and education
Somerville, the daughter of Vice-Admiral Sir William George Fairfax, was related to several prominent Scottish houses through her mother, Margaret Charters. She was born at the manse of Jedburgh, the home of her maternal aunt and the Rev. Dr. Thomas Somerville (1741–1830) (author of My Own Life and Times). Her childhood home was at Burntisland, Fife, where her mother was from.
Somerville was the second of four surviving children (three of her siblings had died in infancy). She was particularly close to her oldest brother Sam. The family lived in genteel poverty as her father's naval pay remained meagre, despite his rise through the ranks. Her mother supplemented the household's income by growing vegetables, maintaining an orchard and keeping cows for milk. Her mother taught her to read the Bible and Calvinist catechisms. When her household chores were done Mary was free to roam among the birds and flowers in the garden.
In her autobiography Somerville recollects that on her father's return from the sea he said to his wife, "This kind of life will never do, Mary must at least know how to write and keep accounts". Ten-year-old Mary was then sent to an expensive boarding school in Musselburgh, where she learned the first principles of writing, rudimentary French and English grammar. Upon returning home, she:
...was no longer amused in the gardens, but wandered about the country. When the tide was out I spent hours on the sands, looking at the star-fish and sea-urchins, or watching the children digging for sand-eels, cockles, and the spouting razor-fish. I made collections of shells, such as were cast asho
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https://en.wikipedia.org/wiki/Gradient%20conjecture
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In mathematics, the gradient conjecture, due to René Thom (1989), was proved in 2000 by three Polish mathematicians, Krzysztof Kurdyka (University of Savoie, France), Tadeusz Mostowski (Warsaw University, Poland) and Adam Parusiński (University of Angers, France).
The conjecture states that given a real-valued analytic function f defined on Rn and a trajectory x(t) of the gradient vector field of f having a limit point x0 ∈ Rn, where f has an isolated critical point at x0, there exists a limit (in the projective space PRn-1) for the secant lines from x(t) to x0, as t tends to zero.
The proof depends on a theorem due to Stanis%C5%82aw %C5%81ojasiewicz.
References
R. Thom (1989) "Problèmes rencontrés dans mon parcours mathématique: un bilan", Publications Math%C3%A9matiques de l%27IH%C3%89S 70: 200 to 214. (This gradient conjecture due to René Thom was in fact well-known among specialists by the early 70's, having been often discussed during that period by Thom during his weekly seminar on singularities at the IHES.)
In 2000 the conjecture was proven correct in Annals of Mathematics 152: 763 to 792. The proof is available here.
Theorems in analysis
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https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20England
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This article concerns football records in England. Unless otherwise stated, records are taken from the Football League or Premier League. Where a different record exists for the top flight (Football League First Division 1888–1992, and Premier League 1992–present), this is also given. This article includes clubs based in Wales that compete in English leagues.
League
The original league saw 12 teams become the founding members of the Football League in 1888–89: Accrington, Blackburn Rovers, Bolton Wanderers, Burnley, Everton, Preston North End, Aston Villa, Derby County, Notts County, Stoke City, West Bromwich Albion and Wolverhampton Wanderers.
Three of the teams (Blackburn Rovers, Everton and Aston Villa) also played in the first Premier League season in 1992–93, but Notts County missed out, finishing in the relegation zone in 1991–92.
A second division was added four years later for the 1892–93 season, resulting in the Football League now becoming the Football League First Division, the top division for the next one hundred years. The Southern League became Division 3 in 1920. A Northern League formed the following year that became Division Three North. In 1958 the regional divisions combined to form the Third Division and a national Fourth Division. The top 12 sides from the Northern and Southern divisions formed the Third Division, whilst the bottom 12 of the respective divisions formed the new fourth tier.
Eight clubs have reached double figures of league titles, with Liverpool and Manchester United leading the chasing pack. Five clubs have managed to win all four divisions, a rare achievement while a further seven clubs need the top title to complete the full set. Luton Town can claim a quadruple of titles when they won the National League, after becoming the non-league champions in 2014.
For the 1919/20 season, the first season after the First World War, Arsenal were controversially elected in to the first division, despite finishing fifth in the last season before the outbreak of war in the second division. However, they have remained at this level ever since. Arsenal had once previously won promotion after finishing second behind Preston North End in the 1903/04 season, staying there until finishing bottom in 1912/13. Other clubs won elections to play in the first division. Blackburn Rovers and Newcastle United in 1898, Bury and Notts County in 1905 and Chelsea (alongside Arsenal) in 1919 were also elected to the top flight. Blackburn later won division 2 in 1938/39, Newcastle United finished runners-up in 1947/48. Notts County became second division champions in 1913/14, while Bury would finish runners-up in 1923/24. In the 1929/30 season Chelsea finished second behind Blackpool.
Holding the record of continuous seasons, Arsenal are some way ahead of other clubs who have suffered relegation, returning by winning the division, finishing in an automatic promotion place or, more recently, via the play offs. Everton spent three years
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https://en.wikipedia.org/wiki/Radial
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Radial is a geometric term of location which may refer to:
Mathematics and Direction
Vector (geometric), a line
Radius, adjective form of
Radial distance (geometry), a directional coordinate in a polar coordinate system
Radial set
A bearing from a waypoint, such as a VHF omnidirectional range
Biology
Radial artery, the main artery of the lateral aspect of the forearm
Radial nerve, supplies the posterior portion of the upper limb
Radial symmetry, one of the types of distribution of body parts or shapes in biology
Radius (bone), a bone of the forearm
Technology
Radial (radio), lines which radiate from a radio antenna
Radial axle, on a locomotive or carriage
Radial compressor
Radial delayed blowback
Radial engine
Radial tire
Radial, Inc., e-commerce business
See also
Axial (disambiguation)
Radiate (disambiguation)
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https://en.wikipedia.org/wiki/H-space
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In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed.
Definition
An H-space consists of a topological space , together with an element of and a continuous map , such that and the maps and are both homotopic to the identity map through maps sending to . This may be thought of as a pointed topological space together with a continuous multiplication for which the basepoint is an identity element up to basepoint-preserving homotopy.
One says that a topological space is an H-space if there exists and such that the triple is an H-space as in the above definition. Alternatively, an H-space may be defined without requiring homotopies to fix the basepoint , or by requiring to be an exact identity, without any consideration of homotopy. In the case of a CW complex, all three of these definitions are in fact equivalent.
Examples and properties
The standard definition of the fundamental group, together with the fact that it is a group, can be rephrased as saying that the loop space of a pointed topological space has the structure of an H-group, as equipped with the standard operations of concatenation and inversion. Furthermore a continuous basepoint preserving map of pointed topological space induces a H-homomorphism of the corresponding loop spaces; this reflects the group homomorphism on fundamental groups induced by a continuous map.
It is straightforward to verify that, given a pointed homotopy equivalence from a H-space to a pointed topological space, there is a natural H-space structure on the latter space. As such, the existence of an H-space structure on a given space is only dependent on pointed homotopy type.
The multiplicative structure of an H-space adds structure to its homology and cohomology groups. For example, the cohomology ring of a path-connected H-space with finitely generated and free cohomology groups is a Hopf algebra. Also, one can define the Pontryagin product on the homology groups of an H-space.
The fundamental group of an H-space is abelian. To see this, let X be an H-space with identity e and let f and g be loops at e. Define a map F: [0,1] × [0,1] → X by F(a,b) = f(a)g(b). Then F(a,0) = F(a,1) = f(a)e is homotopic to f, and F(0,b) = F(1,b) = eg(b) is homotopic to g. It is clear how to define a homotopy from [f][g] to [g][f].
Adams' Hopf invariant one theorem, named after Frank Adams, states that S0, S1, S3, S7 are the only spheres that are H-spaces. Each of these spaces forms an H-space by viewing it as the subset of norm-one elements of the reals, complexes, quaternions, and octonions, respectively, and using the multiplication operations from these algebras. In fact, S0, S1, and S3 are groups (Lie groups) with these multiplications. But S7 is not a group in this way because octonion multiplication is not associative, nor can it be given any other continuous multiplication for whi
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https://en.wikipedia.org/wiki/Minkowski%E2%80%93Bouligand%20dimension
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In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set in a Euclidean space , or more generally in a metric space . It is named after the Polish mathematician Hermann Minkowski and the French mathematician Georges Bouligand.
To calculate this dimension for a fractal , imagine this fractal lying on an evenly spaced grid and count how many boxes are required to cover the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid finer by applying a box-counting algorithm.
Suppose that is the number of boxes of side length required to cover the set. Then the box-counting dimension is defined as
Roughly speaking, this means that the dimension is the exponent such that , which is what one would expect in the trivial case where is a smooth space (a manifold) of integer dimension .
If the above limit does not exist, one may still take the limit superior and limit inferior, which respectively define the upper box dimension and lower box dimension. The upper box dimension is sometimes called the entropy dimension, Kolmogorov dimension, Kolmogorov capacity, limit capacity or upper Minkowski dimension, while the lower box dimension is also called the lower Minkowski dimension.
The upper and lower box dimensions are strongly related to the more popular Hausdorff dimension. Only in very special applications is it important to distinguish between the three (see below). Yet another measure of fractal dimension is the correlation dimension.
Alternative definitions
It is possible to define the box dimensions using balls, with either the covering number or the packing number. The covering number is the minimal number of open balls of radius ε required to cover the fractal, or in other words, such that their union contains the fractal. We can also consider the intrinsic covering number , which is defined the same way but with the additional requirement that the centers of the open balls lie inside the set S. The packing number is the maximal number of disjoint open balls of radius ε one can situate such that their centers would be inside the fractal. While N, Ncovering, N'''covering and Npacking are not exactly identical, they are closely related and give rise to identical definitions of the upper and lower box dimensions. This is easy to prove once the following inequalities are proven:
These, in turn, follow with a little effort from the triangle inequality.
The advantage of using balls rather than squares is that this definition generalizes to any metric space. In other words, the box definition is extrinsic — one assumes the fractal space S is contained in a Euclidean space, and defines boxes according to the external geometry of the containing space. However, the dimension of S should be intrinsic, independent of the environment into which S is placed, and the ball definition can be
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https://en.wikipedia.org/wiki/Pedoe%27s%20inequality
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In geometry, Pedoe's inequality (also Neuberg–Pedoe inequality), named after Daniel Pedoe (1910–1998) and Joseph Jean Baptiste Neuberg (1840–1926), states that if a, b, and c are the lengths of the sides of a triangle with area ƒ, and A, B, and C are the lengths of the sides of a triangle with area F, then
with equality if and only if the two triangles are similar with pairs of corresponding sides (A, a), (B, b), and (C, c).
The expression on the left is not only symmetric under any of the six permutations of the set { (A, a), (B, b), (C, c) } of pairs, but also—perhaps not so obviously—remains the same if a is interchanged with A and b with B and c with C. In other words, it is a symmetric function of the pair of triangles.
Pedoe's inequality is a generalization of Weitzenböck's inequality, which is the case in which one of the triangles is equilateral.
Pedoe discovered the inequality in 1941 and published it subsequently in several articles. Later he learned that the inequality was already known in the 19th century to Neuberg, who however did not prove that the equality implies the similarity of the two triangles.
Proof
By Heron's formula, the area of the two triangles can be expressed as:
and then, using Cauchy-Schwarz inequality we have,
So,
and the proposition is proven.
Equality holds if and only if , that is, the two triangles are similar.
See also
List of triangle inequalities
References
Daniel Pedoe: An Inequality Connecting Any Two Triangles. The Mathematical Gazette, Vol. 25, No. 267 (Dec., 1941), pp. 310-311 (JSTOR)
Daniel Pedoe: A Two-Triangle Inequality. The American Mathematical Monthly, volume 70, number 9, page 1012, November, 1963.
Daniel Pedoe: An Inequality for Two Triangles. Proceedings of the Cambridge Philosophical Society, volume 38, part 4, page 397, 1943.
Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities. MAA, 2009, , p. 108
D.S. Mitrinović, Josip Pečarić: About the Neuberg-Pedoe and the Oppenheim inequalities. Journal of Mathematical Analysis and Applications 129(1):196–210 · January 1988 (online copy)
Triangle inequalities
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https://en.wikipedia.org/wiki/List%20of%20inequalities
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This article lists Wikipedia articles about named mathematical inequalities.
Inequalities in pure mathematics
Analysis
Agmon's inequality
Askey–Gasper inequality
Babenko–Beckner inequality
Bernoulli's inequality
Bernstein's inequality (mathematical analysis)
Bessel's inequality
Bihari–LaSalle inequality
Bohnenblust–Hille inequality
Borell–Brascamp–Lieb inequality
Brezis–Gallouet inequality
Carleman's inequality
Chebyshev–Markov–Stieltjes inequalities
Chebyshev's sum inequality
Clarkson's inequalities
Eilenberg's inequality
Fekete–Szegő inequality
Fenchel's inequality
Friedrichs's inequality
Gagliardo–Nirenberg interpolation inequality
Gårding's inequality
Grothendieck inequality
Grunsky's inequalities
Hanner's inequalities
Hardy's inequality
Hardy–Littlewood inequality
Hardy–Littlewood–Sobolev inequality
Harnack's inequality
Hausdorff–Young inequality
Hermite–Hadamard inequality
Hilbert's inequality
Hölder's inequality
Jackson's inequality
Jensen's inequality
Khabibullin's conjecture on integral inequalities
Kantorovich inequality
Karamata's inequality
Korn's inequality
Ladyzhenskaya's inequality
Landau–Kolmogorov inequality
Lebedev–Milin inequality
Lieb–Thirring inequality
Littlewood's 4/3 inequality
Markov brothers' inequality
Mashreghi–Ransford inequality
Max–min inequality
Minkowski's inequality
Poincaré inequality
Popoviciu's inequality
Prékopa–Leindler inequality
Rayleigh–Faber–Krahn inequality
Remez inequality
Riesz rearrangement inequality
Schur test
Shapiro inequality
Sobolev inequality
Steffensen's inequality
Szegő inequality
Three spheres inequality
Trace inequalities
Trudinger's theorem
Turán's inequalities
Von Neumann's inequality
Wirtinger's inequality for functions
Young's convolution inequality
Young's inequality for products
Inequalities relating to means
Hardy–Littlewood maximal inequality
Inequality of arithmetic and geometric means
Ky Fan inequality
Levinson's inequality
Maclaurin's inequality
Mahler's inequality
Muirhead's inequality
Newton's inequalities
Stein–Strömberg theorem
Combinatorics
Binomial coefficient bounds
Factorial bounds
XYZ inequality
Fisher's inequality
Ingleton's inequality
Lubell–Yamamoto–Meshalkin inequality
Nesbitt's inequality
Rearrangement inequality
Schur's inequality
Shapiro inequality
Stirling's formula (bounds)
Differential equations
Grönwall's inequality
Geometry
Alexandrov–Fenchel inequality
Aristarchus's inequality
Barrow's inequality
Berger–Kazdan comparison theorem
Blaschke–Lebesgue inequality
Blaschke–Santaló inequality
Bishop–Gromov inequality
Bogomolov–Miyaoka–Yau inequality
Bonnesen's inequality
Brascamp–Lieb inequality
Brunn–Minkowski inequality
Castelnuovo–Severi inequality
Cheng's eigenvalue comparison theorem
Clifford's theorem on special divisors
Cohn-Vossen's inequality
Erdős–Mordell inequality
Euler's theorem in geometry
Gromov's inequality for complex projective space
Gro
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https://en.wikipedia.org/wiki/Population%20statistics%20for%20Israeli%20settlements%20in%20the%20Gaza%20Strip
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Population statistics for former Israeli settlements in the Gaza Strip, which were evacuated in 2005 as part of Israel's unilateral disengagement plan.
Israeli settlements in the Gaza Strip
Footnotes
Population Statistic Sources
* Source: List of Localities: Their Population and Codes, 31.12.1999. Jerusalem: Central Bureau of Statistics, 2000.
** Source: List of Localities: Their Population and Codes, 31.12.2000. Jerusalem: Central Bureau of Statistics, 2001.
*** Source: Peace Now Settlement Watch for 31.12.2001
**** After evacuation by the Israeli government, in which all Israeli settlers in Gaza were removed
Settlement Date Sources
The first date is given by the Settlement Division of the Zionist Organization.
The second date is given by the Yesha Council of Jewish Communities in Judea, Samaria and Gaza.
Third dates are from Peace Now.
See also
Israeli settlements
Gush Katif
References
External links
Foundation for Middle East Peace
PeaceNow
Former Israeli settlements in the Gaza Strip
Israel geography-related lists
Gaza Strip
Palestine (region)-related lists
Demographic lists
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https://en.wikipedia.org/wiki/Population%20statistics%20for%20Israeli%20settlements%20in%20the%20West%20Bank
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The population statistics for Israeli settlements in the West Bank are collected by the Israel Central Bureau of Statistics. As such, the data contains only population of settlements recognized by the Israeli authorities. Israeli outposts, which are illegal by Israeli law, are not tracked, and their population is hard to establish. All settlements in the West Bank were advised by the International Court of Justice to be unlawful.
As of January 2023, there are 144 Israeli settlements in the West Bank, including 12 in East Jerusalem. In addition, there are over 100 Israeli illegal outposts in the West Bank. In total, over 450,000 Israeli settlers live in the West Bank excluding East Jerusalem, with an additional 220,000 Jewish settlers residing in East Jerusalem.
The construction of the West Bank barrier keeps a significant number of settlements behind it. The total number of settlers east of the barrier lines in 2012 was at least 79,230. By comparison, the number of Gaza Strip settlers in 2005 who refused to move voluntarily and be compensated, and that were forcibly evicted during the Israeli disengagement from Gaza, was around 9,000.
Statistics
Statistics below refer to the period between 1999 and 2018. For more recent data, see List of Israeli settlements.
Unreported Nahal settlements:
Elisha (population of 753 in 2000)
Gvaot (population of 44 in 2003)
Localities of unknown status:
Bitronot
Doran
Ein Hogla
Mahane Giv'on
Other localities:
Shvut Rachel (est. 1991) – an independently governed settlement which is formally designated as a neighborhood of Shilo. As such, its population is counted within Shilo.
See also
Israeli settlement
Judea and Samaria Area
List of cities administered by the State of Palestine
List of cities in Israel
List of Israeli settlements
List of Israeli settlements with city status in the West Bank
Population statistics for Israeli Gaza Strip settlements
Yesha Council
Notes
External links
Israeli Settlements Population in the West Bank at the Jewish Virtual Library
Yesha Council
FMEP Reports: Settlements in the West Bank
West Bank
Israel geography-related lists
Israeli settlement
Palestine (region)-related lists
Demographic lists
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https://en.wikipedia.org/wiki/Frobenius%20theorem%20%28differential%20topology%29
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In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields. The theorem generalizes the existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of r vector fields mesh into coordinate grids on r-dimensional integral manifolds. The theorem is foundational in differential topology and calculus on manifolds.
Contact geometry studies 1-forms that maximally violates the assumptions of Frobenius' theorem. An example is shown on the right.
Introduction
One-form version
Suppose we are to find the trajectory of a particle in a subset of 3D space, but we do not know its trajectory formula. Instead, we know only that its trajectory satisfies , where are smooth functions of . Then, we can know only for sure that, if at some moment in time, the particle is at location , then its velocity at that moment is restricted within the plane with equation
In other words, we can draw a "local plane" at each point in 3D space, and we know that the particle's trajectory must be tangent to the local plane at all times.
If we have two equationsthen we can draw two local planes at each point, and their intersection is generically a line, allowing us to uniquely solve for the curve starting at any point. In other words, with two 1-forms, we can foliate the domain into curves.
If we have only one equation , then we might be able to foliate into surfaces, in which case, we can be sure that a curve starting at a certain surface must be restricted to wander within that surface. If not, then a curve starting at any point might end up at any other point in . One can imagine starting with a cloud of little planes, and quilting them together to form a full surface. The main danger is that, if we quilt the little planes two at a time, we might go on a cycle and return to where we began, but shifted by a small amount. If this happens, then we would not get a 2-dimensional surface, but a 3-dimensional blob. An example is shown in the diagram on the right.
If the one-form is integrable, then loops exactly close upon themselves, and each surface would be 2-dimensional. Frobenius' theorem states that this happens precisely when over all of the domain, where . The notation is defined in the article on one-forms.
During his development of axiomatic thermodynamics, Carathéodory proved that if is an integrable one-form on an open subset of , then for some scalar functions on the subset. This is usually called Carathé
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https://en.wikipedia.org/wiki/Sober%20space
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In mathematics, a sober space is a topological space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of X: that is, every irreducible closed subset has a unique generic point.
Definitions
Sober spaces have a variety of cryptomorphic definitions, which are documented in this section. All except the definition in terms of nets are described in. In each case below, replacing "unique" with "at most one" gives an equivalent formulation of the T0 axiom. Replacing it with "at least one" is equivalent to the property that the T0 quotient of the space is sober, which is sometimes referred to as having "enough points" in the literature.
In terms of morphisms of frames and locales
A topological space X is sober if every map that preserves all joins and all finite meets from its partially ordered set of open subsets to is the inverse image of a unique continuous function from the one-point space to X.
This may be viewed as a correspondence between the notion of a point in a locale and a point in a topological space, which is the motivating definition.
Using completely prime filters
A filter F of open sets is said to be completely prime if for any family of open sets such that , we have that for some i. A space X is sober if it each completely prime filter is the neighbourhood filter of a unique point in X.
In terms of nets
A net is self-convergent if it converges to every point in , or equivalently if its eventuality filter is completely prime. A net that converges to converges strongly if it can only converge to points in the closure of . A space is sober if every self-convergent net converges strongly to a unique point .
In particular, a space is T1 and sober precisely if every self-convergent net is constant.
With irreducible closed sets
A closed set is irreducible if it cannot be written as the union of two proper closed subsets. A space is sober if every irreducible closed subset is the closure of a unique point.
As a property of sheaves on the space
A space X is sober if every functor from the category of sheaves Sh(X) to Set that preserves all finite limits and all small colimits must be the stalk functor of a unique point x.
Properties and examples
Any Hausdorff (T2) space is sober (the only irreducible subsets being points), and all sober spaces are Kolmogorov (T0), and both implications are strict.
Sobriety is not comparable to the T1 condition:
an example of a T1 space which is not sober is an infinite set with the cofinite topology, the whole space being an irreducible closed subset with no generic point;
an example of a sober space which is not T1 is the Sierpinski space.
Moreover T2 is stronger than T1 and sober, i.e., while every T2 space is at once T1 and sober, there exist spaces that are simultaneously T1 and sober, but not T2. One such example is the following: let X be the set of real numbers, with a new point p adjoined; the open sets being all real open sets,
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https://en.wikipedia.org/wiki/Coherent%20sheaf
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In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.
Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank.
Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf.
Definitions
A quasi-coherent sheaf on a ringed space is a sheaf of -modules which has a local presentation, that is, every point in has an open neighborhood in which there is an exact sequence
for some (possibly infinite) sets and .
A coherent sheaf on a ringed space is a sheaf satisfying the following two properties:
is of finite type over , that is, every point in has an open neighborhood in such that there is a surjective morphism for some natural number ;
for any open set , any natural number , and any morphism of -modules, the kernel of is of finite type.
Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of -modules.
The case of schemes
When is a scheme, the general definitions above are equivalent to more explicit ones. A sheaf of -modules is quasi-coherent if and only if over each open affine subscheme the restriction is isomorphic to the sheaf associated to the module over . When is a locally Noetherian scheme, is coherent if and only if it is quasi-coherent and the modules above can be taken to be finitely generated.
On an affine scheme , there is an equivalence of categories from -modules to quasi-coherent sheaves, taking a module to the associated sheaf . The inverse equivalence takes a quasi-coherent sheaf on to the -module of global sections of .
Here are several further characterizations of quasi-coherent sheaves on a scheme.
Properties
On an arbitrary ringed space quasi-coherent sheaves do not necessarily form an abelian category. On the other hand, the quasi-coherent sheaves on any scheme form an abelian category, and they are extremely useful in that context.
On any ringed space , the coherent sheaves form an abelian category, a full subcategory of the category of -modules. (Analogously, the category of coherent modules over any ring is a full abelian subcategory of the category of all -modules.) So the kernel, image, and cokernel of any map of coherent sheaves are coherent. The direct sum of two coherent sheaves is coherent; more generally, an -module that is an extension of two coherent sheaves is coherent.
A submodule of a coherent sheaf is coherent if it is of finite type. A coherent sheaf is always a
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https://en.wikipedia.org/wiki/Proper%20morphism
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In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.
Some authors call a proper variety over a field k a complete variety. For example, every projective variety over a field k is proper over k. A scheme X of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space X(C) of complex points with the classical (Euclidean) topology is compact and Hausdorff.
A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite.
Definition
A morphism f: X → Y of schemes is called universally closed if for every scheme Z with a morphism Z → Y, the projection from the fiber product
is a closed map of the underlying topological spaces. A morphism of schemes is called proper if it is separated, of finite type, and universally closed ([EGA] II, 5.4.1 ). One also says that X is proper over Y. In particular, a variety X over a field k is said to be proper over k if the morphism X → Spec(k) is proper.
Examples
For any natural number n, projective space Pn over a commutative ring R is proper over R. Projective morphisms are proper, but not all proper morphisms are projective. For example, there is a smooth proper complex variety of dimension 3 which is not projective over C. Affine varieties of positive dimension over a field k are never proper over k. More generally, a proper affine morphism of schemes must be finite. For example, it is not hard to see that the affine line A1 over a field k is not proper over k, because the morphism A1 → Spec(k) is not universally closed. Indeed, the pulled-back morphism
(given by (x,y) ↦ y) is not closed, because the image of the closed subset xy = 1 in A1 × A1 = A2 is A1 − 0, which is not closed in A1.
Properties and characterizations of proper morphisms
In the following, let f: X → Y be a morphism of schemes.
The composition of two proper morphisms is proper.
Any base change of a proper morphism f: X → Y is proper. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×Y Z → Z is proper.
Properness is a local property on the base (in the Zariski topology). That is, if Y is covered by some open subschemes Yi and the restriction of f to all f−1(Yi) is proper, then so is f.
More strongly, properness is local on the base in the fpqc topology. For example, if X is a scheme over a field k and E is a field extension of k, then X is proper over k if and only if the base change XE is proper over E.
Closed immersions are proper.
More generally, finite morphisms are proper. This is a consequence of the going up theorem.
By Deligne, a morphism of schemes is finite if and only if it is proper and quasi-finite. This had been shown by Grothendieck if the morphism f: X → Y is locally of finite presentation, which follows from the other assumptions if Y is noetherian.
For X proper over a scheme S, and Y separated over S, the image of any morphism X → Y over S is
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https://en.wikipedia.org/wiki/K%C3%A4hler%20differential
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In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available.
Definition
Let and be commutative rings and be a ring homomorphism. An important example is for a field and a unital algebra over (such as the coordinate ring of an affine variety). Kähler differentials formalize the observation that the derivatives of polynomials are again polynomial. In this sense, differentiation is a notion which can be expressed in purely algebraic terms. This observation can be turned into a definition of the module
of differentials in different, but equivalent ways.
Definition using derivations
An -linear derivation on is an -module homomorphism to an -module satisfying the Leibniz rule (it automatically follows from this definition that the image of is in the kernel of ). The module of Kähler differentials is defined as the -module for which there is a universal derivation . As with other universal properties, this means that is the best possible derivation in the sense that any other derivation may be obtained from it by composition with an -module homomorphism. In other words, the composition with provides, for every , an -module isomorphism
One construction of and proceeds by constructing a free -module with one formal generator for each in , and imposing the relations
,
,
,
for all in and all and in . The universal derivation sends to . The relations imply that the universal derivation is a homomorphism of -modules.
Definition using the augmentation ideal
Another construction proceeds by letting be the ideal in the tensor product defined as the kernel of the multiplication map
Then the module of Kähler differentials of can be equivalently defined by
and the universal derivation is the homomorphism defined by
This construction is equivalent to the previous one because is the kernel of the projection
Thus we have:
Then may be identified with by the map induced by the complementary projection
This identifies with the -module generated by the formal generators for in , subject to being a homomorphism of -modules which sends each element of to zero. Taking the quotient by precisely imposes the Leibniz rule.
Examples and basic facts
For any commutative ring , the Kähler differentials of the polynomial ring are a free -module of rank n generated by the differentials of the variables:
Kähler differentials are compatible with extension of scalars, in the sense that for a second -algebra and for , there is an isomorphism
As a particular case of this, Kähler differentials are compatible with localizations, meaning that if is a multiplicative set in , then there is an
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https://en.wikipedia.org/wiki/Schinzel%27s%20hypothesis%20H
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In mathematics, Schinzel's hypothesis H is one of the most famous open problems in the topic of number theory. It is a very broad generalization of widely open conjectures such as the twin prime conjecture. The hypothesis is named after Andrzej Schinzel.
Statement
The hypothesis claims that for every finite collection of nonconstant irreducible polynomials over the integers with positive leading coefficients, one of the following conditions holds:
There are infinitely many positive integers such that all of are simultaneously prime numbers, or
There is an integer (called a "fixed divisor"), which depends on the polynomials, which always divides the product . (Or, equivalently: There exists a prime such that for every there is an such that divides .)
The second condition is satisfied by sets such as , since is always divisible by 2. It is easy to see that this condition prevents the first condition from being true. Schinzel's hypothesis essentially claims that condition 2 is the only way condition 1 can fail to hold.
No effective technique is known for determining whether the first condition holds for a given set of polynomials, but the second one is straightforward to check: Let and compute the greatest common divisor of successive values of . One can see by extrapolating with finite differences that this divisor will also divide all other values of too.
Schinzel's hypothesis builds on the earlier Bunyakovsky conjecture, for a single polynomial, and on the Hardy–Littlewood conjectures and Dickson's conjecture for multiple linear polynomials. It is in turn extended by the Bateman–Horn conjecture.
Examples
As a simple example with ,
has no fixed prime divisor. We therefore expect that there are infinitely many primes
This has not been proved, though. It was one of Landau's conjectures and goes back to Euler, who observed in a letter to Goldbach in 1752 that is often prime for up to 1500.
As another example, take with and . The hypothesis then implies the existence of infinitely many twin primes, a basic and notorious open problem.
Variants
As proved by Schinzel and Sierpiński
it is equivalent to the following: if condition 2 does not hold, then there exists at least one positive integer such that all will be simultaneously prime, for any choice of irreducible integral polynomials with positive leading coefficients.
If the leading coefficients were negative, we could expect negative prime values; this is a harmless restriction.
There is probably no real reason to restrict polynomials with integer coefficients, rather than integer-valued polynomials (such as , which takes integer values for all integers even though the coefficients are not integers).
Previous results
The special case of a single linear
polynomial is Dirichlet's theorem on arithmetic progressions, one of the most important results of
number theory. In fact, this special case is the only known instance of Schinzel's Hypothesis H. We do not
know the
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https://en.wikipedia.org/wiki/Stone%27s%20representation%20theorem%20for%20Boolean%20algebras
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In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone. Stone was led to it by his study of the spectral theory of operators on a Hilbert space.
Stone spaces
Each Boolean algebra B has an associated topological space, denoted here S(B), called its Stone space. The points in S(B) are the ultrafilters on B, or equivalently the homomorphisms from B to the two-element Boolean algebra. The topology on S(B) is generated by a (closed) basis consisting of all sets of the form
where b is an element of B. This is the topology of pointwise convergence of nets of homomorphisms into the two-element Boolean algebra.
For every Boolean algebra B, S(B) is a compact totally disconnected Hausdorff space; such spaces are called Stone spaces (also profinite spaces). Conversely, given any topological space X, the collection of subsets of X that are clopen (both closed and open) is a Boolean algebra.
Representation theorem
A simple version of Stone's representation theorem states that every Boolean algebra B is isomorphic to the algebra of clopen subsets of its Stone space S(B). The isomorphism sends an element to the set of all ultrafilters that contain b. This is a clopen set because of the choice of topology on S(B) and because B is a Boolean algebra.
Restating the theorem using the language of category theory; the theorem states that there is a duality between the category of Boolean algebras and the category of Stone spaces. This duality means that in addition to the correspondence between Boolean algebras and their Stone spaces, each homomorphism from a Boolean algebra A to a Boolean algebra B corresponds in a natural way to a continuous function from S(B) to S(A). In other words, there is a contravariant functor that gives an equivalence between the categories. This was an early example of a nontrivial duality of categories.
The theorem is a special case of Stone duality, a more general framework for dualities between topological spaces and partially ordered sets.
The proof requires either the axiom of choice or a weakened form of it. Specifically, the theorem is equivalent to the Boolean prime ideal theorem, a weakened choice principle that states that every Boolean algebra has a prime ideal.
An extension of the classical Stone duality to the category of Boolean spaces (= zero-dimensional locally compact Hausdorff spaces) and continuous maps (respectively, perfect maps) was obtained by G. D. Dimov (respectively, by H. P. Doctor).
See also
Citations
References
General topology
Boolean algebra
Theorems in lattice theory
Categorical logic
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https://en.wikipedia.org/wiki/Multi-index%20notation
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Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.
Definition and basic properties
An n-dimensional multi-index is an -tuple
of non-negative integers (i.e. an element of the -dimensional set of natural numbers, denoted ).
For multi-indices and , one defines:
Componentwise sum and difference
Partial order
Sum of components (absolute value)
Factorial
Binomial coefficient
Multinomial coefficient
where .
Power
.
Higher-order partial derivative
where (see also 4-gradient). Sometimes the notation is also used.
Some applications
The multi-index notation allows the extension of many formulae from elementary calculus to the corresponding multi-variable case. Below are some examples. In all the following, (or ), , and (or ).
Multinomial theorem
Multi-binomial theorem
Note that, since is a vector and is a multi-index, the expression on the left is short for .
Leibniz formula
For smooth functions and ,
Taylor series
For an analytic function in variables one has In fact, for a smooth enough function, we have the similar Taylor expansion where the last term (the remainder) depends on the exact version of Taylor's formula. For instance, for the Cauchy formula (with integral remainder), one gets
General linear partial differential operator
A formal linear -th order partial differential operator in variables is written as
Integration by parts
For smooth functions with compact support in a bounded domain one has This formula is used for the definition of distributions and weak derivatives.
An example theorem
If are multi-indices and , then
Proof
The proof follows from the power rule for the ordinary derivative; if α and β are in , then
Suppose , , and . Then we have that
For each in , the function only depends on . In the above, each partial differentiation therefore reduces to the corresponding ordinary differentiation . Hence, from equation (), it follows that vanishes if for at least one in . If this is not the case, i.e., if as multi-indices, then
for each and the theorem follows. Q.E.D.
See also
Einstein notation
Index notation
Ricci calculus
References
Saint Raymond, Xavier (1991). Elementary Introduction to the Theory of Pseudodifferential Operators. Chap 1.1 . CRC Press.
Combinatorics
Mathematical notation
Articles containing proofs
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https://en.wikipedia.org/wiki/AM-GM%20Inequality
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In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which case they are both that number).
The simplest non-trivial case – i.e., with more than one variable – for two non-negative numbers and , is the statement that
with equality if and only if .
This case can be seen from the fact that the square of a real number is always non-negative (greater than or equal to zero) and from the elementary case of the binomial formula:
Hence , with equality precisely when , i.e. . The AM–GM inequality then follows from taking the positive square root of both sides and then dividing both sides by 2.
For a geometrical interpretation, consider a rectangle with sides of length and , hence it has perimeter and area . Similarly, a square with all sides of length has the perimeter and the same area as the rectangle. The simplest non-trivial case of the AM–GM inequality implies for the perimeters that and that only the square has the smallest perimeter amongst all rectangles of equal area.
The simplest case is implicit in Euclid's Elements, Book 5, Proposition 25.
Extensions of the AM–GM inequality are available to include weights or generalized means.
Background
The arithmetic mean, or less precisely the average, of a list of numbers is the sum of the numbers divided by :
The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division:
If , this is equal to the exponential of the arithmetic mean of the natural logarithms of the numbers:
The inequality
Restating the inequality using mathematical notation, we have that for any list of nonnegative real numbers ,
and that equality holds if and only if .
Geometric interpretation
In two dimensions, is the perimeter of a rectangle with sides of length and . Similarly, is the perimeter of a square with the same area, , as that rectangle. Thus for the AM–GM inequality states that a rectangle of a given area has the smallest perimeter if that rectangle is also a square.
The full inequality is an extension of this idea to dimensions. Every vertex of an -dimensional box is connected to edges. If these edges' lengths are , then is the total length of edges incident to the vertex. There are vertices, so we multiply this by ; since each edge, however, meets two vertices, every edge is counted twice. Therefore, we divide by and conclude that there are edges. There are equally many edges of each length and lengths; hence there are edges of each length and the total of all edge lengths is . On the other hand,
is the total length of edges connected to a vertex on an -dimensional cube of equal volume, since i
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https://en.wikipedia.org/wiki/Charles%20Ehresmann
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Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory.
He was an early member of the Bourbaki group, and is known for his work on the differential geometry of smooth fiber bundles, notably the introduction of the concepts of Ehresmann connection and of jet bundles, and for his seminar on category theory.
Life
Ehresmann was born in Strasbourg (at the time part of the German Empire) to an Alsatian-speaking family; his father was a gardener. After World War I, Alsace returned part of France and Ehresmann was taught in French at Lycée Kléber.
Between 1924 and 1927 he studied at the École Normale Supérieure (ENS) in Paris and obtained agrégation in mathematics. After one year of military service, in 1928-29 he taught at a French school in Rabat, Morocco. He studied further at the University of Göttingen during the years 1930–31, and at Princeton University in 1932–34.
He completed his PhD thesis entitled Sur la topologie de certains espaces homogènes (On the topology of certain homogeneous spaces) at ENS in 1934 under the supervision of Élie Cartan.
From 1935 to 1939 he was a researcher with the Centre national de la recherche scientifique and he contributed to the seminar of Gaston Julia, which was a forerunner of the Bourbaki seminar. In 1939 Ehresmann became a lecturer at the University of Strasbourg, but one year later the whole faculty was evacuated to Clermont-Ferrand due to the German occupation of France. When Germany withdrew in 1945, he returned to Strasbourg.
From 1955 he was Professor of Topology at Sorbonne, and after the reorganization of Parisian universities in 1969 he moved to Paris Diderot University (Paris 7).
Ehresmann was President of the Société Mathématique de France in 1965. He was awarded in 1940 the Prix Francoeur for young researchers in mathematics and in 1967 an honorary doctorate by the University of Bologna. He also held visiting chairs at Yale University, Princeton University, in Brazil (São Paulo, Rio de Janeiro), Buenos Aires, Mexico City, Montreal, and the Tata Institute of Fundamental Research in Bombay.
After his retirement in 1975 and until 1978 he gave lectures at the University of Picardy at Amiens, where he moved because his second wife, Andrée Charles-Ehresmann, was a professor of mathematics there. He died at Amiens in 1979.
Mathematical work
In the first part of his career Ehresmann introduced many new mathematical objects in differential geometry and topology, which gave rise to entire new fields, often developed later by his students.
In his first works he investigated the topology and homology of manifolds associated with classical Lie groups, such as Grassmann manifolds and other homogeneous spaces.
He developed the concept of fiber bundle, and the related notions of Ehresmann connection and solder form, building on the works by Herbert Seifert and Hassler Whitney in the 1930s. Norman Steenrod was
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https://en.wikipedia.org/wiki/Carlo%20Emilio%20Bonferroni
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Carlo Emilio Bonferroni (28 January 1892 – 18 August 1960) was an Italian mathematician who worked on probability theory.
Biography
Bonferroni studied piano and conducting in Turin Conservatory and at University of Turin under Giuseppe Peano and Corrado Segre, where he obtained his laurea. During this time he also studied at University of Vienna and ETH Zurich. During World War I, he was an officer among the engineers. Bonferroni held a position as assistant professor at the Polytechnic University of Turin, and in 1923 took up the chair of financial mathematics at the Economics Institute of the University of Bari. In 1933 he transferred to University of Florence, where he held his chair until his death.
Bonferroni is best known for the Bonferroni inequalities (a generalization of the union bound), and for the Bonferroni correction in statistics (which he did not invent, but which is related to the Bonferroni inequalities).
See also
Bonferroni inequalities
Bonferroni correction
References
Further reading
Material about Bonferroni by Michael Dewey
1892 births
1960 deaths
20th-century Italian mathematicians
Probability theorists
Italian statisticians
University of Turin alumni
Academic staff of the Polytechnic University of Turin
Academic staff of the University of Bari
Academic staff of the University of Florence
People from Bergamo
Econometricians
Turin Conservatory alumni
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https://en.wikipedia.org/wiki/Richard%20Brauer
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Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular representation theory.
Education and career
Alfred Brauer was Richard's brother and seven years older. They were born to a Jewish family. Both were interested in science and mathematics, but Alfred was injured in combat in World War I. As a boy, Richard dreamt of becoming an inventor, and in February 1919 enrolled in Technische Hochschule Berlin-Charlottenburg. He soon transferred to University of Berlin. Except for the summer of 1920 when he studied at University of Freiburg, he studied in Berlin, being awarded his PhD on 16 March 1926. Issai Schur conducted a seminar and posed a problem in 1921 that Alfred and Richard worked on together, and published a result. The problem also was solved by Heinz Hopf at the same time. Richard wrote his thesis under Schur, providing an algebraic approach to irreducible, continuous, finite-dimensional representations of real orthogonal (rotation) groups.
Ilse Karger also studied mathematics at the University of Berlin; she and Brauer were married 17 September 1925. Their sons George Ulrich (born 1927) and Fred Gunther (born 1932) also became mathematicians. Brauer began his teaching career in Königsberg (now Kaliningrad) working as Konrad Knopp’s assistant. Brauer expounded central division algebras over a perfect field while in Königsberg; the isomorphism classes of such algebras form the elements of the Brauer group he introduced.
When the Nazi Party took over in 1933, the Emergency Committee in Aid of Displaced Foreign Scholars took action to help Brauer and other Jewish scientists. Brauer was offered an assistant professorship at University of Kentucky. Brauer accepted the offer, and by the end of 1933 he was in Lexington, Kentucky, teaching in English. Ilse followed the next year with George and Fred; brother Alfred made it to the United States in 1939, but their sister Alice was killed in the Holocaust.
Hermann Weyl invited Brauer to assist him at Princeton's Institute for Advanced Study in 1934. Brauer and Nathan Jacobson edited Weyl's lectures Structure and Representation of Continuous Groups. Through the influence of Emmy Noether, Brauer was invited to University of Toronto to take up a faculty position. With his graduate student Cecil J. Nesbitt he developed modular representation theory, published in 1937. Robert Steinberg, Stephen Arthur Jennings, and Ralph Stanton were also Brauer’s students in Toronto. Brauer also conducted international research with Tadasi Nakayama on representations of algebras. In 1941 University of Wisconsin hosted visiting professor Brauer. The following year he visited the Institute for Advanced Study and Bloomington, Indiana where Emil Artin was teaching.
In 1948, Brauer moved to Ann Arbor, Michigan where he and Robert M. Thrall contributed t
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https://en.wikipedia.org/wiki/Ernesto%20Ces%C3%A0ro
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Ernesto Cesàro (12 March 1859 – 12 September 1906) was an Italian mathematician who worked in the field of differential geometry. He wrote a book, Lezioni di geometria intrinseca (Naples, 1890), on this topic, in which he also describes fractal, space-filling curves, partly covered by the larger class of de Rham curves, but are still known today in his honor as Cesàro curves. He is known also for his 'averaging' method for the 'Cesàro-summation' of divergent series, known as the Cesàro mean.
Biography
After a rather disappointing start of his academic career and a journey through Europe - with the most important stop at Liège, where his older brother Giuseppe Raimondo Pio Cesàro was teaching mineralogy at the local university - Ernesto Cesàro graduated from the University of Rome in 1887, while he was already part of the Royal Science Society of Belgium for the numerous works that he had already published.
The following year, he obtained a mathematics chair at the University of Palermo, which he kept until 1891. He settled in Rome, where he stayed as a professor at the Sapienza University until his accidental death, while trying to rescue his youngest son Manlio from drowning.
Work
Cesàro's main contributions are in the field of differential geometry. Lessons of intrinsic geometry, written in 1894, explains in particular the construction of a fractal curve. After that, Cesàro also studied the "snowflake curve" of von Koch, continuous but not differentiable in any of its points.
Among his other works are Introduction to the mathematical theory of infinitesimal calculus (1893), Algebraic analysis (1894), Elements of infinitesimal calculus (1897). He proposed a possible definition of a limit of divergent sequence, known today as "Cesàro's sum," given by the limit of the mean of the sequence partial terms' sum.
Books by E. Cesàro
Lezioni di geometria intrinseca (Naples, 1896) (trans. into German under the title Vorlesungen über natürliche Geometrie; 1901, 1st edn.; 1926, 2nd edn. trans. and with an appendix by Gerhard Kowalewski)
Elementi di calcolo infinitesimale con numerose applicazioni geometriche (L. Alvano, Naples, 1905)
Corso di analisi algebrica con introduzione al calcolo infinitesimale (Bocca, Torino, 1894)
See also
Stolz–Cesàro theorem
Cesàro's theorem
Cesàro equation
Cesàro mean
Cesàro summation
Cesàro curve
Lévy C curve
Notes
External links
1859 births
1906 deaths
Scientists from Naples
19th-century Italian mathematicians
20th-century Italian mathematicians
Differential geometers
Deaths by drowning
Academic staff of the Sapienza University of Rome
University of Liège alumni
Sapienza University of Rome alumni
Academic staff of the University of Palermo
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https://en.wikipedia.org/wiki/Elwin%20Bruno%20Christoffel
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Elwin Bruno Christoffel (; 10 November 1829 – 15 March 1900) was a German mathematician and physicist. He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later provide the mathematical basis for general relativity.
Life
Christoffel was born on 10 November 1829 in Montjoie (now Monschau) in Prussia in a family of cloth merchants. He was initially educated at home in languages and mathematics, then attended the Jesuit Gymnasium and the Friedrich-Wilhelms Gymnasium in Cologne. In 1850 he went to the University of Berlin, where he studied mathematics with Gustav Dirichlet (which had a strong influence over him) among others, as well as attending courses in physics and chemistry. He received his doctorate in Berlin in 1856 for a thesis on the motion of electricity in homogeneous bodies written under the supervision of Martin Ohm, Ernst Kummer and Heinrich Gustav Magnus.
After receiving his doctorate, Christoffel returned to Montjoie where he spent the following three years in isolation from the academic community. However, he continued to study mathematics (especially mathematical physics) from books by Bernhard Riemann, Dirichlet and Augustin-Louis Cauchy. He also continued his research, publishing two papers in differential geometry.
In 1859 Christoffel returned to Berlin, earning his habilitation and becoming a Privatdozent at the University of Berlin. In 1862 he was appointed to a chair at the Polytechnic School in Zürich left vacant by Dedekind. He organised a new institute of mathematics at the young institution (it had been established only seven years earlier) that was highly appreciated. He also continued to publish research, and in 1868 he was elected a corresponding member of the Prussian Academy of Sciences and of the Istituto Lombardo in Milan. In 1869 Christoffel returned to Berlin as a professor at the Gewerbeakademie (now part of the Technical University of Berlin), with Hermann Schwarz succeeding him in Zürich. However, strong competition from the close proximity to the University of Berlin meant that the Gewerbeakademie could not attract enough students to sustain advanced mathematical courses and Christoffel left Berlin again after three years.
In 1872 Christoffel became a professor at the University of Strasbourg, a centuries-old institution that was being reorganized into a modern university after Prussia's annexation of Alsace-Lorraine in the Franco-Prussian War. Christoffel, together with his colleague Theodor Reye, built a reputable mathematics department at Strasbourg. He continued to publish research and had several doctoral students including Rikitaro Fujisawa, Ludwig Maurer and Paul Epstein. Christoffel retired from the University of Strasbourg in 1894, being succeeded by Heinrich Weber. After retirement he continued to work and publish, with the last treatise finished just before his death and published posthumously.
Christoffel died on 15
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https://en.wikipedia.org/wiki/Census%20geographic%20units%20of%20Canada
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The census geographic units of Canada are the census subdivisions defined and used by Canada's federal government statistics bureau Statistics Canada to conduct the country's quinquennial census. These areas exist solely for the purposes of statistical analysis and presentation; they have no government of their own. They exist on four levels: the top-level (first-level) divisions are Canada's provinces and territories; these are divided into second-level census divisions, which in turn are divided into third-level census subdivisions (often corresponding to municipalities) and fourth-level dissemination areas.
In some provinces, census divisions correspond to the province's second-level administrative divisions such as a county or another similar unit of political organization. In the prairie provinces, census divisions do not correspond to the province's administrative divisions, but rather group multiple administrative divisions together. In Newfoundland and Labrador, the boundaries are chosen arbitrarily as no such level of government exists. Two of Canada's three territories are also divided into census divisions.
Census divisions
In most cases, a census division corresponds to a single unit of the appropriate type listed above. However, in a few cases, Statistics Canada groups two or more units into a single statistical division:
In Ontario, Haldimand County and Norfolk County are grouped as a single census division, as are Brant and Brantford. Additionally, in 2023 the government of Ontario expected to change a few district boundaries. One of which is Peel Region, which will be split by 2025, although it is unclear if the census division boundary will remain.
In Quebec, 93 of 98 census divisions correspond precisely to the territory of one regional county municipality (with the addition of Indian reserves, which do not legally belong to RCMs) or a "territory equivalent to an RCM" (which usually corresponds to a single independent city). However, there are five census divisions consisting of two or three RCMs or equivalent territories each. See List of census divisions of Quebec.
These are the five census divisions:
The CD of Francheville (37) consists of the TE of Trois-Rivières (371) and the RCM of Les Chenaux (372). These two divisions formerly constituted the single RCM of Francheville, which was dissolved in 2002.
The CD of Le Saguenay-et-son-Fjord (94) consists of the TE of Saguenay (941) and the RCM of Le Fjord-du-Saguenay (942). Le Fjord-du-Saguenay formerly included the smaller cities which were amalgamated to create the new city of Saguenay in 2002.
The CD of Sept-Rivières—Caniapiscau (97) consists of the RCMs of Sept-Rivières (971) and Caniapiscau (972).
The CD of Minganie–Le Golfe-du-Saint-Laurent (98) consists of the RCMs of Minganie (981) and Le Golfe-du-Saint-Laurent (982). The latter superseded Basse-Côte-Nord in 2010; Basse-Côte-Nord itself was part of Minganie until 2002.
The CD of Nord-du-Québec (99) is coexten
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https://en.wikipedia.org/wiki/Kronecker%27s%20theorem
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In mathematics, Kronecker's theorem is a theorem about diophantine approximation, introduced by .
Kronecker's approximation theorem had been firstly proved by L. Kronecker in the end of the 19th century. It has been now revealed to relate to the idea of n-torus and Mahler measure since the later half of the 20th century. In terms of physical systems, it has the consequence that planets in circular orbits moving uniformly around a star will, over time, assume all alignments, unless there is an exact dependency between their orbital periods.
Statement
Kronecker's theorem is a result in diophantine approximations applying to several real numbers xi, for 1 ≤ i ≤ n, that generalises Dirichlet's approximation theorem to multiple variables.
The classical Kronecker approximation theorem is formulated as follows.
Given real n-tuples and , the condition:
holds if and only if for any with
the number is also an integer.
In plainer language the first condition states that the tuple can be approximated arbitrarily well by linear combinations of the s (with integer coefficients) and integer vectors.
For the case of a and , Kronecker's Approximation Theorem can be stated as follows. For any with irrational and there exist integers and with , such that
Relation to tori
In the case of N numbers, taken as a single N-tuple and point P of the torus
T = RN/ZN,
the closure of the subgroup <P> generated by P will be finite, or some torus T′ contained in T. The original Kronecker's theorem (Leopold Kronecker, 1884) stated that the necessary condition for
T′ = T,
which is that the numbers xi together with 1 should be linearly independent over the rational numbers, is also sufficient. Here it is easy to see that if some linear combination of the xi and 1 with non-zero rational number coefficients is zero, then the coefficients may be taken as integers, and a character χ of the group T other than the trivial character takes the value 1 on P. By Pontryagin duality we have T′ contained in the kernel of χ, and therefore not equal to T.
In fact a thorough use of Pontryagin duality here shows that the whole Kronecker theorem describes the closure of <P> as the intersection of the kernels of the χ with
χ(P) = 1.
This gives an (antitone) Galois connection between monogenic closed subgroups of T (those with a single generator, in the topological sense), and sets of characters with kernel containing a given point. Not all closed subgroups occur as monogenic; for example a subgroup that has a
torus of dimension ≥ 1 as connected component of the identity element, and that is not connected, cannot be such a subgroup.
The theorem leaves open the question of how well (uniformly) the multiples mP of P fill up the closure. In the one-dimensional case, the distribution is uniform by the equidistribution theorem.
See also
Weyl's criterion
Dirichlet's approximation theorem
References
Diophantine approximation
Topological groups
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https://en.wikipedia.org/wiki/Dienes
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Dienes may refer to:
Dienes (surname), including a list of people with the name
the plural of diene, a class of organic chemical compound
Base ten blocks used in mathematics education, also known as Dienes blocks or simply dienes
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https://en.wikipedia.org/wiki/Lathe%20%28graphics%29
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In 3D computer graphics, a lathed object is a 3D model whose vertex geometry is produced by rotating the points of a spline or other point set around a fixed axis. The lathing may be partial; the amount of rotation is not necessarily a full 360 degrees. The point set providing the initial source data can be thought of as a cross section through the object along a plane containing its axis of radial symmetry.
The reason the lathe has this name is because it creates symmetrical objects around a rotational axis, just like a real lathe would.
Lathes are very similar to surfaces of revolution. However, lathes are constructed by rotating a curve defined by a set of points instead of a function. Note that this means that lathes can be constructed by rotating closed curves or curves that double back on themselves (such as the aforementioned torus), whereas a surface of revolution could not because such curves cannot be described by functions.
See also
Surface of revolution
Solid of revolution
Loft (3D)
Computer-aided design
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https://en.wikipedia.org/wiki/Tits%20group
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In group theory,
the Tits group 2F4(2)′, named for Jacques Tits (), is a finite simple group of order
211 · 33 · 52 · 13 = 17,971,200.
This is the only simple group that is a derivative of a group of Lie type that is not strictly a group of Lie type in any series due to exceptional isomorphism. It is sometimes considered a 27th sporadic group.
History and properties
The Ree groups 2F4(22n+1) were constructed by , who showed that they are simple if n ≥ 1. The first member of this series 2F4(2) is not simple. It was studied by who showed that it is almost simple, its derived subgroup 2F4(2)′ of index 2 being a new simple group, now called the Tits group. The group 2F4(2) is a group of Lie type and has a BN pair, but the Tits group itself does not have a BN pair. The Tits group is member of the infinite family 2F4(22n+1)′ of commutator groups of the Ree groups, and thus by definition not sporadic. But because it is also not strictly a group of Lie type, it is sometimes regarded as a 27th sporadic group.
The Schur multiplier of the Tits group is trivial and its outer automorphism group has order 2, with the full automorphism group being the group 2F4(2).
The Tits group occurs as a maximal subgroup of the Fischer group Fi22. The groups 2F4(2) also occurs as a maximal subgroup of the Rudvalis group, as the point stabilizer of the rank-3 permutation action on 4060 = 1 + 1755 + 2304 points.
The Tits group is one of the simple N-groups, and was overlooked in John G. Thompson's first announcement of the classification of simple N-groups, as it had not been discovered at the time. It is also one of the thin finite groups.
The Tits group was characterized in various ways by and .
Maximal subgroups
and independently found the 8 classes of maximal subgroups of the Tits group as follows:
L3(3):2 Two classes, fused by an outer automorphism. These subgroups fix points of rank 4 permutation representations.
2.[28].5.4 Centralizer of an involution.
L2(25)
22.[28].S3
A6.22 (Two classes, fused by an outer automorphism)
52:4A4
Presentation
The Tits group can be defined in terms of generators and relations by
where [a, b] is the commutator a−1b−1ab. It has an outer automorphism obtained by sending (a, b) to (a, b(ba)5b(ba)5).
Notes
References
External links
ATLAS of Group Representations — The Tits Group
Sporadic groups
de:Gruppe vom Lie-Typ#Die Tits-Gruppe
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https://en.wikipedia.org/wiki/National%20Statistical%20Office%20%28Thailand%29
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The National Statistical Office of Thailand (NSO) (; ) is the government of Thailand's official statistics surveyor. It is an agency of the Ministry of Digital Economy and Society (MDES). One of its tasks is a nationwide census conducted every 10 years, the latest in 2010.
Organization
TNSO has two main administrative branches, central and local, and several other administrative units apart from those two branches. The central administration oversees 11 smaller administrative units: Local administration is composed of 76 Provincial Statistical Offices. The NSO has two other administrative bodies: an Administrative Development Group (กลุ่มงานพัฒนาระบบบริหาร) and an Internal Audit Group (กลุ่มตรวจสอบภายใน).
History
The predecessor of the National Statistical Office was established on 1 April 1915, by King Vajiravudh as the Department of Statistical Forecasting (กรมสถิติพยากรณ์), under the Ministry of Finance. In September 1915, King Vajiravudh expanded the department's responsibilities and renamed it the Department of Commerce and Statistical Forecasting (กรมพาณิชย์และสถิติพยากรณ์), still as part of the Ministry of Finance. The first Statistical Yearbook was published for the year 1916. The Ministry of Commerce was established in August 1920, and regulation over commerce was reassigned to it, with the newly renamed Department of Public Statistical Forecasting (กรมสถิติพยากรณ์สาธารณะ) under the new ministry. However, the statistics department was moved back under the Ministry of Finance in July 1921. In May 1933, the Department of Public Statistical Forecasting was once again reorganized under the Department of Commerce, Ministry of Finance, this time as a division (กอง), a lower administrative unit than a department, with its name changed to the Division of Statistical Forecasting Compilation (กองประมวลสถิติพยากรณ์).
In October 1935, the Division of Statistical Forecasting Compilation was moved under the jurisdiction of the Secretariat of the Cabinet, Office of the Prime Minister (กรมเลขาธิการคณะรัฐมนตรี สำนักนายกรัฐมนตรี). The Statistics Prediction Act of B.E. 2479 (1936) (พระราชบัญญัติการสถิติพยากรณ์ พ.ศ. ๒๔๗๙) reorganized the duties and administration of the division. In May 1942, the division was reassigned to Department of Information under the Ministry of Commerce, but was moved back under the Secretariat of the Cabinet again in May 1943. With increases in the need for and use of official statistics, in February 1950 the Division of Statistical Forecasting Compilation was reassigned to the National Economic Council (สภาเศรษฐกิจแห่งชาติ), which at that time had the status of a subministry (ทบวง), not under any ministry, in order to allow for equal access to statistics-gathering services by all offices of the government. In December 1950, the division was expanded and renamed the Central Statistical Office (สำนักงานสถิติกลาง). The Statistics Act of B.E. 2495 (1952) (พระราชบัญญัติสถิติ พ.ศ. ๒๔๙๕) laid out the expanded authority and duties of
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https://en.wikipedia.org/wiki/Hilbert%27s%20program
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In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early 1920s, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic.
Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of mathematics. In his first theorem, Gödel showed that any consistent system with a computable set of axioms which is capable of expressing arithmetic can never be complete: it is possible to construct a statement that can be shown to be true, but that cannot be derived from the formal rules of the system. In his second theorem, he showed that such a system could not prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger with certainty. This refuted Hilbert's assumption that a finitistic system could be used to prove the consistency of itself, and therefore could not prove everything else.
Statement of Hilbert's program
The main goal of Hilbert's program was to provide secure foundations for all mathematics. In particular, this should include:
A formulation of all mathematics; in other words all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules.
Completeness: a proof that all true mathematical statements can be proved in the formalism.
Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.
Conservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as uncountable sets) can be proved without using ideal objects.
Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.
Gödel's incompleteness theorems
Kurt Gödel showed that most of the goals of Hilbert's program were impossible to achieve, at least if interpreted in the most obvious way. Gödel's second incompleteness theorem shows that any consistent theory powerful enough to encode addition and multiplication of integers cannot prove its own consistency. This presents a challenge to Hilbert's program:
It is not possible to formalize all mathematical true statements within a formal system, as any attempt at such a formalism will omit some true mathematical statements. There is no complete, consistent extension of even Peano arithmetic based on a recursively enumerabl
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https://en.wikipedia.org/wiki/Rhombic%20dodecahedron
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In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Properties
The rhombic dodecahedron is a zonohedron. Its polyhedral dual is the cuboctahedron. The long face-diagonal length is exactly times the short face-diagonal length; thus, the acute angles on each face measure arccos(), or approximately 70.53°.
Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on its set of faces. In elementary terms, this means that for any two faces A and B, there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B.
The rhombic dodecahedron can be viewed as the convex hull of the union of the vertices of a cube and an octahedron. The 6 vertices where 4 rhombi meet correspond to the vertices of the octahedron, while the 8 vertices where 3 rhombi meet correspond to the vertices of the cube.
The rhombic dodecahedron is one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron, and the rhombic triacontahedron.
The rhombic dodecahedron can be used to tessellate three-dimensional space: it can be stacked to fill a space, much like hexagons fill a plane.
This polyhedron in a space-filling tessellation can be seen as the Voronoi tessellation of the face-centered cubic lattice. It is the Brillouin zone of body centered cubic (bcc) crystals. Some minerals such as garnet form a rhombic dodecahedral crystal habit. As Johannes Kepler noted in his 1611 book on snowflakes (Strena seu de Nive Sexangula), honey bees use the geometry of rhombic dodecahedra to form honeycombs from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron. The rhombic dodecahedron also appears in the unit cells of diamond and diamondoids. In these cases, four vertices (alternate threefold ones) are absent, but the chemical bonds lie on the remaining edges.
The graph of the rhombic dodecahedron is nonhamiltonian.
A rhombic dodecahedron can be dissected into 4 obtuse trigonal trapezohedra around its center. These rhombohedra are the cells of a trigonal trapezohedral honeycomb. Analogy: a regular hexagon can be dissected into 3 rhombi around its center. These rhombi are the tiles of a rhombille.
The collections of the Louvre include a die in the shape of a rhombic dodecahedron dating from Ptolemaic Egypt. The faces are inscribed with Greek letters representing the numbers 1 through 12: Α Β Γ Δ Ε Ϛ Z Η Θ Ι ΙΑ ΙΒ. The function of the die is unknown.
Dimensions
Denoting by a the edge length of a rhombic dodecahedron,
the radius of its inscribed sphere (tangent to each of the rhombic dodecahedron's faces) is
(),
the radius of its midsphere is
(),
the radius of the
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https://en.wikipedia.org/wiki/Affine%20connection
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In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. Connections are among the simplest methods of defining differentiation of the sections of vector bundles.
The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity). The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.
On any manifold of positive dimension there are infinitely many affine connections. If the manifold is further endowed with a metric tensor then there is a natural choice of affine connection, called the Levi-Civita connection. The choice of an affine connection is equivalent to prescribing a way of differentiating vector fields which satisfies several reasonable properties (linearity and the Leibniz rule). This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle. A choice of affine connection is also equivalent to a notion of parallel transport, which is a method for transporting tangent vectors along curves. This also defines a parallel transport on the frame bundle. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the affine group or as a principal connection on the frame bundle.
The main invariants of an affine connection are its torsion and its curvature. The torsion measures how closely the Lie bracket of vector fields can be recovered from the affine connection. Affine connections may also be used to define (affine) geodesics on a manifold, generalizing the straight lines of Euclidean space, although the geometry of those straight lines can be very different from usual Euclidean geometry; the main differences are encapsulated in the curvature of the connection.
Motivation and history
A smooth manifold is a mathematical object which looks locally like a smooth deformation of Euclidean space : for example a smooth curve or surface looks locally like a smooth deformation of a line or a plane. Smooth functions and vector fields can be defined on manifolds, just as they can on Euclidean space, and scalar functions on manifolds can be differentiated in a natural way. However, differentiation of vector fields is less straightforward: this is a simple matter in Euclidean space, because the tangent space of based vectors at a point can be identified natu
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https://en.wikipedia.org/wiki/Parallel%20translation
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Parallel translation may refer to:
parallel transport, in mathematics
parallel text, in translation
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https://en.wikipedia.org/wiki/Urban%20areas%20of%20New%20Zealand
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Statistics New Zealand defines urban areas of New Zealand for statistical purposes (they have no administrative or legal basis). The urban areas comprise cities, towns and other conurbations (an aggregation of urban settlements) of a thousand people or more. In combination, the urban areas of the country constitute New Zealand's urban population. As of , the urban population made up % of New Zealand's total population.
The current standard for urban areas is the Statistical Standard for Geographic Areas 2018 (SSGA18), which replaced the New Zealand Standard Areas Classification 1992 (NZSAC92) in 2018.
There are four classes of urban area under SSGA18:
Major urban areas, with a population of 100,000 or more. There are seven major urban areas which combined have a population of (% of the total population).
Large urban areas, with a population of 30,000 to 99,999. There are 13 large urban areas which combined have a population of (% of the total population).
Medium urban areas, with a population of 10,000 to 29,999. There are 23 medium urban areas which combined have a population of (% of the total population).
Small urban areas, with a population of 1,000 to 9,999. There are 152 small urban areas which combined have a population of (% of the total population).
Each urban area consists of one or more level-2 statistical areas (SA2s). Urban areas under SSGA18 do not cross territorial authority boundaries, with one exception (Richmond, which lies in the Tasman District but includes the Daelyn SA2 area from neighbouring Nelson City).
Statistics New Zealand also defines rural settlements with a population of 200 to 999 people or at least 40 dwellings. While these do not fit the standard international definition of an urban population, they serve to distinguish between true rural dwellers and those in rural settlements or towns. There are 402 rural settlements which combined have a population of (% of the total population).
In 2023, Stats NZ updated the 2018 standard for geographical areas with the new NZ Statistical standard for geographic areas 2023. While similar, the new standard has added a new geographical area (SA3), has upgraded Wanaka to a medium urban area, seven rural settlements to small urban areas and has created thirteen new rural settlements.
Statistical Standard for Geographic Areas 2018
The following shows the urban areas as classified under SSGA18 (adjusted according to SSGA23 update).
Major urban areas
Auckland ()
Hamilton ()
Tauranga ()
Lower Hutt ()
Wellington ()
Christchurch ()
Dunedin ()
Large urban areas
Whangārei ()
Hibiscus Coast ()
Rotorua ()
Gisborne ()
Hastings ()
Napier ()
New Plymouth ()
Whanganui ()
Palmerston North ()
Porirua ()
Upper Hutt ()
Nelson ()
Invercargill ()
Medium urban areas
Pukekohe ()
Cambridge ()
Te Awamutu ()
Tokoroa ()
Taupō ()
Whakatāne ()
Havelock North ()
Feilding ()
Levin ()
Waikanae ()
Paraparaumu ()
Masterton ()
Richmond ()
Blenheim ()
Rangiora ()
Kaiapoi ()
Rolleston ()
Ashburto
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https://en.wikipedia.org/wiki/Sunrise%20problem
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The sunrise problem can be expressed as follows: "What is the probability that the sun will rise tomorrow?" The sunrise problem illustrates the difficulty of using probability theory when evaluating the plausibility of statements or beliefs.
According to the Bayesian interpretation of probability, probability theory can be used to evaluate the plausibility of the statement, "The sun will rise tomorrow."
Laplace's approach
The sunrise problem was first introduced in the 18th century by Pierre-Simon Laplace, who treated it by means of his rule of succession. Let p be the long-run frequency of sunrises, i.e., the sun rises on 100 × p% of days. Prior to knowing of any sunrises, one is completely ignorant of the value of p. Laplace represented this prior ignorance by means of a uniform probability distribution on p.
For instance, the probability that p is between 20% and 50% is just 30%. This must not be interpreted to mean that in 30% of all cases, p is between 20% and 50%. Rather, it means that one's state of knowledge (or ignorance) justifies one in being 30% sure that the sun rises between 20% of the time and 50% of the time. Given the value of p, and no other information relevant to the question of whether the sun will rise tomorrow, the probability that the sun will rise tomorrow is p. But we are not "given the value of p". What we are given is the observed data: the sun has risen every day on record. Laplace inferred the number of days by saying that the universe was created about 6000 years ago, based on a young-earth creationist reading of the Bible.
To find the conditional probability distribution of p given the data, one uses Bayes' theorem, which some call the Bayes–Laplace rule. Having found the conditional probability distribution of p given the data, one may then calculate the conditional probability, given the data, that the sun will rise tomorrow. That conditional probability is given by the rule of succession. The plausibility that the sun will rise tomorrow increases with the number of days on which the sun has risen so far. Specifically, assuming p has an a-priori distribution that is uniform over the interval [0,1], and that, given the value of p, the sun independently rises each day with probability p, the desired conditional probability is:
By this formula, if one has observed the sun rising 10000 times previously, the probability it rises the next day is . Expressed as a percentage, this is approximately a chance.
However, Laplace recognized this to be a misapplication of the rule of succession through not taking into account all the prior information available immediately after deriving the result:
E.T. Jaynes noted that Laplace's warning had gone unheeded by workers in the field.
A reference class problem arises: the plausibility inferred will depend on whether we take the past experience of one person, of humanity, or of the earth. A consequence is that each referent would hold different plau
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https://en.wikipedia.org/wiki/Lapponia%20%28book%29
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Lapponia is a book written by Johannes Schefferus (1621 - 1679) in Latin covering a very comprehensive history of Northern Scandinavia topology, environment and Sami living condition, dwelling-places, clothing, gender roles, hunting, child raising, shamanism and pagan religion. It was published in late 1673 and closely followed by English, German, French and Dutch translations. Adapted and abridged versions followed, where only the original chapters on shamanism and religion were preserved, the others being replaced by tales of magic, sorcery, drums and heathenism.
The book uses "Lap" mainly to notice that Samis are still pagan and it is concluded that Lap is a word introduced by the Danish historian Saxo Grammaticus (ca. 1150–1220) to distinguish Sami peoples living near the ocean (coast-fenni) and in the woodland (lapp-fenni).
It was aimed to meet rumors, or as the councillor Magnus De La Gardie saw it, degrading propaganda, from particular German pamphlets claiming the Swedes had used "Sami magic" on European battlefields.
The book was not fully translated into Swedish (as Lappland, Acta Lapponica 8, Uppsala 1956) until 1956. Its references are, however, based on "clergy correspondence", that is, reports made by priests.
A smaller part of the geographical region described in the book is today named Lappland (or Laponia.)
See also
Sápmi (area)
Olaus Sirma, who acted as one of Schefferus' informants.
External links
Johannes Schefferus (1621 - 1679) in the web exhibit «The Northern Lights Route» Tromsø University Library, 1999
1673 books
Sápmi
Sámi history
Swedish books
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https://en.wikipedia.org/wiki/Expression%20%28mathematics%29
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In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.
Many authors distinguish an expression from a formula, the former denoting a mathematical object, and the latter denoting a statement about mathematical objects. For example, is an expression, while is a formula. However, in modern mathematics, and in particular in computer algebra, formulas are viewed as expressions that can be evaluated to true or false, depending on the values that are given to the variables occurring in the expressions. For example takes the value false if is given a value less than –1, and the value true otherwise.
Examples
The use of expressions ranges from the simple:
(linear polynomial)
(quadratic polynomial)
(rational fraction)
to the complex:
Syntax versus semantics
Syntax
An expression is a syntactic construct. It must be well-formed: the allowed operators must have the correct number of inputs in the correct places, the characters that make up these inputs must be valid, have a clear order of operations, etc. Strings of symbols that violate the rules of syntax are not well-formed and are not valid mathematical expressions.
For example, in the usual notation of arithmetic, the expression 1 + 2 × 3 is well-formed, but the following expression is not:
.
Semantics
Semantics is the study of meaning. Formal semantics is about attaching meaning to expressions.
In algebra, an expression may be used to designate a value, which might depend on values assigned to variables occurring in the expression. The determination of this value depends on the semantics attached to the symbols of the expression. The choice of semantics depends on the context of the expression. The same syntactic expression 1 + 2 × 3 can have different values (mathematically 7, but also 9), depending on the order of operations implied by the context (See also Operations § Calculators).
The semantic rules may declare that certain expressions do not designate any value (for instance when they involve division by 0); such expressions are said to have an undefined value, but they are well-formed expressions nonetheless. In general the meaning of expressions is not limited to designating values; for instance, an expression might designate a condition, or an equation that is to be solved, or it can be viewed as an object in its own right that can be manipulated according to certain rules. Certain expressions that designate a value simultaneously express a condition that is assumed to hold, for instance those involving the operator to designate an internal direct sum.
Formal languages and lambda calculus
Formal languages allow formalizing the concept of well-formed expressions.
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https://en.wikipedia.org/wiki/Studentized%20residual
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In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is a form of a Student's t-statistic, with the estimate of error varying between points.
This is an important technique in the detection of outliers. It is among several named in honor of William Sealey Gosset, who wrote under the pseudonym Student. Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing.
Motivation
The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the residuals at different input variable values may differ, even if the variances of the errors at these different input variable values are equal. The issue is the difference between errors and residuals in statistics, particularly the behavior of residuals in regressions.
Consider the simple linear regression model
Given a random sample (Xi, Yi), i = 1, ..., n, each pair (Xi, Yi) satisfies
where the errors , are independent and all have the same variance . The residuals are not the true errors, but estimates, based on the observable data. When the method of least squares is used to estimate and , then the residuals , unlike the errors , cannot be independent since they satisfy the two constraints
and
(Here εi is the ith error, and is the ith residual.)
The residuals, unlike the errors, do not all have the same variance: the variance decreases as the corresponding x-value gets farther from the average x-value. This is not a feature of the data itself, but of the regression better fitting values at the ends of the domain. It is also reflected in the influence functions of various data points on the regression coefficients: endpoints have more influence. This can also be seen because the residuals at endpoints depend greatly on the slope of a fitted line, while the residuals at the middle are relatively insensitive to the slope. The fact that the variances of the residuals differ, even though the variances of the true errors are all equal to each other, is the principal reason for the need for studentization.
It is not simply a matter of the population parameters (mean and standard deviation) being unknown – it is that regressions yield different residual distributions at different data points, unlike point estimators of univariate distributions, which share a common distribution for residuals.
Background
For this simple model, the design matrix is
and the hat matrix H is the matrix of the orthogonal projection onto the column space of the design matrix:
The leverage hii is the ith diagonal entry in the hat matrix. The variance of the ith residual is
In case the design matrix X has only two columns (as in the example above), this is equal to
In the case of an arithmetic mean, the design matrix X has only one column (a vector of ones), and this is simply:
Calculation
Given the definitions above, th
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https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider%20theorem
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In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers.
History
It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
Statement
If a and b are complex algebraic numbers with a ≠ 0, 1, and b not rational, then any value of ab is a transcendental number.
Comments
The values of a and b are not restricted to real numbers; complex numbers are allowed (here complex numbers are not regarded as rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational).
In general, is multivalued, where ln stands for the natural logarithm. This accounts for the phrase "any value of" in the theorem's statement.
An equivalent formulation of the theorem is the following: if α and γ are nonzero algebraic numbers, and we take any non-zero logarithm of α, then is either rational or transcendental. This may be expressed as saying that if , are linearly independent over the rationals, then they are linearly independent over the algebraic numbers. The generalisation of this statement to more general linear forms in logarithms of several algebraic numbers is in the domain of transcendental number theory.
If the restriction that a and b be algebraic is removed, the statement does not remain true in general. For example,
Here, a is , which (as proven by the theorem itself) is transcendental rather than algebraic. Similarly, if and , which is transcendental, then is algebraic. A characterization of the values for a and b which yield a transcendental ab is not known.
Kurt Mahler proved the p-adic analogue of the theorem: if a and b are in Cp, the completion of the algebraic closure of Qp, and they are algebraic over Q, and if and then is either rational or transcendental, where logp is the p-adic logarithm function.
Corollaries
The transcendence of the following numbers follows immediately from the theorem:
Gelfond–Schneider constant and its square root
Gelfond's constant
Applications
The Gelfond–Schneider theorem answers affirmatively Hilbert's seventh problem.
See also
Lindemann–Weierstrass theorem
Baker's theorem; an extension of the result
Schanuel's conjecture; if proven it would imply both the Gelfond–Schneider theorem and the Lindemann–Weierstrass theorem
References
Further reading
External links
A proof of the Gelfond–Schneider theorem
Transcendental numbers
Theorems in number theory
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https://en.wikipedia.org/wiki/Duality%20%28mathematics%29
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In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the dual of is . Such involutions sometimes have fixed points, so that the dual of is itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.
In mathematical contexts, duality has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".
Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.
From a category theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow its dual .
Introductory examples
In the words of Michael Atiyah,
The following list of examples shows the common features of many dualities, but also indicates that the precise meaning of duality may vary from case to case.
Complement of a subset
A simple, maybe the most simple, duality arises from considering subsets of a fixed set . To any subset , the complement consists of all those elements in that are not contained in . It is again a subset of . Taking the complement has the following properties:
Applying it twice gives back the original set, i.e., . This is referred to by saying that the operation of taking the complement is an involution.
An inclusion of sets is turned into an inclusion in the opposite direction .
Given two subsets and of , is contained in if and only if is contained in .
This duality appears in topology as a duality between open and closed subsets of some fixed topological space : a subset of is closed if and only if its complement in is open. Because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of open sets is open, so dually, any intersection of closed sets is closed. The interior of a set is the largest open set contained in it, and the closure of the set is the smallest closed set that contains it. Because of the duality, the complement of the interior of any set is equal to the closure of the complement of .
Dual cone
A duality in geometry is provided by the du
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https://en.wikipedia.org/wiki/Consistency%20%28statistics%29
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In statistics, consistency of procedures, such as computing confidence intervals or conducting hypothesis tests, is a desired property of their behaviour as the number of items in the data set to which they are applied increases indefinitely. In particular, consistency requires that the outcome of the procedure with unlimited data should identify the underlying truth.
Use of the term in statistics derives from Sir Ronald Fisher in 1922.
Use of the terms consistency and consistent in statistics is restricted to cases where essentially the same procedure can be applied to any number of data items. In complicated applications of statistics, there may be several ways in which the number of data items may grow. For example, records for rainfall within an area might increase in three ways: records for additional time periods; records for additional sites with a fixed area; records for extra sites obtained by extending the size of the area. In such cases, the property of consistency may be limited to one or more of the possible ways a sample size can grow.
Estimators
A consistent estimator is one for which, when the estimate is considered as a random variable indexed by the number n of items in the data set, as n increases the estimates converge in probability to the value that the estimator is designed to estimate.
An estimator that has Fisher consistency is one for which, if the estimator were applied to the entire population rather than a sample, the true value of the estimated parameter would be obtained.
Tests
A consistent test is one for which the power of the test for a fixed untrue hypothesis increases to one as the number of data items increases.
Classification
In statistical classification, a consistent classifier is one for which the probability of correct classification, given a training set, approaches, as the size of the training set increases, the best probability theoretically possible if the population distributions were fully known.
Sparsistency
Let be a vector and define the support, , where is the th element of . Let be an estimator for . Then sparsistency is the property that the support of the estimator converges to the true support as the number of samples grows to infinity. More formally, as .
See also
Consistent estimator
Homogeneity (statistics)
Internal consistency
Reliability (statistics)
References
Asymptotic theory (statistics)
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https://en.wikipedia.org/wiki/Cusp%20form
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In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion.
Introduction
A cusp form is distinguished in the case of modular forms for the modular group by the vanishing of the constant coefficient a0 in the Fourier series expansion (see q-expansion)
This Fourier expansion exists as a consequence of the presence in the modular group's action on the upper half-plane via the transformation
For other groups, there may be some translation through several units, in which case the Fourier expansion is in terms of a different parameter. In all cases, though, the limit as q → 0 is the limit in the upper half-plane as the imaginary part of z → ∞. Taking the quotient by the modular group, this limit corresponds to a cusp of a modular curve (in the sense of a point added for compactification). So, the definition amounts to saying that a cusp form is a modular form that vanishes at a cusp. In the case of other groups, there may be several cusps, and the definition becomes a modular form vanishing at all cusps. This may involve several expansions.
Dimension
The dimensions of spaces of cusp forms are, in principle, computable via the Riemann–Roch theorem. For example, the Ramanujan tau function τ(n) arises as the sequence of Fourier coefficients of the cusp form of weight 12 for the modular group, with a1 = 1. The space of such forms has dimension 1, which means this definition is possible; and that accounts for the action of Hecke operators on the space being by scalar multiplication (Mordell's proof of Ramanujan's identities). Explicitly it is the modular discriminant
which represents (up to a normalizing constant) the discriminant of the cubic on the right side of the Weierstrass equation of an elliptic curve; and the 24-th power of the Dedekind eta function. The Fourier coefficients here are written
and called 'Ramanujan's tau function', with the normalization τ(1) = 1.
Related concepts
In the larger picture of automorphic forms, the cusp forms are complementary to Eisenstein series, in a discrete spectrum/continuous spectrum, or discrete series representation/induced representation distinction typical in different parts of spectral theory. That is, Eisenstein series can be 'designed' to take on given values at cusps. There is a large general theory, depending though on the quite intricate theory of parabolic subgroups, and corresponding cuspidal representations.
References
Serre, Jean-Pierre, A Course in Arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, 1978.
Shimura, Goro, An Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, 1994.
Gelbart, Stephen, Automorphic Forms on Adele Groups, Annals of Mathematics Studies, No. 83, Princeton University Press, 1975.
Modular forms
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https://en.wikipedia.org/wiki/Norbert%20Wiener%20Prize%20in%20Applied%20Mathematics
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The Norbert Wiener Prize in Applied Mathematics is a $5000 prize awarded, every three years, for an outstanding contribution to "applied mathematics in the highest and broadest sense." It was endowed in 1967 in honor of Norbert Wiener by MIT's mathematics department and is provided jointly by the American Mathematical Society and Society for Industrial and Applied Mathematics and first issued in 1970. The recipient of the prize has to be a member of one of the awarding societies.
Winners
1970: Richard E. Bellman
1975: Peter D. Lax
1980: Tosio Kato and Gerald B. Whitham
1985: Clifford S. Gardner
1990: Michael Aizenman and Jerrold E. Marsden
1995: Hermann Flaschka and Ciprian Foias
2000: Alexandre J. Chorin and Arthur Winfree
2004: James A. Sethian
2007: Craig Tracy and Harold Widom
2010: David Donoho
2013: Andrew Majda
2016: Constantine M. Dafermos
2019: Marsha Berger and Arkadi Nemirovski
2022: Eitan Tadmor
See also
List of mathematics awards
Prizes named after people
References
External links
AMS webpage for the prize
SIAM webpage for the prize
Awards of the American Mathematical Society
Awards established in 1970
Triennial events
Awards of the Society for Industrial and Applied Mathematics
1970 establishments in the United States
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https://en.wikipedia.org/wiki/Sinc%20function
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In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.
In mathematics, the historical unnormalized sinc function is defined for by
Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(x).
In digital signal processing and information theory, the normalized sinc function is commonly defined for by
In either case, the value at is defined to be the limiting value
for all real (the limit can be proven using the squeeze theorem).
The normalization causes the definite integral of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of ). As a further useful property, the zeros of the normalized sinc function are the nonzero integer values of .
The normalized sinc function is the Fourier transform of the rectangular function with no scaling. It is used in the concept of reconstructing a continuous bandlimited signal from uniformly spaced samples of that signal.
The only difference between the two definitions is in the scaling of the independent variable (the axis) by a factor of . In both cases, the value of the function at the removable singularity at zero is understood to be the limit value 1. The sinc function is then analytic everywhere and hence an entire function.
The function has also been called the cardinal sine or sine cardinal function. The term sinc was introduced by Philip M. Woodward in his 1952 article "Information theory and inverse probability in telecommunication", in which he said that the function "occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own", and his 1953 book Probability and Information Theory, with Applications to Radar.
The function itself was first mathematically derived in this form by Lord Rayleigh in his expression (Rayleigh's Formula) for the zeroth-order spherical Bessel function of the first kind.
Properties
The zero crossings of the unnormalized sinc are at non-zero integer multiples of , while zero crossings of the normalized sinc occur at non-zero integers.
The local maxima and minima of the unnormalized sinc correspond to its intersections with the cosine function. That is, for all points where the derivative of is zero and thus a local extremum is reached. This follows from the derivative of the sinc function:
The first few terms of the infinite series for the coordinate of the -th extremum with positive coordinate are
where
and where odd lead to a local minimum, and even to a local maximum. Because of symmetry around the axis, there exist extrema with coordinates . In addition, there is an absolute maximum at .
The normalized sinc function has a simple representation as the infinite product:
and is related to the gamma function through Euler's reflection formula:
Euler discovered that
and because of the product-to-sum identity
Euler's product ca
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https://en.wikipedia.org/wiki/Localization%20of%20a%20category
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In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent spaces. Calculus of fractions is another name for working in a localized category.
Introduction and motivation
A category C consists of objects and morphisms between these objects. The morphisms reflect relations between the objects. In many situations, it is meaningful to replace C by another category C''' in which certain morphisms are forced to be isomorphisms. This process is called localization.
For example, in the category of R-modules (for some fixed commutative ring R) the multiplication by a fixed element r of R is typically (i.e., unless r is a unit) not an isomorphism:
The category that is most closely related to R-modules, but where this map is an isomorphism turns out to be the category of -modules. Here is the localization of R with respect to the (multiplicatively closed) subset S consisting of all powers of r,
The expression "most closely related" is formalized by two conditions: first, there is a functor
sending any R-module to its localization with respect to S. Moreover, given any category C and any functor
sending the multiplication map by r on any R-module (see above) to an isomorphism of C, there is a unique functor
such that .
Localization of categories
The above examples of localization of R-modules is abstracted in the following definition. In this shape, it applies in many more examples, some of which are sketched below.
Given a category C and some class W of morphisms in C, the localization C[W−1] is another category which is obtained by inverting all the morphisms in W. More formally, it is characterized by a universal property: there is a natural localization functor C → C[W−1] and given another category D, a functor F: C → D factors uniquely over C[W−1] if and only if F sends all arrows in W to isomorphisms.
Thus, the localization of the category is unique up to unique isomorphism of categories, provided that it exists. One construction of the localization is done by declaring that its objects are the same as those in C, but the morphisms are enhanced by adding a formal inverse for each morphism in W. Under suitable hypotheses on W, the morphisms from object X to object Y are given by roofs
(where X' is an arbitrary object of C and f is in the given class W of morphisms), modulo certain equivalence relations. These relations turn the map going in the "wrong" direction into an inverse of f. This "calculus of fractions" can be seen as a generalization of the construction of rational numbers as equivalence classes of pairs of integers.
This procedure, however, in
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https://en.wikipedia.org/wiki/Sobolev%20space
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In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.
Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense.
Motivation
In this section and throughout the article is an open subset of
There are many criteria for smoothness of mathematical functions. The most basic criterion may be that of continuity. A stronger notion of smoothness is that of differentiability (because functions that are differentiable are also continuous) and a yet stronger notion of smoothness is that the derivative also be continuous (these functions are said to be of class — see Differentiability classes). Differentiable functions are important in many areas, and in particular for differential equations. In the twentieth century, however, it was observed that the space (or , etc.) was not exactly the right space to study solutions of differential equations. The Sobolev spaces are the modern replacement for these spaces in which to look for solutions of partial differential equations.
Quantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms. A typical example is measuring the energy of a temperature or velocity distribution by an -norm. It is therefore important to develop a tool for differentiating Lebesgue space functions.
The integration by parts formula yields that for every , where is a natural number, and for all infinitely differentiable functions with compact support
where is a multi-index of order and we are using the notation:
The left-hand side of this equation still makes sense if we only assume to be locally integrable. If there exists a locally integrable function , such that
then we call the weak -th partial derivative of . If there exists a weak -th partial derivative of , then it is uniquely defined almost everywhere, and thus it is uniquely determined as an element of a Lebesgue space. On the other hand, if , then the classical and the weak derivative coincide. Thus, if is a weak -th partial derivative of , we may denote it by .
For example, the function
is not continuous at zero, and not differentiable at −1, 0, or 1. Yet the function
satisfies the definition for being the weak derivative of which then quali
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https://en.wikipedia.org/wiki/Hadamard%27s%20inequality
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In mathematics, Hadamard's inequality (also known as Hadamard's theorem on determinants) is a result first published by Jacques Hadamard in 1893. It is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors. In geometrical terms, when restricted to real numbers, it bounds the volume in Euclidean space of n dimensions marked out by n vectors vi for 1 ≤ i ≤ n in terms of the lengths of these vectors ||vi||.
Specifically, Hadamard's inequality states that if N is the matrix having columns vi, then
If the n vectors are non-zero, equality in Hadamard's inequality is achieved if and only if the vectors are orthogonal.
Alternate forms and corollaries
A corollary is that if the entries of an n by n matrix N are bounded by B, so |Nij|≤B for all i and j, then
In particular, if the entries of N are +1 and −1 only then
In combinatorics, matrices N for which equality holds, i.e. those with orthogonal columns, are called Hadamard matrices.
A positive-semidefinite matrix P can be written as N*N, where N* denotes the conjugate transpose of N (see Decomposition of a semidefinite matrix). Then
So, the determinant of a positive definite matrix is less than or equal to the product of its diagonal entries. Sometimes this is also known as Hadamard's inequality.
Proof
The result is trivial if the matrix N is singular, so assume the columns of N are linearly independent. By dividing each column by its length, it can be seen that the result is equivalent to the special case where each column has length 1, in other words if ei are unit vectors and M is the matrix having the ei as columns then
and equality is achieved if and only if the vectors are an orthogonal set. The general result now follows:
To prove , consider P =M*M and let the eigenvalues of P be λ1, λ2, … λn. Since the length of each column of M is 1, each entry in the diagonal of P is 1, so the trace of P is n. Applying the inequality of arithmetic and geometric means,
so
If there is equality then each of the λi's must all be equal and their sum is n, so they must all be 1. The matrix P is Hermitian, therefore diagonalizable, so it is the identity matrix—in other words the columns of M are an orthonormal set and the columns of N are an orthogonal set. Many other proofs can be found in the literature.
See also
Fischer's inequality
Notes
References
Further reading
Inequalities
Determinants
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https://en.wikipedia.org/wiki/Conchoid
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Conchoid can refer to:
Conchoid (mathematics), an equation of a curve discovered by the mathematician Nicomedes
Conchoidal fracture, a breakage pattern characteristic to certain glasses and crystals
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https://en.wikipedia.org/wiki/Thompson%20groups
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In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted , that were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, F is the most widely studied, and is sometimes referred to as the Thompson group or Thompson's group.
The Thompson groups, and F in particular, have a collection of unusual properties that have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but finitely presented. The groups T and V are (rare) examples of infinite but finitely-presented simple groups. The group F is not simple but its derived subgroup [F,F] is and the quotient of F by its derived subgroup is the free abelian group of rank 2. F is totally ordered, has exponential growth, and does not contain a subgroup isomorphic to the free group of rank 2.
It is conjectured that F is not amenable and hence a further counterexample to the long-standing but recently disproved
von Neumann conjecture for finitely-presented groups: it is known that F is not elementary amenable.
introduced an infinite family of finitely presented simple groups, including Thompson's group V as a special case.
Presentations
A finite presentation of F is given by the following expression:
where [x,y] is the usual group theory commutator, xyx−1y−1.
Although F has a finite presentation with 2 generators and 2 relations,
it is most easily and intuitively described by the infinite presentation:
The two presentations are related by x0=A, xn = A1−nBAn−1 for n>0.
Other representations
The group F also has realizations in terms of operations on ordered rooted binary trees, and as a subgroup of the piecewise linear homeomorphisms of the unit interval that preserve orientation and whose non-differentiable points are dyadic rationals and whose slopes are all powers of 2.
The group F can also be considered as acting on the unit circle by identifying the two endpoints of the unit interval, and the group T is then the group of automorphisms of the unit circle obtained by adding the homeomorphism x→x+1/2 mod 1 to F. On binary trees this corresponds to exchanging the two trees below the root. The group V is obtained from T by adding the discontinuous map that fixes the points of the half-open interval [0,1/2) and exchanges [1/2,3/4) and [3/4,1) in the obvious way. On binary trees this corresponds to exchanging the two trees below the right-hand descendant of the root (if it exists).
The Thompson group F is the group of order-preserving automorphisms of the free Jónsson–Tarski algebra on one generator.
Amenability
The conjecture of Thompson that F is not amenable was further popularized by R. Geoghegan—see also the Cannon–Floyd–Parry article cited in the references below. Its current status is open: E. Shavgulidze published a paper in 2009 in which he claimed to
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https://en.wikipedia.org/wiki/Thompson%20group
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In mathematics, the term Thompson group or Thompson's group can refer to either
The finite Thompson sporadic group Th studied by John G. Thompson
The finite Thompson subgroup of a p-group, the subgroup generated by the abelian subgroups of maximal order.
"Thompson subgroup" can also mean an analogue of the Weyl group used in the classical involution theorem
The infinite Thompson groups F, T and V studied by the logician Richard Thompson.
Outside of mathematics, it may also refer to
Thompson Group Inc.
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https://en.wikipedia.org/wiki/Hans%20Freudenthal
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Hans Freudenthal (17 September 1905 – 13 October 1990) was a Jewish German-born Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education.
Biography
Freudenthal was born in Luckenwalde, Brandenburg, on 17 September 1905, the son of a Jewish teacher. He was interested in both mathematics and literature as a child, and studied mathematics at the University of Berlin beginning in 1923. He met L. E. J. Brouwer in 1927, when Brouwer came to Berlin to give a lecture, and in the same year Freudenthal also visited the University of Paris. He completed his thesis work with Heinz Hopf at Berlin, defended a thesis on the ends of topological groups in 1930, and was officially awarded a degree in October 1931. After defending his thesis in 1930, he moved to Amsterdam to take up a position as assistant to Brouwer. In this pre-war period in Amsterdam, he was promoted to lecturer at the University of Amsterdam, and married his wife, Suus Lutter, a Dutch teacher.
Although he was a German Jew, Freudenthal's position in the Netherlands insulated him from the anti-Jewish laws that had been passed in Germany beginning with the Nazi rise to power in 1933. However, in 1940 the Germans invaded the Netherlands, following which Freudenthal was suspended from duties at the University of Amsterdam by the Nazis. In 1943 Freudenthal was sent to a labor camp in the village of Havelte in the Netherlands, but with the help of his wife (who, as a non-Jew, had not been deported) he escaped in 1944 and went into hiding with his family in occupied Amsterdam. During this period Freudenthal occupied his time in literary pursuits, including winning first prize under a false name in a novel-writing contest.
With the war over, Freudenthal's position at the University of Amsterdam was returned to him, but in 1946 he was given a chair in pure and applied mathematics and foundations of mathematics at Utrecht University, where he remained for the rest of his career. He served as the 8th president of the International Commission on Mathematical Instruction from 1967 to 1970. In 1971 he founded the Institute for the Development of Mathematical Education (IOWO) at Utrecht University, which after his death was renamed the Freudenthal Institute. In 1972 he founded and became editor-in-chief of the journal Geometriae Dedicata. He retired from his professorship in 1975 and from his journal editorship in 1981. He died in Utrecht in 1990, sitting on a bench in a park where he always took a morning walk.
Contributions
In his thesis work, published as a journal article in 1931, Freudenthal introduced the concept of an end of a topological space. Ends are intended to capture the intuitive idea of a direction in which the space extends to infinity, but have a precise mathematical formulation in terms of covers of the space by nested sequences of compact sets. Ends remain of great importance in t
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https://en.wikipedia.org/wiki/Calculus%20of%20constructions
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In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assistants.
Some of its variants include the calculus of inductive constructions (which adds inductive types), the calculus of (co)inductive constructions (which adds coinduction), and the predicative calculus of inductive constructions (which removes some impredicativity).
General traits
The CoC is a higher-order typed lambda calculus, initially developed by Thierry Coquand. It is well known for being at the top of Barendregt's lambda cube. It is possible within CoC to define functions from terms to terms, as well as terms to types, types to types, and types to terms.
The CoC is strongly normalizing, and hence consistent.
Usage
The CoC has been developed alongside the Coq proof assistant. As features were added (or possible liabilities removed) to the theory, they became available in Coq.
Variants of the CoC are used in other proof assistants, such as Matita and Lean.
The basics of the calculus of constructions
The calculus of constructions can be considered an extension of the Curry–Howard isomorphism. The Curry–Howard isomorphism associates a term in the simply typed lambda calculus with each natural-deduction proof in intuitionistic propositional logic. The calculus of constructions extends this isomorphism to proofs in the full intuitionistic predicate calculus, which includes proofs of quantified statements (which we will also call "propositions").
Terms
A term in the calculus of constructions is constructed using the following rules:
is a term (also called type);
is a term (also called prop, the type of all propositions);
Variables () are terms;
If and are terms, then so is ;
If and are terms and is a variable, then the following are also terms:
,
.
In other words, the term syntax, in BNF, is then:
The calculus of constructions has five kinds of objects:
proofs, which are terms whose types are propositions;
propositions, which are also known as small types;
predicates, which are functions that return propositions;
large types, which are the types of predicates ( is an example of a large type);
itself, which is the type of large types.
Judgments
The calculus of constructions allows proving typing judgments:
Which can be read as the implication
If variables have, respectively, types , then term has type .
The valid judgments for the calculus of constructions are derivable from a set of inference rules. In the following, we use to mean a sequence of type assignments
; to mean terms; and to mean either or . We shall write to mean the result of substituting the term for the free variable in the term .
An inference rule is written in the form
which means
If is a valid judgment, the
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https://en.wikipedia.org/wiki/Intransitivity
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In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stronger property of antitransitivity, which describes a relation that is never transitive.
Intransitivity
A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C. Some authors call a relation if it is not transitive, that is, (if the relation in question is named )
This statement is equivalent to
For example, consider the relation R on the integers such that a R b if and only if a is a multiple of b or a divisor of b. This relation is intransitive since, for example, 2 R 6 (2 is a divisor of 6) and 6 R 3 (6 is a multiple of 3), but 2 is neither a multiple nor a divisor of 3. This does not imply that the relation is (see below); for example, 2 R 6, 6 R 12, and 2 R 12 as well.
As another example, in the food chain, wolves feed on deer, and deer feed on grass, but wolves do not feed on grass. Thus, the relation among life forms is intransitive, in this sense.
Another example that does not involve preference loops arises in freemasonry: in some instances lodge A recognizes lodge B, and lodge B recognizes lodge C, but lodge A does not recognize lodge C. Thus the recognition relation among Masonic lodges is intransitive.
Antitransitivity
Often the term is used to refer to the stronger property of antitransitivity.
In the example above, the relation is not transitive, but it still contains some transitivity: for instance, humans feed on rabbits, rabbits feed on carrots, and humans also feed on carrots.
A relation is if this never occurs at all, i.e.
Many authors use the term to mean .
For example, the relation R on the integers, such that a R b if and only if a + b is odd, is intransitive. If a R b and b R c, then either a and c are both odd and b is even, or vice-versa. In either case, a + c is even.
A second example of an antitransitive relation: the defeated relation in knockout tournaments. If player A defeated player B and player B defeated player C, A can have never played C, and therefore, A has not defeated C.
By transposition, each of the following formulas is equivalent to antitransitivity of R:
Properties
An antitransitive relation is always irreflexive.
An antitransitive relation on a set of ≥4 elements is never connex. On a 3-element set, the depicted cycle has both properties.
An irreflexive and left- (or right-) unique relation is always anti-transitive. An example of the former is the mother relation. If A is the mother of B, and B the mother of C, then A cannot be the mother of C.
If a relation R is antitransitive, so is each subset of R.
Cycles
The term is often used when speaking of scenarios in which a relation describes the relative preferences between pairs of options, and weighing several options produces a "loop" of preference:
A is pr
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https://en.wikipedia.org/wiki/Thompson%20sporadic%20group
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In the area of modern algebra known as group theory, the Thompson group Th is a sporadic simple group of order
2153105372131931
= 90745943887872000
≈ 9.
History
Th is one of the 26 sporadic groups and was found by and constructed by . They constructed it as the automorphism group of a certain lattice in the 248-dimensional Lie algebra of E8. It does not preserve the Lie bracket of this lattice, but does preserve the Lie bracket mod 3, so is a subgroup of the Chevalley group E8(3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson group called the Dempwolff group (which unlike the Thompson group is a subgroup of the compact Lie group E8).
Representations
The centralizer of an element of order 3 of type 3C in the Monster group is a product of the Thompson group and a group of order 3, as a result of which the Thompson group acts on a vertex operator algebra over the field with 3 elements. This vertex operator algebra contains the E8 Lie algebra over F3, giving the embedding of Th into E8(3).
The Schur multiplier and the outer automorphism group of the Thompson group are both trivial.
Generalized monstrous moonshine
Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups.
For Th, the relevant McKay-Thompson series is (),
and j(τ) is the j-invariant.
Maximal subgroups
found the 16 conjugacy classes of maximal subgroups of Th as follows:
2+1+8 · A9
25 · L5(2) This is the Dempwolff group
(3 x G2(3)) : 2
(33 × 3+1+2) · 3+1+2 : 2S4
32 · 37 : 2S4
(3 × 34 : 2 · A6) : 2
5+1+2 : 4S4
52 : GL2(5)
72 : (3 × 2S4)
31 : 15
3D4(2) : 3
U3(8) : 6
L2(19)
L3(3)
M10
S5
References
External links
MathWorld: Thompson group
Atlas of Finite Group Representations: Thompson group
Sporadic groups
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https://en.wikipedia.org/wiki/Kandahar%2C%20Saskatchewan
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Kandahar is a hamlet in Rural Municipality of Big Quill No. 308, Saskatchewan, Canada. Listed as a designated place by Statistics Canada, the hamlet had a population of 20 in the Canada 2016 Census. Located on Highway 16 near Wynyard, Saskatchewan, the community was named by Canadian Pacific Railway executives in the late 19th century for a British military victory in Kandahar, Afghanistan. The hamlet is too small to be enumerated on its own, so its population belongs to the Rural Municipality of Big Quill No. 308. It is located near the south shore of Big Quill Lake, the largest inland body of salt water in Canada.
History
Many of the first settlers in the Kandahar district were immigrants from Iceland or of Icelandic descent. A significant number arrived from the Argyle settlement in Manitoba. An Icelandic immigrant, Thorvidur Halldorson (born Þorviður Magnússon), served as the district's first postmaster in 1910. From 1910 to 1913, the spelling of the post office was Candahar. Kandahar became a village in 1913, following a petition from its inhabitants. In 1925, Kandahar was listed as a Canadian Pacific Railway Ltd. Station on the Minnedosa, Saskatoon, Edmonton Section, CPR. Businesses included a printing press, Prentsmiðja A. Helgasonar, run by Andres Helgason (1867-1939), who was a skilled bookbinder and printer.
The one room school house was named Kandahar School District #3333.
Until the 1970s, Kandahar was a thriving town with various stores and attractions, including a popular steak house. However, in the late 1980s the village's only school closed, and the population has steadily decreased since.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Kandahar had a population of 10 living in 6 of its 8 total private dwellings, a change of from its 2016 population of 20. With a land area of , it had a population density of in 2021.
See also
List of communities in Saskatchewan
Hamlets of Saskatchewan
Designated place
References
Big Quill No. 308, Saskatchewan
Designated places in Saskatchewan
Organized hamlets in Saskatchewan
Icelandic settlements in Saskatchewan
Division No. 10, Saskatchewan
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https://en.wikipedia.org/wiki/Hendecagon
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In geometry, a hendecagon (also undecagon or endecagon) or 11-gon is an eleven-sided polygon. (The name hendecagon, from Greek hendeka "eleven" and –gon "corner", is often preferred to the hybrid undecagon, whose first part is formed from Latin undecim "eleven".)
Regular hendecagon
A regular hendecagon is represented by Schläfli symbol {11}.
A regular hendecagon has internal angles of 147.27 degrees (=147 degrees). The area of a regular hendecagon with side length a is given by
As 11 is not a Fermat prime, the regular hendecagon is not constructible with compass and straightedge. Because 11 is not a Pierpont prime, construction of a regular hendecagon is still impossible even with the usage of an angle trisector.
Close approximations to the regular hendecagon can be constructed. For instance, the ancient Greek mathematicians approximated the side length of a hendecagon inscribed in a unit circle as being 14/25 units long.
The hendecagon can be constructed exactly via neusis construction and also via two-fold origami.
Approximate construction
The following construction description is given by T. Drummond from 1800:
"Draw the radius A B, bisect it in C—with an opening of the compasses equal to half the radius, upon A and C as centres describe the arcs C D I and A D—with the distance I D upon I describe the arc D O and draw the line C O, which will be the extent of one side of a hendecagon sufficiently exact for practice."
On a unit circle:
Constructed hendecagon side length
Theoretical hendecagon side length
Absolute error – if is 10 m then this error is approximately 2.3 mm.
Symmetry
The regular hendecagon has Dih11 symmetry, order 22. Since 11 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z11, and Z1.
These 4 symmetries can be seen in 4 distinct symmetries on the hendecagon. John Conway labels these by a letter and group order. Full symmetry of the regular form is r22 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g11 subgroup has no degrees of freedom but can seen as directed edges.
Use in coinage
The Canadian dollar coin, the loonie, is similar to, but not exactly, a regular hendecagonal prism, as are the Indian 2-rupee coin and several other lesser-used coins of other nations. The cross-section of a loonie is actually a Reuleaux hendecagon. The United States Susan B. Anthony dollar has a hendecagonal outline along the inside of its edges.
Related figures
The hendecagon shares the same set of 11 vertices with four regular hendecagrams:
See also
10-simplex - can be seen as a complete graph in a regular hende
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https://en.wikipedia.org/wiki/B%C3%A9la%20Bollob%C3%A1s
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Béla Bollobás FRS (born 3 August 1943) is a Hungarian-born British mathematician who has worked in various areas of mathematics, including functional analysis, combinatorics, graph theory, and percolation. He was strongly influenced by Paul Erdős since the age of 14.
Early life and education
As a student, he took part in the first three International Mathematical Olympiads, winning two gold medals. Paul Erdős invited Bollobás to lunch after hearing about his victories, and they kept in touch afterward. Bollobás' first publication was a joint publication with Erdős on extremal problems in graph theory, written when he was in high school in 1962.
With Erdős's recommendation to Harold Davenport and a long struggle for permission from the Hungarian authorities, Bollobás was able to spend an undergraduate year in Cambridge, England. However, the authorities denied his request to return to Cambridge for doctoral study. A similar scholarship offer from Paris was also quashed. He wrote his first doctorate in discrete geometry under the supervision of László Fejes Tóth and Paul Erdős in Budapest University, 1967, after which he spent a year in Moscow with Israïl Moiseevich Gelfand. After spending a year at Christ Church, Oxford, where Michael Atiyah held the Savilian Chair of Geometry, he vowed never to return to Hungary due to his disillusion with the 1956 Soviet intervention. He then went to Trinity College, Cambridge, where in 1972 he received a second PhD in functional analysis, studying Banach algebras under the supervision of Frank Adams. Bollobás recalled, "By then, I said to myself, 'If I ever manage to leave Hungary, I won't return.'" In 1970, he was awarded a fellowship to the college.
His main area of research is combinatorics, particularly graph theory. His chief interests are in extremal graph theory and random graph theory. In 1996 he resigned his university post, but remained a Fellow of Trinity College, Cambridge.
Career
Bollobás has been a Fellow of Trinity College, Cambridge, since 1970; in 1996 he was appointed to the Jabie Hardin Chair of Excellence at the University of Memphis, and in 2005 he was awarded a senior research fellowship at Trinity College.
Bollobás has proved results on extremal graph theory, functional analysis, the theory of random graphs, graph polynomials and percolation. For example, with Paul Erdős he proved results about the structure of dense graphs; he was the first to prove detailed results about the phase transition in the evolution of random graphs; he proved that the chromatic number of the random graph on n vertices is asymptotically n/2 log n; with Imre Leader he proved basic discrete isoperimetric inequalities; with Richard Arratia and Gregory Sorkin he constructed the interlace polynomial; with Oliver Riordan he introduced the ribbon polynomial (now called the Bollobás–Riordan polynomial); with Andrew Thomason, József Balogh, Miklós Simonovits, Robert Morris and Noga Alon he studied monotone and here
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https://en.wikipedia.org/wiki/Ralph%20Faudree
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Ralph Jasper Faudree (August 23, 1939 – January 13, 2015) was a mathematician, a professor of mathematics and the former provost of the University of Memphis.
Faudree was born in Durant, Oklahoma. He did his undergraduate studies at Oklahoma Baptist University, graduating in 1961, and received his Ph.D. in 1964 from Purdue University under the supervision of Eugene Schenkman (1922–1977). Faudree was an instructor at the University of California, Berkeley and an assistant professor at the University of Illinois before joining the Memphis State University faculty as an associate professor in 1971. Memphis State became renamed as the University of Memphis in 1994, and Faudree was appointed as provost in 2001.
Faudree specialized in combinatorics, and specifically in graph theory and Ramsey theory. He published more than 200 mathematical papers on these topics together with such notable mathematicians as Béla Bollobás, Stefan Burr, Paul Erdős, Ron Gould, András Gyárfás, Brendan McKay, Cecil Rousseau, Richard Schelp, Miklós Simonovits, Joel Spencer, and Vera Sós. He was the 2005 recipient of the Euler Medal for his contributions to combinatorics. His Erdős number was 1: he cowrote 50 joint papers with Paul Erdős beginning in 1976 and was among the three mathematicians who most frequently co-authored with Erdős.
Selected publications
References
External links
Archived version of the professional webpage
1939 births
2015 deaths
20th-century American mathematicians
21st-century American mathematicians
Graph theorists
Oklahoma Baptist University alumni
Purdue University alumni
University of California, Berkeley faculty
University of Illinois faculty
University of Memphis faculty
People from Durant, Oklahoma
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https://en.wikipedia.org/wiki/Alfr%C3%A9d%20R%C3%A9nyi
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Alfréd Rényi (20 March 1921 – 1 February 1970) was a Hungarian mathematician known for his work in probability theory, though he also made contributions in combinatorics, graph theory, and number theory.
Life
Rényi was born in Budapest to Artúr Rényi and Borbála Alexander; his father was a mechanical engineer, while his mother was the daughter of philosopher and literary critic Bernhard Alexander; his uncle was Franz Alexander, a Hungarian-American psychoanalyst and physician.
He was prevented from enrolling in university in 1939 due to the anti-Jewish laws then in force, but enrolled at the University of Budapest in 1940 and finished his studies in 1944. At this point, he was drafted to forced labour service, from which he managed to escape during transportation of his company. He was in hiding with false documents for six months. Biographers tell an incredible story about Rényi: after half of a year in hiding, he managed to get hold of a soldier’s uniform and march his parents out of the Budapest Ghetto, where they were captive. That mission required enormous courage and planning skills.
Rényi then completed his PhD in 1947 at the University of Szeged, under the advisement of Frigyes Riesz. He did his postgraduate in Moscow and Leningrad, where he collaborated with a prominent Soviet mathematician Yuri Linnik.
Rényi married Katalin Schulhof (who used Kató Rényi as her married name), herself a mathematician, in 1946; their daughter Zsuzsanna was born in 1948. After a brief assistant professorship at Budapest, he was appointed Professor Extraordinary at the University of Debrecen in 1949. In 1950, he founded the Mathematics Research Institute of the Hungarian Academy of Sciences, now bearing his name, and directed it until his early death. He also headed the Department of Probability and Mathematical Statistics of the Eötvös Loránd University, from 1952. He was elected a corresponding member (1949), then full member (1956), of the Hungarian Academy of Sciences.
Work
Rényi proved, using the large sieve, that there is a number such that every even number is the sum of a prime number and a number that can be written as the product of at most primes. Chen's theorem, a strengthening of this result, shows that the theorem is true for K = 2, for all sufficiently large even numbers. The case K = 1 is the still-unproven Goldbach conjecture.
In information theory, he introduced the spectrum of Rényi entropies of order α, giving an important generalisation of the Shannon entropy and the Kullback–Leibler divergence. The Rényi entropies give a spectrum of useful diversity indices, and lead to a spectrum of fractal dimensions. The Rényi–Ulam game is a guessing game where some of the answers may be wrong.
In probability theory, he is also known for his parking constants, which characterize the solution to the following problem: given a street of some length and cars of unit length parking on a random free position on the street, what is the mean densi
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https://en.wikipedia.org/wiki/Multivariable%20calculus
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Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one.
Multivariable calculus may be thought of as an elementary part of advanced calculus. For advanced calculus, see calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus.
Typical operations
Limits and continuity
A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions. For example, there are scalar functions of two variables with points in their domain which give different limits when approached along different paths. E.g., the function.
approaches zero whenever the point is approached along lines through the origin (). However, when the origin is approached along a parabola , the function value has a limit of . Since taking different paths toward the same point yields different limit values, a general limit does not exist there.
Continuity in each argument not being sufficient for multivariate continuity can also be seen from the following example. In particular, for a real-valued function with two real-valued parameters, , continuity of in for fixed and continuity of in for fixed does not imply continuity of .
Consider
It is easy to verify that this function is zero by definition on the boundary and outside of the quadrangle . Furthermore, the functions defined for constant and and by
and
are continuous. Specifically,
for all and .
However, the sequence (for natural ) converges to , rendering the function as discontinuous at . Approaching the origin not along parallels to the - and -axis reveals this discontinuity.
Continuity of function composition
If is continuous at and is a single variable function continuous at then the composite function defined by is continuous at
For examples, and
Properties of continuous functions
If and are both continuous at then
(i) are continuous at
(ii) is continuous at for any constant .
(iii) is continuous at point
(iv) is continuous at if
(v) is continuous at
Partial differentiation
The partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a multivariable function is a derivative with respect to one variable with all other variables held constant.
Partial derivatives may be combined in interesting ways to create more complicated expressions of the derivative. In vector calculus, the del operator () is used to define the concepts of gradient, divergence, and curl in terms of partial derivatives. A matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension. The derivative can thus be understood as
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https://en.wikipedia.org/wiki/Seriation
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Seriation is a way of situating an object within a series. It may refer to:
Seriation (archaeology)
Seriation (semiotics)
Seriation (statistics)
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https://en.wikipedia.org/wiki/Residual%20value
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Residual value is one of the constituents of a leasing calculus or operation. It describes the future value of a good in terms of absolute value in monetary terms, and it is sometimes abbreviated into a percentage of the initial price when the item was new.
Example: A car is sold at a list price of $20,000 today. After a usage of 36 months and 50,000 miles (ca. 80,467 km) its value is contractually defined as $10,000 or 50%. The credited amount, on which the interest is applied, thus is $20,000 present value minus the present value of $10,000 future value.
Residual values are contractually dealt with either in terms of closed contracts or open contracts.
In accounting, residual value is another name for salvage value, the remaining value of an asset after it has been fully depreciated, or after deteriorating beyond further use.
The residual value derives its calculation from a base price, calculated after depreciation.
Residual values are calculated using a number of factors, generally a vehicles market value for the term and mileage required is the start point for the calculation, followed by seasonality, monthly adjustment, lifecycle, and disposal performance. The leasing company setting the residual values (RVs) will use their own historical information to insert the adjustment factors within the calculation to set the end value being the residual value.
In accounting, the residual value could be defined as an estimated amount that an entity can obtain when disposing of an asset after its useful life has ended. When doing this, the estimated costs of disposing of the asset should be deducted.
The formula to calculate the residual value can be seen with the next example as follows:
A company owns a machine which was bought for €20,000. This machine has a useful life of five years, which has just ended. The company knows that if it sells the machine now, it will be able to recover 10% of the price of acquisition.
Therefore, the residual value would be:
Business economics
Leasing
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https://en.wikipedia.org/wiki/Andrey%20Tikhonov%20%28mathematician%29
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Andrey Nikolayevich Tikhonov (; 17 October 1906 – 7 October 1993) was a leading Soviet Russian mathematician and geophysicist known for important contributions to topology, functional analysis, mathematical physics, and ill-posed problems. He was also one of the inventors of the magnetotellurics method in geophysics. Other transliterations of his surname include "Tychonoff", "Tychonov", "Tihonov", "Tichonov."
Biography
Born in Gzhatsk, he studied at the Moscow State University where he received a Ph.D. in 1927 under the direction of Pavel Sergeevich Alexandrov. In 1933 he was appointed as a professor at Moscow State University. He became a corresponding member of the USSR Academy of Sciences on 29 January 1939 and a full member of the USSR Academy of Sciences on 1 July 1966.
Research work
Tikhonov worked in a number of different fields in mathematics. He made important contributions to topology, functional analysis, mathematical physics, and certain classes of ill-posed problems. Tikhonov regularization, one of the most widely used methods to solve ill-posed inverse problems, is named in his honor. He is best known for his work on topology, including the metrization theorem he proved in 1926, and the Tychonoff's theorem, which states that every product of arbitrarily many compact topological spaces is again compact. In his honor, completely regular topological spaces are also named Tychonoff spaces.
In mathematical physics, he proved the fundamental uniqueness theorems for the heat equation and studied Volterra integral equations.
He founded the theory of asymptotic analysis for differential equations with small parameter in the leading derivative.
Organizer work
Tikhonov played the leading role in founding the Faculty of Computational Mathematics and Cybernetics of Moscow State University and served as its first dean during the period of 1970–1990.
Awards
Tikhonov received numerous honors and awards for his work, including the Lenin Prize (1966) and the Hero of Socialist Labor (1954, 1986).
Publications
Books
A.G. Sveshnikov, A.N. Tikhonov, The Theory of Functions of a Complex Variable, Mir Publishers, English translation, 1978.
A.N. Tikhonov, V.Y. Arsenin, Solutions of Ill-Posed Problems, Winston, New York, 1977. .
A.N. Tikhonov, A.V. Goncharsky, Ill-posed Problems in the Natural Sciences, Oxford University Press, Oxford, 1987. .
A.N. Tikhonov, A.V. Goncharsky, V.V. Stepanov, A.G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer, Dordrecht, 1995. .
A.N. Tikhonov, A.S. Leonov, A.G. Yagola. Nonlinear Ill-Posed Problems, Chapman and Hall, London, Weinheim, New York, Tokyo, Melbourne, Madras, V. 1–2, 1998. .
Papers
See also
Regularization
Stone–Čech compactification
Tikhonov cube
Tikhonov distribution
Tikhonov plank
Tikhonov space
Tikhonov's theorem on dynamical systems
References
External links
1906 births
1993 deaths
Soviet mathematicians
Topologists
Full Members of the USSR Academy of Sciences
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https://en.wikipedia.org/wiki/Nicolaus%20Tideman
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Thorwald Nicolaus Tideman (, not ; born August 11, 1943, in Chicago, Illinois) is a Georgist economist and professor at Virginia Tech. He received his Bachelor of Arts in economics and mathematics from Reed College in 1965 and his PhD in economics from the University of Chicago in 1969. Tideman was an Assistant Professor of Economics at Harvard University from 1969 to 1973, during which time from 1970 to 1971 he was a Senior Staff Economist for the President's Council of Economic Advisors. Since 1973, he has been at Virginia Tech, with various visiting positions at Harvard Kennedy School (1979-1980), University of Buckingham (1985-1986), and the American Institute for Economic Research (1999-2000).
Research
Tideman's academic interests include taxation of land, voting theory, and political philosophy.
Ranked Pairs
In 1987, he devised the voting system called "ranked pairs" (or the "Tideman method" or simply "RP"), which is a type of Condorcet method. It selects a single winner using votes that express a preference ranking. Ranked pairs can also be used to create a sorted list of winners.
Other research
In 2000, Tideman developed the CPO-STV proportional voting method. Tideman also devised the independence of clones criterion which both of his methods satisfy. He is an associate of the Earth Rights Institute. His book Collective Decisions and Voting: The Potential for Public Choice was published by Ashgate Publishing in November 2006.
References
External links
1943 births
Living people
People from Chicago
Reed College alumni
Economists from Illinois
Virginia Tech faculty
Harvard Kennedy School staff
Academics of the University of Buckingham
Voting theorists
Georgist economists
21st-century American economists
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https://en.wikipedia.org/wiki/IPM
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IPM may refer to:
Organizations
Independence Party of Minnesota, a political party in Minnesota, United States
Institute for Studies in Theoretical Physics and Mathematics, a research institute in Tehran, Iran
Institute of Personnel Management, now the Chartered Institute of Personnel and Development
International Partnership for Microbicides, a non-profit partnership to find a safe and effective microbicide
Science and technology
Imipenem, an antibiotic belonging to the carbapenem class of drugs
Inch per minute, a measure of speed or velocity
Independent-particle model, of nuclear structure (structure of the atomic nucleus)
Integrated power module, a fuse box in an automobile
Interior permanent magnet, the type of motor used in a hybrid electric vehicle
Interior-point method in mathematical programming (optimization)
International prototype metre, a former standard to define the length of a metre
Interplanetary medium, the material which fills the solar system
Intranodal palisaded myofibroblastoma, a rare primary lymph node tumour
Isopropyl myristate, a chemical used in cosmetics and perfumes
InfoPrint Manager, IBM Advanced Function Presentation software
IPM (software), Interactive Policy Making, an online opinion poll management system
Intelligent Power Module, a type of high-performance module designed to drive IGBT devices
Other uses
iPM, a spin-off program of BBC Radio 4's PM
IPM Zmaj, a Serbian company that produces small agricultural machines
Information Processing and Management, academic journal
Inquisition post mortem, an English medieval and post-medieval fiscal record of the death and estate of a tenant-in-chief
Integrated Project Management, a process area in CMMI
Immediate Past Master, the previous Worshipful Master of a masonic lodge
Imperial Porcelain Factory, Saint Petersburg or (IPM)
Indian Police Medal, a medal for gallantry and distinguished service
Integrated pest management, a pest control strategy in agriculture
International Plowing Match, a Canadian agricultural fair
Macao Polytechnic Institute (Instituto Politécnico de Macau)
See also
IIPM (disambiguation)
IPMS (disambiguation)
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https://en.wikipedia.org/wiki/Semicircle
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In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180° (equivalently, radians, or a half-turn). It has only one line of symmetry (reflection symmetry).
In non-technical usage, the term "semicircle" is sometimes used to refer to either a closed curve that also includes the diameter segment from one end of the arc to the other or to the half-disk, which is a two-dimensional geometric region that further includes all the interior points.
By Thales' theorem, any triangle inscribed in a semicircle with a vertex at each of the endpoints of the semicircle and the third vertex elsewhere on the semicircle is a right triangle, with a right angle at the third vertex.
All lines intersecting the semicircle perpendicularly are concurrent at the center of the circle containing the given semicircle.
Uses
A semicircle can be used to construct the arithmetic and geometric means of two lengths using straight-edge and compass. For a semicircle with a diameter of a + b, the length of its radius is the arithmetic mean of a and b (since the radius is half of the diameter).
The geometric mean can be found by dividing the diameter into two segments of lengths a and b, and then connecting their common endpoint to the semicircle with a segment perpendicular to the diameter. The length of the resulting segment is the geometric mean. This can be proven by applying the Pythagorean theorem to three similar right triangles, each having as vertices the point where the perpendicular touches the semicircle and two of the three endpoints of the segments of lengths a and b.
The construction of the geometric mean can be used to transform any rectangle into a square of the same area, a problem called the quadrature of a rectangle. The side length of the square is the geometric mean of the side lengths of the rectangle. More generally, it is used as a lemma in a general method for transforming any polygonal shape into a similar copy of itself with the area of any other given polygonal shape.
Equation
The equation of a semicircle with midpoint on the diameter between its endpoints and which is entirely concave from below is
If it is entirely concave from above, the equation is
Arbelos
An arbelos is a region in the plane bounded by three semicircles connected at the corners, all on the same side of a straight line (the baseline) that contains their diameters.
See also
Amphitheater
Archimedes' twin circles
Archimedes' quadruplets
Salinon
Wigner semicircle distribution
References
External links
Elementary geometry
es:Semicírculo
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https://en.wikipedia.org/wiki/Coset%20enumeration
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In mathematics, coset enumeration is the problem of counting the cosets of a subgroup H of a group G given in terms of a presentation. As a by-product, one obtains a permutation representation for G on the cosets of H. If H has a known finite order, coset enumeration gives the order of G as well.
For small groups it is sometimes possible to perform a coset enumeration by hand. However, for large groups it is time-consuming and error-prone, so it is usually carried out by computer. Coset enumeration is usually considered to be one of the fundamental problems in computational group theory.
The original algorithm for coset enumeration was invented by John Arthur Todd and H. S. M. Coxeter. Various improvements to the original Todd–Coxeter algorithm have been suggested, notably the classical strategies of V. Felsch and HLT (Haselgrove, Leech and Trotter). A practical implementation of these strategies with refinements is available at the ACE website. The Knuth–Bendix algorithm also can perform coset enumeration, and unlike the Todd–Coxeter algorithm, it can sometimes solve the word problem for infinite groups.
The main practical difficulties in producing a coset enumerator are that it is difficult or impossible to predict how much memory or time will be needed to complete the process. If a group is finite, then its coset enumeration must terminate eventually, although it may take arbitrarily long and use an arbitrary amount of memory, even if the group is trivial. Depending on the algorithm used, it may happen that making small changes to the presentation that do not change the group nevertheless have a large impact on the amount of time or memory needed to complete the enumeration. These behaviours are a consequence of the unsolvability of the word problem for groups.
A gentle introduction to coset enumeration is given in Rotman's text on group theory. More detailed information on correctness, efficiency, and practical implementation can be found in the books by Sims and Holt et al.
References
Computational group theory
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https://en.wikipedia.org/wiki/Computational%20group%20theory
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In mathematics, computational group theory is the study of
groups by means of computers. It is concerned
with designing and analysing algorithms and
data structures to compute information about groups. The subject
has attracted interest because for many interesting groups
(including most of the sporadic groups) it is impractical
to perform calculations by hand.
Important algorithms in computational group theory include:
the Schreier–Sims algorithm for finding the order of a permutation group
the Todd–Coxeter algorithm and Knuth–Bendix algorithm for coset enumeration
the product-replacement algorithm for finding random elements of a group
Two important computer algebra systems (CAS) used for group theory are
GAP and Magma. Historically, other systems such as CAS (for character theory) and Cayley (a predecessor of Magma) were important.
Some achievements of the field include:
complete enumeration of all finite groups of order less than 2000
computation of representations for all the sporadic groups
See also
Black box group
References
A survey of the subject by Ákos Seress from Ohio State University, expanded from an article that appeared in the Notices of the American Mathematical Society is available online. There is also a survey by Charles Sims from Rutgers University and an older survey by Joachim Neubüser from RWTH Aachen.
There are three books covering various parts of the subject:
Derek F. Holt, Bettina Eick, Eamonn A. O'Brien, "Handbook of computational group theory", Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, Florida, 2005.
Charles C. Sims, "Computation with Finitely-presented Groups", Encyclopedia of Mathematics and its Applications, vol 48, Cambridge University Press, Cambridge, 1994.
Ákos Seress, "Permutation group algorithms", Cambridge Tracts in Mathematics, vol. 152, Cambridge University Press, Cambridge, 2003. .
Computational fields of study
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https://en.wikipedia.org/wiki/FLT
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FLT may refer to:
Mathematics
Fermat's Last Theorem, in number theory
Fermat's little theorem, using modular arithmetic
Finite Legendre transform, in algebra
Medicine
Alovudine (fluorothymidine), a pharmaceutical drug
Fluorothymidine F-18, a radiolabeled pharmaceutical drug
Places
Finger Lakes Trail, New York, United States
Flitwick railway station, England
Phaeton Airport, Haiti
Organizations
Fairlight (group), a 1980s Commodore warez group
Flight Centre, an Australian travel company (founded 1982; ASX ticker:FLT)
Free Federation of Workers (), a 20th-century Puerto Rican trade union
Liberation Front of Chad (), 1965–1976
Luxembourg Tennis Federation (French: ), a sports governing body (founded 1946)
Other uses
Flutter-tonguing, in music
Foreign Language Teaching, in education
See also
FTL (disambiguation)
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https://en.wikipedia.org/wiki/Deltoid
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Deltoid (delta-shaped) can refer to:
The deltoid muscle, a muscle in the shoulder
Kite (geometry), also known as a deltoid, a type of quadrilateral
A deltoid curve, a three-cusped hypocycloid
A leaf shape
The deltoid tuberosity, a part of the humerus
The deltoid ligament, a ligament in the ankle
See also
Delta (disambiguation)
The Deltoid Pumpkin Seed (1973), a book by John McPhee
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https://en.wikipedia.org/wiki/Non-measurable%20set
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In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Zermelo–Fraenkel set theory, the axiom of choice entails that non-measurable subsets of exist.
The notion of a non-measurable set has been a source of great controversy since its introduction. Historically, this led Borel and Kolmogorov to formulate probability theory on sets which are constrained to be measurable. The measurable sets on the line are iterated countable unions and intersections of intervals (called Borel sets) plus-minus null sets. These sets are rich enough to include every conceivable definition of a set that arises in standard mathematics, but they require a lot of formalism to prove that sets are measurable.
In 1970, Robert M. Solovay constructed the Solovay model, which shows that it is consistent with standard set theory without uncountable choice, that all subsets of the reals are measurable. However, Solovay's result depends on the existence of an inaccessible cardinal, whose existence and consistency cannot be proved within standard set theory.
Historical constructions
The first indication that there might be a problem in defining length for an arbitrary set came from Vitali's theorem. A more recent combinatorial construction which is similar to the construction by Robin Thomas of a non-Lebesgue measurable set with some additional properties appeared in American Mathematical Monthly.
One would expect the measure of the union of two disjoint sets to be the sum of the measure of the two sets. A measure with this natural property is called finitely additive. While a finitely additive measure is sufficient for most intuition of area, and is analogous to Riemann integration, it is considered insufficient for probability, because conventional modern treatments of sequences of events or random variables demand countable additivity.
In this respect, the plane is similar to the line; there is a finitely additive measure, extending Lebesgue measure, which is invariant under all isometries. For higher dimensions the picture gets worse. The Hausdorff paradox and Banach–Tarski paradox show that a three-dimensional ball of radius 1 can be dissected into 5 parts which can be reassembled to form two balls of radius 1.
Example
Consider the set of all points in the unit circle, and the action on by a group consisting of all rational rotations (rotations by angles which are rational multiples of ). Here is countable (more specifically, is isomorphic to ) while is uncountable. Hence breaks up into uncountably many orbits under (the orbit of is the countable set ). Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset with the property that all of the rational translates (translated copies of the form for some rational ) of by are pa
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https://en.wikipedia.org/wiki/Twelfth
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Twelfth can mean:
The Twelfth Amendment to the United States Constitution
The Twelfth, a Protestant celebration originating in Ireland
In mathematics:
12th, an ordinal number; as in the item in an order twelve places from the beginning, following the eleventh and preceding the thirteenth
1/12, a vulgar fraction, one part of a unit divided equally into twelve parts
Music
The note twelve scale degrees from the root (current note, in a chord)
The interval (music) (that is, gap) between the root and the twelfth note: a compound fifth
Currency
Uncia (coin), a Roman coin worth 12th of an As
See also
12 (number)
Eleventh
Thirteenth
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https://en.wikipedia.org/wiki/Landau%E2%80%93Ramanujan%20constant
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In mathematics and the field of number theory, the Landau–Ramanujan constant is the positive real number b that occurs in a theorem proved by Edmund Landau in 1908, stating that for large , the number of positive integers below that are the sum of two square numbers behaves asymptotically as
This constant b was rediscovered in 1913 by Srinivasa Ramanujan, in the first letter he wrote to G.H. Hardy.
Sums of two squares
By the sum of two squares theorem, the numbers that can be expressed as a sum of two squares of integers are the ones for which each prime number congruent to 3 mod 4 appears with an even exponent in their prime factorization. For instance, 45 = 9 + 36 is a sum of two squares; in its prime factorization, 32 × 5, the prime 3 appears with an even exponent, and the prime 5 is congruent to 1 mod 4, so its exponent can be odd.
Landau's theorem states that if is the number of positive integers less than that are the sum of two squares, then
,
where is the Landau–Ramanujan constant.
The Landau-Ramanujan constant can also be written as an infinite product:
History
This constant was stated by Landau in the limit form above; Ramanujan instead approximated as an integral, with the same constant of proportionality, and with a slowly growing error term.
References
Additive number theory
Analytic number theory
Mathematical constants
Srinivasa Ramanujan
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https://en.wikipedia.org/wiki/Eugenio%20Calabi
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Eugenio Calabi (May 11, 1923 – September 25, 2023) was an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, specializing in differential geometry, partial differential equations and their applications.
Early life and education
Calabi was born in Milan, Italy on May 11, 1923, into a Jewish family. His sister was the journalist Tullia Zevi Calabi. In 1938, the family left Italy because of the racial laws, and in 1939 arrived in the United States.
In the fall of 1939, aged only 16, Calabi enrolled at the Massachusetts Institute of Technology, studying chemical engineering. His studies were interrupted when he was drafted in the US military in 1943 and served during World War II. Upon his discharge in 1946, Calabi was able to finish his bachelor's degree under the G.I. Bill, and was a Putnam Fellow. He received a master's degree in mathematics from the University of Illinois Urbana-Champaign in 1947 and his PhD in mathematics from Princeton University in 1950. His doctoral dissertation, titled "Isometric complex analytic imbedding of Kähler manifolds", was done under the supervision of Salomon Bochner.
Academic career
From 1951 to 1955 he was an assistant professor at Louisiana State University, and he moved to the University of Minnesota in 1955, where he become a full professor in 1960. In 1964, Calabi joined the mathematics faculty at the University of Pennsylvania. Following the retirement of Hans Rademacher, he was appointed to the Thomas A. Scott Professorship of Mathematics at the University of Pennsylvania in 1968. In 1994, Calabi assumed emeritus status, and in 2014 the university awarded him an honorary doctorate of science.
In 1982, Calabi was elected to the National Academy of Sciences. He won the Leroy P. Steele Prize from the American Mathematical Society in 1991, where his "fundamental work on global differential geometry, especially complex differential geometry" was cited as having "profoundly changed the landscape of the field". In 2012, he became a fellow of the American Mathematical Society. In 2021, he was awarded Commander of the Order of Merit of the Italian Republic.
Calabi married Giuliana Segre in 1952, with whom he had a son and a daughter. He turned 100 on May 11, 2023, and died on September 25.
Research
Calabi made a number of contributions to the field of differential geometry. Other contributions, not discussed here, include the construction of a holomorphic version of the long line with Maxwell Rosenlicht, a study of the moduli space of space forms, a characterization of when a metric can be found so that a given differential form is harmonic, and various works on affine geometry. In the comments on his collected works in 2021, Calabi cited his article "Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens" as that which he was "most proud of".
Kähler geometry
At the 1954 International Congress of Mathema
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https://en.wikipedia.org/wiki/List%20of%20computer%20graphics%20and%20descriptive%20geometry%20topics
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This is a list of computer graphics and descriptive geometry topics, by article name.
2D computer graphics
2D geometric model
3D computer graphics
3D projection
Alpha compositing
Anisotropic filtering
Anti-aliasing
Axis-aligned bounding box
Axonometric projection
Bézier curve
Bézier surface
Bicubic interpolation
Bilinear interpolation
Binary space partitioning
Bitmap graphics editor
Bounding volume
Bresenham's line algorithm
Bump mapping
Collision detection
Color space
Colour banding
Computational geometry
Computer animation
Computer-generated art
Computer painting
Convex hull
Curvilinear perspective
Cylindrical perspective
Data compression
Digital raster graphic
Dimetric projection
Distance fog
Dithering
Elevation
Engineering drawing
Flat shading
Flood fill
Geometric model
Geometric primitive
Global illumination
Gouraud shading
Graphical projection
Graphics suite
Heightfield
Hidden face removal
Hidden line removal
High-dynamic-range rendering
Isometric projection
Lathe (graphics)
Line drawing algorithm
Linear perspective
Mesh generation
Motion blur
Orthographic projection
Orthographic projection (geometry)
Orthogonal projection
Perspective (graphical)
Phong reflection model
Phong shading
Pixel shaders
Polygon (computer graphics)
Procedural surface
Projection
Projective geometry
Quadtree
Radiosity
Raster graphics
Raytracing
Rendering (computer graphics)
Reverse perspective
Scan line rendering
Scrolling
Technical drawing
Texture mapping
Trimetric projection
Vanishing point
Vector graphics
Vector graphics editor
Vertex shaders
Volume rendering
Voxel
See also
List of geometry topics
List of graphical methods
Computing-related lists
Mathematics-related lists
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https://en.wikipedia.org/wiki/Cartan%27s%20theorems%20A%20and%20B
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In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf on a Stein manifold . They are significant both as applied to several complex variables, and in the general development of sheaf cohomology.
Theorem B is stated in cohomological terms (a formulation that Cartan (1953, p. 51) attributes to J.-P. Serre):
Analogous properties were established by Serre (1957) for coherent sheaves in algebraic geometry, when is an affine scheme. The analogue of Theorem B in this context is as follows :
These theorems have many important applications. For instance, they imply that a holomorphic function on a closed complex submanifold, , of a Stein manifold can be extended to a holomorphic function on all of . At a deeper level, these theorems were used by Jean-Pierre Serre to prove the GAGA theorem.
Theorem B is sharp in the sense that if for all coherent sheaves on a complex manifold (resp. quasi-coherent sheaves on a noetherian scheme ), then is Stein (resp. affine); see (resp. and ).
See also
Cousin problems
References
.
.
.
Several complex variables
Topological methods of algebraic geometry
Theorems in algebraic geometry
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https://en.wikipedia.org/wiki/Geometric%20topology
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In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of simple homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently.
Differences between low-dimensional and high-dimensional topology
Manifolds differ radically in behavior in high and low dimension.
High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2.
Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as exotic differentiable structures on R4. Thus the topological classification of 4-manifolds is in principle tractable, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the generalized Poincaré conjecture; see Gluck twists.
The distinction is because surgery theory works in dimension 5 and above (in fact, in many cases, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above may be studied using the surgery theory program. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work.
Indeed, one approach to discussing low-dimensional manifolds is to ask "what would surgery theory predict to be true, were it to work?" – and then understand low-dimensional phenomena as deviations from this.
The precise reason for the difference at dimension 5 is because the Whitney embedding theorem, the key technical trick which underlies surgery theory, requires 2+1 dimensions. Roughly, the Whitney trick allows one to "unknot" knotted spheres – more precisely, remove self-intersections of immersions; it does this via a homotopy of a disk – the disk has 2 dimensions, and the homotopy adds 1 more – and thus in codimension greater than 2, this can be done without intersecting itself; hence embeddings in codimension greater than 2 can be understood by surgery. In surgery theory, the key step is in the middle dimension, and thus when the middle dimension has codimension more than 2 (loosely, 2½ is enough, hence total dimension 5 is enough), the Whitney trick works. The key consequence of this is Smale's h-cobordism theorem, which works in dimension 5 and above, and forms the basis for
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https://en.wikipedia.org/wiki/Marjorie%20Rice
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Marjorie Ruth Rice (née Jeuck; 1923–2017) was an American amateur mathematician most famous for her discoveries of pentagonal tilings in geometry.
Background
Rice was born February 16, 1923, in St. Petersburg, Florida.
Marjorie Rice was a San Diego mother of five, who had become an ardent follower of Martin Gardner's long-running column, "Mathematical Games", which appeared monthly, 1957–1986, in the pages of Scientific American magazine. By the 1970s, Gardner was a popular science writer and amateur mathematician. Rice said later that she would rush to grab each issue from the mail before anyone else could get it, especially her son who subscribed to the magazine.
In 1975, Rice read Gardner's July column, "On Tessellating the Plane with Convex Polygon Tiles", that discussed what kinds of convex polygons can fit together perfectly without any overlaps or gaps to fill the plane. In his column, Gardner indicated that "the task of finding all convex polygons that tile the plane …. was not completed until 1967 when Richard Brandon Kershner … found three pentagonal tilers that had been missed by all predecessors who had worked on the problem". Gardner was repeating Kershner's claim that the list of convex pentagon tilers was complete. But within a month, Gardner received an example, by one of his readers, Richard James III, of a new convex pentagon tiler, and published this news in his December 1975 column.
Discoveries of pentagonal tilings
Inspired by this new discovery, Rice decided to try to find other new pentagon tilers. Despite having only a high-school education, but a keen interest in art, she began devoting her free time to discovering new pentagonal tilings, ways to tile a plane using pentagons. She worked on the problem in her free time and through the 1975 holiday season "by drawing diagrams on the kitchen table when no one was around and hiding them when her husband and children came home, or when friends stopped by". She even developed her own system of notation to represent the constraints on and relationships between the sides and angles of the pentagons.
By February 1976, she had discovered a new pentagon type and its variations in shape and drew up several tessellations by these pentagon tiles. She mailed her discoveries to Gardner using her own home-made notation. He, in turn, sent Rice's work to Doris Schattschneider, an expert in tiling patterns, who was skeptical at first, saying that Rice's peculiar notation system seemed odd, like "hieroglyphics". But with careful examination, she was able to validate Rice's results.
By October 1976, Rice had discovered 58 pentagon tilings that needed two pentagons stuck together in order to tile "transitively" (most of them previously unknown), which she arranged into 12 classes. By December 1976, she had discovered two additional new types of tessellating pentagons and over 75 distinct tessellations by pentagons that were in blocks that could be seen as "double hexagons". In December
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https://en.wikipedia.org/wiki/List%20of%20amateur%20mathematicians
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This is a list of amateur mathematicians—people whose primary vocation did not involve mathematics (or any similar discipline) yet made notable, and sometimes important, contributions to the field of mathematics.
Ahmes (scribe)
Ashutosh Mukherjee (lawyer)
Robert Ammann (programmer and postal worker)
John Arbuthnot (surgeon and author)
Jean-Robert Argand (shopkeeper)
Leon Bankoff (Beverly Hills dentist)
Rev. Thomas Bayes (Presbyterian minister)
Andrew Beal (businessman)
Isaac Beeckman (candlemaker)
Chester Ittner Bliss (biologist)
Napoléon Bonaparte (general)
Mary Everest Boole (homemaker, librarian)
William Bourne (innkeeper)
Nathaniel Bowditch (indentured bookkeeper)
Achille Brocot (clockmaker)
Jost Bürgi (clockmaker)
Marvin Ray Burns (veteran)
Gerolamo Cardano (medical doctor)
D. G. Champernowne (college student)
Thomas Clausen (technical assistant)
Sir James Cockle (judge)
Federico Commandino (medical doctor)
William Crabtree (merchant)
Nathan Daboll (cooper)
Felix Delastelle (bonded warehouseman)
Martin Demaine (goldsmith and glass artist)
Humphry Ditton (minister)
Harvey Dubner (engineer)
Henry Dudeney (civil servant)
Albrecht Dürer (painter)
Greg Egan (writer)
M. C. Escher (graphic artist)
Eugène Ehrhart (mathematics teacher)
John Ernest (painter)
Pasquale Joseph Federico (patent attorney)
Pierre de Fermat (lawyer)
Sarah Flannery (high school student)
Reo Fortune (anthropologist)
John G.F. Francis (research assistant)
Benjamin Franklin (printer and diplomat)
Bernard Frenicle de Bessy
Gemma Frisius (medical doctor)
Britney Gallivan (high school student)
James Garfield (United States President)
Antoine Gombaud (essayist)
Thorold Gosset (lawyer)
Jørgen Pedersen Gram (actuary)
Hermann Grassmann (school teacher)
John Graunt (haberdasher)
George Green (miller)
Aubrey de Grey (gerontologist)
André-Michel Guerry (lawyer)
Charles James Hargreave (judge)
Oliver Heaviside (telegraph operator)
Kurt Heegner (private scholar)
John R. Hendricks (meteorologist)
Anthony Hill (painter)
Paul Jaccard (botanist)
Alfred Bray Kempe (lawyer)
Thomas Kirkman (church rector)
Laurence Monroe Klauber (herpetologist)
Harry Lindgren (civil servant)
Ada Lovelace (countess)
Lu Jiaxi (high school physics teacher)
Kenneth McIntyre (lawyer)
Danica McKellar (actress)
Anderson Gray McKendrick (medical doctor)
Marin Mersenne (theologian)
Florence Nightingale (nurse and statistician)
George Phillips Odom Jr. (artist)
B. Nicolò I. Paganini (schoolboy)
Pāṇini (linguist)
Blaise Pascal (heir, private scholar)
Padmakumar (technician)
Henry Perigal (stockbroker)
Kenneth Perko (lawyer)
Ivan Pervushin (priest)
Piero della Francesca (painter)
Pingala (musician)
William Playfair (draftsman)
Henry Cabourn Pocklington (schoolmaster)
François Proth (farmer)
Ramchundra (head master)
Marjorie Rice (homemaker)
Olinde Rodrigues (banker, social reformer)
Lee Sallows (engineer)
Robert Schlaifer (classics scholar)
Robert Schneider (musician and record producer)
William Shanks (landlord)
Abraham Sh
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https://en.wikipedia.org/wiki/Low-dimensional%20topology
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In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.
History
A number of advances starting in the 1960s had the effect of emphasising low dimensions in topology. The solution by Stephen Smale, in 1961, of the Poincaré conjecture in five or more dimensions made dimensions three and four seem the hardest; and indeed they required new methods, while the freedom of higher dimensions meant that questions could be reduced to computational methods available in surgery theory. Thurston's geometrization conjecture, formulated in the late 1970s, offered a framework that suggested geometry and topology were closely intertwined in low dimensions, and Thurston's proof of geometrization for Haken manifolds utilized a variety of tools from previously only weakly linked areas of mathematics. Vaughan Jones' discovery of the Jones polynomial in the early 1980s not only led knot theory in new directions but gave rise to still mysterious connections between low-dimensional topology and mathematical physics. In 2002, Grigori Perelman announced a proof of the three-dimensional Poincaré conjecture, using Richard S. Hamilton's Ricci flow, an idea belonging to the field of geometric analysis.
Overall, this progress has led to better integration of the field into the rest of mathematics.
Two dimensions
A surface is a two-dimensional, topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3—for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections.
Classification of surfaces
The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families:
the sphere;
the connected sum of g tori, for ;
the connected sum of k real projective planes, for .
The surfaces in the first two families are orientable. It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number g of tori involved is called the genus of the surface. The sphere and the torus have Euler characteristics 2 and 0, respectively, and in general the Euler characteristic of the connected sum of g tori is .
The surfaces in the third family are nonorientable. The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of k of them is .
Teichmüller space
In mat
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https://en.wikipedia.org/wiki/Kurt%20Heegner
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Kurt Heegner (; 16 December 1893 – 2 February 1965) was a German private scholar from Berlin, who specialized in
radio engineering and mathematics. He is famous for his mathematical discoveries in number theory and, in particular, the Stark–Heegner theorem.
Life and career
Heegner was born and died in Berlin. In 1952, he published the Stark–Heegner theorem which he claimed was the solution to a classic number theory problem proposed by the great mathematician Gauss, the class number 1 problem. Heegner's work was not accepted for years, mainly due to his quoting of a portion of Heinrich Martin Weber's work that was known to be incorrect (though he never used this result in the proof).
Heegner's proof was accepted as essentially correct after a 1967 announcement by Bryan Birch, and definitively resolved by a paper by Harold Stark that had been delayed in publication until 1969 (Stark had independently arrived at a similar proof, but disagrees with the common notion that his proof is "more or less the same" as Heegner's). Stark attributed Heegner's mistakes to the fact he used a textbook by Weber that contained some results with incomplete proofs.
The book The Legacy of Leonhard Euler: A Tricentennial Tribute by Lokenath Debnath claims on page 64, that Heegner was a "retired Swiss mathematician", but he appears to have been neither Swiss nor retired at the time of his 1952 paper.
See also
List of amateur mathematicians
Stark–Heegner theorem
Heegner number
Heegner point
Heegner's lemma
Literature
References
External links
Heegner's entry in "Foundation for German communication and related technologies"
1893 births
1965 deaths
20th-century German mathematicians
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https://en.wikipedia.org/wiki/Homogeneous%20function
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In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the degree; that is, if is an integer, a function of variables is homogeneous of degree if
for every and
For example, a homogeneous polynomial of degree defines a homogeneous function of degree .
The above definition extends to functions whose domain and codomain are vector spaces over a field : a function between two -vector spaces is homogeneous of degree if
for all nonzero and This definition is often further generalized to functions whose domain is not , but a cone in , that is, a subset of such that implies for every nonzero scalar .
In the case of functions of several real variables and real vector spaces, a slightly more general form of homogeneity called positive homogeneity is often considered, by requiring only that the above identities hold for and allowing any real number as a degree of homogeneity. Every homogeneous real function is positively homogeneous. The converse is not true, but is locally true in the sense that (for integer degrees) the two kinds of homogeneity cannot be distinguished by considering the behavior of a function near a given point.
A norm over a real vector space is an example of a positively homogeneous function that is not homogeneous. A special case is the absolute value of real numbers. The quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero. This example is fundamental in the definition of projective schemes.
Definitions
The concept of a homogeneous function was originally introduced for functions of several real variables. With the definition of vector spaces at the end of 19th century, the concept has been naturally extended to functions between vector spaces, since a tuple of variable values can be considered as a coordinate vector. It is this more general point of view that is described in this article.
There are two commonly used definitions. The general one works for vector spaces over arbitrary fields, and is restricted to degrees of homogeneity that are integers.
The second one supposes to work over the field of real numbers, or, more generally, over an ordered field. This definition restricts to positive values the scaling factor that occurs in the definition, and is therefore called positive homogeneity, the qualificative positive being often omitted when there is no risk of confusion. Positive homogeneity leads to consider more functions as homogeneous. For example, the absolute value and all norms are positively homogeneous functions that are not homogeneous.
The restriction of the scaling factor to real positive values allows also considering homogeneous functions whose degree of homogeneity is any real number.
General homogeneity
Let and be two vector spaces ove
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https://en.wikipedia.org/wiki/Cousin%20problems
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In mathematics, the Cousin problems are two questions in several complex variables, concerning the existence of meromorphic functions that are specified in terms of local data. They were introduced in special cases by Pierre Cousin in 1895. They are now posed, and solved, for any complex manifold M, in terms of conditions on M.
For both problems, an open cover of M by sets Ui is given, along with a meromorphic function fi on each Ui.
First Cousin problem
The first Cousin problem or additive Cousin problem assumes that each difference
is a holomorphic function, where it is defined. It asks for a meromorphic function f on M such that
is holomorphic on Ui; in other words, that f shares the singular behaviour of the given local function. The given condition on the is evidently necessary for this; so the problem amounts to asking if it is sufficient. The case of one variable is the Mittag-Leffler theorem on prescribing poles, when M is an open subset of the complex plane. Riemann surface theory shows that some restriction on M will be required. The problem can always be solved on a Stein manifold.
The first Cousin problem may be understood in terms of sheaf cohomology as follows. Let K be the sheaf of meromorphic functions and O the sheaf of holomorphic functions on M. A global section of K passes to a global section of the quotient sheaf K/O. The converse question is the first Cousin problem: given a global section of K/O, is there a global section of K from which it arises? The problem is thus to characterize the image of the map
By the long exact cohomology sequence,
is exact, and so the first Cousin problem is always solvable provided that the first cohomology group H1(M,O) vanishes. In particular, by Cartan's theorem B, the Cousin problem is always solvable if M is a Stein manifold.
Second Cousin problem
The second Cousin problem or multiplicative Cousin problem assumes that each ratio
is a non-vanishing holomorphic function, where it is defined. It asks for a meromorphic function f on M such that
is holomorphic and non-vanishing. The second Cousin problem is a multi-dimensional generalization of the Weierstrass theorem on the existence of a holomorphic function of one variable with prescribed zeros.
The attack on this problem by means of taking logarithms, to reduce it to the additive problem, meets an obstruction in the form of the first Chern class (see also exponential sheaf sequence). In terms of sheaf theory, let be the sheaf of holomorphic functions that vanish nowhere, and the sheaf of meromorphic functions that are not identically zero. These are both then sheaves of abelian groups, and the quotient sheaf is well-defined. The multiplicative Cousin problem then seeks to identify the image of quotient map
The long exact sheaf cohomology sequence associated to the quotient is
so the second Cousin problem is solvable in all cases provided that The quotient sheaf is the sheaf of germs of Cartier divisors on M. The quest
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https://en.wikipedia.org/wiki/Henry%20William%20Watson
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Rev. Henry William Watson FRS (25 February 1827, Marylebone, London11 January 1903, Berkswell near Coventry) was a mathematician and author of a number of mathematics books. He was an ordained priest and Cambridge Apostle.
Life
He was born at Marylebone on 25 Feb. 1827.
He was the son of Thomas Watson, R.N., and Eleanor Mary Kingston.
He was educated at King's College London and at Trinity College, Cambridge.
He graduated as second wrangler and Smith's prizeman in 1850, Dr. W. H. Besant being senior wrangler.
He became fellow in 1851, and from 1851 to 1853 was assistant tutor.
Watson formed a close friendship with James Fitzjames Stephen, who entered Trinity in 1847.
He was made a Fellow of the Royal Society in 1881. He and Francis Galton introduced the Galton–Watson process in 1875.
Books by H. W. Watson
The mathematical theory of electricity and magnetism (Volume 1: electrostatics) (Clarendon, Oxford, 1885–1889)
The mathematical theory of electricity and magnetism (Volume 2: magnetism & electrodynamics) (Clarendon, Oxford, 1885–1889)
A treatise on the application of generalised coordinates to the kinetics of a material system (Clarendon, Oxford, 1879)
A treatise on the kinetic theory of gases (Clarendon, Oxford, 1893)
References
External links
1827 births
1903 deaths
Alumni of King's College London
Alumni of Trinity College, Cambridge
Second Wranglers
Fellows of the Royal Society
19th-century English mathematicians
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https://en.wikipedia.org/wiki/QN
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QN or qn may refer to:
Qn, one of several robust measures of scale in statistics
ATCvet code QN Nervous system, a section of the Anatomical Therapeutic Chemical Classification System for veterinary medicinal products
QN connector, a type of coaxial RF connector
Queen's Nurse (QN), an honorary title awarded by the Queen's Nursing Institute (QNI) to community nurses
Queen regnant (Qn.), in the Christian Church, following the name of a Christian saint who was a Queen
Queer Nation (QN), a United States LGBT social movement
Quintillion (qn), a large number
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https://en.wikipedia.org/wiki/Tadatoshi%20Akiba
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is a Japanese mathematician and politician and served as the mayor of the city of Hiroshima, Japan from 1999 to 2011.
Early life
He studied mathematics at the University of Tokyo, receiving a B.S. in 1966 and an M.S. in 1968. He continued his studies under John Milnor at the Massachusetts Institute of Technology, earning his PhD in mathematics in 1970. He took teaching jobs at a series of universities: State University of New York at Stony Brook (1970), Tufts University (1972–1986), and Hiroshima Shudo University (1986–1997). His research was on topology, with an interest in homotopy groups.
While at Tufts, Akiba established the Hibakusha Travel Grant program, which brought several American print and broadcast journalists annually to Hiroshima in August, to craft stories about the city (and typically about the experiences of those exposed to the atomic bomb in 1945).
Political career
As a member of the Social Democratic Party, he was elected to the House of Representatives, and served from 1990 to 1999. He assumed office as mayor of Hiroshima in February, 1999, and was reelected to this position in 2003 and in April 2007.
Peace activities
As mayor, he has been a visible peace activist. He is active in the Mayors for Peace organization, serving as the president of their World Conference. The 2020 Vision Campaign launched in 2003, which aims to eliminate nuclear weapons, has earned Mayors for Peace the "World Citizenship Award" from the Nuclear Age Peace Foundation in 2004, the "Sean McBride" Award from the International Peace Bureau in 2006, and the Nuclear-Free Future Award from the Franz-Moll Foundation in 2007. He has also been an advocate of the abolition of nuclear weapons, and a vocal critic of George W. Bush. Since May 2007 he is also Councillor at the World Future Council.
In 2007 he received the Nuclear-Free Future Award in the solutions category.
He was longlisted for the 2008 World Mayor award, but failed to win, as the award went to Mayor Helen Zille of Cape Town, Republic of South Africa.
On January 21, 2010, he attended the 78th winter meeting of the U.S. Conference of Mayors as a special guest speaker, and in that capacity attended the reception held at the White House and met US President Barack Obama. He is so far the only serving Mayor of Hiroshima who has officially met a serving US president.
In August 2010, he received the Ramon Magsaysay Award for his advocacy for nuclear disarmament, and in April 2013 he was awarded the Otto Hahn Peace Medal from the United Nations Association of Germany (DGVN) and the Governing Mayor of Berlin.
Akiba is on the Board of Advisors of the Global Security Institute.
Hiroshima is one place outside the United States where Martin Luther King Jr. Day is observed with equal importance as in the United States. Tadatoshi Akiba holds a special banquet at the mayor's office as an act of unifying his city's call for peace with King's message of human rights.
In August 2022, he was awarded the
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https://en.wikipedia.org/wiki/Bertrand%20paradox%20%28probability%29
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The Bertrand paradox is a problem within the classical interpretation of probability theory. Joseph Bertrand introduced it in his work Calcul des probabilités (1889), as an example to show that the principle of indifference may not produce definite, well-defined results for probabilities if it is applied uncritically when the domain of possibilities is infinite.
Bertrand's formulation of the problem
The Bertrand paradox is generally presented as follows: Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?
Bertrand gave three arguments (each using the principle of indifference), all apparently valid, yet yielding different results:
The "random endpoints" method: Choose two random points on the circumference of the circle and draw the chord joining them. To calculate the probability in question imagine the triangle rotated so its vertex coincides with one of the chord endpoints. Observe that if the other chord endpoint lies on the arc between the endpoints of the triangle side opposite the first point, the chord is longer than a side of the triangle. The length of the arc is one third of the circumference of the circle, therefore the probability that a random chord is longer than a side of the inscribed triangle is .
The "random radial point" method: Choose a radius of the circle, choose a point on the radius and construct the chord through this point and perpendicular to the radius. To calculate the probability in question imagine the triangle rotated so a side is perpendicular to the radius. The chord is longer than a side of the triangle if the chosen point is nearer the center of the circle than the point where the side of the triangle intersects the radius. The side of the triangle bisects the radius, therefore the probability a random chord is longer than a side of the inscribed triangle is .
The "random midpoint" method: Choose a point anywhere within the circle and construct a chord with the chosen point as its midpoint. The chord is longer than a side of the inscribed triangle if the chosen point falls within a concentric circle of radius the radius of the larger circle. The area of the smaller circle is one fourth the area of the larger circle, therefore the probability a random chord is longer than a side of the inscribed triangle is .
These three selection methods differ as to the weight they give to chords which are diameters. This issue can be avoided by "regularizing" the problem so as to exclude diameters, without affecting the resulting probabilities. But as presented above, in method 1, each chord can be chosen in exactly one way, regardless of whether or not it is a diameter; in method 2, each diameter can be chosen in two ways, whereas each other chord can be chosen in only one way; and in method 3, each choice of midpoint corresponds to a single chord, except the center of the circle,
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