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https://en.wikipedia.org/wiki/Hadamard%20regularization | In mathematics, Hadamard regularization (also called Hadamard finite part or Hadamard's partie finie) is a method of regularizing divergent integrals by dropping some divergent terms and keeping the finite part, introduced by . showed that this can be interpreted as taking the meromorphic continuation of a convergent integral.
If the Cauchy principal value integral
exists, then it may be differentiated with respect to to obtain the Hadamard finite part integral as follows:
Note that the symbols and are used here to denote Cauchy principal value and Hadamard finite-part integrals respectively.
The Hadamard finite part integral above (for ) may also be given by the following equivalent definitions:
The definitions above may be derived by assuming that the function is differentiable infinitely many times at , that is, by assuming that can be represented by its Taylor series about . For details, see . (Note that the term in the second equivalent definition above is missing in but this is corrected in the errata sheet of the book.)
Integral equations containing Hadamard finite part integrals (with unknown) are termed hypersingular integral equations. Hypersingular integral equations arise in the formulation of many problems in mechanics, such as in fracture analysis.
Example
Consider the divergent integral
Its Cauchy principal value also diverges since
To assign a finite value to this divergent integral, we may consider
The inner Cauchy principal value is given by
Therefore,
Note that this value does not represent the area under the curve , which is clearly always positive.
References
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Integrals
Summability methods |
https://en.wikipedia.org/wiki/La%20Estrella%2C%20Chile | La Estrella is a Chilean town and commune in Cardenal Caro Province, O'Higgins Region.
Demographics
According to the 2002 census of the National Statistics Institute, La Estrella spans an area of and has 4,221 inhabitants (2,766 men and 1,455 women). Of these, 1,380 (32.7%) lived in urban areas and 2,841 (67.3%) in rural areas. The population grew by 51.9% (1,442 persons) between the 1992 and 2002 censuses.
Administration
As a commune, La Estrella is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2021-2024 alcalde is Angélica Silva Arrué.
References
External links
Municipality of La Estrella
Communes of Chile
Populated places in Cardenal Caro Province |
https://en.wikipedia.org/wiki/Paredones | Paredones is a Chilean town and commune in Cardenal Caro Province, O'Higgins Region.
Demographics
According to the 2002 census of the National Statistics Institute, Paredones spans an area of and has 6,695 inhabitants (3,562 men and 3,133 women). Of these, 2,195 (32.8%) lived in urban areas and 4,500 (67.2%) in rural areas. The population grew by 1.1% (73 persons) between the 1992 and 2002 censuses.
Administration
As a commune, Paredones is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2021-2024 alcalde is Antonio Carvacho Vargas.
See also
Bucalemu
References
External links
Municipality of Paredones
Communes of Chile
Populated places in Cardenal Caro Province |
https://en.wikipedia.org/wiki/Manolo%20Jim%C3%A9nez%20%28footballer%2C%20born%201976%29 | Manuel "Manolo" Jiménez Soria (born 12 August 1976) is an Andorran footballer. He currently plays for Santa Coloma and the Andorra national football team.
International statistics
Updated 28 September 2014."
International goalScores and results list Andorra's goal tally first.''
References
External links
1976 births
Living people
Andorran men's footballers
FC Andorra players
Andorra men's international footballers
FC Santa Coloma players
Men's association football midfielders
Primera Divisió players |
https://en.wikipedia.org/wiki/Proof%20by%20intimidation | Proof by intimidation (or argumentum verbosum) is a jocular phrase used mainly in mathematics to refer to a specific form of hand-waving, whereby one attempts to advance an argument by marking it as obvious or trivial, or by giving an argument loaded with jargon and obscure results. It attempts to intimidate the audience into simply accepting the result without evidence, by appealing to their ignorance and lack of understanding.
The phrase is often used when the author is an authority in their field, presenting their proof to people who respect a priori the author's insistence of the validity of the proof, while in other cases, the author might simply claim that their statement is true because it is trivial or because they say so. Usage of this phrase is for the most part in good humour, though it can also appear in serious criticism. A proof by intimidation is often associated with phrases such as:
"Clearly..."
"It is self-evident that..."
"It can be easily shown that..."
"... does not warrant a proof."
"The proof is left as an exercise for the reader."
Outside mathematics, "proof by intimidation" is also cited by critics of junk science, to describe cases in which scientific evidence is thrown aside in favour of dubious arguments—such as those presented to the public by articulate advocates who pose as experts in their field.
Proof by intimidation may also back valid assertions. Ronald A. Fisher claimed in the book credited with the new evolutionary synthesis, "...by the analogy of compound interest the present value of the future offspring of persons aged x is easily seen to be...", thence presenting a novel integral-laden definition of reproductive value. At this, Hal Caswell remarked, "With all due respect to Fisher, I have yet to meet anyone who finds this equation 'easily seen.'" Valid proofs were provided by subsequent researchers such as Leo A. Goodman (1968).
In a memoir, Gian-Carlo Rota claimed that the expression "proof by intimidation" was coined by Mark Kac, to describe a technique used by William Feller in his lectures:
See also
References
Professional humor
Mathematical proofs
In-jokes
Mathematical humor
English phrases |
https://en.wikipedia.org/wiki/Trivialization | Trivialization or trivialisation may refer to:
Trivialization (mathematics), a trivialization of a fiber bundle
Trivialization (psychology), a form of minimization, a cognitive distortion |
https://en.wikipedia.org/wiki/Underdetermined%20system | In mathematics, a system of linear equations or a system of polynomial equations is considered underdetermined if there are fewer equations than unknowns (in contrast to an overdetermined system, where there are more equations than unknowns). The terminology can be explained using the concept of constraint counting. Each unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom.
Therefore, the critical case (between overdetermined and underdetermined) occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint removing a degree of freedom. The underdetermined case, by contrast, occurs when the system has been underconstrained—that is, when the unknowns outnumber the equations.
Solutions of underdetermined systems
An underdetermined linear system has either no solution or infinitely many solutions.
For example,
is an underdetermined system without any solution; any system of equations having no solution is said to be inconsistent. On the other hand, the system
is consistent and has an infinitude of solutions, such as , , and . All of these solutions can be characterized by first subtracting the first equation from the second, to show that all solutions obey ; using this in either equation shows that any value of y is possible, with .
More specifically, according to the Rouché–Capelli theorem, any system of linear equations (underdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution; since in an underdetermined system this rank is necessarily less than the number of unknowns, there are indeed an infinitude of solutions, with the general solution having k free parameters where k is the difference between the number of variables and the rank.
There are algorithms to decide whether an underdetermined system has solutions, and if it has any, to express all solutions as linear functions of k of the variables (same k as above). The simplest one is Gaussian elimination. See System of linear equations for more details.
Homogeneous case
The homogeneous (with all constant terms equal to zero) underdetermined linear system always has non-trivial solutions (in addition to the trivial solution where all the unknowns are zero). There are an infinity of such solutions, which form a vector space, whose dimension is the difference between the number of unknowns and the rank of the matrix of the system.
Underdetermined polynomial systems
The main property of linear underdetermined systems, of having either no solution or infinitely many, extends to systems of polynomial equations in the following way.
A system of polynomial equations which has fewer equations than |
https://en.wikipedia.org/wiki/Bing%27s%20recognition%20theorem | In topology, a branch of mathematics, Bing's recognition theorem, named for R. H. Bing, asserts that a necessary and sufficient condition for a 3-manifold M to be homeomorphic to the 3-sphere is that every Jordan curve in M be contained within a topological ball.
References
3-manifolds
Geometric topology
Theorems in topology |
https://en.wikipedia.org/wiki/Prescribed%20Ricci%20curvature%20problem | In Riemannian geometry, a branch of mathematics, the prescribed Ricci curvature problem is as follows: given a smooth manifold M and a symmetric 2-tensor h, construct a metric on M whose Ricci curvature tensor equals h.
See also
Prescribed scalar curvature problem
References
Thierry Aubin, Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics, 1998.
Arthur L. Besse. Einstein manifolds. Reprint of the 1987 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2008. xii+516 pp.
Dennis M. DeTurck, Existence of metrics with prescribed Ricci curvature: local theory. Invent. Math. 65 (1981/82), no. 1, 179–207.
Riemannian geometry
Mathematical problems
ricci curvature |
https://en.wikipedia.org/wiki/Ellis%20Kolchin | Ellis Robert Kolchin (April 18, 1916 – October 30, 1991) was an American mathematician at Columbia University. Kolchin earned a doctorate in mathematics from Columbia University in 1941 under supervision of Joseph Ritt. He was awarded a Guggenheim Fellowship in 1954 and 1961.
Kolchin worked on differential algebra and its relation to differential equations, and founded the modern theory of linear algebraic groups. His doctoral students include Azriel Rosenfeld and Irving Adler.
See also
Kolchin topology
Lie–Kolchin theorem
Picard–Vessiot theory
Publications
References
External links
1916 births
1991 deaths
20th-century American mathematicians
Columbia University alumni
Columbia University faculty
Institute for Advanced Study visiting scholars |
https://en.wikipedia.org/wiki/Prescribed%20scalar%20curvature%20problem | In Riemannian geometry, a branch of mathematics, the prescribed scalar curvature problem is as follows: given a closed, smooth manifold M and a smooth, real-valued function ƒ on M, construct a Riemannian metric on M whose scalar curvature equals ƒ. Due primarily to the work of J. Kazdan and F. Warner in the 1970s, this problem is well understood.
The solution in higher dimensions
If the dimension of M is three or greater, then any smooth function ƒ which takes on a negative value somewhere is the scalar curvature of some Riemannian metric. The assumption that ƒ be negative somewhere is needed in general, since not all manifolds admit metrics which have strictly positive scalar curvature. (For example, the three-dimensional torus is such a manifold.) However, Kazdan and Warner proved that if M does admit some metric with strictly positive scalar curvature, then any smooth function ƒ is the scalar curvature of some Riemannian metric.
See also
Prescribed Ricci curvature problem
Yamabe problem
References
Aubin, Thierry. Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics, 1998.
Kazdan, J., and Warner F. Scalar curvature and conformal deformation of Riemannian structure. Journal of Differential Geometry. 10 (1975). 113–134.
Riemannian geometry
Mathematical problems
Scalar curvature |
https://en.wikipedia.org/wiki/Half-normal%20distribution | In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.
Let follow an ordinary normal distribution, . Then, follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero.
Properties
Using the parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by
where .
Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if is near zero), obtained by setting , the probability density function is given by
where .
The cumulative distribution function (CDF) is given by
Using the change-of-variables , the CDF can be written as
where erf is the error function, a standard function in many mathematical software packages.
The quantile function (or inverse CDF) is written:
where and is the inverse error function
The expectation is then given by
The variance is given by
Since this is proportional to the variance σ2 of X, σ can be seen as a scale parameter of the new distribution.
The differential entropy of the half-normal distribution is exactly one bit less the differential entropy of a zero-mean normal distribution with the same second moment about 0. This can be understood intuitively since the magnitude operator reduces information by one bit (if the probability distribution at its input is even). Alternatively, since a half-normal distribution is always positive, the one bit it would take to record whether a standard normal random variable were positive (say, a 1) or negative (say, a 0) is no longer necessary. Thus,
Applications
The half-normal distribution is commonly utilized as a prior probability distribution for variance parameters in Bayesian inference applications.
Parameter estimation
Given numbers drawn from a half-normal distribution, the unknown parameter of that distribution can be estimated by the method of maximum likelihood, giving
The bias is equal to
which yields the bias-corrected maximum likelihood estimator
Related distributions
The distribution is a special case of the folded normal distribution with μ = 0.
It also coincides with a zero-mean normal distribution truncated from below at zero (see truncated normal distribution)
If Y has a half-normal distribution, then (Y/σ)2 has a chi square distribution with 1 degree of freedom, i.e. Y/σ has a chi distribution with 1 degree of freedom.
The half-normal distribution is a special case of the generalized gamma distribution with d = 1, p = 2, a = .
If Y has a half-normal distribution, Y -2 has a Levy distribution
The Rayleigh distribution is a moment-tilted and scaled generalization of the half-normal distribution.
Modified half-normal distribution with the pdf on is given as , where denotes the Fox–Wright Psi function.
See also
Half-t distribution
Truncated normal distribution
Folded no |
https://en.wikipedia.org/wiki/Ratner%27s%20theorems | In mathematics, Ratner's theorems are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The theorems grew out of Ratner's earlier work on horocycle flows. The study of the dynamics of unipotent flows played a decisive role in the proof of the Oppenheim conjecture by Grigory Margulis. Ratner's theorems have guided key advances in the understanding of the dynamics of unipotent flows. Their later generalizations provide ways to both sharpen the results and extend the theory to the setting of arbitrary semisimple algebraic groups over a local field.
Short description
The Ratner orbit closure theorem asserts that the closures of orbits of unipotent flows on the quotient of a Lie group by a lattice are nice, geometric subsets. The Ratner equidistribution theorem further asserts that each such orbit is equidistributed in its closure. The Ratner measure classification theorem is the weaker statement that every ergodic invariant probability measure is homogeneous, or algebraic: this turns out to be an important step towards proving the more general equidistribution property. There is no universal agreement on the names of these theorems: they are variously known as the "measure rigidity theorem", the "theorem on invariant measures" and its "topological version", and so on.
The formal statement of such a result is as follows. Let be a Lie group, a lattice in , and a one-parameter subgroup of consisting of unipotent elements, with the associated flow on . Then the closure of every orbit of is homogeneous. This means that there exists a connected, closed subgroup of such that the image of the orbit for the action of by right translations on under the canonical projection to is closed, has a finite -invariant measure, and contains the closure of the -orbit of as a dense subset.
Example:
The simplest case to which the statement above applies is . In this case it takes the following more explicit form; let be a lattice in and a closed subset which is invariant under all maps where . Then either there exists an such that (where ) or .
In geometric terms is a cofinite Fuchsian group, so the quotient of the hyperbolic plane by is a hyperbolic orbifold of finite volume. The theorem above implies that every horocycle of has an image in which is either a closed curve (a horocycle around a cusp of ) or dense in .
See also
Equidistribution theorem
References
Expositions
Selected original articles
Ergodic theory
Lie groups
Theorems in dynamical systems |
https://en.wikipedia.org/wiki/Elongation | Elongation may refer to:
Elongation (astronomy)
Elongation (geometry)
Elongation (plasma physics)
Part of transcription of DNA into RNA of all types, including mRNA, tRNA, rRNA, etc.
Part of translation (biology) of mRNA into proteins
Elongated organisms
Elongation (mechanics), linear deformation
See also |
https://en.wikipedia.org/wiki/Conway%20triangle%20notation | In geometry, the Conway triangle notation, named after John Horton Conway, allows trigonometric functions of a triangle to be managed algebraically. Given a reference triangle whose sides are a, b and c and whose corresponding internal angles are A, B, and C then the Conway triangle notation is simply represented as follows:
where S = 2 × area of reference triangle and
in particular
where is the Brocard angle. The law of cosines is used: .
for values of where
Furthermore the convention uses a shorthand notation for and
Hence:
Some important identities:
where R is the circumradius and abc = 2SR and where r is the incenter, and
Some useful trigonometric conversions:
Some useful formulas:
Some examples using Conway triangle notation:
Let D be the distance between two points P and Q whose trilinear coordinates are pa : pb : pc and qa : qb : qc. Let Kp = apa + bpb + cpc and let Kq = aqa + bqb + cqc. Then D is given by the formula:
Using this formula it is possible to determine OH, the distance between the circumcenter and the orthocenter as follows:
For the circumcenter pa = aSA and for the orthocenter qa = SBSC/a
Hence:
This gives:
References
Triangle geometry
Trigonometry
John Horton Conway |
https://en.wikipedia.org/wiki/Best%20linear%20unbiased%20prediction | In statistics, best linear unbiased prediction (BLUP) is used in linear mixed models for the estimation of random effects. BLUP was derived by Charles Roy Henderson in 1950 but the term "best linear unbiased predictor" (or "prediction") seems not to have been used until 1962. "Best linear unbiased predictions" (BLUPs) of random effects are similar to best linear unbiased estimates (BLUEs) (see Gauss–Markov theorem) of fixed effects. The distinction arises because it is conventional to talk not about estimating fixed effects but rather about predicting random effects, but the two terms are otherwise equivalent. (This is a bit strange since the random effects have already been "realized"; they already exist. The use of the term "prediction" may be because in the field of animal breeding in which Henderson worked, the random effects were usually genetic merit, which could be used to predict the quality of offspring (Robinson page 28)). However, the equations for the "fixed" effects and for the random effects are different.
In practice, it is often the case that the parameters associated with the random effect(s) term(s) are unknown; these parameters are the variances of the random effects and residuals. Typically the parameters are estimated and plugged into the predictor, leading to the empirical best linear unbiased predictor (EBLUP). Notice that by simply plugging in the estimated parameter into the predictor, additional variability is unaccounted for, leading to overly optimistic prediction variances for the EBLUP.
Best linear unbiased predictions are similar to empirical Bayes estimates of random effects in linear mixed models, except that in the latter case, where weights depend on unknown values of components of variance, these unknown variances are replaced by sample-based estimates.
Example
Suppose that the model for observations {Yj ; j = 1, ..., n} is written as
where is the mean of all observations , and ξj and εj represent the random effect and observation error for observation j, and suppose they are uncorrelated and have known variances σξ2 and σε2, respectively. Further, xj is a vector of independent variables for the jth observation and is a vector of regression parameters.
The BLUP problem of providing an estimate of the observation-error-free value for the kth observation,
can be formulated as requiring that the coefficients of a linear predictor, defined as
should be chosen so as to minimise the variance of the prediction error,
subject to the condition that the predictor is unbiased,
BLUP vs BLUE
In contrast to the case of best linear unbiased estimation, the "quantity to be estimated", , not only has a contribution from a random element but one of the observed quantities, specifically which contributes to , also has a contribution from this same random element.
In contrast to BLUE, BLUP takes into account known or estimated variances.
History of BLUP in breeding
Henderson explored breeding from a statistical poi |
https://en.wikipedia.org/wiki/Malcolm%20Ludvigsen | Malcolm Ludvigsen (born 14 February 1946) is a British mathematician and plein air painter. He is a former research fellow and visiting lecturer in mathematics at the University of York and the author of a book on general relativity. Many of his paintings depict the beaches of the Yorkshire coast.
References
External links
Official site
1946 births
Living people
20th-century British mathematicians
21st-century British mathematicians
20th-century British painters
British male painters
21st-century British painters
British relativity theorists
20th-century British male artists
21st-century British male artists |
https://en.wikipedia.org/wiki/Cochran%27s%20Q%20test | In statistics, in the analysis of two-way randomized block designs where the response variable can take only two possible outcomes (coded as 0 and 1), Cochran's Q test is a non-parametric statistical test to verify whether k treatments have identical effects. It is named after William Gemmell Cochran. Cochran's Q test should not be confused with Cochran's C test, which is a variance outlier test. Put in simple technical terms, Cochran's Q test requires that there only be a binary response (e.g. success/failure or 1/0) and that there be more than 2 groups of the same size. The test assesses whether the proportion of successes is the same between groups. Often it is used to assess if different observers of the same phenomenon have consistent results (interobserver variability).
Background
Cochran's Q test assumes that there are k > 2 experimental treatments and that the observations are arranged in b blocks; that is,
The "blocks" here might be individual people or other organisms. For example, if b respondents in a survey had each been asked k Yes/No questions, the Q test could be used to test the null hypothesis that all questions were equally likely to elicit the answer "Yes".
Description
Cochran's Q test is
Null hypothesis (H0): the treatments are equally effective.
Alternative hypothesis (Ha): there is a difference in effectiveness between treatments.
The Cochran's Q test statistic is
where
k is the number of treatments
X• j is the column total for the jth treatment
b is the number of blocks
Xi • is the row total for the ith block
N is the grand total
Critical region
For significance level α, the asymptotic critical region is
where Χ21 − α,k − 1 is the (1 − α)-quantile of the chi-squared distribution with k − 1 degrees of freedom. The null hypothesis is rejected if the test statistic is in the critical region. If the Cochran test rejects the null hypothesis of equally effective treatments, pairwise multiple comparisons can be made by applying Cochran's Q test on the two treatments of interest.
The exact distribution of the T statistic may be computed for small samples. This allows obtaining an exact critical region. A first algorithm had been suggested in 1975 by Patil and a second one has been made available by Fahmy and Bellétoile in 2017.
Assumptions
Cochran's Q test is based on the following assumptions:
If the large sample approximation is used (and not the exact distribution), b is required to be "large".
The blocks were randomly selected from the population of all possible blocks.
The outcomes of the treatments can be coded as binary responses (i.e., a "0" or "1") in a way that is common to all treatments within each block.
Related tests
The Friedman test or Durbin test can be used when the response is not binary but ordinal or continuous.
When there are exactly two treatments the Cochran Q test is equivalent to McNemar's test, which is itself equivalent to a two-tailed sign test.
References
Statistical tests
Nonparametri |
https://en.wikipedia.org/wiki/List%20of%20Crewe%20Alexandra%20F.C.%20records%20and%20statistics | This article details Crewe Alexandra F.C. records since the club's establishment in 1877.
Honours and achievements
In 100 seasons, Crewe Alexandra have never won a division title, and have only been division runners-up twice.
Football League Second Division (3rd tier)
2nd place promotion: 2002–03
Play-off winners: 1997
Football League Fourth Division / League Two (4th tier)
2nd place promotion: 2019–20
3rd place promotion: 1962–63, 1993–94
4th place promotion: 1967–68, 1988–89
Play-off winners: 2012
Football League Trophy
Winners: 2013
Welsh Cup
Winners: 1936, 1937
Cheshire Senior Cup
Winners: 1910, 1912, 1913, 1923, 2002, 2003, 2017
Cheshire Premier Cup
Winners: 2009, 2010
Milk Cup
Premier Section Winners: 1987, 1999
Junior Section Winners: 1990, 1998
Club records
Football League record
Including the 2023-2024 season, Crewe Alexandra has played in 101 Football League seasons. The club has never played in the top flight, and has played only 12 seasons in the second tier (four, from 1892 to 1896, when there were only two divisions); 48 seasons have been in the third tier (30 when there were only three tiers), and 41 in the fourth tier. In total, Crewe has played in the bottom division of the Football League in 75 out of 101 seasons.
Aggregate league record from 1888-89 to 2022-23
Cup records
Wins
Biggest league win - 8–0 vs Rotherham United (Division 3 North, 1 October 1932) - six different Crewe scorers.
Biggest cup win - 9–1 against Northwich Victoria (FA Cup, 16 November 1889)Eight goal margins also achieved (8-0 wins) against:
Hartlepool United (Auto Windscreens Shield 1st Rd, 17 October 1995) - seven different Crewe scorers (plus an own goal).
Doncaster Rovers (LDV Vans Trophy 3rd Rd, 10 November 2002)
Defeats
Biggest league defeat - 1–11 vs Lincoln City (Division 3 North, 29 September 1951)
Biggest cup defeat - 2–13 vs Tottenham Hotspur (FA Cup 4th Rd Replay, 3 February 1960)
Sequences
Longest undefeated run in league - 19 (in 2011-12 season, 24 February 2012 to 17 May 2012 inclusive, including three play-off matches.)
Longest league run without a win - 30 (22 September 1956 to 6 April 1957)
Longest run without an away win - 56 (25 December 1954 to 24 April 1957)
Longest run of home defeats - 8 (29 January to 2 April 2022)
League points
Most league points in a season
Three points for win - 86 (2002-2003, Division 2)
Two points for win - 59 (1962-1963, Division 4 - under 'three points for win' system, total would have been 83)
Least league points in a season
Two points for win - 10 (1894-1895, Division 2 - 16 teams; Crewe lost all 15 away games that season)
Two points for win - 21 (1956-1957, Division 3 North - 24 teams)
Three points for win - 27 (1981-1982, Division 4 - 24 teams; tally would have been 21 under old points system)
Others
Most goals in a season - 95 (in 40 games), 1931-32 Division 3 North
Most clean sheets in a season - 24 (2002-2003, Clayton Ince 20 and Ademola Bankole 4)
Most away wins in |
https://en.wikipedia.org/wiki/Statistical%20graphics | Statistical graphics, also known as statistical graphical techniques, are graphics used in the field of statistics for data visualization.
Overview
Whereas statistics and data analysis procedures generally yield their output in numeric or tabular form, graphical techniques allow such results to be displayed in some sort of pictorial form. They include plots such as scatter plots, histograms, probability plots, spaghetti plots, residual plots, box plots, block plots and biplots.
Exploratory data analysis (EDA) relies heavily on such techniques. They can also provide insight into a data set to help with testing assumptions, model selection and regression model validation, estimator selection, relationship identification, factor effect determination, and outlier detection. In addition, the choice of appropriate statistical graphics can provide a convincing means of communicating the underlying message that is present in the data to others.
Graphical statistical methods have four objectives:
The exploration of the content of a data set
The use to find structure in data
Checking assumptions in statistical models
Communicate the results of an analysis.
If one is not using statistical graphics, then one is forfeiting insight into one or more aspects of the underlying structure of the data.
History
Statistical graphics have been central to the development of science and date to the earliest attempts to analyse data. Many familiar forms, including bivariate plots, statistical maps, bar charts, and coordinate paper were used in the 18th century. Statistical graphics developed through attention to four problems:
Spatial organization in the 17th and 18th century
Discrete comparison in the 18th and early 19th century
Continuous distribution in the 19th century and
Multivariate distribution and correlation in the late 19th and 20th century.
Since the 1970s statistical graphics have been re-emerging as an important analytic tool with the revitalisation of computer graphics and related technologies.
Examples
Famous graphics were designed by:
William Playfair who produced what could be called the first line, bar, pie, and area charts. For example, in 1786 he published the well known diagram that depicts the evolution of England's imports and exports,
Florence Nightingale, who used statistical graphics to persuade the British Government to improve army hygiene,
John Snow who plotted deaths from cholera in London in 1854 to detect the source of the disease, and
Charles Joseph Minard who designed a large portfolio of maps of which the one depicting Napoleon's campaign in Russia is the best known.
See the plots page for many more examples of statistical graphics.
See also
Data Presentation Architecture
List of graphical methods
Visual inspection
Chart
List of charting software
References
Citations
Attribution
Further reading
External links
Trend Compass
Alphabetic gallery of graphical techniques
DataScope a website devoted to data |
https://en.wikipedia.org/wiki/Sinusoidal%20model | In statistics, signal processing, and time series analysis, a sinusoidal model is used to approximate a sequence Yi to a sine function:
where C is constant defining a mean level, α is an amplitude for the sine, ω is the angular frequency, Ti is a time variable, φ is the phase-shift, and Ei is the error sequence.
This sinusoidal model can be fit using nonlinear least squares; to obtain a good fit, routines may require good starting values for the unknown parameters.
Fitting a model with a single sinusoid is a special case of spectral density estimation and least-squares spectral analysis.
Good starting values
Good starting value for the mean
A good starting value for C can be obtained by calculating the mean of the data. If the data show a trend, i.e., the assumption of constant location is violated, one can replace C with a linear or quadratic least squares fit. That is, the model becomes
or
Good starting value for frequency
The starting value for the frequency can be obtained from the dominant frequency in a periodogram. A complex demodulation phase plot can be used to refine this initial estimate for the frequency.
Good starting values for amplitude
The root mean square of the detrended data can be scaled by the square root of two to obtain an estimate of the sinusoid amplitude. A complex demodulation amplitude plot can be used to find a good starting value for the amplitude. In addition, this plot can indicate whether or not the amplitude is constant over the entire range of the data or if it varies. If the plot is essentially flat, i.e., zero slope, then it is reasonable to assume a constant amplitude in the non-linear model. However, if the slope varies over the range of the plot, one may need to adjust the model to be:
That is, one may replace α with a function of time. A linear fit is specified in the model above, but this can be replaced with a more elaborate function if needed.
Model validation
As with any statistical model, the fit should be subjected to graphical and quantitative techniques of model validation. For example, a run sequence plot to check for significant shifts in location, scale, start-up effects and outliers. A lag plot can be used to verify the residuals are independent. The outliers also appear in the lag plot, and a histogram and normal probability plot to check for skewness or other non-normality in the residuals.
Extensions
A different method consists in transforming the non-linear regression to a linear regression thanks to a convenient integral equation. Then, there is no need for initial guess and no need for iterative process : the fitting is directly obtained.
See also
Pitch detection algorithm
References
External links
Beam deflection case study
Regression with time series structure
Regression models |
https://en.wikipedia.org/wiki/Regression%20validation | In statistics, regression validation is the process of deciding whether the numerical results quantifying hypothesized relationships between variables, obtained from regression analysis, are acceptable as descriptions of the data. The validation process can involve analyzing the goodness of fit of the regression, analyzing whether the regression residuals are random, and checking whether the model's predictive performance deteriorates substantially when applied to data that were not used in model estimation.
Goodness of fit
One measure of goodness of fit is the R2 (coefficient of determination), which in ordinary least squares with an intercept ranges between 0 and 1. However, an R2 close to 1 does not guarantee that the model fits the data well: as Anscombe's quartet shows, a high R2 can occur in the presence of misspecification of the functional form of a relationship or in the presence of outliers that distort the true relationship.
One problem with the R2 as a measure of model validity is that it can always be increased by adding more variables into the model, except in the unlikely event that the additional variables are exactly uncorrelated with the dependent variable in the data sample being used. This problem can be avoided by doing an F-test of the statistical significance of the increase in the R2, or by instead using the adjusted R2.
Analysis of residuals
The residuals from a fitted model are the differences between the responses observed at each combination of values of the explanatory variables and the corresponding prediction of the response computed using the regression function. Mathematically, the definition of the residual for the ith observation in the data set is written
with yi denoting the ith response in the data set and xi the vector of explanatory variables, each set at the corresponding values found in the ith observation in the data set.
If the model fit to the data were correct, the residuals would approximate the random errors that make the relationship between the explanatory variables and the response variable a statistical relationship. Therefore, if the residuals appear to behave randomly, it suggests that the model fits the data well. On the other hand, if non-random structure is evident in the residuals, it is a clear sign that the model fits the data poorly. The next section details the types of plots to use to test different aspects of a model and gives the correct interpretations of different results that could be observed for each type of plot.
Graphical analysis of residuals
A basic, though not quantitatively precise, way to check for problems that render a model inadequate is to conduct a visual examination of the residuals (the mispredictions of the data used in quantifying the model) to look for obvious deviations from randomness. If a visual examination suggests, for example, the possible presence of heteroskedasticity (a relationship between the variance of the model errors and the size of an |
https://en.wikipedia.org/wiki/Lack-of-fit%20test | In statistics, a lack-of-fit test is any of many tests of a null hypothesis that a proposed statistical model fits well. See:
Goodness of fit
Lack-of-fit sum of squares |
https://en.wikipedia.org/wiki/Statistical%20Science | Statistical Science is a review journal published by the Institute of Mathematical Statistics. The founding editor was Morris H. DeGroot, who explained the mission of the journal in his 1986 editorial:
"A central purpose of Statistical Science is to convey the richness, breadth and unity of the field by presenting
the full range of contemporary statistical thought at a modest technical level accessible to the wide community
of practitioners, teachers, researchers and students of statistics and probability."
Editors
2017-2019 Cun-Hui Zhang
2014-2016 Peter Green
2011-2013 Jon Wellner
2008-2010 David Madigan
2005-2007 Ed George
2002-2004 George Casella
2001 Morris Eaton
2001 Richard Tweedie
1998-2000 Leon Gleser
1995-1997 Paul Switzer
1992-1994 Robert E. Kass
1989-1991 Carl N. Morris
1985-1989 Morris H. DeGroot
References
Further reading
External links
Statistical Science home page
Institute of Mathematical Statistics academic journals
Statistics journals
English-language journals
Academic journals established in 1986 |
https://en.wikipedia.org/wiki/Bitonic%20tour | In computational geometry, a bitonic tour of a set of point sites in the Euclidean plane is a closed polygonal chain that has each site as one of its vertices, such that any vertical line crosses the chain at most twice.
Optimal bitonic tours
The optimal bitonic tour is a bitonic tour of minimum total length. It is a standard exercise in dynamic programming to devise a polynomial time algorithm that constructs the optimal bitonic tour. Although the usual method for solving it in this way takes time , a faster algorithm with time is known.
The problem of constructing optimal bitonic tours is often credited to Jon L. Bentley, who published in 1990 an experimental comparison of many heuristics for the traveling salesman problem; however, Bentley's experiments do not include bitonic tours. The first publication that describes the bitonic tour problem appears to be a different 1990 publication, the first edition of the textbook Introduction to Algorithms by Thomas H. Cormen, Charles E. Leiserson, and Ron Rivest, which lists Bentley as the originator of the problem.
Properties
The optimal bitonic tour has no self-crossings, because any two edges that cross can be replaced by an uncrossed pair of edges with shorter total length due to the triangle inequality. Therefore, it forms a polygonalization of the input.
When compared to other tours that might not be bitonic,
the optimal bitonic tour is the one that minimizes the total amount of horizontal motion, with ties broken by Euclidean distance.
For points in the plane with distinct integer -coordinates and with real-number -coordinates that lie within an interval of length or less, the optimal bitonic tour is an optimal traveling salesperson tour.
Other optimization criteria
The same dynamic programming algorithm that finds the optimal bitonic tour may be used to solve other variants of the traveling salesman problem that minimize lexicographic combinations of motion in a fixed number of coordinate directions.
At the 5th International Olympiad in Informatics, in Mendoza, Argentina in 1993, one of the contest problems involved bitonic tours: the contestants were to devise an algorithm that took as input a set of sites and a collection of allowed edges between sites and construct a bitonic tour using those edges that included as many sites as possible. As with the optimal bitonic tour, this problem may be solved by dynamic programming.
References
Geometric algorithms
Dynamic programming |
https://en.wikipedia.org/wiki/Pocket%20set%20theory | Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ0 (aleph-naught, the cardinality of the set of all natural numbers) and c (the cardinality of the continuum). The theory was first suggested by Rudy Rucker in his Infinity and the Mind. The details set out in this entry are due to the American mathematician Randall M. Holmes.
Arguments supporting PST
There are at least two independent arguments in favor of a small set theory like PST.
One can get the impression from mathematical practice outside set theory that there are “only two infinite cardinals which demonstrably ‘occur in nature’ (the cardinality of the natural numbers and the cardinality of the continuum),” therefore “set theory produces far more superstructure than is needed to support classical mathematics.” Although it may be an exaggeration (one can get into a situation in which one has to talk about arbitrary sets of real numbers or real functions), with some technical tricks a considerable portion of mathematics can be reconstructed within PST; certainly enough for most of its practical applications.
A second argument arises from foundational considerations. Most of mathematics can be implemented in standard set theory or one of its large alternatives. Set theories, on the other hand, are introduced in terms of a logical system; in most cases it is first-order logic. The syntax and semantics of first-order logic, on the other hand, is built on set-theoretical grounds. Thus, there is a foundational circularity, which forces us to choose as weak a theory as possible for bootstrapping. This line of thought, again, leads to small set theories.
Thus, there are reasons to think that Cantor's infinite hierarchy of the infinites is superfluous. Pocket set theory is a “minimalistic” set theory that allows for only two infinites: the cardinality of the (standard) natural numbers and the cardinality of the (standard) reals.
Theory
PST uses standard first-order language with identity and the binary relation symbol . Ordinary variables are upper case X, Y, etc. In the intended interpretation, the variables these stand for classes, and the atomic formula means "class X is an element of class Y". A set is a class that is an element of a class. Small case variables x, y, etc. stand for sets. A proper class is a class that is not a set. Two classes are equinumerous iff a bijection exists between them. A class is infinite iff it is equinumerous with one of its proper subclasses. The axioms of PST are
(A1) (extensionality) — Classes that have the same elements are the same.
(A2) (class comprehension) — If is a formula, then there exists a class the elements of which are exactly those sets x that satisfy .
(A3) (axiom of infinity) — There is an infinite set, and all infinite sets are equinumerous.
(inf(x) stands for “x is infinite”; abbreviates that x is equinumerous with y.)
(A4) (limitation of size) – A class is a proper class i |
https://en.wikipedia.org/wiki/David%20P.%20Dobkin | David Paul Dobkin is an American computer scientist and the Phillip Y. Goldman '86 Professor of Computer Science at Princeton University. His research has concerned computational geometry and computer graphics.
Early life and education
Dobkin was born February 29, 1948, in Pittsburgh, Pennsylvania. He received a B.S. from the Massachusetts Institute of Technology in 1970 and then moved to Harvard University for his graduate studies, receiving a Ph.D. in applied mathematics in 1973 under the supervision of Roger W. Brockett.
Career
He taught at Yale University and the University of Arizona before moving to Princeton in 1981. He was initially appointed to the Department of Electrical Engineering and Computer Science at Princeton and was subsequently named one of the first professors of Computer Science when that department was formed in 1985. In 1999, he became the first holder of the Goldman chair after its namesake donated two million dollars to the university. He was chair of the Computer Science Department at Princeton from 1994 to 2003, and in 2003 was appointed Dean of the Faculty. David Dobkin also chaired the governing board of The Geometry Center, a NSF-established research and education center at the University of Minnesota.
Dobkin has been on the editorial boards of eight journals.
Recognition
In 1997 he was selected as a Fellow of the Association for Computing Machinery for his contributions to both fields.
References
Further reading
Dobkin keeps pace with faculty interests, Princeton Weekly Bulletin, January 9, 2006
External links
Dobkin's web site at the Princeton Computer Science department
Dobkin's publications at DBLP
1948 births
Living people
Massachusetts Institute of Technology alumni
Harvard University alumni
Yale University faculty
University of Arizona faculty
Princeton University faculty
American computer scientists
Researchers in geometric algorithms
Computer graphics researchers
Fellows of the Association for Computing Machinery
Taylor Allderdice High School alumni
Fulbright alumni |
https://en.wikipedia.org/wiki/Harvey%20Goldstein | Harvey Goldstein (30 October 1939 – 9 April 2020) was a British statistician known for his contributions to multilevel modelling methodology, statistical software, social statistics, and for applying this to educational assessment and league tables.
Goldstein was born in Whitechapel, London to a Jewish family. He was professor of social statistics in the Centre for Multilevel Modelling at the University of Bristol. From 1977 to 2005, he was professor of statistical methods at the Institute of Education of the University of London. He was author of a monograph on multilevel statistical models.
He came from a left-wing family, and as a teenager he briefly joined the Young Communist League. He was elected a fellow of the British Academy in 1996 and awarded the Guy Medal in silver by the Royal Statistical Society in 1998.
He died on 9 April 2020.
It was reported that his death was due to COVID-19 during the COVID-19 pandemic in England.
See also
MLwiN (software)
References
External links
Prof Harvey Goldstein, FBA at Debrett's People of Today
Full text of 2nd edition (1995) at author's website
1939 births
2020 deaths
British Jews
British statisticians
Academics of the University of Bristol
Academics of the UCL Institute of Education
Fellows of the British Academy
Deaths from the COVID-19 pandemic in England |
https://en.wikipedia.org/wiki/David%20Clayton | David George Clayton (born 13 June 1944), is a British statistician and epidemiologist. He is titular Professor of Biostatistics in the University of Cambridge and Wellcome Trust and Juvenile Diabetes Research Foundation Principal Research Fellow in the Diabetes and Inflammation Laboratory, where he chairs the statistics group. Clayton is an ISI highly cited researcher placing him in the top 250 most cited scientists in the mathematics world over the last 20 years.
Career
Clayton has worked in theoretical and applied statistics, both frequentist and Bayesian. With Norman Breslow he has published important work on generalized linear mixed models. Clayton was a pioneer in the application of MCMC methods to problems in biostatistics. More recently, he has worked in genetic epidemiology.
Clayton read Natural Sciences at King's College, Cambridge and following this worked as a researcher in ergonomics and cybernetics. He then worked as a statistician at the London School of Hygiene and Tropical Medicine, the University of Leicester and the MRC Biostatistics Unit in Cambridge before taking up his present position.
He was awarded the Guy Medal in Silver of the Royal Statistical Society in 1990 and, with Norman Breslow, the Snedecor Prize of the Committee of Presidents of Statistical Societies of North America (COPSS) in 1995. He was a lead statistician for the Wellcome Trust Case Control Consortium, a genome-wide association study.
References
Clayton, David and Michael Hills (1993) Statistical Models in Epidemiology Oxford University Press.
Living people
British statisticians
Genetic epidemiologists
British epidemiologists
1944 births
Wellcome Trust Principal Research Fellows |
https://en.wikipedia.org/wiki/Michael%20Healy%20%28statistician%29 | Michael John Romer Healy (26 November 1923 – 17 July 2016) was a British statistician known for his contributions to statistical computing, auxology, laboratory statistics and quality control, and methods for analysing longitudinal data, among other areas. He was professor of medical statistics at the London School of Hygiene and Tropical Medicine from 1977 until his retirement. The Royal Statistical Society awarded him the Guy Medal in Silver in 1979 and Gold in 1999, and he also acted as chairman of its medical section. He was the author or co-author of three books and over 200 scientific papers.
He died on 17 July 2016 at the age of 92.
Books
Assessment of Skeletal Maturity and Prediction of Adult Height (TW2Method) (with J. M. Tanner, R. H. Whitehouse, W. A. Marshall and H. Goldstein), Academic Press, London, 1975 (2nd edn, 1983, additionally with N. Cameron).
Matrices for Statistics, Oxford University Press, Oxford, 1986.
GLIM: an Introduction, Oxford University Press, Oxford, 1988.
References
1923 births
2016 deaths
British statisticians
Auxologists
Academics of the London School of Hygiene and Tropical Medicine
People from Paignton
Alumni of Trinity College, Cambridge |
https://en.wikipedia.org/wiki/Splitting%20principle | In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles.
In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computations are well understood for line bundles and for direct sums of line bundles. In this case the splitting principle can be quite useful.
The theorem above holds for complex vector bundles and integer coefficients or for real vector bundles with coefficients. In the complex case, the line bundles or their first characteristic classes are called Chern roots.
The fact that is injective means that any equation which holds in (say between various Chern classes) also holds in .
The point is that these equations are easier to understand for direct sums of line bundles than for arbitrary vector bundles, so equations should be understood in and then pushed down to .
Since vector bundles on are used to define the K-theory group , it is important to note that is also injective for the map in the above theorem.
The splitting principle admits many variations. The following, in particular, concerns real vector bundles and their complexifications:
Symmetric polynomial
Under the splitting principle, characteristic classes for complex vector bundles correspond to symmetric polynomials in the first Chern classes of complex line bundles; these are the Chern classes.
See also
K-theory
Grothendieck splitting principle for holomorphic vector bundles on the complex projective line
References
section 3.1
Raoul Bott and Loring Tu. Differential Forms in Algebraic Topology, section 21.
Characteristic classes
Vector bundles
Mathematical principles |
https://en.wikipedia.org/wiki/Infinitary%20combinatorics | In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets.
Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom.
Recent developments concern combinatorics of the continuum and combinatorics on successors of singular cardinals.
Ramsey theory for infinite sets
Write κ, λ for ordinals, m for a cardinal number and n for a natural number. introduced the notation
as a shorthand way of saying that every partition of the set [κ]n of n-element subsets of into m pieces has a homogeneous set of order type λ. A homogeneous set is in this case a subset of κ such that every n-element subset is in the same element of the partition. When m is 2 it is often omitted.
Assuming the axiom of choice, there are no ordinals κ with κ→(ω)ω, so n is usually taken to be finite. An extension where n is almost allowed to be infinite is
the notation
which is a shorthand way of saying that every partition of the set of finite subsets of κ into m pieces has a subset of order type λ such that for any finite n, all subsets of size n are in the same element of the partition. When m is 2 it is often omitted.
Another variation is the notation
which is a shorthand way of saying that every coloring of the set [κ]n of n-element subsets of κ with 2 colors has a subset of order type λ such that all elements of [λ]n have the first color, or a subset of order type μ such that all elements of [μ]n have the second color.
Some properties of this include: (in what follows is a cardinal)
for all finite n and k (Ramsey's theorem).
(Erdős–Rado theorem.)
(Sierpiński theorem)
(Erdős–Dushnik–Miller theorem).
In choiceless universes, partition properties with infinite exponents may hold, and some of them are obtained as consequences of the axiom of determinacy (AD). For example, Donald A. Martin proved that AD implies
Large cardinals
Several large cardinal properties can be defined using this notation. In particular:
Weakly compact cardinals κ are those that satisfy κ→(κ)2
α-Erdős cardinals κ are the smallest that satisfy κ→(α)<ω
Ramsey cardinals κ are those that satisfy κ→(κ)<ω
Notes
References
Set theory
Combinatorics |
https://en.wikipedia.org/wiki/Yusuke%20Tanaka%20%28footballer%2C%20born%20February%201986%29 | is a Japanese football player who plays for Kataller Toyama.
Club statistics
Updated to 23 February 2020.
References
External links
Profile at Ventforet Kofu
1986 births
Living people
Association football people from Fukuoka Prefecture
People from Yame, Fukuoka
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Avispa Fukuoka players
JEF United Chiba players
Ventforet Kofu players
Kataller Toyama players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Liouville%20surface | In the mathematical field of differential geometry a Liouville surface is a type of surface which in local coordinates may be written as a graph in R3
such that the first fundamental form is of the form
Sometimes a metric of this form is called a Liouville metric. Every surface of revolution is a Liouville surface.
References
(Translated from the Russian by R. Silverman.)
Surfaces |
https://en.wikipedia.org/wiki/Wilks%20Memorial%20Award | The Wilks Memorial Award is awarded by the American Statistical Association to recognize outstanding contributions to statistics. It was established in 1964 and is awarded yearly. It is named in memory of the statistician Samuel S. Wilks. The award consists of a medal, a citation and a cash honorarium of US$1500 (as of 2008).
Recipients
1964 Frank E. Grubbs
1965 John W. Tukey
1966 Leslie E. Simon
1967 William G. Cochran
1968 Jerzy Neyman
1969 W. J. Youden
1970 George W. Snedecor
1971 Harold F. Dodge
1972 George E.P. Box
1973 Herman Otto Hartley
1974 Cuthbert Daniel
1975 Herbert Solomon
1976 Solomon Kullback
1977 Churchill Eisenhart
1978 William Kruskal
1979 Alexander M. Mood
1980 W. Allen Wallis
1981 Holbrook Working
1982 Frank Proschan
1983 W. Edwards Deming
1984 Z. W. Birnbaum
1985 Leo A. Goodman
1986 Frederick Mosteller
1987 Herman Chernoff
1988 Theodore W. Anderson
1989 C. R. Rao
1990 Bradley Efron
1991 Ingram Olkin
1992 Wilfrid Dixon
1993 Norman L. Johnson
1994 Emanuel Parzen
1995 Donald Rubin
1996 Erich L. Lehmann
1997 Leslie Kish
1998 David O. Siegmund
1999 Lynne Billard
2000 Stephen Fienberg
2001 George C. Tiao
2002 Lawrence D. Brown
2003 David L. Wallace
2004 Paul Meier
2005 Roderick J. A. Little
2006 Marvin Zelen
2007 Colin L. Mallows
2008 Scott Zeger
2009 Lee-Jen Wei
2010 Pranab K. Sen
2011 Nan Laird
2012 Peter Gavin Hall
2013 Kanti Mardia
2014 Madan L. Puri
2015 James O. Berger
2016 David Donoho
2017 Wayne Fuller
2018 Peter J. Bickel
2019 Alan E. Gelfand
2020 Malay Ghosh
2021 Sallie Ann Keller
2022 Jessica Utts
References
Statistical awards
American awards
Awards established in 1964
1964 establishments in the United States |
https://en.wikipedia.org/wiki/Hadamard%27s%20lemma | In mathematics, Hadamard's lemma, named after Jacques Hadamard, is essentially a first-order form of Taylor's theorem, in which we can express a smooth, real-valued function exactly in a convenient manner.
Statement
Proof
Consequences and applications
See also
Citations
References
Real analysis
Theorems in analysis |
https://en.wikipedia.org/wiki/Affine%20geometry%20of%20curves | In the mathematical field of differential geometry, the affine geometry of curves is the study of curves in an affine space, and specifically the properties of such curves which are invariant under the special affine group
In the classical Euclidean geometry of curves, the fundamental tool is the Frenet–Serret frame. In affine geometry, the Frenet–Serret frame is no longer well-defined, but it is possible to define another canonical moving frame along a curve which plays a similar decisive role. The theory was developed in the early 20th century, largely from the efforts of Wilhelm Blaschke and Jean Favard.
The affine frame
Let x(t) be a curve in . Assume, as one does in the Euclidean case, that the first n derivatives of x(t) are linearly independent so that, in particular, x(t) does not lie in any lower-dimensional affine subspace of . Then the curve parameter t can be normalized by setting determinant
Such a curve is said to be parametrized by its affine arclength. For such a parameterization,
determines a mapping into the special affine group, known as a special affine frame for the curve. That is, at each point of the quantities define a special affine frame for the affine space , consisting of a point x of the space and a special linear basis attached to the point at x. The pullback of the Maurer–Cartan form along this map gives a complete set of affine structural invariants of the curve. In the plane, this gives a single scalar invariant, the affine curvature of the curve.
Discrete invariant
The normalization of the curve parameter s was selected above so that
If n≡0 (mod 4) or n≡3 (mod 4) then the sign of this determinant is a discrete invariant of the curve. A curve is called dextrorse (right winding, frequently weinwendig in German) if it is +1, and sinistrorse (left winding, frequently hopfenwendig in German) if it is −1.
In three-dimensions, a right-handed helix is dextrorse, and a left-handed helix is sinistrorse.
Curvature
Suppose that the curve x in is parameterized by affine arclength. Then the affine curvatures, k1, …, kn−1 of x are defined by
That such an expression is possible follows by computing the derivative of the determinant
so that x(n+1) is a linear combination of x′, …, x(n−1).
Consider the matrix
whose columns are the first n derivatives of x (still parameterized by special affine arclength). Then,
In concrete terms, the matrix C is the pullback of the Maurer–Cartan form of the special linear group along the frame given by the first n derivatives of x.
See also
Moving frame
Affine sphere
References
Curves
Differential geometry
Affine geometry |
https://en.wikipedia.org/wiki/Jerzy%20Giedymin | Jerzy Giedymin (September 18, 1925 – June 24, 1993) was a philosopher and historian of mathematics and science.
Life
Giedymin, of Polish origin, was born in 1925.
He studied at the University of Poznań under Kazimierz Ajdukiewicz. In 1953 Jerzy Giedymin succeeded Adam Wiegner at the Chair of Logic at the Faculty of Philosophy.
The so-called Poznań School was a Marxist current of philosophy marked by an idealisational theory of science which emphasised the scientific features of Marxism in close confrontation with contemporary logic and epistemology.
In 1968 Giedymin moved to England and attended seminars by Karl Popper at the London School of Economics.
In 1971 he came to Sussex to become Professor at the School of Mathematical and Physical Sciences of the University of Sussex.
Giedymin died during a trip to Poland on 24 June 1993.
Work
Giedymin was convinced that Henri Poincaré's conventionalist philosophy was fundamentally misunderstood and thus underestimated. Giedymin argues that Poincaré was at the origin of much of the 20th century's innovations in relativity theory and quantum physics.
Giedymin's standpoint was much influenced by his exposure to Kazimierz Ajdukiewicz's perception of the history of ideas which in defiance of traditional empiricism reviews the philosophy of science of the early 20th century in the light of pragmatic conventionalism.
Bibliography
Books
Jerzy Giedymin, Z problemow logicznych analizy historycznej [Some Logical Problems of Historical Analysis], Poznanskie towarzystwo przyjaciol nauk. Wydzial filologiczno-filozoficzny. Prace Komisji filozoficznej. tom 10. zesz. 3., Poznań, 1961.
Jerzy Giedymin, Problemy, zalozenia, rozstrzygniecia. Studia nad logicznymi podstawami nauk spolecznych [Questions, assumptions, decidability. Essays concerning the logical functions of the social sciences], Polskie Towarzystwo Ekonomiczne. Oddzial w Poznaniu. Rozprawy i monografie. No. 10, Poznań, 1964.
Jerzy Giedymin ed., Kazimierz Ajdukiewicz: The scientific world-perspective and other essays, 1931-1963, Dordrecht: D. Reidel Publishing Co., 1974
Jerzy Giedymin, Science and convention: essays on Henri Poincaré’s philosophy of science and the conventionalist tradition, Oxford: Pergamon, 1982
Articles (selection)
Jerzy Giedymin, "Confirmation, critical region and empirical content of hypotheses", in Studia Logica, Volume 10, Number 1 (1960)
Jerzy Giedymin, "A generalization of the refutability postulate", in Studia Logica, Volume 10, Number 1 (1960)
Jerzy Giedymin, "Authorship hypotheses and reliability of informants", in Studia Logica, Volume 12, Number 1 (1961)
Jerzy Giedymin, "Reliability of Informants", in British Journal for the Philosophy of Science, XIII (1963)
Jerzy Giedymin, "The Paradox of Meaning Variance", in British Journal for the Philosophy of Science, 21 (1970)
Jerzy Giedymin, "Consolations for the Irrationalist", in British Journal for the Philosophy of Science, 22 (1971)
Jerzy Giedymin, "Antipositivism in C |
https://en.wikipedia.org/wiki/2006%20Sport%20Club%20Internacional%20season | The Sport Club Internacional won two important titles in the year 2006: Copa Libertadores and Fifa Club World Cup.
Squad
Squad statistics
Libertadores 2006 squad
FIFA Club World Cup 2006 squad
Transfers in
Transfers out
Season 2006
Friendly
Matches
Campeonato Gaúcho
I Fase
Matches
Classification
Group
Results summary
Pld = Matches played; W = Matches won; D = Matches drawn; L = Matches lost;
II Fase
Matches
Classification
Group
Results summary
Pld = Matches played; W = Matches won; D = Matches drawn; L = Matches lost;
Finals
Matches
Copa Libertadores
Group stage
Round of 16
Quarter-finals
Semi-finals
Finals
Campeonato Brasileiro
Matches
Classification
Results summary
Pld = Matches played; W = Matches won; D = Matches drawn; L = Matches lost;
Fifa World Club Cup
Semi-finals
Finals
2006
Internacional
2006 |
https://en.wikipedia.org/wiki/Haagerup%20property | In mathematics, the Haagerup property, named after Uffe Haagerup and also known as Gromov's a-T-menability, is a property of groups that is a strong negation of Kazhdan's property (T). Property (T) is considered a representation-theoretic form of rigidity, so the Haagerup property may be considered a form of strong nonrigidity; see below for details.
The Haagerup property is interesting to many fields of mathematics, including harmonic analysis, representation theory, operator K-theory, and geometric group theory.
Perhaps its most impressive consequence is that groups with the Haagerup Property satisfy the Baum–Connes conjecture and the related Novikov conjecture. Groups with the Haagerup property are also uniformly embeddable into a Hilbert space.
Definitions
Let be a second countable locally compact group. The following properties are all equivalent, and any of them may be taken to be definitions of the Haagerup property:
There is a proper continuous conditionally negative definite function .
has the Haagerup approximation property, also known as Property : there is a sequence of normalized continuous positive-definite functions which vanish at infinity on and converge to 1 uniformly on compact subsets of .
There is a strongly continuous unitary representation of which weakly contains the trivial representation and whose matrix coefficients vanish at infinity on .
There is a proper continuous affine isometric action of on a Hilbert space.
Examples
There are many examples of groups with the Haagerup property, most of which are geometric in origin. The list includes:
All compact groups (trivially). Note all compact groups also have property (T). The converse holds as well: if a group has both property (T) and the Haagerup property, then it is compact.
SO(n,1)
SU(n,1)
Groups acting properly on trees or on -trees
Coxeter groups
Amenable groups
Groups acting properly on CAT(0) cubical complexes
Sources
Representation theory
Geometric group theory |
https://en.wikipedia.org/wiki/Martin%27s%20maximum | In set theory, a branch of mathematical logic, Martin's maximum, introduced by and named after Donald Martin, is a generalization of the proper forcing axiom, itself a generalization of Martin's axiom. It represents the broadest class of forcings for which a forcing axiom is consistent.
Martin's maximum states that if D is a collection of dense subsets of a notion of forcing that preserves stationary subsets of ω1, then there is a D-generic filter. Forcing with a ccc notion of forcing preserves stationary subsets of ω1, thus extends . If (P,≤) is not a stationary set preserving notion of forcing, i.e., there is a stationary subset of ω1, which becomes nonstationary when forcing with (P,≤), then there is a collection D of dense subsets of (P,≤), such that there is no D-generic filter. This is why is called the maximal extension of Martin's axiom.
The existence of a supercompact cardinal implies the consistency of Martin's maximum. The proof uses Shelah's theories of semiproper forcing and iteration with revised countable supports.
implies that the value of the continuum is and that the ideal of nonstationary sets on ω1 is -saturated. It further implies stationary reflection, i.e., if S is a stationary subset of some regular cardinal κ ≥ ω2 and every element of S has countable cofinality, then there is an ordinal α < κ such that S ∩ α is stationary in α. In fact, S contains a closed subset of order type ω1.
Notes
References
correction
See also
Transfinite number
Forcing (mathematics) |
https://en.wikipedia.org/wiki/J%C3%B8rgen%20Horn | Jørgen Horn (born 7 June 1987) is a Norwegian footballer. He currently plays for Norwegian club Sarpsborg 08.
Career statistics
Honours
Club
Strømsgodset
Tippeligaen (1): 2013
References
1987 births
Living people
Footballers from Oslo
Norwegian men's footballers
Norwegian expatriate men's footballers
Norway men's international footballers
Norway men's under-21 international footballers
Kjelsås Fotball players
Vålerenga Fotball players
Moss FK players
Viking FK players
Fredrikstad FK players
Strømsgodset Toppfotball players
IF Elfsborg players
Sarpsborg 08 FF players
Allsvenskan players
Eliteserien players
Norwegian First Division players
Expatriate men's footballers in Sweden
Norwegian expatriate sportspeople in Sweden
Men's association football defenders |
https://en.wikipedia.org/wiki/St%C3%B8rmer%20number | In mathematics, a Størmer number or arc-cotangent irreducible number is a positive integer for which the greatest prime factor of is greater than or equal to . They are named after Carl Størmer.
Sequence
The first few Størmer numbers are:
Density
John Todd proved that this sequence is neither finite nor cofinite.
More precisely, the natural density of the Størmer numbers lies between 0.5324 and 0.905.
It has been conjectured that their natural density is the natural logarithm of 2, approximately 0.693, but this remains unproven.
Because the Størmer numbers have positive density, the Størmer numbers form a large set.
Application
The Størmer numbers arise in connection with the problem of representing the Gregory numbers (arctangents of rational numbers) as sums of Gregory numbers for integers (arctangents of unit fractions). The Gregory number may be decomposed by repeatedly multiplying the Gaussian integer by numbers of the form , in order to cancel prime factors from the imaginary part; here is chosen to be a Størmer number such that is divisible by .
References
Eponymous numbers in mathematics
Integer sequences |
https://en.wikipedia.org/wiki/Gregory%20number | In mathematics, a Gregory number, named after James Gregory, is a real number of the form:
where x is any rational number greater or equal to 1. Considering the power series expansion for arctangent, we have
Setting x = 1 gives the well-known Leibniz formula for pi. Thus, in particular,
is a Gregory number.
Properties
See also
Størmer number
References
Sets of real numbers |
https://en.wikipedia.org/wiki/Amrita%20Learning | Amrita Learning is a computer-based 'adaptive learning' program in English and Mathematics. It provides interactive, audio-visual supplementary education for children of three-and-a-half years to 7th grade, for home and classroom use. Developed in collaboration with Amrita University, the content of this research-based software is in line with the NCERT/CBSE syllabi.
Adaptive Assessment
Under the 'Adaptive Assessment' concept, as a student learns, s/he is continually assessed by the programme and the software adapts to their level. "The programme is also being used at some schools to administer the CBSE mandated Continuous and Comprehensive Evaluation (CCE). Teachers can replace a large portion of paper-based CCE with computer based assessment. After the teacher teaches a set of topics in the classroom, students are immediately taken to the computer and evaluated. Students who have mastered one topic move on to the next level and those who have not mastered a topic are provided with more practice."
History
Based on research conducted by Amrita University, Amrita Learning was founded in Kochi, Kerala, India by Raghu Raman and is directed by Prema Nedungadi in 2005. In August 2007, Amrita Learning partnered with broadband providers BSNL to provide educational content as part of their subscriber package. In February 2008, Former President of India Dr A. P. J. Abdul Kalam inaugurated the Amrita Vidyalayam e-learning Network which connects 55 schools throughout India and will feature content from Amrita Learning.
References
External links
Amrita Learning Website
Amrita University Website
MoU between Amrita University and US Universities
Online companies of India
Distance education in India
Education companies of India |
https://en.wikipedia.org/wiki/Cha%20Ji-ho | Cha Ji-ho (; born 23 March 1983 in Seoul) is a retired South Korean football player.
Cha mostly played for Roasso Kumamoto.
Club statistics
References
External links
Living people
1983 births
Men's association football midfielders
South Korean men's footballers
South Korean expatriate men's footballers
Lyn Fotball players
Kongsvinger IL Toppfotball players
Busan IPark players
Melbourne Knights FC players
Roasso Kumamoto players
Pohang Steelers players
Eliteserien players
Norwegian First Division players
K League 1 players
Victorian Premier League players
J2 League players
Expatriate men's footballers in Norway
Expatriate men's soccer players in Australia
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Norway
South Korean expatriate sportspeople in Australia
South Korean expatriate sportspeople in Japan
Footballers from Seoul |
https://en.wikipedia.org/wiki/Littlewood%E2%80%93Richardson%20rule | In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occur in many other mathematical contexts, for instance as multiplicity in the decomposition of tensor products of finite-dimensional representations of general linear groups, or in the decomposition of certain induced representations in the representation theory of the symmetric group, or in the area of algebraic combinatorics dealing with Young tableaux and symmetric polynomials.
Littlewood–Richardson coefficients depend on three partitions, say , of which and describe the Schur functions being multiplied, and gives the Schur function of which this is the coefficient in the linear combination; in other words they are the coefficients such that
The Littlewood–Richardson rule states that is equal to the number of Littlewood–Richardson tableaux of skew shape and of weight .
History
The Littlewood–Richardson rule was first stated by but though they claimed it as a theorem they only proved it in some fairly simple special cases.
claimed to complete their proof, but his argument had gaps, though it was so obscurely written that these gaps were not noticed for some time, and his argument is reproduced in the book . Some of the gaps were later filled by . The first rigorous proofs of the rule were given four decades after it was found, by and , after the necessary combinatorial theory was developed by , , and in their work on the Robinson–Schensted correspondence.
There are now several short proofs of the rule, such as , and using Bender-Knuth involutions.
used the Littelmann path model to generalize the Littlewood–Richardson rule to other semisimple Lie groups.
The Littlewood–Richardson rule is notorious for the number of errors that appeared prior to its complete, published proof. Several published attempts to prove it are incomplete, and it is particularly difficult to avoid errors when doing hand calculations with it: even the original example in contains an error.
Littlewood–Richardson tableaux
A Littlewood–Richardson tableau is a skew semistandard tableau with the additional property that the sequence obtained by concatenating its reversed rows is a lattice word (or lattice permutation), which means that in every initial part of the sequence any number occurs at least as often as the number . Another equivalent (though not quite obviously so) characterization is that the tableau itself, and any tableau obtained from it by removing some number of its leftmost columns, has a weakly decreasing weight. Many other combinatorial notions have been found that turn out to be in bijection with Littlewood–Richardson tableaux, and can therefore also be used to define the Littlewood–Richardson coeffic |
https://en.wikipedia.org/wiki/Philadelphia%2076ers%20all-time%20roster | The following is a list of players, both past and current, who appeared at least in one game for the Philadelphia 76ers NBA franchise.
Players
Note: Statistics are correct through the end of the season.
A to B
|-
|align="left"| || align="center"|F/C || align="left"|Duke || align="center"|1 || align="center"| || 3 || 30 || 8 || 0 || 2 || 10.0 || 2.7 || 0.0 || 0.7 || align=center|
|-
|align="left"| || align="center"|G || align="left"|Western Kentucky || align="center"|1 || align="center"| || 1 || 1 || 1 || 1 || 0 || 1.0 || 1.0 || 1.0 || 0.0 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|Turkey || align="center"|1 || align="center"| || 41 || 540 || 176 || 28 || 93 || 13.2 || 4.3 || 0.7 || 2.3 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|Temple || align="center"|3 || align="center"|– || 171 || 3,253 || 843 || 172 || 887 || 19.0 || 4.9 || 1.0 || 5.2 || align=center|
|-
|align="left"| || align="center"|F || align="left"|Duquesne || align="center"|2 || align="center"|– || 137 || 2,646 || 521 || 94 || 751 || 19.3 || 3.8 || 0.7 || 5.5 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|UNLV || align="center"|2 || align="center"|– || 26 || 151 || 40 || 1 || 34 || 5.8 || 1.5 || 0.0 || 1.3 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|Saint Joseph's || align="center"|1 || align="center"| || 5 || 27 || 11 || 4 || 7 || 5.4 || 2.2 || 0.8 || 1.4 || align=center|
|-
|align="left"| || align="center"|F || align="left"|Bradley || align="center"|1 || align="center"| || 13 || 48 || 12 || 1 || 17 || 3.7 || 0.9 || 0.1 || 1.3 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|Oklahoma State || align="center"|1 || align="center"| || 80 || 2,309 || 300 || 149 || 810 || 28.9 || 3.8 || 1.9 || 10.1 || align=center|
|-
|align="left"| || align="center"|G || align="left"|Virginia || align="center"|2 || align="center"|– || 62 || 1,037 || 189 || 59 || 439 || 16.7 || 3.0 || 1.0 || 7.1 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|Fresno State || align="center"|5 || align="center"|– || 393 || 10,742 || 1,530 || 625 || 5,138 || 27.3 || 3.9 || 1.6 || 13.1 || align=center|
|-
|align="left"| || align="center"|F || align="left"|Alabama || align="center"|1 || align="center"| || 8 || 32 || 4 || 2 || 15 || 4.0 || 0.5 || 0.3 || 1.9 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|Memphis || align="center"|1 || align="center"| || 14 || 234 || 22 || 33 || 52 || 16.7 || 1.6 || 2.4 || 3.7 || align=center|
|-
|align="left"| || align="center"|C || align="left"|Arizona State || align="center"|1 || align="center"| || 14 || 201 || 69 || 17 || 72 || 14.4 || 4.9 || 1.2 || 5.1 || align=center|
|-
|align="left"| || align="center"|C || align="left"|Santa Clara || align="center"|3 || align="center"|– || 131 || 2,123 || 692 || 142 || 756 || 16.2 || 5.3 || 1.1 || 5.8 || align=center|
|-
|align="left"| || align="center"|F/C || alig |
https://en.wikipedia.org/wiki/Detroit%20Pistons%20all-time%20roster | The following is a list of players, both past and current, who appeared at least in one game for the Detroit Pistons NBA franchise.
Players
Note: Statistics are correct through the end of the season.
A to B
|-
|align="left"| || align="center"|G || align="left"|Pepperdine || align="center"|2 || align="center"| || 12 || 55 || 7 || 5 || 18 || 4.6 || 0.6 || 0.4 || 1.5 || align=center|
|-
|align="left"| || align="center"|F || align="left"|Northwestern || align="center"|3 || align="center"|– || 195 || 5,472 || 1,111 || 328 || 1,710 || 28.1 || 5.7 || 1.7 || 8.8 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|Syracuse || align="center"|1 || align="center"| || 79 || 1,776 || 242 || 109 || 656 || 22.5 || 3.1 || 1.4 || 8.3 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|UCLA || align="center"|2 || align="center"|– || 149 || 2,204 || 271 || 96 || 639 || 14.8 || 1.8 || 0.6 || 4.3 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|DePaul || align="center"|5 || align="center"|– || 318 || 7,717 || 1,218 || 604 || 4,115 || 24.3 || 3.8 || 1.9 || 12.9 || align=center|
|-
|align="left"| || align="center"|C || align="left"|Fresno State || align="center"|1 || align="center"| || 58 || 670 || 279 || 22 || 230 || 11.6 || 4.8 || 0.4 || 4.0 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|Iowa State || align="center"|1 || align="center"| || 15 || 97 || 29 || 6 || 40 || 6.5 || 1.9 || 0.4 || 2.7 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|Houston || align="center"|1 || align="center"| || 77 || 1,624 || 571 || 51 || 491 || 21.1 || 7.4 || 0.7 || 6.4 || align=center|
|-
|align="left"| || align="center"|C || align="left"|UNLV || align="center"|2 || align="center"|– || 68 || 502 || 114 || 6 || 105 || 7.4 || 1.7 || 0.1 || 1.5 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|Indiana || align="center"|3 || align="center"|– || 153 || || 89 || 358 || 1,040 || || 2.3 || 2.3 || 6.8 || align=center|
|-
|align="left"| || align="center"|F || align="left"|Penn State || align="center"|1 || align="center"| || 31 || 409 || 170 || 18 || 147 || 13.2 || 5.5 || 0.6 || 4.7 || align=center|
|-
|align="left"| || align="center"|G || align="left"|FIU || align="center"|2 || align="center"|– || 90 || 1,306 || 128 || 283 || 373 || 14.5 || 1.4 || 3.1 || 4.1 || align=center|
|-
|align="left"| || align="center"|G || align="left"|Indiana State || align="center"|1 || align="center"| || 18 || 160 || 24 || 19 || 44 || 8.9 || 1.3 || 1.1 || 2.4 || align=center|
|-
|align="left"| || align="center"|G || align="left"|South Florida || align="center"|5 || align="center"|– || 305 || 7,441 || 504 || 954 || 2,800 || 24.4 || 1.7 || 3.1 || 9.2 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|UNLV || align="center"|1 || align="center"| || 20 || 292 || 49 || 15 || 90 || 14.6 || 2.5 || 0.8 || 4.5 || align=center|
|-
|align="left"| || align="center"|G |
https://en.wikipedia.org/wiki/Elizeu%20%28footballer%2C%20born%201979%29 | Elizeu Ferreira Marciano (born October 21, 1979, in Brazil) is a Brazilian football player
Club statistics
References
External links
Galo Digital
1979 births
Living people
Brazilian men's footballers
Criciúma Esporte Clube players
Joinville Esporte Clube players
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
J1 League players
J2 League players
Expatriate men's footballers in China
Yokohama FC players
Vegalta Sendai players
Tokushima Vortis players
Brazilian expatriate sportspeople in Japan
Chongqing Liangjiang Athletic F.C. players
Brazilian expatriate sportspeople in China
Men's association football defenders
Footballers from São Paulo |
https://en.wikipedia.org/wiki/Bruno%20Rodrigues%20%28footballer%2C%20born%201989%29 | Bruno Alexandre Rodrigues, or simply Bruno (born February 6, 1989 in São Paulo), is a Brazilian footballer who plays as a striker.
Club statistics
References
External links
1989 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
J2 League players
Ventforet Kofu players
Gainare Tottori players
América Futebol Clube (SP) players
Marília Atlético Clube players
Clube Esportivo Lajeadense players
Sociedade Esportiva Palmeiras players
São José Esporte Clube players
Lemense Futebol Clube players
Esporte Clube Noroeste players
Rio Preto Esporte Clube players
Men's association football forwards
Brazilian expatriate sportspeople in Japan
Brazilian expatriate sportspeople in Portugal
Expatriate men's footballers in Japan
Expatriate men's footballers in Portugal
Footballers from São Paulo (state) |
https://en.wikipedia.org/wiki/Sergey%20Kostyuk | Sergey Kostyuk (; born 30 November 1978) is a retired Kazakhstani football midfielder.
Club statistics
Last update: 28 October 2012
References
External links
1978 births
Living people
Footballers from Odesa
Ukrainian men's footballers
Ukrainian expatriate men's footballers
Expatriate men's footballers in Kazakhstan
Ukrainian expatriate sportspeople in Kazakhstan
Kazakhstani men's footballers
Kazakhstan men's international footballers
Men's association football forwards
Ukrainian emigrants to Kazakhstan
FC SKA-Lotto Odesa players
SC Odesa players
FC Chornomorets Odesa players
FC Chornomorets-2 Odesa players
FC Vorskla Poltava players
FC Vorskla-2 Poltava players
FC Polissya Zhytomyr players
FC Odesa players
FC Atyrau players
FC Shakhter Karagandy players
FC Vostok players
FC Zhetysu players
FC Tobol players
FC Yelimay players
FC Real Pharma Odesa players
Ukrainian Premier League players
Ukrainian First League players
Ukrainian Second League players
Kazakhstan Premier League players |
https://en.wikipedia.org/wiki/%C5%BDeljko%20Kova%C4%8Devi%C4%87 | Željko Kovačević (Serbian Cyrillic: Жељко Ковачевић; born 30 October 1981) is a Serbian professional footballer who plays as a defender.
Statistics
Honours
Rabotnički
Macedonian First League: 2007–08
Macedonian Cup: 2007–08
External links
HLSZ profile
Men's association football defenders
Expatriate men's footballers in Bosnia and Herzegovina
Expatriate men's footballers in Hungary
Expatriate men's footballers in North Macedonia
First League of Serbia and Montenegro players
FK Borac Čačak players
FK Rabotnički players
FK Slavija Sarajevo players
FK Smederevo 1924 players
FK Vardar players
Nemzeti Bajnokság I players
Nyíregyháza Spartacus FC players
Serbian expatriate men's footballers
Serbian expatriate sportspeople in Bosnia and Herzegovina
Serbian expatriate sportspeople in Hungary
Serbian expatriate sportspeople in North Macedonia
Serbian men's footballers
Serbian SuperLiga players
Footballers from Čačak
1981 births
Living people |
https://en.wikipedia.org/wiki/St.%20Petersburg%20Department%20of%20Steklov%20Mathematical%20Institute%20of%20the%20Russian%20Academy%20of%20Sciences | The St. Petersburg Department of Steklov Institute of Mathematics of the Russian Academy of Sciences (, abbreviated ПОМИ (PDMI) for "Петербургское отделение Математического института", Petersburg Department of the Mathematical Institute; PDMI) is a mathematical research institute in St. Petersburg, part of the Russian Academy of Sciences. Until 1992 it was known as Leningrad Department of Steklov Institute of Mathematics of the USSR Academy of Sciences (ЛОМИ, LOMI).
The name of the institution is a historical tradition and since 1995 it has no subordination to the Steklov Institute of Mathematics.
The institute was established in 1940 as a department of the Steklov Institute and is named after Vladimir Andreevich Steklov, a Soviet/Russian mathematician, mechanician and physicist.
Directors
V. A. Tartakovskii (1940-1941)
Andrey Markov, Jr. (1942-1953)
Nikolai Erugin (1953-1957)
Georgii Petrashen' (1957-1976)
Ludvig Faddeev (1976-2000)
Ildar Abdulovich Ibragimov (2000-2006)
Sergei Kislyakov (2007-2021)
Maxim Vsemirnov (2021 - )
Notable researchers
Aleksandr Danilovich Aleksandrov
Yuri Burago
Nikolai Durov
Dmitry Konstantinovich Faddeev
Vera Faddeeva
Fedor Fomin
Leonid Kantorovich
Vladimir Korepin
Olga Ladyzhenskaya
Yuri Linnik
Yuri Matiyasevich
Grigori Perelman worked at this institution when he proved the Poincaré conjecture.
Nicolai Reshetikhin
Nikolai Aleksandrovich Shanin
Samson Shatashvili
Andrei Suslin
Leon Takhtajan
Anatoly Vershik
Alexander Volberg
Oleg Viro
Victor Zalgaller
References
External links
Institute's website
Mathematical institutes
Buildings and structures in Saint Petersburg
Institutes of the Russian Academy of Sciences
Universities and institutes established in the Soviet Union
Research institutes in the Soviet Union |
https://en.wikipedia.org/wiki/STATEC | STATEC (officially in French: Institut national de la statistique et des études économiques) is the government statistics service of Luxembourg. It is headquartered in the Kirchberg quarter of Luxembourg City.
Footnotes
External links
Luxembourg Statistics Portal
Government of Luxembourg
Luxembourg
Organisations based in Luxembourg City
goliki (10) |
https://en.wikipedia.org/wiki/BCK%20algebra | In mathematics, BCI and BCK algebras are algebraic structures in universal algebra, which were introduced by Y. Imai, K. Iséki and S. Tanaka in 1966, that describe fragments of the propositional calculus involving implication known as BCI and BCK logics.
Definition
BCI algebra
An algebra (in the sense of universal algebra) of type is called a BCI-algebra if, for any , it satisfies the following conditions. (Informally, we may read as "truth" and as " implies ".)
BCI-1
BCI-2
BCI-3
BCI-4
BCI-5
BCK algebra
A BCI-algebra is called a BCK-algebra if it
satisfies the following condition:
BCK-1
A partial order can then be defined as x ≤ y iff x * y = 0.
A BCK-algebra is said to be commutative if it satisfies:
In a commutative BCK-algebra x * (x * y) = x ∧ y is the greatest lower bound of x and y under the partial order ≤.
A BCK-algebra is said to be bounded if it has a largest element, usually denoted by 1. In a bounded commutative BCK-algebra the least upper bound of two elements satisfies x ∨ y = 1 * ((1 * x) ∧ (1 * y)); that makes it a distributive lattice.
Examples
Every abelian group is a BCI-algebra, with * defined as group subtraction and 0 defined as the group identity.
The subsets of a set form a BCK-algebra, where A*B is the difference A\B (the elements in A but not in B), and 0 is the empty set.
A Boolean algebra is a BCK algebra if A*B is defined to be A∧¬B (A does not imply B).
The bounded commutative BCK-algebras are precisely the MV-algebras.
References
Y. Huang, BCI-algebra, Science Press, Beijing, 2006.
Algebraic structures
Universal algebra |
https://en.wikipedia.org/wiki/P-adic%20valuation | In number theory, the valuation or -adic order of an integer is the exponent of the highest power of the prime number that divides .
It is denoted .
Equivalently, is the exponent to which appears in the prime factorization of .
The -adic valuation is a valuation and gives rise to an analogue of the usual absolute value.
Whereas the completion of the rational numbers with respect to the usual absolute value results in the real numbers , the completion of the rational numbers with respect to the -adic absolute value results in the numbers .
Definition and properties
Let be a prime number.
Integers
The -adic valuation of an integer is defined to be
where denotes the set of natural numbers and denotes divisibility of by . In particular, is a function .
For example, , , and since .
The notation is sometimes used to mean .
If is a positive integer, then
;
this follows directly from .
Rational numbers
The -adic valuation can be extended to the rational numbers as the function
defined by
For example, and since .
Some properties are:
Moreover, if , then
where is the minimum (i.e. the smaller of the two).
-adic absolute value
The -adic absolute value on is the function
defined by
Thereby, for all and
for example, and
The -adic absolute value satisfies the following properties.
{| class="wikitable"
|-
|Non-negativity ||
|-
|Positive-definiteness ||
|-
|Multiplicativity ||
|-
|Non-Archimedean ||
|}
From the multiplicativity it follows that for the roots of unity and and consequently also
The subadditivity follows from the non-Archimedean triangle inequality .
The choice of base in the exponentiation makes no difference for most of the properties, but supports the product formula:
where the product is taken over all primes and the usual absolute value, denoted . This follows from simply taking the prime factorization: each prime power factor contributes its reciprocal to its -adic absolute value, and then the usual Archimedean absolute value cancels all of them.
The -adic absolute value is sometimes referred to as the "-adic norm", although it is not actually a norm because it does not satisfy the requirement of homogeneity.
A metric space can be formed on the set with a (non-Archimedean, translation-invariant) metric
defined by
The completion of with respect to this metric leads to the set of -adic numbers.
See also
-adic number
Archimedean property
Multiplicity (mathematics)
Ostrowski's theorem
Legendre's formula
References
Algebraic number theory
p-adic numbers |
https://en.wikipedia.org/wiki/2007%E2%80%9308%20Euroleague%20Individual%20Statistics | Euroleague 2007–08 Individual Statistics is the statistics about players, playing in Euroleague 2007-08. Top 5 for each category.
Regular season
Points
Rebounds
Assists
Top 16
Points
Rebounds
Assists
Playoffs
Points
Rebounds
Assists
Final four
Points
Rebounds
Assists
Full Season
Points
Rebounds
Assists
References
Statistics
2007–08 Euroleague |
https://en.wikipedia.org/wiki/Integration%20by%20reduction%20formulae | In integral calculus, integration by reduction formulae is a method relying on recurrence relations. It is used when an expression containing an integer parameter, usually in the form of powers of elementary functions, or products of transcendental functions and polynomials of arbitrary degree, can't be integrated directly. But using other methods of integration a reduction formula can be set up to obtain the integral of the same or similar expression with a lower integer parameter, progressively simplifying the integral until it can be evaluated. This method of integration is one of the earliest used.
How to find the reduction formula
The reduction formula can be derived using any of the common methods of integration, like integration by substitution, integration by parts, integration by trigonometric substitution, integration by partial fractions, etc. The main idea is to express an integral involving an integer parameter (e.g. power) of a function, represented by In, in terms of an integral that involves a lower value of the parameter (lower power) of that function, for example In-1 or In-2. This makes the reduction formula a type of recurrence relation. In other words, the reduction formula expresses the integral
in terms of
where
How to compute the integral
To compute the integral, we set n to its value and use the reduction formula to express it in terms of the (n – 1) or (n – 2) integral. The lower index integral can be used to calculate the higher index ones; the process is continued repeatedly until we reach a point where the function to be integrated can be computed, usually when its index is 0 or 1. Then we back-substitute the previous results until we have computed In.
Examples
Below are examples of the procedure.
Cosine integral
Typically, integrals like
can be evaluated by a reduction formula.
Start by setting:
Now re-write as:
Integrating by this substitution:
Now integrating by parts:
solving for In:
so the reduction formula is:
To supplement the example, the above can be used to evaluate the integral for (say) n = 5;
Calculating lower indices:
back-substituting:
where C is a constant.
Exponential integral
Another typical example is:
Start by setting:
Integrating by substitution:
Now integrating by parts:
shifting indices back by 1 (so n + 1 → n, n → n – 1):
solving for In:
so the reduction formula is:
An alternative way in which the derivation could be done starts by substituting .
Integration by substitution:
Now integrating by parts:
which gives the reduction formula when substituting back:
which is equivalent to:
Another alternative way in which the derivation could be done by integrating by parts:
Remember:
which gives the reduction formula when substituting back:
which is equivalent to:
Tables of integral reduction formulas
Rational functions
The following integrals contain:
Factors of the linear radical
Linear factors and the linear radical
Quadratic factors
Quadratic f |
https://en.wikipedia.org/wiki/Kylemore%2C%20Saskatchewan | Kylemore is an unincorporated community in the Rural Municipality of Sasman No. 336, Saskatchewan, Canada. Listed as a designated place by Statistics Canada, the community had a population of 0 in the Canada 2016 Census. It was designated as an organized hamlet prior to 2018. The community is located 12 km east of Wadena, and approximately 250 km east of Saskatoon.
Canada's first aboriginal urban reserve was established here in 1981.
History
Kylemore relinquished its organized hamlet designation on December 31, 2017.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Kylemore had a population of 5 living in 2 of its 2 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 communities in Saskatchewan
References
Sasman No. 336, Saskatchewan
Designated places in Saskatchewan
Unincorporated communities in Saskatchewan
Ethnic enclaves in Saskatchewan
Division No. 10, Saskatchewan |
https://en.wikipedia.org/wiki/List%20of%20busiest%20passenger%20air%20routes | These are lists of the busiest air routes by the number of passengers flown, by seat capacity and by aircraft movements.
Global statistics
By number of passengers
Top 10 busiest air routes
Busiest international air routes by origin-and-destination passenger volume (airport pairs)
Busiest routes by region
By aircraft movements
The following are the lists of the world's busiest air routes based on the number of scheduled flights in both directions. Note that these statistics do not consider the number of passengers carried.
Domestic
International
The only international route to make the Top 50 busiest routes in 2021 was between Saint Barthélemy (SBH) and Sint Maarten (SXM).
Regional statistics
Europe (2011-2022)
Busiest air routes inside the EU, United Kingdom, Switzerland, Iceland and Norway*
* Figures are only available for EU countries, Switzerland, Iceland and Norway.
United Kingdom has not reported for 2021 and parts of 2020, due to Brexit, but data are available from EU/Eurostat countries for flights between them and UK, but not for UK domestic or UK to outside Europe. The multiple airports of London lower the airport-to-airport figures, and the busiest UK domestic was in 2019 Heathrow-Edinburgh with 1,196,921.
Among other European countries, based on airport statistics, no other country than Russia, Ukraine and Turkey can have domestic routes with more than 1 million passengers.
The St-Petersburg Pulkovo Airport - Moscow (three airports) route was used by 2,965,331 passengers in 2013.
**Madrid-Barcelona had 4,627,000 passengers in 2007, meaning a reduction by 34% in 2010, partly because of a new high-speed railway. The decline has continued from 560 weekly flights in 2010 to 380 in 2012.
Busiest air routes between an airport in Europe (EU, UK, Switzerland, Iceland and Norway) and outside Europe
For routes from EU, UK, Switzerland, Iceland and Norway to other countries inside Europe except to Turkey, the busiest was in 2019 Paris/de Gaulle–Moscow/Sheremetyevo with 830,980
Busiest air routes in or from Europe by city pairs
London includes City, Gatwick, Heathrow, Luton, Stansted and Southend airports
New York includes Kennedy and Newark airports
Paris includes Charles De Gaulle and Orly airports
Barcelona includes El Prat and Girona airports
Berlin includes Schönefeld (from 2020 Brandenburg) and Tegel airports
Moscow includes Domodedovo, Sheremetyevo and Vnukovo airports
Turkey is not included even if Istanbul–Izmir has (as of 2019) more passengers than London–Dublin, because all its domestic routes are mostly or entirely inside Asia
Brazil (2022)
Busiest domestic air routes in Brazil
Argentina (2017)
Busiest international air routes in Argentina
Busiest domestic air routes in Argentina
Australia (2019)
Busiest domestic air routes in Australia (all passengers including connecting)
Busiest international air routes in Australia (all passengers including connecting)
Colombia (2017)
Busiest domestic air routes in Colom |
https://en.wikipedia.org/wiki/Tarski%27s%20plank%20problem | In mathematics, Tarski's plank problem is a question about coverings of convex regions in n-dimensional Euclidean space by "planks": regions between two hyperplanes. Tarski asked if the sum of the widths of the planks must be at least the minimum width of the convex region. The question was answered affirmatively by
.
Statement
Given a convex body C in Rn and a hyperplane H, the width of C parallel to H, w(C,H), is the distance between the two supporting hyperplanes of C that are parallel to H. The smallest such distance (i.e. the infimum over all possible hyperplanes) is called the minimal width of C, w(C).
The (closed) set of points P between two distinct, parallel hyperplanes in Rn is called a plank, and the distance between the two hyperplanes is called the width of the plank, w(P). Tarski conjectured that if a convex body C of minimal width w(C) was covered by a collection of planks, then the sum of the widths of those planks must be at least w(C). That is, if P1,…,Pm are planks such that
then
Bang proved this is indeed the case.
Nomenclature
The name of the problem, specifically for the sets of points between parallel hyperplanes, comes from the visualisation of the problem in R2. Here, hyperplanes are just straight lines and so planks become the space between two parallel lines. Thus the planks can be thought of as (infinitely long) planks of wood, and the question becomes how many planks does one need to completely cover a convex tabletop of minimal width w? Bang's theorem shows that, for example, a circular table of diameter d feet can't be covered by fewer than d planks of wood of width one foot each.
References
Geometry |
https://en.wikipedia.org/wiki/Akurdet%20Subregion | Akurdet Subregion is a subregion in the western Gash-Barka region (Zoba Gash-Barka) of Eritrea. Its capital lies at Akurdet.
References
Awate.com: Martyr Statistics
Gash-Barka Region
Subregions of Eritrea |
https://en.wikipedia.org/wiki/Gogne%20Subregion | Gogne Subregion is a subregion in the western Gash-Barka region of Eritrea. Its capital lies at Gogne.
Towns and villages
Ad Casub
Gogne
Hambok
Markaughe
References
Awate.com: Martyr Statistics
Gash-Barka Region
Subregions of Eritrea |
https://en.wikipedia.org/wiki/Forto%20Subregion | Forto Subregion is a subregion in the Gash-Barka region of western Eritrea. Its capital is Forto.
Towns and villages
Algheden
References
Awate.com: Martyr Statistics
Gash-Barka Region
Subregions of Eritrea |
https://en.wikipedia.org/wiki/Mensura%20Subregion | Mensura Subregion is a subregion in the Gash Barka region of western Eritrea. The capital lies at Mensura.
References
Awate.com: Martyr Statistics
Gash-Barka Region
Subregions of Eritrea |
https://en.wikipedia.org/wiki/Mogolo%20Subregion | Mogolo Subregion is a subregion in the Gash-Barka region of western Eritrea. Its capital lies at Mogolo.
Towns and villages
Aredda
Attai
Chibabo
Mescul
Mogolo
References
Awate.com: Martyr Statistics
Gash-Barka Region
Subregions of Eritrea |
https://en.wikipedia.org/wiki/Molki%20Subregion | Molki Subregion is a subregion in the Gash-Barka region (Zoba Gash-Barka) of western Eritrea. Its capital lies at Molki.
References
Awate.com: Martyr Statistics
Subregions of Eritrea |
https://en.wikipedia.org/wiki/Teseney%20Subregion | Teseney Subregion is a subregion in the western Gash-Barka region (Zoba Gash-Barka) of Eritrea. Its capital lies at Teseney.
References
Awate.com: Martyr Statistics
Subregions of Eritrea |
https://en.wikipedia.org/wiki/Afabet%20Subregion | Afabet Subregion is a subregion in the Northern Red Sea (Zoba Semienawi Keyih Bahri) region of Eritrea. Its capital lies at Afabet.
References
Awate.com: Martyr Statistics
Northern Red Sea Region
Subregions of Eritrea |
https://en.wikipedia.org/wiki/She%27eb%20Subregion | She'eb Subregion is a subregion in the Northern Red Sea region (Zoba Semienawi Keyih Bahri) of Eritrea. Its capital lies at She'eb.
References
Awate.com: Martyr Statistics
Northern Red Sea Region
Subregions of Eritrea |
https://en.wikipedia.org/wiki/Areza%20Subregion | Areza Subregion is a subregion in the Debub (Southern) region of Eritrea. Its capital lies at Areza.
References
Awate.com: Martyr Statistics
Southern Region (Eritrea)
Subregions of Eritrea |
https://en.wikipedia.org/wiki/Debarwa%20Subregion | Debarwa Subregion is a subregion in the southern Debub region (Zoba Debarwa) of Eritrea. Its capital lies at Debarwa.
References
Awate.com: Martyr Statistics
Southern Region (Eritrea)
Subregions of Eritrea |
https://en.wikipedia.org/wiki/Segheneyti%20Subregion | Segheneyti Subregion is a subregion in the Debub (Southern) region (Zoba Debub) of Eritrea. Its capital lies at Segheneyti.
References
Awate.com: Martyr Statistics
Southern Region (Eritrea)
Subregions of Eritrea |
https://en.wikipedia.org/wiki/Struve%20function | In mathematics, the Struve functions , are solutions of the non-homogeneous Bessel's differential equation:
introduced by . The complex number α is the order of the Struve function, and is often an integer.
And further defined its second-kind version as .
The modified Struve functions are equal to , are solutions of the non-homogeneous Bessel's differential equation:
And further defined its second-kind version as .
Definitions
Since this is a non-homogeneous equation, solutions can be constructed from a single particular solution by adding the solutions of the homogeneous problem. In this case, the homogeneous solutions are the Bessel functions, and the particular solution may be chosen as the corresponding Struve function.
Power series expansion
Struve functions, denoted as have the power series form
where is the gamma function.
The modified Struve functions, denoted , have the following power series form
Integral form
Another definition of the Struve function, for values of satisfying , is possible expressing in term of the Poisson's integral representation:
Asymptotic forms
For small , the power series expansion is given above.
For large , one obtains:
where is the Neumann function.
Properties
The Struve functions satisfy the following recurrence relations:
Relation to other functions
Struve functions of integer order can be expressed in terms of Weber functions and vice versa: if is a non-negative integer then
Struve functions of order where is an integer can be expressed in terms of elementary functions. In particular if is a non-negative integer then
where the right hand side is a spherical Bessel function.
Struve functions (of any order) can be expressed in terms of the generalized hypergeometric function :
Applications
The Struve and Weber functions were shown to have an application to beamforming in., and in describing the effect of confining interface on Brownian motion of colloidal particles at low Reynolds numbers.
References
External links
Struve functions at the Wolfram functions site.
Special functions
Struve family |
https://en.wikipedia.org/wiki/Pignistic%20probability | In decision theory, a pignistic probability is a probability that a rational person will assign to an option when required to make a decision.
A person may have, at one level certain beliefs or a lack of knowledge, or uncertainty, about the options and their actual likelihoods. However, when it is necessary to make a decision (such as deciding whether to place a bet), the behaviour of the rational person would suggest that the person has assigned a set of regular probabilities to the options. These are the pignistic probabilities.
The term was coined by Philippe Smets, and stems from the Latin pignus, a bet. He contrasts the pignistic level, where one might take action, with the credal level, where one interprets the state of the world:
The transferable belief model is based on the assumption that beliefs manifest themselves at two mental levels: the ‘credal’ level where beliefs are entertained and the ‘pignistic’ level where beliefs are used to make decisions (from ‘credo’ I believe and ‘pignus’ a bet, both in Latin). Usually these two levels are not distinguished and probability functions are used to quantify beliefs at both levels. The justification for the use of probability functions is usually linked to “rational” behavior to be held by an ideal agent involved in some decision contexts.
A pignistic probability transform will calculate these pignistic probabilities from a structure that describes belief structures.
Notes
Further reading
P. Smets and R. Kennes, “The Transferable Belief Model", Artificial Intelligence (v.66, 1994) pp. 191–243
Decision theory
Probability interpretations |
https://en.wikipedia.org/wiki/Husimi%20Q%20representation | The Husimi Q representation, introduced by Kôdi Husimi in 1940, is a quasiprobability distribution commonly used in quantum mechanics to represent the phase space distribution of a quantum state such as light in the phase space formulation. It is used in the field of quantum optics and particularly for tomographic purposes. It is also applied in the study of quantum effects in superconductors.
Definition and properties
The Husimi Q distribution (called Q-function in the context of quantum optics) is one of the simplest distributions of quasiprobability in phase space. It is constructed in such a way that observables written in anti-normal order follow the optical equivalence theorem. This means that it is essentially the density matrix put into normal order. This makes it relatively easy to calculate compared to other quasiprobability distributions through the formula
which is proportional to a trace of the operator involving the projection to the coherent state . It produces a pictorial representation of the state ρ to illustrate several of its mathematical properties. Its relative ease of calculation is related to its smoothness compared to other quasiprobability distributions. In fact, it can be understood as the Weierstrass transform of the Wigner quasiprobability distribution, i.e. a smoothing by a Gaussian filter,
Such Gauss transforms being essentially invertible in the Fourier domain via the convolution theorem, Q provides an equivalent description of quantum mechanics in phase space to that furnished by the Wigner distribution.
Alternatively, one can compute the Husimi Q distribution by taking the Segal–Bargmann transform of the wave function and then computing the associated probability density.
Q is normalized to unity,
and is non-negative definite and bounded:
Despite the fact that is non-negative definite and bounded like a standard joint probability distribution, this similarity may be misleading, because different coherent states are not orthogonal. Two different points do not represent disjoint physical contingencies; thus, Q(α) does not represent the probability of mutually exclusive states, as needed in the third axiom of probability theory.
may also be obtained by a different Weierstrass transform of the Glauber–Sudarshan P representation,
given , and the standard inner product of coherent states.
See also
Nonclassical light
Glauber–Sudarshan P-representation
Wehrl entropy
References
Quantum optics
Particle statistics |
https://en.wikipedia.org/wiki/Support%20of%20a%20module | In commutative algebra, the support of a module M over a commutative ring A is the set of all prime ideals of A such that (that is, the localization of M at is not equal to zero). It is denoted by . The support is, by definition, a subset of the spectrum of A.
Properties
if and only if its support is empty.
Let be a short exact sequence of A-modules. Then
Note that this union may not be a disjoint union.
If is a sum of submodules , then
If is a finitely generated A-module, then is the set of all prime ideals containing the annihilator of M. In particular, it is closed in the Zariski topology on Spec A.
If are finitely generated A-modules, then
If is a finitely generated A-module and I is an ideal of A, then is the set of all prime ideals containing This is .
Support of a quasicoherent sheaf
If F is a quasicoherent sheaf on a scheme X, the support of F is the set of all points x in X such that the stalk Fx is nonzero. This definition is similar to the definition of the support of a function on a space X, and this is the motivation for using the word "support". Most properties of the support generalize from modules to quasicoherent sheaves word for word. For example, the support of a coherent sheaf (or more generally, a finite type sheaf) is a closed subspace of X.
If M is a module over a ring A, then the support of M as a module coincides with the support of the associated quasicoherent sheaf on the affine scheme Spec A. Moreover, if is an affine cover of a scheme X, then the support of a quasicoherent sheaf F is equal to the union of supports of the associated modules Mα over each Aα.
Examples
As noted above, a prime ideal is in the support if and only if it contains the annihilator of . For example, over , the annihilator of the module
is the ideal . This implies that , the vanishing locus of the polynomial f. Looking at the short exact sequence
we might mistakenly conjecture that the support of I = (f) is Spec(R(f)), which is the complement of the vanishing locus of the polynomial f. In fact, since R is an integral domain, the ideal I = (f) = Rf is isomorphic to R as a module, so its support is the entire space: Supp(I) = Spec(R).
The support of a finite module over a Noetherian ring is always closed under specialization.
Now, if we take two polynomials in an integral domain which form a complete intersection ideal , the tensor property shows us that
See also
Annihilator (ring theory)
Associated prime
Support (mathematics)
References
Atiyah, M. F., and I. G. Macdonald, Introduction to Commutative Algebra, Perseus Books, 1969,
Module theory |
https://en.wikipedia.org/wiki/Lagrange%20number | In mathematics, the Lagrange numbers are a sequence of numbers that appear in bounds relating to the approximation of irrational numbers by rational numbers. They are linked to Hurwitz's theorem.
Definition
Hurwitz improved Peter Gustav Lejeune Dirichlet's criterion on irrationality to the statement that a real number α is irrational if and only if there are infinitely many rational numbers p/q, written in lowest terms, such that
This was an improvement on Dirichlet's result which had 1/q2 on the right hand side. The above result is best possible since the golden ratio φ is irrational but if we replace by any larger number in the above expression then we will only be able to find finitely many rational numbers that satisfy the inequality for α = φ.
However, Hurwitz also showed that if we omit the number φ, and numbers derived from it, then we can increase the number . In fact he showed we may replace it with 2. Again this new bound is best possible in the new setting, but this time the number is the problem. If we don't allow then we can increase the number on the right hand side of the inequality from 2 to /5. Repeating this process we get an infinite sequence of numbers , 2, /5, ... which converge to 3. These numbers are called the Lagrange numbers, and are named after Joseph Louis Lagrange.
Relation to Markov numbers
The nth Lagrange number Ln is given by
where mn is the nth Markov number, that is the nth smallest integer m such that the equation
has a solution in positive integers x and y.
References
External links
Lagrange number. From MathWorld at Wolfram Research.
Introduction to Diophantine methods irrationality and transcendence - Online lecture notes by Michel Waldschmidt, Lagrange Numbers on pp. 24–26.
Diophantine approximation |
https://en.wikipedia.org/wiki/Saint-Basile%2C%20Quebec | Saint-Basile is a municipality situated in Portneuf Regional County Municipality in the Canadian province of Quebec.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Basile 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: 2463 (2006 to 2011 population change: -3.8%)
Population in 2006: 2560
Population in 2001: 2575
Population in 1996:
Saint-Basile (parish): 840
Saint-Basile-Sud (village): 1684
Population in 1991:
Saint-Basile (parish): 823
Saint-Basile-Sud (village): 1733
Mother tongue:
English as first language: 2.4%
French as first language: 95.4%
English and French as first language: 0.4%
Other as first language: 1.8%
References
External links
Territoire de St-Basile - MRC de Portneuf
Cities and towns in Quebec
Incorporated places in Capitale-Nationale
Portneuf Regional County Municipality |
https://en.wikipedia.org/wiki/Michael%20Thelv%C3%A9n | Arne Michael "Tellus" Thelvén (born January 7, 1961) is a Swedish former professional ice hockey defenceman who played 207 games in the National Hockey League for the Boston Bruins.
Career statistics
Regular season and playoffs
International
External links
Michael Thelvén Biography and Statistics - Olympics at Sports-Reference.com
1961 births
Boston Bruins draft picks
Boston Bruins players
Djurgårdens IF Hockey players
Living people
Ice hockey people from Stockholm
Swedish ice hockey defencemen
Ice hockey players at the 1984 Winter Olympics
Medalists at the 1984 Winter Olympics
Olympic bronze medalists for Sweden
Olympic ice hockey players for Sweden
Olympic medalists in ice hockey |
https://en.wikipedia.org/wiki/Imaginary%20element | In model theory, a branch of mathematics, an imaginary element of a structure is roughly a definable equivalence class. These were introduced by , and elimination of imaginaries was introduced by .
Definitions
M is a model of some theory.
x and y stand for n-tuples of variables, for some natural number n.
An equivalence formula is a formula φ(x, y) that is a symmetric and transitive relation. Its domain is the set of elements a of Mn such that φ(a, a); it is an equivalence relation on its domain.
An imaginary element a/φ of M is an equivalence formula φ together with an equivalence class a.
M has elimination of imaginaries if for every imaginary element a/φ there is a formula θ(x, y) such that there is a unique tuple b so that the equivalence class of a consists of the tuples x such that θ(x, b).
A model has uniform elimination of imaginaries if the formula θ can be chosen independently of a.
A theory has elimination of imaginaries if every model of that theory does (and similarly for uniform elimination).
Examples
ZFC set theory has elimination of imaginaries.
Peano arithmetic has uniform elimination of imaginaries.
A vector space of dimension at least 2 over a finite field with at least 3 elements does not have elimination of imaginaries.
References
Model theory |
https://en.wikipedia.org/wiki/Michael%20J.%20D.%20Powell | Michael James David Powell (29 July 193619 April 2015) was a British mathematician, who worked in the Department of Applied Mathematics and Theoretical Physics (DAMTP) at the University of Cambridge.
Education and early life
Born in London, Powell was educated at Frensham Heights School and Eastbourne College. He earned his Bachelor of Arts degree followed by a Doctor of Science (DSc) degree in 1979 at the University of Cambridge.
Career and research
Powell was known for his extensive work in numerical analysis, especially nonlinear optimisation and approximation. He was a founding member of the Institute of Mathematics and its Applications and a founding Managing Editor of the Journal for Numerical Analysis. His mathematical contributions include quasi-Newton methods, particularly the Davidon-Fletcher-Powell formula and the Powell's Symmetric Broyden formula, augmented Lagrangian function (also called Powell-Rockafellar penalty function), sequential quadratic programming method (also called as Wilson-Han-Powell method), trust region algorithms (Powell's dog leg method), conjugate direction method (also called Powell's method), and radial basis function. He had been working on derivative-free optimization algorithms in recent years, the resultant algorithms including COBYLA, UOBYQA, NEWUOA, BOBYQA, and LINCOA. He was the author of numerous scientific papers and of several books, most notably Approximation Theory and Methods.
Awards and honours
Powell won several awards, including the George B. Dantzig Prize from the Mathematical Programming Society/Society for Industrial and Applied Mathematics (SIAM) and the Naylor Prize from the London Mathematical Society. Powell was elected a Foreign Associate of the National Academy of Sciences of the United States in 2001 and as a corresponding fellow to the Australian Academy of Science in 2007.
References
1936 births
2015 deaths
Alumni of Peterhouse, Cambridge
Fellows of Pembroke College, Cambridge
Mathematicians from London
Academic journal editors
Cambridge mathematicians
Fellows of the Institute of Mathematics and its Applications
Foreign associates of the National Academy of Sciences
Fellows of the Australian Academy of Science
People educated at Frensham Heights School
People educated at Eastbourne College
John Humphrey Plummer Professors |
https://en.wikipedia.org/wiki/Boston%20Celtics%20all-time%20roster | The following is a list of players, both past and current, who appeared in at least one regular season or playoff game for the Boston Celtics NBA franchise.
Players
Note: Statistics are correct through the end of the season.
A to B
|-
|align="left"| || align="center"|F/C || align="left"|Duke || align="center"|2 || align="center"|– || 76 || 1,311 || 346 || 20 || 578 || 17.3 || 4.6 || 0.3 || 7.6 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|Iowa State || align="center"|1 || align="center"| || 2 || 24 || 15 || 3 || 8 || 12.0 || 7.5 || 1.5 || 4.0 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|Oral Roberts || align="center"|2 || align="center"|– || 141 || 1,783 || 416 || 61 || 424 || 12.6 || 3.0 || 0.4 || 3.0 || align=center|
|-
|align="left" bgcolor="#FFCC00"|+ || align="center"|G || align="left"|BYU || align="center"|8 || align="center"|– || 556 || 15,603 || 1,534 || 2,422 || 6,257 || 28.1 || 2.8 || 4.4 || 11.3 || align=center|
|-
|align="left"| || align="center"|G || align="left"|Arizona || align="center"|1 || align="center"| || 18 || 107 || 11 || 12 || 19 || 5.9 || 0.6 || 0.7 || 1.1 || align=center|
|-
|align="left" bgcolor="#FFFF99"|^ || align="center"|G || align="left"|UConn || align="center"|5 || align="center"|– || 358 || 12,774 || 1,215 || 981 || 5,987 || 35.7 || 3.4 || 2.7 || 16.7 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|Oklahoma State || align="center"|6 || align="center"|– || 336 || 6,194 || 873 || 439 || 2,423 || 18.4 || 2.6 || 1.3 || 7.2 || align=center|
|-
|align="left"| || align="center"|G || align="left"|West Virginia || align="center"|1 || align="center"| || 22 || 126 || 13 || 6 || 61 || 5.7 || 0.6 || 0.3 || 2.8 || align=center|
|-
|align="left"| || align="center"|G || align="left"|Georgia Tech || align="center"|5 || align="center"|– || 241 || 7,268 || 715 || 1,250 || 2,717 || 30.2 || 3.0 || 5.2 || 11.3 || align=center|
|-
|align="left"| || align="center"|C || align="left"|UNLV || align="center"|1 || align="center"| || 21 || 149 || 31 || 2 || 22 || 7.1 || 1.5 || 0.1 || 1.0 || align=center|
|-
|align="left" bgcolor="#FFFF99"|^ || align="center"|G || align="left"|UTEP || align="center"|5 || align="center"|– || 363 || 11,324 || 683 || 2,563 || 4,550 || 31.2 || 1.9 || 7.1 || 12.5 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|Cincinnati || align="center"|4 || align="center"|– || 204 || 2,550 || 788 || 142 || 753 || 12.5 || 3.9 || 0.7 || 3.7 || align=center|
|-
|align="left"| || align="center"|G || align="left"|FIU || align="center"|1 || align="center"| || 15 || 190 || 23 || 25 || 36 || 12.7 || 1.5 || 1.7 || 2.4 || align=center|
|-
|align="left"| || align="center"|G || align="left"|South Florida || align="center"|1 || align="center"| || 24 || 793 || 45 || 128 || 289 || 33.0 || 1.9 || 5.3 || 12.0 || align=center|
|-
|align="left"| || align="center"|C || align="left"|Santa Clara || align="center"|1 || align="center"| || 23 || |
https://en.wikipedia.org/wiki/Baily%E2%80%93Borel%20compactification | In mathematics, the Baily–Borel compactification is a compactification of a quotient of a Hermitian symmetric space by an arithmetic group, introduced by .
Example
If C is the quotient of the upper half plane by a congruence subgroup of SL2(Z), then the Baily–Borel compactification of C is formed by adding a finite number of cusps to it.
See also
L² cohomology
References
Algebraic geometry
Compactification (mathematics) |
https://en.wikipedia.org/wiki/Nonrecursive%20ordinal | In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using recursive ordinal notations.
The Church–Kleene ordinal and variants
The smallest non-recursive ordinal is the Church Kleene ordinal, , named after Alonzo Church and S. C. Kleene; its order type is the set of all recursive ordinals. Since the successor of a recursive ordinal is recursive, the Church–Kleene ordinal is a limit ordinal. It is also the smallest ordinal that is not hyperarithmetical, and the smallest admissible ordinal after (an ordinal is called admissible if .) The -recursive subsets of are exactly the subsets of .
The notation is in reference to , the first uncountable ordinal, which is the set of all countable ordinals, analogously to how the Church-Kleene ordinal is the set of all recursive ordinals. Some old sources use to denote the Church-Kleene ordinal.
For a set , A set is x-computable if it is computable from a Turing machine with an oracle state that queries x. The relativized Church–Kleene ordinal is the supremum of the order types of x-computable relations. The Friedman-Jensen-Sacks theorem states that for every countable admissible ordinal , there exists a set x such that .
, first defined by Stephen G. Simpson is an extension of the Church–Kleene ordinal. This is the smallest limit of admissible ordinals, yet this ordinal is not admissible. Alternatively, this is the smallest α such that is a model of -comprehension.
Recursively ordinals
The th admissible ordinal is sometimes denoted by .
Recursively "x" ordinals, where "x" typically represents a large cardinal property, are kinds of nonrecursive ordinals.
An ordinal is called recursively inaccessible if it is admissible and a limit of admissibles. Alternatively, is recursively inaccessible iff is the th admissible ordinal, or iff , an extension of Kripke–Platek set theory stating that each set is contained in a model of Kripke–Platek set theory. Under the condition that ("every set is hereditarily countable"), is recursively inaccessible iff is a model of -comprehension.
An ordinal is called recursively hyperinaccessible if it is recursively inaccessible and a limit of recursively inaccessibles, or where is the th recursively inaccessible. Like "hyper-inaccessible cardinal", different authors conflict on this terminology.
An ordinal is called recursively Mahlo if it is admissible and for any -recursive function there is an admissible such that (that is, is closed under ). Mirroring the Mahloness hierarchy, is recursively -Mahlo for an ordinal if it is admissible and for any -recursive function there is an admissible ordinal such that is closed under , and is recursively -Mahlo for all .
An ordinal is called recursively weakly compact if it is -reflecting, or equivalently, 2-admissible. These ordinals have strong recursive Mahloness properties, if α is -refl |
https://en.wikipedia.org/wiki/Feferman%E2%80%93Sch%C3%BCtte%20ordinal | In mathematics, the Feferman–Schütte ordinal Γ0 is a large countable ordinal.
It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion.
It is named after Solomon Feferman and Kurt Schütte, the former of whom suggested the name Γ0.
There is no standard notation for ordinals beyond the Feferman–Schütte ordinal. There are several ways of representing the Feferman–Schütte ordinal, some of which use ordinal collapsing functions: , , , or .
Definition
The Feferman–Schütte ordinal can be defined as the smallest ordinal that cannot be obtained by starting with 0 and using the operations of ordinal addition and the Veblen functions φα(β). That is, it is the smallest α such that φα(0) = α.
Properties
This ordinal is sometimes said to be the first impredicative ordinal, though this is controversial, partly because there is no generally accepted precise definition of "predicative". Sometimes an ordinal is said to be predicative if it is less than Γ0.
Any recursive path ordering whose function symbols are well-founded with order type less than that of itself has order type .
References
Proof theory
Ordinal numbers |
https://en.wikipedia.org/wiki/Piste%20de%20Bobsleigh%20des%20Pellerins | The Piste de Bobsleigh des Pellerins (Pellerins Bobsleigh Track in ) was a bobsleigh track constructed for the 1924 Winter Olympics in Chamonix, France.
Track statistics
The track was 1369.88 metres long with 19 curves and an elevation difference of 156.29 metres. Turns 1-2, 4-6, 8, 10, 12, and 14 have no names attached to them. Average grade for the track was 11.4%.
During the bobsleigh competitions, a total of nine teams from five countries competed. Two teams withdrew after the first run while another team withdrew after the third run. The three medalist teams were from Switzerland (gold), Great Britain (silver), and Belgium (bronze). Bobsleigh speeds reached up to 115 km/h (71.5 mph) during competition.
The track is no longer in use. It was located near the Ligne du téléférique de l'Aiguille du Midi aerial tramway area, near the village of Les Pèlerins.
References
Venues of the 1924 Winter Olympics
Bobsleigh, luge, and skeleton tracks
Defunct sports venues in France
Olympic bobsleigh venues
Sports venues in Haute-Savoie |
https://en.wikipedia.org/wiki/Multinomial%20test | In statistics, the multinomial test is the test of the null hypothesis that the parameters of a multinomial distribution equal specified values; it is used for categorical data.
Beginning with a sample of items each of which has been observed to fall into one of categories. It is possible to define as the observed numbers of items in each cell. Hence
Next, defining a vector of parameters where:
These are the parameter values under the null hypothesis.
The exact probability of the observed configuration under the null hypothesis is given by
The significance probability for the test is the probability of occurrence of the data set observed, or of a data set less likely than that observed, if the null hypothesis is true. Using an exact test, this is calculated as
where the sum ranges over all outcomes as likely as, or less likely than, that observed. In practice this becomes computationally onerous as and increase so it is probably only worth using exact tests for small samples. For larger samples, asymptotic approximations are accurate enough and easier to calculate.
One of these approximations is the likelihood ratio. An alternative hypothesis can be defined under which each value is replaced by its maximum likelihood estimate The exact probability of the observed configuration under the alternative hypothesis is given by
The natural logarithm of the likelihood ratio, between these two probabilities, multiplied by is then the statistic for the likelihood ratio test
(The factor is chosen to make the statistic asymptotically chi-squared distributed, for convenient comparison to a familiar statistic commonly used for the same application.)
If the null hypothesis is true, then as increases, the distribution of converges to that of chi-squared with degrees of freedom. However it has long been known (e.g. Lawley) that for finite sample sizes, the moments of are greater than those of chi-squared, thus inflating the probability of type I errors (false positives). The difference between the moments of chi-squared and those of the test statistic are a function of Williams showed that the first moment can be matched as far as if the test statistic is divided by a factor given by
In the special case where the null hypothesis is that all the values are equal to (i.e. it stipulates a uniform distribution), this simplifies to
Subsequently, Smith et al. derived a dividing factor which matches the first moment as far as For the case of equal values of this factor is
The null hypothesis can also be tested by using Pearson's chi-squared test
where is the expected number of cases in category under the null hypothesis. This statistic also converges to a chi-squared distribution with degrees of freedom when the null hypothesis is true but does so from below, as it were, rather than from above as does, so may be preferable to the uncorrected version of for small samples.
References
Categorical variable interactions |
https://en.wikipedia.org/wiki/Projective%20harmonic%20conjugate | In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following construction:
Given three collinear points , let be a point not lying on their join and let any line through meet at respectively. If and meet at , and meets at , then is called the harmonic conjugate of with respect to and .
The point does not depend on what point is taken initially, nor upon what line through is used to find and . This fact follows from Desargues theorem.
In real projective geometry, harmonic conjugacy can also be defined in terms of the cross-ratio as .
Cross-ratio criterion
The four points are sometimes called a harmonic range (on the real projective line) as it is found that always divides the segment internally in the same proportion as divides externally. That is:
If these segments are now endowed with the ordinary metric interpretation of real numbers they will be signed and form a double proportion known as the cross ratio (sometimes double ratio)
for which a harmonic range is characterized by a value of −1. We therefore write:
The value of a cross ratio in general is not unique, as it depends on the order of selection of segments (and there are six such selections possible). But for a harmonic range in particular there are just three values of cross ratio: since −1 is self-inverse – so exchanging the last two points merely reciprocates each of these values but produces no new value, and is known classically as the harmonic cross-ratio.
In terms of a double ratio, given points on an affine line, the division ratio of a point is
Note that when , then is negative, and that it is positive outside of the interval.
The cross-ratio is a ratio of division ratios, or a double ratio. Setting the double ratio to minus one means that when , then and are harmonic conjugates with respect to and . So the division ratio criterion is that they be additive inverses.
Harmonic division of a line segment is a special case of Apollonius' definition of the circle.
In some school studies the configuration of a harmonic range is called harmonic division.
Of midpoint
When is the midpoint of the segment from to , then
By the cross-ratio criterion, the harmonic conjugate of will be when . But there is no finite solution for on the line through and . Nevertheless,
thus motivating inclusion of a point at infinity in the projective line. This point at infinity serves as the harmonic conjugate of the midpoint .
From complete quadrangle
Another approach to the harmonic conjugate is through the concept of a complete quadrangle such as in the above diagram. Based on four points, the complete quadrangle has pairs of opposite sides and diagonals. In the expression of harmonic conjugates by H. S. M. Coxeter, the diagonals are considered a pair of opposite sides:
is the harmonic conjugate of with respect to and , which means that there is a quadrangle su |
https://en.wikipedia.org/wiki/FIBA%20European%20Champions%20Cup%20and%20EuroLeague%20records%20and%20statistics | This page details statistics of the FIBA European Champions Cup and EuroLeague.
General performances
By club
Maccabi Elite Tel Aviv beat Panathinaikos in the 2000-01 FIBA SuproLeague final. The league did not contain all of the European champions.
Kinder Bologna beat Tau Cerámica in the 2000–01 Euroleague final. The league did not contain all of the European champions.
By nation
By head coach
Players with the most championships
By city
Number of participating clubs of the EuroLeague Basketball era
The following is a list of clubs that have played or will be playing in the EuroLeague group stages, during the Euroleague Basketball era.
Clubs
By semi-final appearances (FIBA European Champions Cup and EuroLeague Basketball)
EuroLeague Final Four
The history of the EuroLeague Final Four system, which was permanently introduced in the 1987–88 season.
By season
Performance by club
References
External links
EuroLeague Official Website
Basketball |
https://en.wikipedia.org/wiki/Erwin%20Voellmy | Erwin Voellmy (9 September 1886, Herzogenbuchsee – 15 January 1951, Basel) was a Swiss chess master.
Voellmy, a mathematics teacher by profession, edited the chess column in Basler Nachrichten for 40 years, and was an author of several chess books.
He was Swiss Champion three times; in 1911 (jointly), 1920 and 1922. Voellmy represented Switzerland at:
the 1st unofficial Chess Olympiad at Paris 1924 (+6 –2 =5),
the 2nd Chess Olympiad at The Hague 1928 (+5 –2 =4),
the 3rd unofficial Chess Olympiad at Munich 1936 (+5 –7 =5).
He won team bronze medal at Paris 1924.
He shared 1st with Alexander Alekhine and Oskar Naegeli at Bern 1932 (Qudrangular).
References
External links
Erwin Voellmy´s games
1886 births
1951 deaths
Swiss chess players
Chess Olympiad competitors |
https://en.wikipedia.org/wiki/Kleene%27s%20O | In set theory and computability theory, Kleene's is a canonical subset of the natural numbers when regarded as ordinal notations. It contains ordinal notations for every computable ordinal, that is, ordinals below Church–Kleene ordinal, . Since is the first ordinal not representable in a computable system of ordinal notations the elements of can be regarded as the canonical ordinal notations.
Kleene (1938) described a system of notation for all computable ordinals (those less than the Church–Kleene ordinal). It uses a subset of the natural numbers instead of finite strings of symbols. Unfortunately, there is in general no effective way to tell whether some natural number represents an ordinal, or whether two numbers represent the same ordinal. However, one can effectively find notations which represent the ordinal sum, product, and power (see ordinal arithmetic) of any two given notations in Kleene's ; and given any notation for an ordinal, there is a computably enumerable set of notations which contains one element for each smaller ordinal and is effectively ordered.
Definition
The basic idea of Kleene's system of ordinal notations is to build up ordinals in an effective manner. For members of , the ordinal for which is a notation is . and (a partial ordering of Kleene's ) are the smallest sets such that the following holds.
.
Suppose is the -th partial computable function. If is total and , then
This definition has the advantages that one can computably enumerate the predecessors of a given ordinal (though not in the ordering) and that the notations are downward closed, i.e., if there is a notation for and then there is a notation for . There are alternate definitions, such as the set of indices of (partial) well-orderings of the natural numbers.
Basic properties of <O
If and and then ; but the converse may fail to hold.
induces a tree structure on , so is well-founded.
only branches at limit ordinals; and at each notation of a limit ordinal, is infinitely branching.
Since every computable function has countably many indices, each infinite ordinal receives countably many notations; the finite ordinals have unique notations, usually denoted .
The first ordinal that doesn't receive a notation is called the Church–Kleene ordinal and is denoted by . Since there are only countably many computable functions, the ordinal is evidently countable.
The ordinals with a notation in Kleene's are exactly the computable ordinals. (The fact that every computable ordinal has a notation follows from the closure of this system of ordinal notations under successor and effective limits.)
is not computably enumerable, but there is a computably enumerable relation which agrees with precisely on members of .
For any notation , the set of notations below is computably enumerable. However, Kleene's , when taken as a whole, is (see analytical hierarchy) and not arithmetical because of the following:
is -complete (i.e. |
https://en.wikipedia.org/wiki/Doublesix | Doublesix was a subsidiary of Kuju Entertainment based in Guildford that develops video games for the digital download market. The studio was formed from the team that made Geometry Wars: Galaxies. They also made the zombie themed shooter, Burn Zombie Burn!. The company has received awards and nominations since its inception; notably that of the develop "Best New UK Studio 2008" and a nomination for best hand-held game (Geometry Wars: Galaxies) at the 2009 BAFTAs. They also worked on a successor to Burn Zombie Burn!, entitled All Zombies Must Die!.
References
External links
British companies established in 2007
Defunct video game companies of the United Kingdom
Video game development companies
Video game companies established in 2007
Companies based in Guildford |
https://en.wikipedia.org/wiki/Small%20Veblen%20ordinal | In mathematics, the small Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. It is occasionally called the Ackermann ordinal, though the Ackermann ordinal described by is somewhat smaller than the small Veblen ordinal.
There is no standard notation for ordinals beyond the Feferman–Schütte ordinal . Most systems of notation use symbols such as , , , some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions".
The small Veblen ordinal or is the limit of ordinals that can be described using a version of Veblen functions with finitely many arguments. It is the ordinal that measures the strength of Kruskal's theorem. It is also the ordinal type of a certain ordering of rooted trees .
References
Ordinal numbers |
https://en.wikipedia.org/wiki/Veblen%20ordinal | In mathematics, the Veblen ordinal is either of two large countable ordinals:
The small Veblen ordinal
The large Veblen ordinal
See also
Veblen function |
https://en.wikipedia.org/wiki/Large%20Veblen%20ordinal | In mathematics, the large Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen.
There is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are ordinal collapsing functions.
The large Veblen ordinal is sometimes denoted by or or . It was constructed by Veblen using an extension of Veblen functions allowing infinitely many arguments.
References
Ordinal numbers |
https://en.wikipedia.org/wiki/Aragua%20Municipality | The Aragua Municipality is one of the 21 municipalities (municipios) that makes up the eastern Venezuelan state of Anzoátegui and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 29,268. The town of Aragua de Barcelona is the shire town of the Aragua Municipality.
Demographics
The Aragua Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 32,558 (up from 28,723 in 2000). This amounts to 2.2% of the state's population. The municipality's population density is .
Government
The mayor of the Aragua Municipality is Juan de Dios Figueredo, re-elected on 23 November 2008 with 52% of the vote. The municipality is divided into two parishes; Capital Aragua and Cachipo.
See also
Anzoátegui
Municipalities of Venezuela
References
External links
aragua-anzoategui.gob.ve
Municipalities of Anzoategui |
https://en.wikipedia.org/wiki/Veblen%20function | In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal function, then for any non-zero ordinal α, φα is the function enumerating the common fixed points of φβ for β<α. These functions are all normal.
Veblen hierarchy
In the special case when φ0(α)=ωα
this family of functions is known as the Veblen hierarchy.
The function φ1 is the same as the ε function: φ1(α)= εα. If then . From this and the fact that φβ is strictly increasing we get the ordering: if and only if either ( and ) or ( and ) or ( and ).
Fundamental sequences for the Veblen hierarchy
The fundamental sequence for an ordinal with cofinality ω is a distinguished strictly increasing ω-sequence which has the ordinal as its limit. If one has fundamental sequences for α and all smaller limit ordinals, then one can create an explicit constructive bijection between ω and α, (i.e. one not using the axiom of choice). Here we will describe fundamental sequences for the Veblen hierarchy of ordinals. The image of n under the fundamental sequence for α will be indicated by α[n].
A variation of Cantor normal form used in connection with the Veblen hierarchy is — every nonzero ordinal number α can be uniquely written as , where k>0 is a natural number and each term after the first is less than or equal to the previous term, and each If a fundamental sequence can be provided for the last term, then that term can be replaced by such a sequence to get
For any β, if γ is a limit with then let
No such sequence can be provided for = ω0 = 1 because it does not have cofinality ω.
For we choose
For we use and i.e. 0, , , etc..
For , we use and
Now suppose that β is a limit:
If , then let
For , use
Otherwise, the ordinal cannot be described in terms of smaller ordinals using and this scheme does not apply to it.
The Γ function
The function Γ enumerates the ordinals α such that φα(0) = α.
Γ0 is the Feferman–Schütte ordinal, i.e. it is the smallest α such that φα(0) = α.
For Γ0, a fundamental sequence could be chosen to be and
For Γβ+1, let and
For Γβ where is a limit, let
Generalizations
Finitely many variables
To build the Veblen function of a finite number of arguments (finitary Veblen function), let the binary function be as defined above.
Let be an empty string or a string consisting of one or more comma-separated zeros and be an empty string or a string consisting of one or more comma-separated ordinals with . The binary function can be written as where both and are empty strings.
The finitary Veblen functions are defined as follows:
if , then denotes the -th common fixed point of the functions for each
For example, is the -th fixed point of the functions , namely ; then enumerates the fixed points of that function, i.e., of the function; and enumerates the fixed points of all the . Each instance |
https://en.wikipedia.org/wiki/Anaco%20Municipality | The Anaco Municipality is one of the 21 municipalities (municipios) that makes up the eastern Venezuelan state of Anzoátegui and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 122,634. The town of Anaco is the shire town of the Anaco Municipality.
Economy
Anaco is an industrial town/municipality, connected to the natural gas and petroleum industries.
Demographics
The Anaco Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 124,431 (up from 106,720 in 2000). This amounts to 8.4% of the state's population. The municipality's population density is .
Government
The mayor of the Anaco Municipality is Francisco Solorzano, elected on 23 November 2008 with 60% of the vote. He replaced Jacinto Romero Luna shortly after the elections. The municipality is divided into two parishes; Capital Anaco and San Joaquín.
See also
Anaco
Anzoátegui
Municipalities of Venezuela
References
External links
anaco-anzoategui.gob.ve
Municipalities of Anzoategui |
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