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https://en.wikipedia.org/wiki/Homogeneous%20distribution | In mathematics, a homogeneous distribution is a distribution S on Euclidean space Rn or } that is homogeneous in the sense that, roughly speaking,
for all t > 0.
More precisely, let be the scalar division operator on Rn. A distribution S on Rn or } is homogeneous of degree m provided that
for all positive real t and all test functions φ. The additional factor of t−n is needed to reproduce the usual notion of homogeneity for locally integrable functions, and comes about from the Jacobian change of variables. The number m can be real or complex.
It can be a non-trivial problem to extend a given homogeneous distribution from Rn \ {0} to a distribution on Rn, although this is necessary for many of the techniques of Fourier analysis, in particular the Fourier transform, to be brought to bear. Such an extension exists in most cases, however, although it may not be unique.
Properties
If S is a homogeneous distribution on Rn \ {0} of degree α, then the weak first partial derivative of S
has degree α−1. Furthermore, a version of Euler's homogeneous function theorem holds: a distribution S is homogeneous of degree α if and only if
One dimension
A complete classification of homogeneous distributions in one dimension is possible. The homogeneous distributions on } are given by various power functions. In addition to the power functions, homogeneous distributions on R include the Dirac delta function and its derivatives.
The Dirac delta function is homogeneous of degree −1. Intuitively,
by making a change of variables y = tx in the "integral". Moreover, the kth weak derivative of the delta function δ(k) is homogeneous of degree −k−1. These distributions all have support consisting only of the origin: when localized over }, these distributions are all identically zero.
x
In one dimension, the function
is locally integrable on }, and thus defines a distribution. The distribution is homogeneous of degree α. Similarly and are homogeneous distributions of degree α.
However, each of these distributions is only locally integrable on all of R provided Re(α) > −1. But although the function naively defined by the above formula fails to be locally integrable for Re α ≤ −1, the mapping
is a holomorphic function from the right half-plane to the topological vector space of tempered distributions. It admits a unique meromorphic extension with simple poles at each negative integer . The resulting extension is homogeneous of degree α, provided α is not a negative integer, since on the one hand the relation
holds and is holomorphic in α > 0. On the other hand, both sides extend meromorphically in α, and so remain equal throughout the domain of definition.
Throughout the domain of definition, x also satisfies the following properties:
Other extensions
There are several distinct ways to extend the definition of power functions to homogeneous distributions on R at the negative integers.
χ
The poles in x at the negative integers can be removed |
https://en.wikipedia.org/wiki/Tertiary%20ideal | In mathematics, a tertiary ideal is a two-sided ideal in a perhaps noncommutative ring that cannot be expressed as a nontrivial intersection of a right fractional ideal with another ideal. Tertiary ideals generalize primary ideals to the case of noncommutative rings. Although primary decompositions do not exist in general for ideals in noncommutative rings, tertiary decompositions do, at least if the ring is Noetherian.
Every primary ideal is tertiary. Tertiary ideals and primary ideals coincide for commutative rings. To any (two-sided) ideal, a tertiary ideal can be associated called the tertiary radical, defined as
Then t(I) always contains I.
If R is a (not necessarily commutative) Noetherian ring and I a right ideal in R, then I has a unique irredundant decomposition into tertiary ideals
.
See also
Primary ideal
Lasker–Noether theorem
References
Tertiary ideal, Encyclopedia of Mathematics, Springer Online Reference Works.
Algebra |
https://en.wikipedia.org/wiki/Christoffel%E2%80%93Darboux%20formula | In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by and . It states that
where fj(x) is the jth term of a set of orthogonal polynomials of squared norm hj and leading coefficient kj.
There is also a "confluent form" of this identity by taking limit:
Proof
Let be a sequence of polynomials orthonormal with respect to a probability measure , and define(they are called the "Jacobi parameters"), then we have the three-term recurrence
Proof:
By definition, , so if , then is a linear combination of , and thus . So, to construct , it suffices to perform Gram-Schmidt process on using , which yields the desired recurrence.
Proof of Christoffel–Darboux formula:
Since both sides are unchanged by multiplying with a constant, we can scale each to .
Since is a degree polynomial, it is perpendicular to , and so .
Now the Christoffel-Darboux formula is proved by induction, using the three-term recurrence.
Specific cases
Hermite polynomials:
Associated Legendre polynomials:
See also
Turán's inequalities
Sturm Chain
References
(Hardback, Paperback)
Orthogonal polynomials
Functional analysis |
https://en.wikipedia.org/wiki/Hurwitz%27s%20theorem%20%28composition%20algebras%29 | In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras.
The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields. Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwitz problem, solved also in . Subsequent proofs of the restrictions on the dimension have been given by using the representation theory of finite groups and by and using Clifford algebras. Hurwitz's theorem has been applied in algebraic topology to problems on vector fields on spheres and the homotopy groups of the classical groups and in quantum mechanics to the classification of simple Jordan algebras.
Euclidean Hurwitz algebras
Definition
A Hurwitz algebra or composition algebra is a finite-dimensional not necessarily associative algebra with identity endowed with a nondegenerate quadratic form such that . If the underlying coefficient field is the reals and is positive-definite, so that is an inner product, then is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra.
If is a Euclidean Hurwitz algebra and is in , define the involution and right and left multiplication operators by
Evidently the involution has period two and preserves the inner product and norm. These operators have the following properties:
the involution is an antiautomorphism, i.e.
, , so that the involution on the algebra corresponds to taking adjoints
if
, , so that is an alternative algebra.
These properties are proved starting from the polarized version of the identity :
Setting or yields and .
Hence .
Similarly .
Hence , so that .
By the polarized identity so . Applied to 1 this gives . Replacing by gives the other identity.
Substituting the formula for in gives . The formula is proved analogically.
Classification
It is routine to check that the real numbers , the complex numbers and the quaternions are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions. There are moreover natural inclusions .
Analysing such an inclusion leads to the Cayley–Dickson construction, formalized by A.A. Albert. Let be a Euclidean Hurwitz algebra and a proper unital subalgebra, so a Euclidean Hurwitz algebra in its own right. Pick a unit vector in orthogonal to . Since , it follows tha |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Fibonacci | The Fibonacci numbers are the best known concept named after Leonardo of Pisa, known as Fibonacci. Among others are the following.
Concepts in mathematics and computing
A professional association and a scholarly journal that it publishes
The Fibonacci Association
Fibonacci Quarterly
An asteroid
6765 Fibonacci
An art rock band
The Fibonaccis
Fibonacci |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Joseph-Louis%20Lagrange | Several concepts from mathematics and physics are named after the mathematician and astronomer Joseph-Louis Lagrange, as are a crater on the Moon and a street in Paris.
Lagrangian
Lagrangian analysis
Lagrangian coordinates
Lagrangian derivative
Lagrangian drifter
Lagrangian foliation
Lagrangian Grassmannian
Lagrangian intersection Floer homology
Lagrangian mechanics
Relativistic Lagrangian mechanics
Lagrangian (field theory)
Lagrangian system
Lagrangian mixing
Lagrangian point
Lagrangian relaxation
Lagrangian submanifold
Lagrangian subspace
Nonlocal Lagrangian
Proca lagrangian
Special Lagrangian submanifold
Lagrange
Euler–Lagrange equation
Green–Lagrange strain
Lagrange bracket
Lagrange–Bürmann formula
Lagrange–d'Alembert principle
Lagrange error bound
Lagrange form
Lagrange form of the remainder
Lagrange interpolation
Lagrange invariant
Lagrange inversion theorem
Lagrange multiplier
Augmented Lagrangian method
Lagrange number
Lagrange point colonization
Lagrange polynomial
Lagrange property
Lagrange reversion theorem
Lagrange resolvent
Lagrange spectrum
Lagrange stability
Lagrange stream function
Lagrange top
Lagrange−Sylvester interpolation
Lagrange's
Lagrange's approximation theorem
Lagrange's formula
Lagrange's identity
Lagrange's identity (boundary value problem)
Lagrange's mean value theorem
Lagrange's notation
Lagrange's theorem (group theory)
Lagrange's theorem (number theory)
Lagrange's four-square theorem
Lagrange's trigonometric identities
Non-mathematical
Lagrange point colonization
Lagrange (crater)
Lagrange Island, Antarctica
Lagrange Island (Australia)
Rue Lagrange, a street in Paris
Via Giuseppe Luigi Lagrange, in Turin, the street where the house of his birth still stands.
Lagrange, a character from the 2017 rhythm game Arcaea
Lagrange, Joseph Louis
L |
https://en.wikipedia.org/wiki/List%20of%20Sidecarcross%20World%20Championship%20records%20and%20statistics | The FIM Sidecarcross World Championship is an annual event, organised by the Fédération Internationale de Motocyclisme (FIM) and held since 1980.
Previous to that, a European competition was held from 1971 onwards, first the FIM Cup and then, from 1975, the FIM European Championship.
All races held since the competitions interception in 1971 were staged in Europe and almost all riders hail from this continent. As World Championship winning riders is concerned, the competition is dominated by a small number of countries, with World Champions coming exclusively from Germany, the Netherlands, Switzerland, Latvia and Belgium.
Champions
The top-three teams season-by-season were:
FIM Cup
The FIM Cup was the first incarnation of the competition, held from 1971 to 1974:
FIM European Championship
From 1975 to 1979, the competition was called the FIM European Championship:
FIM World Championship
Since 1980, the competition runs under the name of FIM World Championship:
Passengers in italics.
Statistics
GP winning drivers
The number of races has historically. This leads to the Grand Prix winners table being somewhat in favour of the more recent riders, early seasons having had only a small number of races.
Every race weekend now consists of two races, with the best team out of the two becoming the week ends Grand Prix winner. In some earlier seasons, race weekends were also staged in three separate races.
Listed are all Grand Prix winning drivers up until the end of the 2015 season:
Bold denotes driver was active in the 2015 World Championship.
Results are for 1971 to 2015.
Countries
The overwhelming majority of Grand Prix winners hail from the five countries that also have provided a World Champion. These five also sit at the top of the list of countries having held Grand Prix, alongside France and the United Kingdom, who are without a World Championship:
GP wins per country
Results are for 1971 to 2015.
GP's held by country
GP's figures are for 1971 to 2015.
GP Locations
In the history of the competition from 1971 to 2015, races have been held in 157 different locations, with Malpartida de Cáceres and Stelpe having been new additions for 2015:
Figures are for 1971 to 2015.
Top-ten finishers in the World Championship
The drivers (excluding passengers) finishing in the top-ten and their season-by-season finish:
1980 to 1999
The first twenty seasons of the World Championship:
2000 to present
The seasons of the World Championship since 2000:
Key
References
External links
The official FIM website
The John Davey Grand Prix Pages
The World Championship on Sidecarcross.com
Sidecarcross World Championship
Sidecarcross World Championship records and statistics |
https://en.wikipedia.org/wiki/Saint-Didace%2C%20Quebec | Saint-Didace is a parish municipality in the D'Autray Regional County Municipality in the Lanaudière region of Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Didace had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Mother tongue:
English as first language: 0%
French as first language: 98.5%
English and French as first language: 0%
Other as first language: 1.5%
Education
Commission scolaire des Samares operates Francophone schools:
École Germain-Caron
See also
List of parish municipalities in Quebec
References
External links
Saint-Didace - MRC d'Autray
Incorporated places in Lanaudière
Parish municipalities in Quebec |
https://en.wikipedia.org/wiki/Gaussian%20q-distribution | In mathematical physics and probability and statistics, the Gaussian q-distribution is a family of probability distributions that includes, as limiting cases, the uniform distribution and the normal (Gaussian) distribution. It was introduced by Diaz and Teruel. It is a q-analog of the Gaussian or normal distribution.
The distribution is symmetric about zero and is bounded, except for the limiting case of the normal distribution. The limiting uniform distribution is on the range -1 to +1.
Definition
Let q be a real number in the interval [0, 1). The probability density function of the Gaussian q-distribution is given by
where
The q-analogue [t]q of the real number is given by
The q-analogue of the exponential function is the q-exponential, E, which is given by
where the q-analogue of the factorial is the q-factorial, [n]q!, which is in turn given by
for an integer n > 2 and [1]q! = [0]q! = 1.
The cumulative distribution function of the Gaussian q-distribution is given by
where the integration symbol denotes the Jackson integral.
The function Gq is given explicitly by
where
Moments
The moments of the Gaussian q-distribution are given by
where the symbol [2n − 1]!! is the q-analogue of the double factorial given by
See also
Q-Gaussian process
References
Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, , ,
Continuous distributions
Q-analogs |
https://en.wikipedia.org/wiki/Shadowstats.com | Shadowstats.com is a website that analyzes and offers alternatives to government economic statistics for the United States. Shadowstats primarily focuses on inflation, but also keeps track of the money supply, unemployment and GDP by utilizing methodologies abandoned by previous administrations from the Clinton era to the Great Depression.
The site is authored by consulting economist Walter J. Williams, who holds a BA in economics and an MBA from Dartmouth College. He is popularly known as "John Williams."
Claims
Shadowstats is perhaps best known for its alternative inflation statistics. Williams says that major changes to the Consumer Price Index were made between 1997 and 1999 in an effort to reduce Social Security outlays, using controversial changes by Alan Greenspan that include "hedonic regression", or the increased quality of goods. Some other investors have echoed Williams' views, most prominently Bill Gross, who reportedly called the US CPI an "haute con job". John S. Greenlees and Robert B. McClelland, staff economists at the US Bureau of Labor Statistics, wrote a paper to address CPI misconceptions, such as those of Williams.
Williams points out that under President Lyndon B. Johnson, the U-3 unemployment rate series was created, which excludes people who stopped looking for work for more than a year ago as well as part-time workers who are seeking full-time employment. Although the old unemployment rate series', which include part-time workers looking for full-time work and unemployed who stopped looking over a year ago, is still published monthly by BLS, the U-3 series is generally considered more meaningful and is the headline rate picked up by most media outlets. Williams calculates the U-6 rate as it was calculated until December 1993.
Shadowstats also tracks alternative growth statistics, and Williams has characterized the official numbers for U.S. Gross Domestic Product (GDP) and jobs growth range as "deceptive", “rigged", and "manipulated".
On July 24, 2008, Williams testified before the United States House Committee on Financial Services on the "Implications of a Weaker Dollar for Oil Prices and the U.S. Economy."
Reception
Positive
Economist and former Assistant Secretary of the Treasury for Economic Policy Paul Craig Roberts has cited John Williams' estimates in a review of unemployment rates in 2013. Williams' work has also been cited by author Wayne Allyn Root. Williams has been featured on Fox Business Network and his work has been cited by CNBC.
Negative
A number of economists and finance experts have claimed that the Shadowstats CPI is conceptually wrong and that their usage leads to easily disproven and absurd conclusions.
University of Maryland Professor Katharine Abraham, who previously headed the Bureau of Labor Statistics, the agency responsible for publishing official unemployment and inflation data, says of Williams' claims about data manipulation that "[t]he culture of the Bureau of Labor Statistics |
https://en.wikipedia.org/wiki/Unorganized%20Division%20No.%201%2C%20Manitoba | Division No. 1, Unorganized, or Whiteshell Unorganized, is a Statistics Canada census subdivision of its Division No. 1, Manitoba that consists of a part of the division that is not organized into either incorporated municipalities or Indian reserves. Unlike in some other provinces, census divisions do not reflect the organization of local government in Manitoba.
It is located at the southeast corner of Manitoba, along the border of both the Province of Ontario and the U.S. State of Minnesota. The northern half of the subdivision consists of Manitoba's Whiteshell Provincial Park. The 2006 Census reported a population of 1,130, a 68.66% increase from the 670 reported in the 2001 Census. Several Indian reserves are located within the territory of the southern portion, although they are not administratively a part of it. They include the Buffalo Point First Nation, Reed River 36A First Nation, Iskatewizaagegan 39 Independent First Nation, and Shoal Lake 40 First Nation Indian reserves.
Geography
According to Statistics Canada, the census subdivision has an area of 4,129.58 km2 (1,594.44 sq mi).
Adjacent rural municipalities and counties
Rural Municipality of Reynolds - (west)
Rural Municipality of Whitemouth - (west)
Pinawa, Manitoba - (west)
Rural Municipality of Lac du Bonnet - (west)
Rural Municipality of Alexander - (northwest)
Unorganized Division No. 19 - (north)
Unorganized Kenora - (east)
Shoal Lake 39A - (east)
Shoal Lake 40 First Nation - (east)
Lake of the Woods County, Minnesota - (southeast)
Buffalo Point First Nation - (south)
Rural Municipality of Piney - (southwest)
References
Unorganized areas in Manitoba
Unincorporated communities in Eastman Region, Manitoba |
https://en.wikipedia.org/wiki/Keza | Keza is a ward in Ngara District of the Kagera Region in west-central Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,810 people in the ward, from 9,525 in 2012.
Villages
The ward has 15 villages.
Keza
Rukira
Nyakabanda
Kibirizi
Rubanga I
Kazingati
Rubanga II
References
Populated places in Tanzania |
https://en.wikipedia.org/wiki/Convolution%20for%20optical%20broad-beam%20responses%20in%20scattering%20media | Photon transport theories in Physics, Medicine, and Statistics (such as the Monte Carlo method), are commonly used to model light propagation in tissue. The responses to a pencil beam incident on a scattering medium are referred to as Green's functions or impulse responses. Photon transport methods can be directly used to compute broad-beam responses by distributing photons over the cross section of the beam. However, convolution can be used in certain cases to improve computational efficiency.
General convolution formulas
In order for convolution to be used to calculate a broad-beam response, a system must be time invariant, linear, and translation invariant. Time invariance implies that a photon beam delayed by a given time produces a response shifted by the same delay. Linearity indicates that a given response will increase by the same amount if the input is scaled and obeys the property of superposition. Translational invariance means that if a beam is shifted to a new location on the tissue surface, its response is also shifted in the same direction by the same distance. Here, only spatial convolution is considered.
Responses from photon transport methods can be physical quantities such as absorption, fluence, reflectance, or transmittance. Given a specific physical quantity, G(x,y,z), from a pencil beam in Cartesian space and a collimated light source with beam profile S(x,y), a broad-beam response can be calculated using the following 2-D convolution formula:
Similar to 1-D convolution, 2-D convolution is commutative between G and S with a change of variables and :
Because the broad-beam response has cylindrical symmetry, its convolution integrals can be rewritten as:
where . Because the inner integration of Equation 4 is independent of z, it only needs to be calculated once for all depths. Thus this form of the broad-beam response is more computationally advantageous.
Common beam profiles
Gaussian beam
For a Gaussian beam, the intensity profile is given by
Here, R denotes the radius of the beam, and S0 denotes the intensity at the center of the beam. S0 is related to the total power P0 by
Substituting Eq. 5 into Eq. 4, we obtain
where I0 is the zeroth-order modified Bessel function.
Top-hat beam
For a top-hat beam of radius R, the source function becomes
where S0 denotes the intensity inside the beam. S0 is related to the total beam power P0 by
Substituting Eq. 8 into Eq. 4, we obtain
where
Errors in numerical evaluation
First interactions
First photon-tissue interactions always occur on the z axis and hence contribute to the specific absorption or related physical quantities as a Dirac delta function. Errors will result if absorption due to the first interactions is not recorded separately from absorption due to subsequent interactions. The total impulse response can be expressed in two parts:
where the first term results from the first interactions and the second, from subsequent interactions.
For a Ga |
https://en.wikipedia.org/wiki/Generalized%20symmetric%20group | In mathematics, the generalized symmetric group is the wreath product of the cyclic group of order m and the symmetric group of order n.
Examples
For the generalized symmetric group is exactly the ordinary symmetric group:
For one can consider the cyclic group of order 2 as positives and negatives () and identify the generalized symmetric group with the signed symmetric group.
Representation theory
There is a natural representation of elements of as generalized permutation matrices, where the nonzero entries are m-th roots of unity:
The representation theory has been studied since ; see references in . As with the symmetric group, the representations can be constructed in terms of Specht modules; see .
Homology
The first group homology group (concretely, the abelianization) is (for m odd this is isomorphic to ): the factors (which are all conjugate, hence must map identically in an abelian group, since conjugation is trivial in an abelian group) can be mapped to (concretely, by taking the product of all the values), while the sign map on the symmetric group yields the These are independent, and generate the group, hence are the abelianization.
The second homology group (in classical terms, the Schur multiplier) is given by :
Note that it depends on n and the parity of m: and which are the Schur multipliers of the symmetric group and signed symmetric group.
References
Permutation groups |
https://en.wikipedia.org/wiki/Distortion%20%28mathematics%29 | In mathematics, the distortion is a measure of the amount by which a function from the Euclidean plane to itself distorts circles to ellipses. If the distortion of a function is equal to one, then it is conformal; if the distortion is bounded and the function is a homeomorphism, then it is quasiconformal. The distortion of a function ƒ of the plane is given by
which is the limiting eccentricity of the ellipse produced by applying ƒ to small circles centered at z. This geometrical definition is often very difficult to work with, and the necessary analytical features can be extrapolated to the following definition. A mapping ƒ : Ω → R2 from an open domain in the plane to the plane has finite distortion at a point x ∈ Ω if ƒ is in the Sobolev space W(Ω, R2), the Jacobian determinant J(x,ƒ) is locally integrable and does not change sign in Ω, and there is a measurable function K(x) ≥ 1 such that
almost everywhere. Here Df is the weak derivative of ƒ, and |Df| is the Hilbert–Schmidt norm.
For functions on a higher-dimensional Euclidean space Rn, there are more measures of distortion because there are more than two principal axes of a symmetric tensor. The pointwise information is contained in the distortion tensor
The outer distortion KO and inner distortion KI are defined via the Rayleigh quotients
The outer distortion can also be characterized by means of an inequality similar to that given in the two-dimensional case. If Ω is an open set in Rn, then a function has finite distortion if its Jacobian is locally integrable and does not change sign, and there is a measurable function KO (the outer distortion) such that
almost everywhere.
See also
Deformation (mechanics)
References
.
.
Conformal mappings
Real analysis
Complex analysis
Topology
Measure theory
Euclidean geometry |
https://en.wikipedia.org/wiki/Alexandra%2C%20Prince%20Edward%20Island | Alexandra is a rural municipality located in Queens County, Prince Edward Island, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Alexandra had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Communities in Queens County, Prince Edward Island
Rural municipalities in Prince Edward Island |
https://en.wikipedia.org/wiki/Central%20Kings | Central Kings (population: 329) is a rural municipality in Prince Edward Island, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Central Kings had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Communities in Kings County, Prince Edward Island
Rural municipalities in Prince Edward Island |
https://en.wikipedia.org/wiki/Darlington%2C%20Prince%20Edward%20Island | Darlington (population: 99) is a rural municipality in Prince Edward Island, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Darlington had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Communities in Queens County, Prince Edward Island
Rural municipalities in Prince Edward Island |
https://en.wikipedia.org/wiki/Greenmount-Montrose | Greenmount (population: 262) is a rural municipality in Prince Edward Island, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Greenmount-Montrose had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Communities in Prince County, Prince Edward Island
Rural municipalities in Prince Edward Island |
https://en.wikipedia.org/wiki/Hampshire%2C%20Prince%20Edward%20Island | Hampshire is a rural municipality in Prince Edward Island, Canada. It was incorporated in 1974 and has a population of 339.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Hampshire had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Communities in Queens County, Prince Edward Island
Rural municipalities in Prince Edward Island |
https://en.wikipedia.org/wiki/Hazelbrook%2C%20Prince%20Edward%20Island | Hazelbrook (Population: 220) is a rural municipality in Prince Edward Island, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Hazelbrook had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Communities in Queens County, Prince Edward Island
Rural municipalities in Prince Edward Island |
https://en.wikipedia.org/wiki/Kinkora%2C%20Prince%20Edward%20Island | Kinkora is a rural municipality in Prince Edward Island, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Kinkora had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Communities in Prince County, Prince Edward Island
Rural municipalities in Prince Edward Island |
https://en.wikipedia.org/wiki/Northport%2C%20Prince%20Edward%20Island | Northport (population: 188) is a rural municipality in Prince Edward Island, Canada. It is located in Lot 5 township.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Northport had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Communities in Prince County, Prince Edward Island
Rural municipalities in Prince Edward Island |
https://en.wikipedia.org/wiki/Sherbrooke%2C%20Prince%20Edward%20Island | Sherbrooke is a rural municipality in Prince Edward Island, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Sherbrooke had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Communities in Prince County, Prince Edward Island
Rural municipalities in Prince Edward Island |
https://en.wikipedia.org/wiki/Tignish%20Shore | Tignish Shore (population: 57) is a rural municipality in Prince Edward Island, Canada. It is located in the Lot 1 township.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Tignish Shore had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Communities in Prince County, Prince Edward Island
Rural municipalities in Prince Edward Island |
https://en.wikipedia.org/wiki/Erik%20Mellevold%20Br%C3%A5then | Erik Mellevold Bråthen (born 16 September 1987) is a retired Norwegian football goalkeeper.
Career statistics
References
1987 births
Living people
Norwegian men's footballers
Eliteserien players
Kvik Halden FK players
Fredrikstad FK players
Rosenborg BK players
Footballers from Fredrikstad
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Vidar%20Martinsen | Vidar Martinsen (born 11 March 1982) is a Norwegian footballer currently playing for Råde IL after eight seasons in the Norwegian Fredrikstad.
Career statistics
References
External links
Guardian's Stats Centre
1981 births
Living people
People from Råde
Norwegian men's footballers
Moss FK players
Fredrikstad FK players
Eliteserien players
Norwegian First Division players
Men's association football defenders
Footballers from Viken (county) |
https://en.wikipedia.org/wiki/Galois%20geometry | Galois geometry (so named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or Galois field). More narrowly, a Galois geometry may be defined as a projective space over a finite field.
Objects of study include affine and projective spaces over finite fields and various structures that are contained in them. In particular, arcs, ovals, hyperovals, unitals, blocking sets, ovoids, caps, spreads and all finite analogues of structures found in non-finite geometries. Vector spaces defined over finite fields play a significant role, especially in construction methods.
Projective spaces over finite fields
Notation
Although the generic notation of projective geometry is sometimes used, it is more common to denote projective spaces over finite fields by , where is the "geometric" dimension (see below), and is the order of the finite field (or Galois field) , which must be an integer that is a prime or prime power.
The geometric dimension in the above notation refers to the system whereby lines are 1-dimensional, planes are 2-dimensional, points are 0-dimensional, etc. The modifier, sometimes the term projective instead of geometric is used, is necessary since this concept of dimension differs from the concept used for vector spaces (that is, the number of elements in a basis). Normally having two different concepts with the same name does not cause much difficulty in separate areas due to context, but in this subject both vector spaces and projective spaces play important roles and confusion is highly likely. The vector space concept is at times referred to as the algebraic dimension.
Construction
Let denote the vector space of (algebraic) dimension defined over the finite field . The projective space consists of all the positive (algebraic) dimensional vector subspaces of . An alternate way to view the construction is to define the points of as the equivalence classes of the non-zero vectors of under the equivalence relation whereby two vectors are equivalent if one is a scalar multiple of the other. Subspaces are then built up from the points using the definition of linear independence of sets of points.
Subspaces
A vector subspace of algebraic dimension of is a (projective) subspace of of geometric dimension . The projective subspaces are given common geometric names; points, lines, planes and solids are the 0,1,2 and 3-dimensional subspaces, respectively. The whole space is an -dimensional subspace and an ()-dimensional subspace is called a hyperplane (or prime).
The number of vector subspaces of algebraic dimension in vector space is given by the Gaussian binomial coefficient,
Therefore, the number of dimensional projective subspaces in is given by
Thus, for example, the number of lines ( = 1) in PG(3,2) is
It follows that the total number of points ( = 0) of is
This also equals the number of hyperplanes of |
https://en.wikipedia.org/wiki/Pizza%20theorem | In elementary geometry, the pizza theorem states the equality of two areas that arise when one partitions a disk in a certain way.
The theorem is so called because it mimics a traditional pizza slicing technique. It shows that if two people share a pizza sliced into 8 pieces (or any multiple of 4 greater than 8), and take alternating slices, then they will each get an equal amount of pizza, irrespective of the central cutting point.
Statement
Let p be an interior point of the disk, and let n be a multiple of 4 that is greater than or equal to 8. Form n sectors of the disk with equal angles by choosing an arbitrary line through p, rotating the line times by an angle of radians, and slicing the disk on each of the resulting lines. Number the sectors consecutively in a clockwise or anti-clockwise fashion. Then the pizza theorem states that:
The sum of the areas of the odd-numbered sectors equals the sum of the areas of the even-numbered sectors .
History
The pizza theorem was originally proposed as a challenge problem by . The published solution to this problem, by Michael Goldberg, involved direct manipulation of the algebraic expressions for the areas of the sectors.
provide an alternative proof by dissection. They show how to partition the sectors into smaller pieces so that each piece in an odd-numbered sector has a congruent piece in an even-numbered sector, and vice versa. gave a family of dissection proofs for all cases (in which the number of sectors ).
Generalizations
The requirement that the number of sectors be a multiple of four is necessary: as Don Coppersmith showed, dividing a disk into four sectors, or a number of sectors that is not divisible by four, does not in general produce equal areas. answered a problem of by providing a more precise version of the theorem that determines which of the two sets of sectors has greater area in the cases that the areas are unequal. Specifically, if the number of sectors is 2 (mod 8) and no slice passes through the center of the disk, then the subset of slices containing the center has smaller area than the other subset, while if the number of sectors is 6 (mod 8) and no slice passes through the center, then the subset of slices containing the center has larger area. An odd number of sectors is not possible with straight-line cuts, and a slice through the center causes the two subsets to be equal regardless of the number of sectors.
also observe that, when the pizza is divided evenly, then so is its crust (the crust may be interpreted as either the perimeter of the disk or the area between the boundary of the disk and a smaller circle having the same center, with the cut-point lying in the latter's interior), and since the disks bounded by both circles are partitioned evenly so is their difference. However, when the pizza is divided unevenly, the diner who gets the most pizza area actually gets the least crust.
As note, an equal division of the pizza also leads to an equal divis |
https://en.wikipedia.org/wiki/Simen%20Skappel | Simen Skappel (7 March 1866 – 15 July 1945) was a Norwegian historian and statistician.
He specialized in agricultural history and agricultural statistics, and worked for Statistics Norway for 34 years, leading the Department of agricultural statistics for parts of that period. He was hired in Statistics Norway in 1902, was promoted to secretary in 1911 and department head in 1928. His most prominent publication was Om husmannsvesenet i Norge, published in 1922. A husmann in Norway was a type of crofter. In the book, Skappel distinguished between crofters who received some land in exchange for work, often on short-term contracts, and crofters who received land in exchange for rent, often on lifetime contracts. Skappel's theory has been discussed as late as around 2000, but has been found (by historian Ståle Dyrvik) models to lack important complexity. Other prominent works by Skappel include Hedemarkens amt 1814–1914 and Ringsaker Sparebank 1847–1927, but he mostly published in journals. He died in the summer of 1945.
References
1866 births
1945 deaths
20th-century Norwegian historians
Norwegian statisticians |
https://en.wikipedia.org/wiki/Lac-Saint-Joseph | Lac-Saint-Joseph is a town in Quebec, Canada, located on the namesake Saint-Joseph Lake.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Lac-Saint-Joseph had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Mother tongue:
English as first language: 0%
French as first language: 98.4%
English and French as first language: 0%
Other as first language: 0%
List of mayors
J. Gérald Coote, 1936
J. A. Saucier, 1936-1952
Henri Giguère, 1952-1961
Guy Desrivières, lawyer, 1961-1967
Fernand Grenier, Professor University Laval 1967-1974
J.-Arthur Bédard, 1974-1982
Raymond Blouin, 1982-1990 and 1994-2005
Robert Simard, 1990-1994
O'Donnell Bédard, 2005-
Political representation
Provincially it is part of the riding of La Peltrie. In the 2022 Quebec general election the incumbent MNA Éric Caire, of the Coalition Avenir Québec, was re-elected to represent the population of Lac-Saint-Joseph in the National Assembly of Quebec.
Federally, Lac-Saint-Joseph is part of the federal riding of Portneuf—Jacques-Cartier. In the 2021 Canadian federal election, the incumbent Joël Godin of the Conservative Party was re-elected to represent the population Lac-Saint-Joseph in the House of Commons of Canada.
See also
List of cities in Quebec
References
External links
Incorporated places in Capitale-Nationale
Cities and towns in Quebec |
https://en.wikipedia.org/wiki/Lac-Delage | Lac-Delage is a town in the Capitale-Nationale region of Quebec, Canada, located on the eponymous Lake Delage.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Lac-Delage had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Mother tongue:
English as first language: 1.3%
French as first language: 92.9%
English and French as first language: 1.9%
Other as first language: 2.6%
Local government
Lac-Delage forms part of the federal electoral district of Portneuf—Jacques-Cartier and has been represented by Joël Godin of the Conservative Party since 2015. Provincially, Lac-Delage is part of the Chauveau electoral district and is represented by Sylvain Lévesque of the Coalition Avenir Québec since 2018.
Municipal council
Mayor: Guy Rochette
Councillors: Jannys Landry, Alexandre Morin, Marc Boiteau, Isabelle Coulombe, Christiane Gosselin, Jonathan Baker
See also
List of cities in Quebec
References
Incorporated places in Capitale-Nationale
Cities and towns in Quebec |
https://en.wikipedia.org/wiki/Saint-Z%C3%A9non-du-Lac-Humqui | Saint-Zénon-du-Lac-Humqui is a parish municipality in Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Zénon-du-Lac-Humqui had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
See also
List of parish municipalities in Quebec
References
Parish municipalities in Quebec
Incorporated places in Bas-Saint-Laurent
La Matapédia Regional County Municipality
Canada geography articles needing translation from French Wikipedia |
https://en.wikipedia.org/wiki/Saint-Alexandre-des-Lacs | Saint-Alexandre-des-Lacs is a parish municipality in Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Alexandre-des-Lacs had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Mother tongue:
English as first language: 0%
French as first language: 100%
English and French as first language: 0%
Other as first language: 0%
See also
List of parish municipalities in Quebec
References
Parish municipalities in Quebec
Incorporated places in Bas-Saint-Laurent
La Matapédia Regional County Municipality
Canada geography articles needing translation from French Wikipedia |
https://en.wikipedia.org/wiki/Saint-Mo%C3%AFse%2C%20Quebec | Saint-Moïse is a parish municipality in Quebec, Canada. It is located at the intersection of routes 132 and 297.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Moïse had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Notable residents
Joseph Kaeble - Saint-Moïse born recipient of the Victoria Cross for actions in France during the First World War
See also
List of parish municipalities in Quebec
References
Parish municipalities in Quebec
Incorporated places in Bas-Saint-Laurent
La Matapédia Regional County Municipality |
https://en.wikipedia.org/wiki/Saint-Damase%2C%20Bas-Saint-Laurent%2C%20Quebec | Saint-Damase is a parish municipality in Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Damase had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Mother tongue:
English as first language: 0%
French as first language: 98.7%
English and French as first language: 0%
Other as first language: 0%
See also
List of parish municipalities in Quebec
References
Parish municipalities in Quebec
Incorporated places in Bas-Saint-Laurent
La Matapédia Regional County Municipality
Canada geography articles needing translation from French Wikipedia |
https://en.wikipedia.org/wiki/Grosses-Roches | Grosses-Roches is a municipality in Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Grosses-Roches had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
See also
List of municipalities in Quebec
References
Incorporated places in Bas-Saint-Laurent
Municipalities in Quebec |
https://en.wikipedia.org/wiki/Saint-Charles-Garnier%2C%20Quebec | Saint-Charles-Garnier is a parish municipality in Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Charles-Garnier had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
See also
List of parish municipalities in Quebec
References
External links
Parish municipalities in Quebec
Incorporated places in Bas-Saint-Laurent |
https://en.wikipedia.org/wiki/Padoue%2C%20Quebec | Padoue is a municipality in Quebec, Canada.
Demographics
In the 2021 Census, Statistics Canada reported that Padoue had a population of 250 living in 111 of its 123 total dwellings, a 2% change from its 2016 population of 245.
See also
List of municipalities in Quebec
References
External links
Municipalities in Quebec
Incorporated places in Bas-Saint-Laurent |
https://en.wikipedia.org/wiki/Saint-Octave-de-M%C3%A9tis%2C%20Quebec | Saint-Octave-de-Métis is a parish municipality in Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Octave-de-Métis had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Notable people
Jules-André Brillant, born 1888, baptized and educated in Saint-Octave-de-Métis, French Canadian entrepreneur
Hormisdas Langlais, born 1890, Canadian politician and a seven-term Member of the Legislative Assembly of Quebec
Marguerite Ruest-Pitre, born 1908, Canadian murderer also known as "Madame le Corbeau", last woman criminal executed in Canada (1953)
Marie-Thérèse Fortin, born 1959, Canadian actress
Anaïs Favron, born 1977, radio and TV host, actress, and improviser
See also
List of parish municipalities in Quebec
References
External links
Parish municipalities in Quebec
Incorporated places in Bas-Saint-Laurent |
https://en.wikipedia.org/wiki/Sainte-Fran%C3%A7oise%2C%20Bas-Saint-Laurent%2C%20Quebec | Sainte-Françoise is a parish municipality in the Bas-Saint-Laurent region of Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Sainte-Françoise had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
See also
List of parish municipalities in Quebec
References
External links
Parish municipalities in Quebec
Incorporated places in Bas-Saint-Laurent |
https://en.wikipedia.org/wiki/Saint-Ars%C3%A8ne%2C%20Quebec | Saint-Arsène () is a parish municipality in Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Arsène had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
See also
List of parish municipalities in Quebec
References
External links
Parish municipalities in Quebec
Incorporated places in Bas-Saint-Laurent |
https://en.wikipedia.org/wiki/Saint-Eus%C3%A8be%2C%20Quebec | Saint-Eusèbe is a parish municipality in Quebec, Canada. It was established in 1911.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Eusèbe had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
See also
Branche à Jerry, a stream
List of parish municipalities in Quebec
References
External links
Parish municipalities in Quebec
Incorporated places in Bas-Saint-Laurent |
https://en.wikipedia.org/wiki/Parallelohedron | In geometry, a parallelohedron is a polyhedron that can be translated without rotations in 3-dimensional Euclidean space to fill space with a honeycomb in which all copies of the polyhedron meet face-to-face. There are five types of parallelohedron, first identified by Evgraf Fedorov in 1885 in his studies of crystallographic systems: the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron.
Classification
Every parallelohedron is a zonohedron, a centrally symmetric polyhedron with centrally symmetric faces. Like any zonohedron, it can be constructed as the Minkowski sum of line segments, one segment for each parallel class of edges of the polyhedron. For parallelohedra, there are between three and six of these parallel classes. The lengths of the segments can be adjusted arbitrarily; doing so extends or shrinks the corresponding edges of the parallelohedron, without changing its combinatorial type or its property of tiling space. As a limiting case, for a parallelohedron with more than three parallel classes of edges, the length of any one of these classes can be adjusted to zero, producing another parallelohedron of a simpler form, with one fewer class of parallel edges. As with all zonohedra, these shapes automatically have 2 Ci central inversion symmetry, but additional symmetries are possible with an appropriate choice of the generating segments.
The five types of parallelohedron are:
A parallelepiped, generated from three line segments that are not all parallel to a common plane. Its most symmetric form is the cube, generated by three perpendicular unit-length line segments. It tiles space to form the cubic honeycomb.
A hexagonal prism, generated from four line segments, three of them parallel to a common plane and the fourth not. Its most symmetric form is the right prism over a regular hexagon. It tiles space to form the hexagonal prismatic honeycomb.
The rhombic dodecahedron, generated from four line segments, no two of which are parallel to a common plane. Its most symmetric form is generated by the four long diagonals of a cube. It tiles space to form the rhombic dodecahedral honeycomb.
The elongated dodecahedron, generated from five line segments, with two triples of coplanar segments. It can be generated by using an edge of the cube and its four long diagonals as generators. It tiles space to form the elongated dodecahedral honeycomb.
The truncated octahedron, generated from six line segments with four triples of coplanar segments. It can be embedded in four-dimensional space as the 4-permutahedron, whose vertices are all permutations of the counting numbers (1,2,3,4). In three-dimensional space, its most symmetric form is generated from six line segments parallel to the face diagonals of a cube. It tiles space to form the bitruncated cubic honeycomb.
Any zonohedron whose faces have the same combinatorial structure as one of these five shapes is a parallelohedron, regardless of its particul |
https://en.wikipedia.org/wiki/Saint-Cyrille-de-Lessard | Saint-Cyrille-de-Lessard is a parish municipality in Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Cyrille-de-Lessard had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
See also
List of municipalities in Quebec
References
External links
Parish municipalities in Quebec
Incorporated places in Chaudière-Appalaches |
https://en.wikipedia.org/wiki/Sainte-Louise%2C%20Quebec | Sainte-Louise is a parish municipality in Quebec.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Sainte-Louise had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Notable people
The artist Michèle Lorrain resides in Sainte-Louise.
See also
List of municipalities in Quebec
References
Parish municipalities in Quebec
Incorporated places in Chaudière-Appalaches |
https://en.wikipedia.org/wiki/Saint-Fabien-de-Panet%2C%20Quebec | Saint-Fabien-de-Panet is a parish municipality in Quebec.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Fabien-de-Panet had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
See also
List of parish municipalities in Quebec
References
Parish municipalities in Quebec
Incorporated places in Chaudière-Appalaches |
https://en.wikipedia.org/wiki/Saint-Pierre-de-la-Rivi%C3%A8re-du-Sud%2C%20Quebec | Saint-Pierre-de-la-Rivière-du-Sud is a parish municipality in Quebec.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Pierre-de-la-Rivière-du-Sud had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
See also
List of parish municipalities in Quebec
Amable Bélanger
References
Parish municipalities in Quebec
Incorporated places in Chaudière-Appalaches
Canada geography articles needing translation from French Wikipedia |
https://en.wikipedia.org/wiki/Saint-Pierre-Baptiste%2C%20Quebec | Saint-Pierre-Baptiste is a parish municipality in Quebec.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Pierre-Baptiste had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Parish municipalities in Quebec
Incorporated places in Centre-du-Québec
Canada geography articles needing translation from French Wikipedia |
https://en.wikipedia.org/wiki/Notre-Dame-de-Lourdes%2C%20Centre-du-Qu%C3%A9bec%2C%20Quebec | Notre-Dame-de-Lourdes is a parish municipality in Quebec.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Notre-Dame-de-Lourdes had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Parish municipalities in Quebec
Incorporated places in Centre-du-Québec |
https://en.wikipedia.org/wiki/Saint-Thuribe%2C%20Quebec | Saint-Thuribe is a parish municipality in the Capitale-Nationale region of Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Thuribe had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Population trend:
Population in 2011: 288 (2006 to 2011 population change: -5.0%)
Population in 2006: 303
Population in 2001: 313
Population in 1996: 360
Population in 1991: 410
Mother tongue:
English as first language: 0%
French as first language: 100%
English and French as first language: 0%
Other as first language: 0%
References
Incorporated places in Capitale-Nationale
Parish municipalities in Quebec |
https://en.wikipedia.org/wiki/Saints-Martyrs-Canadiens | Saints-Martyrs-Canadiens is a parish municipality located in the Centre-du-Québec region of Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saints-Martyrs-Canadiens had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Parish municipalities in Quebec
Incorporated places in Centre-du-Québec |
https://en.wikipedia.org/wiki/Saint-Christophe-d%27Arthabaska | Saint-Christophe-d'Arthabaska is a parish municipality located in the Centre-du-Québec region of Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Christophe-d'Arthabaska had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Parish municipalities in Quebec
Incorporated places in Centre-du-Québec |
https://en.wikipedia.org/wiki/Sainte-S%C3%A9raphine%2C%20Quebec | Sainte-Séraphine is a parish municipality located in the Centre-du-Québec region of Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Sainte-Séraphine had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Parish municipalities in Quebec
Incorporated places in Centre-du-Québec |
https://en.wikipedia.org/wiki/Saint-Rosaire%2C%20Quebec | Saint-Rosaire is a parish municipality located in the Centre-du-Québec region of Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Rosaire had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
External links
Parish municipalities in Quebec
Incorporated places in Centre-du-Québec |
https://en.wikipedia.org/wiki/Redouane%20Akniouene | Redouane Akniouene (born 15 January 1982) is an Algerian footballer who plays for CR Belouizdad in Algeria.
National team statistics
External links
1982 births
Living people
Algerian men's footballers
Algeria men's international footballers
CA Bordj Bou Arréridj players
Algerian Ligue Professionnelle 1 players
CR Belouizdad players
OMR El Annasser players
Footballers from Algiers
Men's association football defenders
21st-century Algerian people |
https://en.wikipedia.org/wiki/Loun%C3%A9s%20Bendahmane | Lounés Bendahmane (born 3 April 1977 in Baghlia) is an Algerian former footballer who last played for O Médéa in Algeria. He has been retired since 1 July 2015.
National team statistics
Honours
Won the CAF Cup three times with JS Kabylie in 2000, 2001 and 2002
Won the Algerian League once with JS Kabylie in 2004
Participated in the 2002 African Cup of Nations in Mali
Has 5 caps for the Algerian National Team
References
External links
1977 births
Living people
People from Baghlia
Kabyle people
Algerian men's footballers
Algeria men's international footballers
JS Kabylie players
Men's association football midfielders
USM Annaba players
MC Saïda players
RC Kouba players
CR Belouizdad players
OMR El Annasser players
CA Bordj Bou Arréridj players
2002 African Cup of Nations players
Algerian Ligue 2 players
Algerian Ligue Professionnelle 1 players
JS Bordj Ménaïel players
21st-century Algerian people |
https://en.wikipedia.org/wiki/Additive%20K-theory | In mathematics, additive K-theory means some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl. It is not, therefore, one theory but a way of creating additive or infinitesimal analogues of multiplicative theories.
Formulation
Following Boris Feigin and Boris Tsygan, let be an algebra over a field of characteristic zero and let be the algebra of infinite matrices over with only finitely many nonzero entries. Then the Lie algebra homology
has a natural structure of a Hopf algebra. The space of its primitive elements of degree is denoted by and called the -th additive K-functor of A.
The additive K-functors are related to cyclic homology groups by the isomorphism
References
K-theory |
https://en.wikipedia.org/wiki/Saint-Z%C3%A9non%2C%20Quebec | Saint-Zénon is a municipality in the Lanaudière region of Quebec, part of the Matawinie Regional County Municipality.
Demographics
Population
In the 2021 Census, Statistics Canada reported that Saint-Zénon had a population of 1,317 living in 721 of its 1,357 total dwellings, an 17.6% change from its 2016 population of 1,120. With a land area of , it had a population density of in 2021.
Language
Mother tongue:
English as first language: 1.5%
French as first language: 94.3%
English and French as first language: 0.8%
Other as first language: 2.7%
Education
Commission scolaire des Samares operates francophone public schools:
École Bérard
The Sir Wilfrid Laurier School Board operates anglophone public schools serving the community at the secondary level, including:
Joliette High School in Joliette
See also
List of municipalities in Quebec
References
Incorporated places in Lanaudière
Municipalities in Quebec
Matawinie Regional County Municipality |
https://en.wikipedia.org/wiki/Laid%20Belhamel | Laid Belhamel (born November 12, 1977 in El Eulma, Algeria) is a former Algerian football player.
National team statistics
External links
1977 births
Algerian men's footballers
Living people
Algeria men's international footballers
Algerian expatriate men's footballers
CR Belouizdad players
ES Sétif players
CS Constantine players
CA Bordj Bou Arréridj players
Expatriate men's footballers in Tunisia
Algerian expatriate sportspeople in Tunisia
People from El Eulma
MC El Eulma players
EGS Gafsa players
Algeria men's under-23 international footballers
Men's association football midfielders
21st-century Algerian people |
https://en.wikipedia.org/wiki/Atiyah%20conjecture | In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of -Betti numbers.
History
In 1976, Michael Atiyah introduced -cohomology of manifolds with a free co-compact action of a discrete countable group (e.g. the universal cover of a compact manifold together with the action of the fundamental group by deck transformations.) Atiyah defined also numbers as von Neumann dimensions of the resulting groups, and computed several examples, which all turned out to be rational numbers. He therefore asked if it is possible for -Betti numbers to be irrational.
Since then, various researchers asked more refined questions about possible values of -Betti numbers, all of which are customarily referred to as "Atiyah conjecture".
Results
Many positive results were proven by Peter Linnell. For example, if the group acting is a free group, then the -Betti numbers are integers.
The most general question open as of late 2011 is whether -Betti numbers are rational if there is a bound on the orders of finite subgroups of the group which acts. In fact, a precise relationship between possible denominators and the orders in question is conjectured; in the case of torsion-free groups, this statement generalizes the zero-divisors conjecture. For a discussion see
the article of B. Eckmann.
In the case there is no such bound, Tim Austin showed in 2009 that -Betti numbers can assume transcendental values. Later it was shown that in that case they can be any non-negative real numbers.
References
Conjectures
Cohomology theories
Differential geometry
Differential topology |
https://en.wikipedia.org/wiki/Ground%20axiom | In set theory, the ground axiom states that the universe of set theory is not a nontrivial set-forcing extension of an inner model. The axiom was introduced by and .
References
Axioms of set theory |
https://en.wikipedia.org/wiki/Patrick%20Dehornoy | Patrick Dehornoy (11 September 1952 – 4 September 2019) was a mathematician at the University of Caen Normandy who worked on set theory and group theory.
Early life and education
Dehornoy was born on 11 September 1952 in Rouen, France. He graduated from the Lycée Pierre-Corneille in 1971. He studied at the École normale supérieure from 1971 to 1975 and completed his Ph.D. in 1978 at the University of Paris, with a thesis written under the direction of Kenneth Walter McAloon.
Career
Dehornoy was a researcher at the French National Centre for Scientific Research (CNRS) from 1975 to 1982. He was at the University of Caen Normandy as a Professor from 1983 to 2017 and as an Emeritus Professor from 2017 until his death. From 2009 to 2013, he was an adjunct scientific director of the (INSMI) at the CNRS. Dehornoy died on 4 September 2019 in Villejuif, France at the age of 66.
Research
Dehornoy found one of the first applications of large cardinals to algebra by constructing a certain left-invariant total order, called the Dehornoy order, on the braid group. In his later career, he was a major contributor to the theory of braid groups, including creating a fast algorithm for comparing braids, and was one of the main contributors to the development of Garside methods.
Awards
In 1999, Dehornoy received the Ferran Sunyer i Balaguer Prize. In 2002, he was elected a senior member of the Institut Universitaire de France (renewed in 2007). In 2005, he received the of the French Academy of Sciences. In 2014, he received the EMS Monograph Award for his book Foundations of Garside Theory.
Selected publications
References
External links
1952 births
2019 deaths
20th-century French mathematicians
21st-century French mathematicians
École Normale Supérieure alumni
Academic staff of the University of Caen Normandy
Scientists from Rouen
Group theorists
Set theorists
Research directors of the French National Centre for Scientific Research
University of Paris alumni |
https://en.wikipedia.org/wiki/Ellerman | Ellerman is a surname. Notable people with the surname include:
David Ellerman (born 1943), philosopher working in economics, political economy, social theory, philosophy and mathematics
Derek Ellerman (born 1978), the co-founder and board chair of Polaris Project, a Washington DC-based organization
Ferdinand Ellerman (1869–1940), American astronomer and photographer
John Ellerman, CH (1862–1933), English shipowner and investor
Sir John Ellerman, 2nd Baronet (1910–1973), English shipowner, natural historian and philanthropist
Juul Ellerman (born 1965), Dutch former international footballer
See also
Ellermann, surname
Ellerman (crater), lunar crater on the far side of the Moon
Ellerman bombs, micro solar flares named after Ferdinand Ellerman
Ellerman Lines, cargo and passenger shipping company that operated from the late 19th century into the 20th century |
https://en.wikipedia.org/wiki/Kamel%20Kherkhache | Kamel Kherkhache born December 2, 1976 in Mostaganem is an Algerian football player currently playing for AB Mérouana.
Career statistics
Club
External links
1976 births
Living people
Algerian men's footballers
Algeria men's international footballers
USM Blida players
People from Mostaganem
2002 African Cup of Nations players
Men's association football forwards
MO Constantine players
21st-century Algerian people |
https://en.wikipedia.org/wiki/Connes%20embedding%20problem | Connes' embedding problem, formulated by Alain Connes in the 1970s, is a major problem in von Neumann algebra theory. During that time, the problem was reformulated in several different areas of mathematics. Dan Voiculescu developing his free entropy theory found that Connes’ embedding problem is related to the existence of microstates. Some results of von Neumann algebra theory can be obtained assuming positive solution to the problem. The problem is connected to some basic questions in quantum theory, which led to the realization that it also has important implications in computer science.
The problem admits a number of equivalent formulations. Notably, it is equivalent to the following long standing problems:
Kirchberg's QWEP conjecture in C*-algebra theory
Tsirelson's problem in quantum information theory
The predual of any (separable) von Neumann algebra is finitely representable in the trace class.
In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen announced a result in quantum complexity theory that implies a negative answer to Connes' embedding problem. However, an error was discovered in September 2020 in an earlier result they used; a new proof avoiding the earlier result was published as a preprint in September,. A broad outline was published in Communications of the ACM in November 2021, and an article explaining the connection between MIP*=RE and the Connes Embedding Problem appeared in October 2022.
Statement
Let be a free ultrafilter on the natural numbers and let R be the hyperfinite type II1 factor with trace . One can construct the ultrapower as follows: let be the von Neumann algebra of norm-bounded sequences and let . The quotient turns out to be a II1 factor with trace , where is any representative sequence of .
Connes' embedding problem asks whether every type II1 factor on a separable Hilbert space can be embedded into some .
A positive solution to the problem would imply that invariant subspaces exist for a large class of operators in type II1 factors (Uffe Haagerup); all countable discrete groups are hyperlinear. A positive solution to the problem would be implied by equality between free entropy and free entropy defined by microstates (Dan Voiculescu). In January 2020, a group of researchers claimed to have resolved the problem in the negative, i.e., there exist type II1 von Neumann factors that do not embed in an ultrapower of the hyperfinite II1 factor.
The isomorphism class of is independent of the ultrafilter if and only if the continuum hypothesis is true (Ge-Hadwin and Farah-Hart-Sherman), but such an embedding property does not depend on the ultrafilter because von Neumann algebras acting on separable Hilbert spaces are, roughly speaking, very small.
The problem admits a number of equivalent formulations.
Conferences dedicated to Connes' embedding problem
Connes' embedding problem and quantum information theory workshop; Vanderbilt University in Nashville Tennessee; May 1–7, 2020 (postpo |
https://en.wikipedia.org/wiki/Nehari%20manifold | In the calculus of variations, a branch of mathematics, a Nehari manifold is a manifold of functions, whose definition is motivated by the work of . It is a differentiable manifold associated to the Dirichlet problem for the semilinear elliptic partial differential equation
Here Δ is the Laplacian on a bounded domain Ω in Rn.
There are infinitely many solutions to this problem. Solutions are precisely the critical points for the energy functional
on the Sobolev space . The Nehari manifold is defined to be the set of such that
Solutions to the original variational problem that lie in the Nehari manifold are (constrained) minimizers of the energy, and so direct methods in the calculus of variations can be brought to bear.
More generally, given a suitable functional J, the associated Nehari manifold is defined as the set of functions u in an appropriate function space for which
Here J′ is the functional derivative of J.
References
A. Bahri and P. L. Lions (1988), Morse Index of Some Min-Max Critical Points. I. Applications to Multiplicity Results. Communications on Pure and Applied Mathematics. (XLI) 1027–1037.
Calculus of variations |
https://en.wikipedia.org/wiki/Generalized%20p-value | In statistics, a generalized p-value is an extended version of the classical p-value, which except in a limited number of applications, provides only approximate solutions.
Conventional statistical methods do not provide exact solutions to many statistical problems, such as those arising in mixed models and MANOVA, especially when the problem involves a number of nuisance parameters. As a result, practitioners often resort to approximate statistical methods or asymptotic statistical methods that are valid only when the sample size is large. With small samples, such methods often have poor performance. Use of approximate and asymptotic methods may lead to misleading conclusions or may fail to detect truly significant results from experiments.
Tests based on generalized p-values are exact statistical methods in that they are based on exact probability statements. While conventional statistical methods do not provide exact solutions to such problems as testing variance components or ANOVA under unequal variances, exact tests for such problems can be obtained based on generalized p-values.
In order to overcome the shortcomings of the classical p-values, Tsui and Weerahandi extended the classical definition so that one can obtain exact solutions for such problems as the Behrens–Fisher problem and testing variance components. This is accomplished by allowing test variables to depend on observable random vectors as well as their observed values, as in the Bayesian treatment of the problem, but without having to treat constant parameters as random variables.
Example
To describe the idea of generalized p-values in a simple example, consider a situation of sampling from a normal population with the mean , and the variance . Let and be the sample mean and the sample variance. Inferences on all unknown parameters can be based on the distributional results
and
Now suppose we need to test the coefficient of variation, . While the problem is not trivial with conventional p-values, the task can be easily accomplished based on the generalized test variable
where is the observed value of and is the observed value of . Note that the distribution of and its observed value are both free of nuisance parameters. Therefore, a test of a hypothesis with a one-sided alternative such as can be based on the generalized p-value , a quantity that can be easily evaluated via Monte Carlo simulation or using the non-central t-distribution.
Notes
References
Gamage J, Mathew T, and Weerahandi S. (2013). Generalized prediction intervals for BLUPs in mixed models, Journal of Multivariate Analysis}, 220, 226-233.
Hamada, M., and Weerahandi, S. (2000). Measurement System Assessment via Generalized Inference. Journal of Quality Technology, 32, 241-253.
Krishnamoorthy, K. and Tian, L. (2007), “Inferences on the ratio of means of two inverse Gaussian distributions: the generalized variable approach”, Journal of Statistical Planning and Inferences, Volume 138, Issue 7, 1 |
https://en.wikipedia.org/wiki/Hellmuth%20Stachel | Hellmuth Stachel (born October 6, 1942, in Graz, Austria) is an Austrian mathematician, a professor of geometry at the Technical University of Vienna, who is known due to his contributions to geometry, kinematics and computer-aided design.
Biography
Stachel was born on October 6, 1942, in Graz to the family of primary school teachers. He graduated from an elementary school in Trofaiach and secondary school (gymnasium) in Leoben. In 1965 he graduated from the University of Graz and Graz University of Technology, obtaining a diploma of a school teacher in “mathematics” and “descriptive geometry.” As a Ph.D. student of Graz University he majored in “mathematics, astronomy and philosophy.” His doctoral advisor was Professor Fritz Hohenberg. Hellmuth Stachel obtained his doctorate in 1969 from Graz University and passed thought habilitation in 1971 at the Technical University of Graz. In 1980 he moved to the Vienna University of Technology.
Stachel held several visiting positions in China: in August 1984 he was a visiting professor at South China University of Technology in Guangzhou and, in October 1989, at Tongji University in Shanghai.
Stachel takes an active part in the work of “International Society for Geometry and Graphics” (ISGG). In 1990 he was elected a chairman and, in 1994, a vice-president of the society. In 1996 he has created the scientific journal Journal for Geometry and Graphics and since then serves it as the Editor-in-Chief.
Research
Stachel wrote three books (in cooperation with other scholars) and approximately 120 scientific articles on classical and descriptive geometry, kinematics and the theory of mechanisms, as well as on computer aided design. He studied flexible polyhedra in the 4-dimensional Euclidean space and 3-dimensional Lobachevsky space.
Awards and prizes
In 1991 Stachel was elected a corresponding member of the Austrian Academy of Sciences. In 1993 he received the “German-Austrian University-Software Award” for the development of educational Cad-3D programs. In 2004 he received The Steve M. Slaby Award. On November, 1st 2010 Hellmuth Stachel received an honorary doctorate from the Dresden University of Technology.
Books
G. Glaeser and H. Stachel. Open Geometry: OpenGL + Advanced Geometry. Springer: New York, 1999, 392 p. .
T. Arens, F. Hettlich, Ch. Karpfinger, U. Kockelkorn, K. Lichtenegger, and H. Stachel. Mathematik, 1.Aufl. Spektrum: Heidelberg, 2008, 1500 p. .
T. Arens, F. Hettlich, Ch. Karpfinger, U. Kockelkorn, K. Lichtenegger, and H. Stachel. Arbeitsbuch Mathematik. Spektrum: Heidelberg, 2009, 683 p. .
G. Glaeser, H. Stachel and B. Odehnal. The Universe of Conics: From the ancient Greeks to 21st century developments. Springer Spektrum: Heidelberg, 1st ed. 2016, 488 p. .
B. Odehnal, G. Glaeser and H. Stachel. The Universe of Quadrics. Springer: NY, 1st ed. 2020, 600 p.
Articles on flexible polyhedra
H. Stachel. Flexible octahedra in the hyperbolic space. In the book: Non-Euclidean geome |
https://en.wikipedia.org/wiki/Radial%20function | In mathematics, a radial function is a real-valued function defined on a Euclidean space Rn whose value at each point depends only on the distance between that point and the origin. The distance is usually the Euclidean distance. For example, a radial function Φ in two dimensions has the form
where φ is a function of a single non-negative real variable. Radial functions are contrasted with spherical functions, and any descent function (e.g., continuous and rapidly decreasing) on Euclidean space can be decomposed into a series consisting of radial and spherical parts: the solid spherical harmonic expansion.
A function is radial if and only if it is invariant under all rotations leaving the origin fixed. That is, ƒ is radial if and only if
for all , the special orthogonal group in n dimensions. This characterization of radial functions makes it possible also to define radial distributions. These are distributions S on Rn such that
for every test function φ and rotation ρ.
Given any (locally integrable) function ƒ, its radial part is given by averaging over spheres centered at the origin. To wit,
where ωn−1 is the surface area of the (n−1)-sphere Sn−1, and , . It follows essentially by Fubini's theorem that a locally integrable function has a well-defined radial part at almost every r.
The Fourier transform of a radial function is also radial, and so radial functions play a vital role in Fourier analysis. Furthermore, the Fourier transform of a radial function typically has stronger decay behavior at infinity than non-radial functions: for radial functions bounded in a neighborhood of the origin, the Fourier transform decays faster than R−(n−1)/2. The Bessel functions are a special class of radial function that arise naturally in Fourier analysis as the radial eigenfunctions of the Laplacian; as such they appear naturally as the radial portion of the Fourier transform.
See also
Radial basis function
References
.
Harmonic analysis
Rotational symmetry
Types of functions |
https://en.wikipedia.org/wiki/Fibonorial | In mathematics, the Fibonorial , also called the Fibonacci factorial, where is a nonnegative integer, is defined as the product of the first positive Fibonacci numbers, i.e.
where is the th Fibonacci number, and gives the empty product (defined as the multiplicative identity, i.e. 1).
The Fibonorial is defined analogously to the factorial . The Fibonorial numbers are used in the definition of Fibonomial coefficients (or Fibonacci-binomial coefficients) similarly as the factorial numbers are used in the definition of binomial coefficients.
Asymptotic behaviour
The series of fibonorials is asymptotic to a function of the golden ratio : .
Here the fibonorial constant (also called the fibonacci factorial constant) is defined by , where and is the golden ratio.
An approximate truncated value of is 1.226742010720 (see for more digits).
Almost-Fibonorial numbers
Almost-Fibonorial numbers: .
Almost-Fibonorial primes: prime numbers among the almost-Fibonorial numbers.
Quasi-Fibonorial numbers
Quasi-Fibonorial numbers: .
Quasi-Fibonorial primes: prime numbers among the quasi-Fibonorial numbers.
Connection with the q-Factorial
The fibonorial can be expressed in terms of the q-factorial and the golden ratio :
Sequences
Product of first nonzero Fibonacci numbers .
and for such that and are primes, respectively.
References
Fibonacci numbers
fr:Analogues de la factorielle#Factorielle de Fibonacci |
https://en.wikipedia.org/wiki/David%20G.%20Grigoryan | David G. Grigoryan (born 17 July 1989), is a retired Armenian football midfielder who made one appearance for the Armenia national football team.
Career statistics
International
References
External links
Living people
1989 births
Armenian men's footballers
Armenia men's international footballers
FC Ararat Yerevan players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Hovhannes%20Demirchyan | Hovhannes Demirchyan (born 15 August 1975) is a retired Armenian football player.
National team statistics
External links
Living people
1975 births
Armenian men's footballers
Armenia men's international footballers
Shirak SC players
FC Stal Alchevsk players
Expatriate men's footballers in Ukraine
Armenian expatriate sportspeople in Ukraine
Ukrainian Premier League players
Men's association football defenders
Soviet men's footballers |
https://en.wikipedia.org/wiki/Hovhannes%20Tahmazyan | Hovhannes Tahmazyan (; born on 11 January 1970) is a retired Armenian international footballer who played for Shirak and Mika.
Career statistics
International
References
External links
Profile at ffa.am
Profile at armfootball.tripod.com
Living people
1970 births
Armenian men's footballers
Armenia men's international footballers
Shirak SC players
FC Mika players
Armenian Premier League players
Place of birth missing (living people)
Men's association football defenders
Soviet men's footballers |
https://en.wikipedia.org/wiki/Boundary-incompressible%20surface | In low-dimensional topology, a boundary-incompressible surface is a two-dimensional surface within a three-dimensional manifold whose topology cannot be made simpler by a certain type of operation known as boundary compression.
Suppose M is a 3-manifold with boundary. Suppose also that S is a compact surface with boundary that is properly embedded in M,
meaning that the boundary of S is a subset of the boundary of M and the interior points of S are a subset of the interior points of M.
A boundary-compressing disk for S in M is defined to be a disk D in M such that and are arcs in , with , , and is an essential arc in S ( does not cobound a disk in S with another arc in ).
The surface S is said to be boundary-compressible if either S is a disk that cobounds a ball with a disk in or there exists a boundary-compressing disk for S in M. Otherwise, S is boundary-incompressible.
Alternatively, one can relax this definition by dropping the requirement that the surface be properly embedded. Suppose now that S is a compact surface (with boundary) embedded in the boundary of a 3-manifold M. Suppose further that D is a properly embedded disk in M such that D intersects S in an essential arc (one that does not cobound a disk in S with another arc in ). Then D is called a boundary-compressing disk for S in M. As above, S is said to be boundary-compressible if either S is a disk in or there exists a boundary-compressing disk for S in M. Otherwise, S is boundary-incompressible.
For instance, if K is a trefoil knot embedded in the boundary of a solid torus V and S is the closure of a small annular neighborhood of K in , then S is not properly embedded in V since the interior of S is not contained in the interior of V. However, S is embedded in and there does not exist a boundary-compressing disk for S in V, so S is boundary-incompressible by the second definition.
See also
Incompressible surface
References
W. Jaco, Lectures on Three-Manifold Topology, volume 43 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, R.I., 1980.
T. Kobayashi, A construction of 3-manifolds whose homeomorphism classes of Heegaard splittings have polynomial growth, Osaka J. Math. 29 (1992), no. 4, 653–674. .
Manifolds |
https://en.wikipedia.org/wiki/Compact%20Lie%20algebra | In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite; this definition is more restrictive and excludes tori,. A compact Lie algebra can be seen as the smallest real form of a corresponding complex Lie algebra, namely the complexification.
Definition
Formally, one may define a compact Lie algebra either as the Lie algebra of a compact Lie group, or as a real Lie algebra whose Killing form is negative definite. These definitions do not quite agree:
The Killing form on the Lie algebra of a compact Lie group is negative semidefinite, not negative definite in general.
If the Killing form of a Lie algebra is negative definite, then the Lie algebra is the Lie algebra of a compact semisimple Lie group.
In general, the Lie algebra of a compact Lie group decomposes as the Lie algebra direct sum of a commutative summand (for which the corresponding subgroup is a torus) and a summand on which the Killing form is negative definite.
It is important to note that the converse of the first result above is false: Even if the Killing form of a Lie algebra is negative semidefinite, this does not mean that the Lie algebra is the Lie algebra of some compact group. For example, the Killing form on the Lie algebra of the Heisenberg group is identically zero, hence negative semidefinite, but this Lie algebra is not the Lie algebra of any compact group.
Properties
Compact Lie algebras are reductive; note that the analogous result is true for compact groups in general.
The Lie algebra for the compact Lie group G admits an Ad(G)-invariant inner product,. Conversely, if admits an Ad-invariant inner product, then is the Lie algebra of some compact group. If is semisimple, this inner product can be taken to be the negative of the Killing form. Thus relative to this inner product, Ad(G) acts by orthogonal transformations () and acts by skew-symmetric matrices (). It is possible to develop the theory of complex semisimple Lie algebras by viewing them as the complexifications of Lie algebras of compact groups; the existence of an Ad-invariant inner product on the compact real form greatly simplifies the development.
This can be seen as a compact analog of Ado's theorem on the representability of Lie algebras: just as every finite-dimensional Lie algebra in characteristic 0 embeds in every compact Lie algebra embeds in
The Satake diagram of a compact Lie algebra is the Dynkin diagram of the complex Lie algebra with all vertices blackened.
Compact Lie algebras are opposite to split real Lie algebras among real forms, split Lie algebras being "as far as possible" from being compact.
Classification
The compact Lie algebras are classified and named according to the compact real for |
https://en.wikipedia.org/wiki/Exceptional%20isomorphism | In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families, usually infinite, of mathematical objects, which is incidental, in that it is not an instance of a general pattern of such isomorphisms. These coincidences are at times considered a matter of trivia, but in other respects they can give rise to consequential phenomena, such as exceptional objects. In the following, coincidences are organized according to the structures where they occur.
Groups
Finite simple groups
The exceptional isomorphisms between the series of finite simple groups mostly involve projective special linear groups and alternating groups, and are:
the smallest non-abelian simple group (order 60) – icosahedral symmetry;
the second-smallest non-abelian simple group (order 168) – PSL(2,7);
between a projective special orthogonal group and a projective symplectic group.
Alternating groups and symmetric groups
There are coincidences between symmetric/alternating groups and small groups of Lie type/polyhedral groups:
Dihedral group of order 6,
tetrahedral group,
full tetrahedral group octahedral group,
icosahedral group,
,
These can all be explained in a systematic way by using linear algebra (and the action of on affine -space) to define the isomorphism going from the right side to the left side. (The above isomorphisms for and are linked via the exceptional isomorphism .)
There are also some coincidences with symmetries of regular polyhedra: the alternating group A5 agrees with the icosahedral group (itself an exceptional object), and the double cover of the alternating group A5 is the binary icosahedral group.
Trivial group
The trivial group arises in numerous ways. The trivial group is often omitted from the beginning of a classical family. For instance:
, the cyclic group of order 1;
, the alternating group on 0, 1, or 2 letters;
, the symmetric group on 0 or 1 letters;
, linear groups of a 0-dimensional vector space;
, linear groups of a 1-dimensional vector space
and many others.
Spheres
The spheres S0, S1, and S3 admit group structures, which can be described in many ways:
, the last being the group of units of the integers;
circle group;
unit quaternions.
Spin groups
In addition to , and above, there are isomorphisms for higher dimensional spin groups:
Also, Spin(8) has an exceptional order 3 triality automorphism.
Coxeter–Dynkin diagrams
There are some exceptional isomorphisms of Dynkin diagrams, yielding isomorphisms of the corresponding Coxeter groups and of polytopes realizing the symmetries, as well as isomorphisms of Lie algebras whose root systems are described by the same diagrams. These are:
See also
Exceptional object
Mathematical coincidence, for numerical coincidences
Notes
References
Mathematical relations |
https://en.wikipedia.org/wiki/Bechir%20Mogaadi | Bechir Mogaadi (born April 11, 1978) is a Tunisian footballer who plays for Zanaco FC.
Azerbaijan career statistics
References
External links
Tunisian men's footballers
Tunisian expatriate men's footballers
Tunisia men's international footballers
1978 births
Living people
Expatriate men's footballers in Chile
Expatriate men's footballers in Azerbaijan
Expatriate men's footballers in Spain
Expatriate men's footballers in Zambia
Zanaco F.C. players
ES Hammam Sousse players
Étoile Sportive du Sahel players
Espérance Sportive de Tunis players
CD Numancia players
El Makarem de Mahdia players
Karvan FK players
Shuvalan FK players
Club Deportivo Palestino footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Split%20Lie%20algebra | In the mathematical field of Lie theory, a split Lie algebra is a pair where is a Lie algebra and is a splitting Cartan subalgebra, where "splitting" means that for all , is triangularizable. If a Lie algebra admits a splitting, it is called a splittable Lie algebra. Note that for reductive Lie algebras, the Cartan subalgebra is required to contain the center.
Over an algebraically closed field such as the complex numbers, all semisimple Lie algebras are splittable (indeed, not only does the Cartan subalgebra act by triangularizable matrices, but even stronger, it acts by diagonalizable ones) and all splittings are conjugate; thus split Lie algebras are of most interest for non-algebraically closed fields.
Split Lie algebras are of interest both because they formalize the split real form of a complex Lie algebra, and because split semisimple Lie algebras (more generally, split reductive Lie algebras) over any field share many properties with semisimple Lie algebras over algebraically closed fields – having essentially the same representation theory, for instance – the splitting Cartan subalgebra playing the same role as the Cartan subalgebra plays over algebraically closed fields. This is the approach followed in , for instance.
Properties
Over an algebraically closed field, all Cartan subalgebras are conjugate. Over a non-algebraically closed field, not all Cartan subalgebras are conjugate in general; however, in a splittable semisimple Lie algebra all splitting Cartan algebras are conjugate.
Over an algebraically closed field, all semisimple Lie algebras are splittable.
Over a non-algebraically closed field, there exist non-splittable semisimple Lie algebras.
In a splittable Lie algebra, there may exist Cartan subalgebras that are not splitting.
Direct sums of splittable Lie algebras and ideals in splittable Lie algebras are splittable.
Split real Lie algebras
For a real Lie algebra, splittable is equivalent to either of these conditions:
The real rank equals the complex rank.
The Satake diagram has neither black vertices nor arrows.
Every complex semisimple Lie algebra has a unique (up to isomorphism) split real Lie algebra, which is also semisimple, and is simple if and only if the complex Lie algebra is.
For real semisimple Lie algebras, split Lie algebras are opposite to compact Lie algebras – the corresponding Lie group is "as far as possible" from being compact.
Examples
The split real forms for the complex semisimple Lie algebras are:
Exceptional Lie algebras: have split real forms EI, EV, EVIII, FI, G.
These are the Lie algebras of the split real groups of the complex Lie groups.
Note that for and , the real form is the real points of (the Lie algebra of) the same algebraic group, while for one must use the split forms (of maximally indefinite index), as the group SO is compact.
See also
Compact Lie algebra
Real form
Split-complex number
Split orthogonal group
References
Properties of Lie |
https://en.wikipedia.org/wiki/Commuting%20matrices | In linear algebra, two matrices and are said to commute if , or equivalently if their commutator is zero. A set of matrices is said to commute if they commute pairwise, meaning that every pair of matrices in the set commute with each other.
Characterizations and properties
Commuting matrices preserve each other's eigenspaces. As a consequence, commuting matrices over an algebraically closed field are simultaneously triangularizable; that is, there are bases over which they are both upper triangular. In other words, if commute, there exists a similarity matrix such that is upper triangular for all . The converse is not necessarily true, as the following counterexample shows:
However, if the square of the commutator of two matrices is zero, that is, , then the converse is true.
Two diagonalizable matrices and commute () if they are simultaneously diagonalizable (that is, there exists an invertible matrix such that both and are diagonal). The converse is also true; that is, if two diagonalizable matrices commute, they are simultaneously diagonalizable. But if you take any two matrices that commute (and do not assume they are two diagonalizable matrices) they are simultaneously diagonalizable already if one of the matrices has no multiple eigenvalues.
If and commute, they have a common eigenvector. If has distinct eigenvalues, and and commute, then 's eigenvectors are 's eigenvectors.
If one of the matrices has the property that its minimal polynomial coincides with its characteristic polynomial (that is, it has the maximal degree), which happens in particular whenever the characteristic polynomial has only simple roots, then the other matrix can be written as a polynomial in the first.
As a direct consequence of simultaneous triangulizability, the eigenvalues of two commuting complex matrices A, B with their algebraic multiplicities (the multisets of roots of their characteristic polynomials) can be matched up as in such a way that the multiset of eigenvalues of any polynomial in the two matrices is the multiset of the values . This theorem is due to Frobenius.
Two Hermitian matrices commute if their eigenspaces coincide. In particular, two Hermitian matrices without multiple eigenvalues commute if they share the same set of eigenvectors. This follows by considering the eigenvalue decompositions of both matrices. Let and be two Hermitian matrices. and have common eigenspaces when they can be written as and . It then follows that
The property of two matrices commuting is not transitive: A matrix may commute with both and , and still and do not commute with each other. As an example, the identity matrix commutes with all matrices, which between them do not all commute. If the set of matrices considered is restricted to Hermitian matrices without multiple eigenvalues, then commutativity is transitive, as a consequence of the characterization in terms of eigenvectors.
Lie's theorem, which shows that any r |
https://en.wikipedia.org/wiki/Bogomol%27nyi%E2%80%93Prasad%E2%80%93Sommerfield%20state | In theoretical physics, massive representations of an extended supersymmetry algebra called BPS states have mass equal to the supersymmetry central charge Z. Quantum mechanically, if the supersymmetry remains unbroken, exact equality to the modulus of Z exists. Their importance arises as the supermultiplets shorten for generic massive representations, with stability and mass formula exact.
d = 4 N = 2
The generators for the odd part of the superalgebra have relations:
where: are the Lorentz group indices, A and B are R-symmetry indices.
Take linear combinations of the above generators as follows:
Consider a state ψ which has 4 momentum . Applying the following operator to this state gives:
But because this is the square of a Hermitian operator, the right hand side coefficient must be positive for all .
In particular the strongest result from this is
Example applications
Supersymmetric black hole entropies
See also
Bogomol'nyi–Prasad–Sommerfield bound
Short supermultiplet
Wall-crossing
References
Supersymmetry |
https://en.wikipedia.org/wiki/Arturo%20Carbonaro | Arturo Carbonaro (born 4 March 1986) is an Italian football defender who plays for Chiavari Caperana.
External links
Career statistics
Living people
1986 births
Italian men's footballers
US Salernitana 1919 players
Men's association football defenders
Sportspeople from Salerno
Footballers from the Province of Salerno |
https://en.wikipedia.org/wiki/Monomial%20conjecture | In commutative algebra, a field of mathematics, the monomial conjecture of Melvin Hochster says the following:
Let A be a Noetherian local ring of Krull dimension d and let x1, ..., xd be a system of parameters for A (so that A/(x1, ..., xd) is an Artinian ring). Then for all positive integers t, we have
The statement can relatively easily be shown in characteristic zero.
References
See also
Homological conjectures in commutative algebra
Commutative algebra
Conjectures |
https://en.wikipedia.org/wiki/Malcev%20Lie%20algebra | In mathematics, a Malcev Lie algebra, or Mal'tsev Lie algebra, is a generalization of a rational nilpotent Lie algebra, and Malcev groups are similar. Both were introduced by , based on the work of .
Definition
According to a Malcev Lie algebra is a rational Lie algebra together with a complete, descending -vector space filtration , such that:
the associated graded Lie algebra is generated by elements of degree one.
Applications
Relation to Hopf algebras
showed that Malcev Lie algebras and Malcev groups are both equivalent to complete Hopf algebras, i.e., Hopf algebras H endowed with a filtration so that H is isomorphic to . The functors involved in these equivalences are as follows: a Malcev group G is mapped to the completion (with respect to the augmentation ideal) of its group ring QG, with inverse given by the group of grouplike elements of a Hopf algebra H, essentially those elements 1 + x such that . From complete Hopf algebras to Malcev Lie algebras one gets by taking the (completion of) primitive elements, with inverse functor given by the completion of the universal enveloping algebra.
This equivalence of categories was used by to prove that, after tensoring with Q, relative K-theory K(A, I), for a nilpotent ideal I, is isomorphic to relative cyclic homology HC(A, I). This theorem was a pioneering result in the area of trace methods.
Hodge theory
Malcev Lie algebras also arise in the theory of mixed Hodge structures.
References
Hodge theory
Lie algebras |
https://en.wikipedia.org/wiki/Crime%20in%20Oregon |
Crime statistics (1960–2009)
Reported cases of crime in the state of Oregon between 1960 and 2009:
Capital punishment laws
The Oregon Constitution originally had no provision for a death penalty. A statute was enacted in 1864 allowing for the death penalty in cases of first degree murder. Authority to conduct executions was initially granted to local sheriffs, but in 1903, the Oregon Legislative Assembly passed a law requiring all executions to be conducted at the Oregon State Penitentiary in Salem, the first state prison in Oregon which opened in 1866.
Oregon voters amended the Constitution in 1914 to repeal the death penalty, with 50.04% of the vote. The repeal was an initiative of Governor Oswald West. The death penalty was restored, again by constitutional amendment, in 1920.
Initially, all executions were performed by hanging; lethal gas was adopted as the method after 1931.
Voters outlawed the death penalty in the general election of 1964, with 60% of the vote. Governor Mark Hatfield commuted the sentences of three death row inmates two days later.
Notable cases
Criminals
Crimes
References
Crimes in Oregon
History of Oregon |
https://en.wikipedia.org/wiki/Gabriel%27s%20theorem | In mathematics, Gabriel's theorem, proved by Pierre Gabriel, classifies the quivers of finite type in terms of Dynkin diagrams.
Statement
A quiver is of finite type if it has only finitely many isomorphism classes of indecomposable representations. classified all quivers of finite type, and also their indecomposable representations. More precisely, Gabriel's theorem states that:
A (connected) quiver is of finite type if and only if its underlying graph (when the directions of the arrows are ignored) is one of the ADE Dynkin diagrams: , , , , .
The indecomposable representations are in a one-to-one correspondence with the positive roots of the root system of the Dynkin diagram.
found a generalization of Gabriel's theorem in which all Dynkin diagrams of finite-dimensional semisimple Lie algebras occur.
References
Theorems in representation theory |
https://en.wikipedia.org/wiki/Anthony%20Morse | Anthony Perry Morse (21 August 1911 – 6 March 1984) was an American mathematician who worked in both analysis, especially measure theory, and in the foundations of mathematics. He is best known as the co-creator, together with John L. Kelley, of Morse–Kelley set theory. This theory first appeared in print in Kelley's General Topology. Morse's own version appeared later in A Theory of Sets.
He is also known for his work on the Morse–Sard theorem and the Federer–Morse theorem.
Anthony Morse should not be confused with Marston Morse, known for developing Morse theory.
Career
He received his PhD in 1937 at Brown University with C. R. Adams as thesis advisor. After two years at the Institute for Advanced Study he joined the mathematics faculty at Berkeley where except for two interruptions he worked for the rest of his life on mathematics. In the first of these, from 1943 until the end of World War II, he worked on ballistics at the Aberdeen Proving Ground.
In 1950 his life was interrupted by the McCarthyist loyalty oath controversy. He was one of 29
"non-signers".
But he was also one of 6 who took advantage of a 10-day grace period to sign, while continuing to refer to
the remaining non-signers as "patriots."
His doctoral students include Herbert Federer, Woody Bledsoe, and Maurice Sion.
References
External links
Obituary
20th-century American mathematicians
University of California, Berkeley faculty
1911 births
1984 deaths
Brown University alumni |
https://en.wikipedia.org/wiki/Solovay%20model | In the mathematical field of set theory, the Solovay model is a model constructed by in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measurable. The construction relies on the existence of an inaccessible cardinal.
In this way Solovay showed that in the proof of the existence of a non-measurable set from ZFC (Zermelo–Fraenkel set theory plus the axiom of choice), the axiom of choice is essential, at least granted that the existence of an inaccessible cardinal is consistent with ZFC.
Statement
ZF stands for Zermelo–Fraenkel set theory, and DC for the axiom of dependent choice.
Solovay's theorem is as follows.
Assuming the existence of an inaccessible cardinal, there is an inner model of ZF + DC of a suitable forcing extension V[G] such that every set of reals is Lebesgue measurable, has the perfect set property, and has the Baire property.
Construction
Solovay constructed his model in two steps, starting with a model M of ZFC containing an inaccessible cardinal κ.
The first step is to take a Levy collapse M[G] of M by adding a generic set G for the notion of forcing that collapses all cardinals less than κ to ω. Then M[G] is a model of ZFC with the property that every set of reals that is definable over a countable sequence of ordinals is Lebesgue measurable, and has the Baire and perfect set properties. (This includes all definable and projective sets of reals; however for reasons related to Tarski's undefinability theorem the notion of a definable set of reals cannot be defined in the language of set theory, while the notion of a set of reals definable over a countable sequence of ordinals can be.)
The second step is to construct Solovay's model N as the class of all sets in M[G] that are hereditarily definable over a countable sequence of ordinals. The model N is an inner model of M[G] satisfying ZF + DC such that every set of reals is Lebesgue measurable, has the perfect set property, and has the Baire property. The proof of this uses the fact that every real in M[G] is definable over a countable sequence of ordinals, and hence N and M[G] have the same reals.
Instead of using Solovay's model N, one can also use the smaller inner model L(R) of M[G], consisting of the constructible closure of the real numbers, which has similar properties.
Complements
Solovay suggested in his paper that the use of an inaccessible cardinal might not be necessary. Several authors proved weaker versions of Solovay's result without assuming the existence of an inaccessible cardinal. In particular showed there was a model of ZFC in which every ordinal-definable set of reals is measurable, Solovay showed there is a model of ZF + DC in which there is some translation-invariant extension of Lebesgue measure to all subsets of the reals, and showed that there is a model in which all sets of reals have the Baire property (so that the inaccessible cardinal |
https://en.wikipedia.org/wiki/Sylvio%20Breleur | Sylvio Breleur (born 13 October 1978) is a French former professional footballer who played as a striker. He represented the French Guiana national team at international level.
Career statistics
Scores and results list French Guiana's goal tally first, score column indicates score after each Breleur goal.
References
External links
1978 births
Living people
Sportspeople from Cayenne
Men's association football forwards
French Guianan men's footballers
French men's footballers
French Guiana men's international footballers
Levallois SC players
Entente SSG players
K.S.K. Ronse players
S.V. Zulte Waregem players
R.F.C. Tournai players
K.V. Oostende players
French Guianan expatriate men's footballers
French expatriate sportspeople in Belgium
Expatriate men's footballers in Belgium
Black French sportspeople |
https://en.wikipedia.org/wiki/K-distribution | In probability and statistics, the generalized K-distribution is a three-parameter family of continuous probability distributions.
The distribution arises by compounding two gamma distributions. In each case, a re-parametrization of the usual form of the family of gamma distributions is used, such that the parameters are:
the mean of the distribution,
the usual shape parameter.
K-distribution is a special case of variance-gamma distribution, which in turn is a special case of generalised hyperbolic distribution. A simpler special case of the generalized K-distribution is often referred as the K-distribution.
Density
Suppose that a random variable has gamma distribution with mean and shape parameter , with being treated as a random variable having another gamma distribution, this time with mean and shape parameter . The result is that has the following probability density function (pdf) for :
where is a modified Bessel function of the second kind. Note that for the modified Bessel function of the second kind, we have . In this derivation, the K-distribution is a compound probability distribution. It is also a product distribution: it is the distribution of the product of two independent random variables, one having a gamma distribution with mean 1 and shape parameter , the second having a gamma distribution with mean and shape parameter .
A simpler two parameter formalization of the K-distribution can be obtained by setting as
where is the shape factor, is the scale factor, and is the modified Bessel function of second kind. The above two parameter formalization can also be obtained by setting , , and , albeit with different physical interpretation of and parameters. This two parameter formalization is often referred to as the K-distribution, while the three parameter formalization is referred to as the generalized K-distribution.
This distribution derives from a paper by Eric Jakeman and Peter Pusey (1978) who used it to model microwave sea echo. Jakeman and Tough (1987) derived the distribution from a biased random walk model. Keith D. Ward (1981) derived the distribution from the product for two random variables, z = a y, where a has a chi distribution and y a complex Gaussian distribution. The modulus of z, |z|, then has K-distribution.
Moments
The moment generating function is given by
where and is the Whittaker function.
The n-th moments of K-distribution is given by
So the mean and variance are given by
Other properties
All the properties of the distribution are symmetric in and
Applications
K-distribution arises as the consequence of a statistical or probabilistic model used in synthetic-aperture radar (SAR) imagery. The K-distribution is formed by compounding two separate probability distributions, one representing the radar cross-section, and the other representing speckle that is a characteristic of coherent imaging. It is also used in wireless communication to model composite fast fading and shadowing |
https://en.wikipedia.org/wiki/Self-financing%20portfolio | In financial mathematics, a self-financing portfolio is a portfolio having the feature that, if there is no exogenous infusion or withdrawal of money, the purchase of a new asset must be financed by the sale of an old one. This concept is used to define for example admissible strategies and replicating portfolios, the latter being fundamental for arbitrage-free derivative pricing.
Mathematical definition
Discrete time
Assume we are given a discrete filtered probability space , and let be the solvency cone (with or without transaction costs) at time t for the market. Denote by . Then a portfolio (in physical units, i.e. the number of each stock) is self-financing (with trading on a finite set of times only) if
for all we have that with the convention that .
If we are only concerned with the set that the portfolio can be at some future time then we can say that .
If there are transaction costs then only discrete trading should be considered, and in continuous time then the above calculations should be taken to the limit such that .
Continuous time
Let be a d-dimensional semimartingale frictionless market market and a d-dimensional predictable stochastic process such that the stochastic integrals exist . The process denote the number of shares of stock number in the portfolio at time , and the price of stock number . Denote the value process of the trading strategy by
Then the portfolio/the trading strategy is called self-financing if
.
The term corresponds to the initial wealth of the portfolio, while is the cumulated gain or loss from trading up to time . This means in particular that there have been no infusion of money in or withdrawal of money from the portfolio.
See also
Replicating portfolio
References
Mathematical finance |
https://en.wikipedia.org/wiki/Hessenberg%20variety | In geometry, Hessenberg varieties, first studied by Filippo De Mari, Claudio Procesi, and Mark A. Shayman, are a family of subvarieties of the full flag variety which are defined by a Hessenberg function h and a linear transformation X. The study of Hessenberg varieties was first motivated by questions in numerical analysis in relation to algorithms for computing eigenvalues and eigenspaces of the linear operator X. Later work by T. A. Springer, Dale Peterson, Bertram Kostant, among others, found connections with combinatorics, representation theory and cohomology.
Definitions
A Hessenberg function is a map
such that
for each i. For example, the function that sends the numbers 1 to 5 (in order) to 2, 3, 3, 4, and 5 is a Hessenberg function.
For any Hessenberg function h and a linear transformation
the Hessenberg variety is the set of all flags such that
for all i.
Examples
Some examples of Hessenberg varieties (with their function) include:
The Full Flag variety: h(i) = n for all i
The Peterson variety: for
The Springer variety: for all .
References
Bertram Kostant, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight , Selecta Mathematica (N.S.) 2, 1996, 43–91.
Julianna Tymoczko, Linear conditions imposed on flag varieties, American Journal of Mathematics 128 (2006), 1587–1604.
Algebraic geometry
Algebraic combinatorics |
https://en.wikipedia.org/wiki/University%20of%20Sydney%20School%20of%20Mathematics%20and%20Statistics | The School of Mathematics and Statistics is a constituent body of the Faculty of Science at the University of Sydney, Australia. It was established in its present form in 1991.
As of 29 August 2022, the Head of School is Professor Dingxuan Zhou, and the Deputy Head of School is Professor Mary Myerscough.
The Magma computer algebra system is produced and distributed by the Computational Algebra Group within the School.
History
Mathematics has been taught at the university since its establishment. The first Professor of Mathematics was Morris Pell, one of the university's three foundation professors. Pell gave the first lecture in mathematics on 13 October 1852, two days after the university's inauguration, to all 24 students at the time. At least two years of mathematics were still required of all the university's students at the time of Pell's retirement in 1877.
The School of Mathematics and Statistics was established on 1 January 1991 with the merging of the departments of Applied Mathematics, Pure Mathematics, and Mathematical Statistics.
The School is located in the Carslaw Building, which was completed in the early 1960s, and is named after mathematician Horatio Carslaw (1870-1954), who was once Professor of Mathematics at the university.
References
External links
School of Mathematics and Statistics at the Faculty of Science, University of Sydney
Educational institutions established in 1991
Schools of mathematics
Mathematics and Statistics, School of
1991 establishments in Australia |
https://en.wikipedia.org/wiki/Journal%20of%20Epidemiology%20and%20Biostatistics | The Journal of Epidemiology and Biostatistics is a peer reviewed journal for epidemiological and biostatistical studies. It is published by Martin Dunitz Ltd, part of Taylor & Francis Group. It covers Biology, Mathematics and Statistics and Public Health. In 2002 the title was changed to Journal of Cancer Epidemiology and Prevention.
References
Epidemiology journals |
https://en.wikipedia.org/wiki/Integral%20polytope | In geometry and polyhedral combinatorics, an integral polytope is a convex polytope whose vertices all have integer Cartesian coordinates. That is, it is a polytope that equals the convex hull of its integer points.
Integral polytopes are also called lattice polytopes or Z-polytopes. The special cases of two- and three-dimensional integral polytopes may be called polygons or polyhedra instead of polytopes, respectively.
Examples
An -dimensional regular simplex can be represented as an integer polytope in , the convex hull of the integer points for which one coordinate is one and the rest are zero. Another important type of integral simplex, the orthoscheme, can be formed as the convex hull of integer points whose coordinates begin with some number of consecutive ones followed by zeros in all remaining coordinates. The -dimensional unit cube in has as its vertices all integer points whose coordinates are zero or one. A permutahedron has vertices whose coordinates are obtained by applying all possible permutations to the vector . An associahedron in Loday's convex realization is also an integer polytope and a deformation of the permutahedron.
In optimization
In the context of linear programming and related problems in mathematical optimization, convex polytopes are often described by a system of linear inequalities that their points must obey. When a polytope is integral, linear programming can be used to solve integer programming problems for the given system of inequalities, a problem that can otherwise be more difficult.
Some polyhedra arising from combinatorial optimization problems are automatically integral. For instance, this is true of the order polytope of any partially ordered set, a polytope defined by pairwise inequalities between coordinates corresponding to comparable elements in the set. Another well-known polytope in combinatorial optimization is the matching polytope. Clearly, one seeks for finding matchings algorithmically and one technique is linear programming. The polytope described by the linear program upper bounding the sum of edges taken per vertex is integral in the case of bipartite graphs, that is, it exactly describes the matching polytope, while for general graphs it is non-integral. Hence, for bipartite graphs, it suffices to solve the corresponding linear program to obtain a valid matching. For general graphs, however, there are two other characterizations of the matching polytope one of which makes use of the blossom inequality for odd subsets of vertices and hence allows to relax the integer program to a linear program while still obtaining valid matchings. These characterizations are of further interest in Edmonds' famous blossom algorithm used for finding such matchings in general graphs.
Computational complexity
For a polytope described by linear inequalities, when the polytope is non-integral, one can prove its non-integrality by finding a vertex whose coordinates are not integers. Such a vertex can be des |
https://en.wikipedia.org/wiki/VSH | VSH may refer to:
Mathematics
Vector spherical harmonics
Very smooth hash, in cryptography
Media and entertainment
VSH News, a Pakistani television station
XrossMediaBar (Sony codename: VSH)
Other uses
VSH project, to develop suborbital spacecraft
Varroa sensitive hygiene, in bees
company stock code for Vishay Intertechnology |
https://en.wikipedia.org/wiki/Morris%20Birkbeck%20Pell | Morris Birkbeck Pell (31 March 1827 – 7 May 1879) was an American-Australian mathematician, professor, lawyer and actuary. He became the inaugural Professor of Mathematics and Natural Philosophy at the University of Sydney in 1852, and continued in the role until ill health enforced his retirement in 1877. He was for many years a member of the University Senate, and councillor and secretary of the Royal Society of New South Wales.
Early life
Pell's mother Eliza Birkbeck (1797-1880) was a daughter of Morris Birkbeck (1764-1825), the English agricultural innovator, social reformer and antislavery campaigner. In 1817-18 Birkbeck, with George Flower, had founded a utopian colony, the English Settlement, in the Illinois Territory of the United States, and Birkbeck laid out the new town there of Albion, Illinois. A widower since 1804, Birkbeck had brought his seven children with him to America, and it was there that his daughter Eliza met and married Gilbert Titus Pell (1796-1860), who came from a prominent family of New York politicians. Gilbert Pell was descended from Sir John Pell (1643-1702), Lord of Pelham Manor, New York—who was the son of English mathematician Dr. John Pell (1611-1685), and nephew and heir of early American pioneer and settler Thomas Pell. Gilbert Pell served as a representative in the Illinois legislature, and in the 1850s was appointed United States envoy to Mexico.
Morris Pell was born of this union in the new settlement of Albion in 1827, their third child and only son. In 1835 the family separated and Mrs Pell took her children first to Poughkeepsie, New York, then to Plymouth, England, in 1841, where Morris attended the New Grammar School. In 1849 he graduated as Senior Wrangler in mathematics at Cambridge University—a position once regarded as "the greatest intellectual achievement attainable in Britain."
Career
In 1852, aged 24, Pell was chosen from twenty-six candidates to become the first Professor of Mathematics and Natural Philosophy at the newly opened University of Sydney, in the British colony of New South Wales, Australia. With his new wife Jane Juliana (née Rusden), his mother and two sisters he sailed from England to Australia on the Asiatic and became one of the university's three foundation professors. Professor Pell gave the first lecture in Mathematics on 13 October 1852, two days after the university's inauguration, to all 24 students of the university. One of them, William Windeyer, later to become Chancellor of the university, wrote in his diary: "Went to a lecture at 10 with Mr Pell, who amused as well as instructed, I think I shall like him ...".
In 1854, in evidence to a New South Wales Legislative Council select committee on education, Pell advocated the opening of a secular grammar school. In 1859 he testified to the New South Wales Legislative Assembly select committees on the Sydney Grammar School and the University of Sydney, regarding the composition of the University Senate, the adverse |
https://en.wikipedia.org/wiki/Special%20values%20of%20L-functions | In mathematics, the study of special values of -functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely
by the recognition that expression on the left-hand side is also where is the Dirichlet -function for the field of Gaussian rational numbers. This formula is a special case of the analytic class number formula, and in those terms reads that the Gaussian field has class number 1. The factor on the right hand side of the formula corresponds to the fact that this field contains four roots of unity.
Conjectures
There are two families of conjectures, formulated for general classes of -functions (the very general setting being for -functions associated to Chow motives over number fields), the division into two reflecting the questions of:
how to replace in the Leibniz formula by some other "transcendental" number (regardless of whether it is currently possible for transcendental number theory to provide a proof of the transcendence); and
how to generalise the rational factor in the formula (class number divided by number of roots of unity) by some algebraic construction of a rational number that will represent the ratio of the -function value to the "transcendental" factor.
Subsidiary explanations are given for the integer values of for which a formulae of this sort involving can be expected to hold.
The conjectures for (a) are called Beilinson's conjectures, for Alexander Beilinson. The idea is to abstract from the regulator of a number field to some "higher regulator" (the Beilinson regulator), a determinant constructed on a real vector space that comes from algebraic K-theory.
The conjectures for (b) are called the Bloch–Kato conjectures for special values (for Spencer Bloch and Kazuya Kato; this circle of ideas is distinct from the Bloch–Kato conjecture of K-theory, extending the Milnor conjecture, a proof of which was announced in 2009). For the sake of greater clarity, they are also called the Tamagawa number conjecture, a name arising via the Birch–Swinnerton-Dyer conjecture and its formulation as an elliptic curve analogue of the Tamagawa number problem for linear algebraic groups. In a further extension, the equivariant Tamagawa number conjecture (ETNC) has been formulated, to consolidate the connection of these ideas with Iwasawa theory, and its so-called Main Conjecture.
Current status
All of these conjectures are known to be true only in special cases.
See also
Brumer–Stark conjecture
Notes
References
External links
L-funktionen und die Vermutingen von Deligne und Beilinson (L-functions and the conjectures of Deligne and Beilsnson)
Zeta and L-functions |
https://en.wikipedia.org/wiki/MTV%20Europe%20Music%20Award%20for%20Best%20Artist | The following is a list of the MTV Europe Music Award winners and nominees for Best Artist.
Winners and nominees
2000s
2010s
2020s
Statistics
The statistic holder of the 2018 award.
References
MTV Europe Music Awards
Awards established in 2007 |
https://en.wikipedia.org/wiki/David%20Mount | David Mount is a professor at the University of Maryland, College Park department of computer science whose research is in computational geometry.
Biography
Mount received a B.S. in Computer Science at Purdue University in 1977 and received his Ph.D. in Computer Science at Purdue University in 1983 under the advisement of Christoph Hoffmann.
He began teaching at the University of Maryland in 1984 and is Professor in the department of Computer Science there.
As a teacher, he has won the University of Maryland, College of Computer Mathematical and Physical Sciences Dean's Award for Excellence in Teaching in 2005 and 1997 as well as other teaching awards including the Hong Kong Science and Technology, School of Engineering Award for Teaching Excellence Appreciation in 2001.
Research
Mounts's main area of research is computational geometry, which is the branch of algorithms devoted to solving problems of a geometric nature. This field includes problems from classic geometry, like the closest pair of points problem, as well as more recent applied problems, such as computer representation and modeling of curves and surfaces. In particular, Mount has worked on the k-means clustering problem, nearest neighbor search, and point location problem.
Mount has worked on developing practical algorithms for k-means clustering, a problem known to be NP-hard. The most common algorithm used is Lloyd's algorithm, which is heuristic in nature but performs well in practice. He and others later showed how k-d trees could be used to speed up Lloyd's algorithm. They have implemented this algorithm, along with some additional improvements, in the software library Kmeans.
Mount has worked on the nearest neighbor and approximate nearest neighbor search problems. By allowing the algorithm to return an approximate solution to the nearest neighbor query, a significant speedup in space and time complexity can be obtained. One class of approximate algorithms takes as input the error distance, , and forms a data structure that can be stored efficiently (low space complexity) and that returns the -approximate nearest neighbor quickly (low time complexity). In co-authored work with Arya, Netanyahu, R. Silverman and A. Wu, Mount showed that the approximate nearest neighbor problem could be solved efficiently in spaces of low dimension. The data structure described in that paper formed the basis of the ANN open-source library for approximate nearest neighbor searching. In subsequent work, he investigated the computational complexity of approximate nearest neighbor searching. Together with co-authors Arya and Malamatos, he provided efficient space–time tradeoffs for approximate nearest neighbor searching, based on a data structure called the AVD (or approximate Voronoi diagram).
Mount has also worked on point location, which involves preprocessing a planar polygonal subdivision S of size to determine the cell of a subdivision that a query point is in. The paper give |
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