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https://en.wikipedia.org/wiki/Continuous-time%20stochastic%20process | In probability theory and statistics, a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process for which the index variable takes a continuous set of values, as contrasted with a discrete-time process for which the index variable takes only distinct values. An alternative terminology uses continuous parameter as being more inclusive.
A more restricted class of processes are the continuous stochastic processes; here the term often (but not always) implies both that the index variable is continuous and that sample paths of the process are continuous. Given the possible confusion, caution is needed.
Continuous-time stochastic processes that are constructed from discrete-time processes via a waiting time distribution are called continuous-time random walks.
Examples
An example of a continuous-time stochastic process for which sample paths are not continuous is a Poisson process. An example with continuous paths is the Ornstein–Uhlenbeck process.
See also
Continuous signal
References
Stochastic processes |
https://en.wikipedia.org/wiki/K.%20C.%20Sreedharan%20Pillai | K C Sreedharan Pillai (1920–1985) was an Indian statistician who was known for his works on multivariate analysis and probability distributions.
Pillai studied at the University of Travancore in Trivandrum. He graduated in 1941 and obtained his master's degree in 1945. He was appointed a lecturer at the University of Kerala in 1945 and worked there for six years until he went to the United States in 1951. After studying for one year at Princeton University, he went to the University of North Carolina where he was awarded a doctorate in statistics in 1954.
His first post was as a statistician with the United Nations, a post he held from 1954 until 1962. One of his achievements at that post was the founding of the Statistical Center of the University of the Philippines. He was a visiting Professor and Advisor to the University of Philippines for a number of years and supervised graduate students there. In 1962 Pillai was appointed Professor of Statistics and Mathematics at Purdue University. Pillai's research was in statistics, in particular in multivariate statistical analysis. Pillai was honoured by being elected a Fellow of the American Statistical Association and a Fellow of the Institute of Mathematical Statistics. He was an elected member of the International Statistical Institute.
He was a keen golfer too.
He died on 5 June 1985 in Lafayette, Indiana, USA.
External links
Indian emigrants to the United States
Indian statisticians
1920 births
1985 deaths
20th-century Indian mathematicians
Scientists from Thiruvananthapuram
Malayali people
Princeton University alumni
American statisticians
University of North Carolina at Chapel Hill alumni
American Hindus
Fellows of the American Statistical Association
University of Kerala alumni
Elected Members of the International Statistical Institute
American academics of Indian descent |
https://en.wikipedia.org/wiki/Hadamard%20manifold | In mathematics, a Hadamard manifold, named after Jacques Hadamard — more often called a Cartan–Hadamard manifold, after Élie Cartan — is a Riemannian manifold that is complete and simply connected and has everywhere non-positive sectional curvature. By Cartan–Hadamard theorem all Cartan–Hadamard manifolds are diffeomorphic to the Euclidean space Furthermore it follows from the Hopf–Rinow theorem that every pairs of points in a Cartan–Hadamard manifold may be connected by a unique geodesic segment. Thus Cartan–Hadamard manifolds are some of the closest relatives of
Examples
The Euclidean space with its usual metric is a Cartan–Hadamard manifold with constant sectional curvature equal to
Standard -dimensional hyperbolic space is a Cartan–Hadamard manifold with constant sectional curvature equal to
Properties
In Cartan-Hadamard manifolds, the map is a diffeomorphism for all
See also
References
Riemannian manifolds |
https://en.wikipedia.org/wiki/Koreans%20in%20Peru | Koreans in Peru (; ) formed Latin America's seventh-largest Korean diaspora community , according to the statistics of South Korea's Ministry of Foreign Affairs and Trade. They are relatively small in size compared to the other Asian communities in Peru.
Migration history
The first Korean migrant to Peru is believed to have been Park Man-bok, who was invited to Peru to coach the women's national volleyball team in 1972. Under his tutelage, the team would go on to a variety of successes in the 1980s, culminating in the winning of a silver medal for their country at the 1988 Summer Olympics in Seoul.
However, few of Park's countrymen joined him in Peru; as late as 1985, there were only nine Korean families resident in the country, totalling 27 individuals. A large portion did not come directly from Korea, but had instead first settled in Bolivia, Paraguay, or Chile. The population began to increase in 1993, as the economic and social situation in Peru stabilised; during the 1990s, roughly two or three new Korean families arrived in Peru every month. However, after 1997, their population fell by nearly 56% from 1,774 to just 788 by 2005, largely due to outward migration to Mexico and Guatemala in 1998 and 1999; some of those who had arrived via Chile also returned there.
By 2011, Peru's Korean population had recovered slightly to 1,305. According to South Korean government statistics, 24 took up Peruvian nationality, 342 stayed in Peru as permanent residents, 30 were international students, and the remaining 909 had other kinds of visas.
Professions
The economic profile of the Korean community in Peru is widely varied and has continued to shift over the years. In the 1980s, many were involved in calamari fishing. The roughly 900 Koreans resident in Peru in 2001 included among their number 500 business people, 90 representatives of the South Korean government, 48 factory owners, 39 religious workers, and 25 sportspeople. Many businesspeople are involved in the import of products from South Korea, especially used cars, computers, and construction equipment; however, the largest portion of Koreans in Peru are involved with the textile industry.
Aside from Park Man-bok, other Koreans have made notable contributions to sport in Peru. Lee Ki-Hyung, a 1973 taekwondo world champion, went on to work as a martial arts instructor in the Peruvian Air Force. Chung Eui-Hwang, 9th dan black belt, arrived in Peru in 1979, and from then until 1989 trained the Military Academy of Chorrillos, the Peruvian Commando Special Forces, the Technical Academy of the Military, and the First Brigade of the Special Forces. These efforts began the trend of popularisation of taekwondo in Peru, which grew to 30,000 practitioners .
Politics
Despite the small size of its Korean population, Peru was the site of a major precedent for Korean immigrants' political integration when the city of Chanchamayo, Junín elected Mario Jung (정흥원, also spelled Mario Yung) as its mayor in 2011. |
https://en.wikipedia.org/wiki/Van%20Wijngaarden%20transformation | In mathematics and numerical analysis, the van Wijngaarden transformation is a variant on the Euler transform used to accelerate the convergence of an alternating series.
One algorithm to compute Euler's transform runs as follows: Compute a row of partial sums and form rows of averages between neighbors The first column then contains the partial sums of the Euler transform.
Adriaan van Wijngaarden's contribution was to point out that it is better not to carry this procedure through to the very end, but to stop two-thirds of the way. If are available, then is almost always a better approximation to the sum than . In many cases the diagonal terms do not converge in one cycle so process of averaging is to be repeated with diagonal terms by bringing them in a row. (For example, this will be needed in a geometric series with ratio .) This process of successive averaging of the average of partial sum can be replaced by using the formula to calculate the diagonal term.
For a simple-but-concrete example, recall the Leibniz formula for pi The algorithm described above produces the following table:
These correspond to the following algorithmic outputs:
References
See also
Euler summation
Mathematical series
Numerical analysis |
https://en.wikipedia.org/wiki/List%20of%20Leicester%20City%20F.C.%20records%20and%20statistics | This article collates key records and statistics relating to Leicester City F.C., including information on honours, player appearances and goals, matches, sequences, internationals, season records, opponents and attendances.
Honours
League
First Division / Premier League (level 1)
Champions: 2015–16
Runners-up: 1928–29
Second Division / First Division / Championship (level 2)
Champions (7, joint record): 1924–25, 1936–37, 1953–54, 1956–57, 1970–71, 1979–80, 2013–14
Runners-up: 1907–08, 2002–03
Play-off winners: 1994, 1996
League One (level 3)
Champions: 2008–09
Cup
FA Cup
Winners: 2020–21
Runners-up: 1948–49, 1960–61, 1962–63, 1968–69
League Cup
Winners: 1963–64, 1996–97, 1999–2000
Runners-up: 1964–65, 1998–99
FA Charity Shield / FA Community Shield
Winners: 1971, 2021
Runners-up: 2016
Appearances
Most appearances
All-time most appearances (Does not include wartime appearances)
Current players in bold.
Most appearances – 600 by Graham Cross (29 April 1961 – 23 August 1975)
Most league appearances – 528 by Adam Black (24 January 1920 – 9 February 1935)
Most appearances in the first tier (Premier League and predecessors) – 414 by Graham Cross
Most appearances in the second tier (Championship and predecessors) – 304 by Mal Griffiths
Most appearances in the third tier (League One and predecessors) – 46 by Matty Fryatt
Most FA Cup appearances – 59 by Graham Cross (8 January 1963 – 24 February 1975)
Most League Cup appearances – 40 by Graham Cross (26 September 1962 – 8 October 1974) and Steve Walsh (23 September 1986 – 25 January 2000)
Most appearances in a single season – 61 by Gary Mills (46 in FL, 3 PO, 2 FAC, 4 FLC, 6 FMC) (1991–92)
Consecutive appearances
Most consecutive appearances – 331 by Mark Wallington (4 January 1975 – 6 March 1982)
Most consecutive League appearances – 294 by Mark Wallington (11 January 1975 – 2 March 1982)
Most consecutive FA Cup appearances – 52 by Graham Cross – (14 January 1965 – 24 February 1975)
Most consecutive League Cup appearances – 21 by John Sjoberg (15 January 1964 – 4 September 1968) and Mark Wallington (9 September 1975 – 9 October 1984)
Youngest and oldest appearances
Longest Spell at club – 19 years 249 days by Sep Smith (31 August 1929 – 7 May 1949))
Youngest first-team player – 15 years 203 days by Ashley Chambers (v Blackpool, 15 September 2005)
Oldest first-team player – 43 years 21 days by Mark Schwarzer (v Hull City, 27 October 2015)
Oldest debutant – 42 years 111 days by Mark Schwarzer (v Tottenham, 24 January 2015)
Goalscorers
Top goalscorers
Top 10 all-time top goalscorers (Does not include wartime appearances.) Current players in bold.
Most goals – 273 by Arthur Chandler
Most league goals – 259 by Arthur Chandler
Most goals in the first tier (Premier League and predecessors) – 203 by Arthur Chandler
Most goals in the second tier (Championship and predecessors) – 208 by Arthur Rowley
Most goals in the third tier (League One and predecessors) - 27 by Matty Fryatt
Most |
https://en.wikipedia.org/wiki/Point%20reflection | In geometry, a point reflection (also called a point inversion or central inversion) is a transformation of affine space in which every point is reflected across a specific fixed point. When dealing with crystal structures and in the physical sciences the terms inversion symmetry, inversion center or centrosymmetric are more commonly used.
A point reflection is an involution: applying it twice is the identity transformation. It is equivalent to a homothetic transformation with scale factor . The point of inversion is also called homothetic center.
An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric. A point group including a point reflection among its symmetries is called centrosymmetric.
In Euclidean space, a point reflection is an isometry (preserves distance). In the Euclidean plane, a point reflection is the same as a half-turn rotation (180° or radians); a point reflection through the object's centroid is the same as a half-turn spin.
Terminology
The term reflection is loose, and considered by some an abuse of language, with inversion preferred; however, point reflection is widely used. Such maps are involutions, meaning that they have order 2 – they are their own inverse: applying them twice yields the identity map – which is also true of other maps called reflections. More narrowly, a reflection refers to a reflection in a hyperplane ( dimensional affine subspace – a point on the line, a line in the plane, a plane in 3-space), with the hyperplane being fixed, but more broadly reflection is applied to any involution of Euclidean space, and the fixed set (an affine space of dimension k, where ) is called the mirror. In dimension 1 these coincide, as a point is a hyperplane in the line.
In terms of linear algebra, assuming the origin is fixed, involutions are exactly the diagonalizable maps with all eigenvalues either 1 or −1. Reflection in a hyperplane has a single −1 eigenvalue (and multiplicity on the 1 eigenvalue), while point reflection has only the −1 eigenvalue (with multiplicity n).
The term inversion should not be confused with inversive geometry, where inversion is defined with respect to a circle.
Examples
In two dimensions, a point reflection is the same as a rotation of 180 degrees. In three dimensions, a point reflection can be described as a 180-degree rotation composed with reflection across the plane of rotation, perpendicular to the axis of rotation. In dimension n, point reflections are orientation-preserving if n is even, and orientation-reversing if n is odd.
Formula
Given a vector a in the Euclidean space Rn, the formula for the reflection of a across the point p is
In the case where p is the origin, point reflection is simply the negation of the vector a.
In Euclidean geometry, the inversion of a point X with respect to a point P is a poin |
https://en.wikipedia.org/wiki/Risk%20measure | In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator. In recent years attention has turned towards convex and coherent risk measurement.
Mathematically
A risk measure is defined as a mapping from a set of random variables to the real numbers. This set of random variables represents portfolio returns. The common notation for a risk measure associated with a random variable is . A risk measure should have certain properties:
Normalized
Translative
Monotone
Set-valued
In a situation with -valued portfolios such that risk can be measured in of the assets, then a set of portfolios is the proper way to depict risk. Set-valued risk measures are useful for markets with transaction costs.
Mathematically
A set-valued risk measure is a function , where is a -dimensional Lp space, , and where is a constant solvency cone and is the set of portfolios of the reference assets. must have the following properties:
Normalized
Translative in M
Monotone
Examples
Value at risk
Expected shortfall
Superposed risk measures
Entropic value at risk
Drawdown
Tail conditional expectation
Entropic risk measure
Superhedging price
Expectile
Variance
Variance (or standard deviation) is not a risk measure in the above sense. This can be seen since it has neither the translation property nor monotonicity. That is, for all , and a simple counterexample for monotonicity can be found. The standard deviation is a deviation risk measure. To avoid any confusion, note that deviation risk measures, such as variance and standard deviation are sometimes called risk measures in different fields.
Relation to acceptance set
There is a one-to-one correspondence between an acceptance set and a corresponding risk measure. As defined below it can be shown that and .
Risk measure to acceptance set
If is a (scalar) risk measure then is an acceptance set.
If is a set-valued risk measure then is an acceptance set.
Acceptance set to risk measure
If is an acceptance set (in 1-d) then defines a (scalar) risk measure.
If is an acceptance set then is a set-valued risk measure.
Relation with deviation risk measure
There is a one-to-one relationship between a deviation risk measure D and an expectation-bounded risk measure where for any
.
is called expectation bounded if it satisfies for any nonconstant X and for any constant X.
See also
References
Further reading
Actuarial science
Financial risk modeling |
https://en.wikipedia.org/wiki/Sayed%20Abdel%20Hafeez | Sayed Mohamed Mohamed Abdel Hafeez (; born 27 October 1977) is an Egyptian retired professional footballer who played as a winger.
Career statistics
International
International Goals
Scores and results list Egypt's goal tally first.
Honours
Club
Al Ahly
Egyptian Premier League: 1996–97, 1997–98, 1998–99, 1999–00, 2004–05, 2005–06
Egypt Cup: 2001, 2003
Egyptian Super Cup: 2002, 2005
CAF Champions League: 2001, 2005
CAF Super Cup: 2002
External links
1977 births
Living people
People from Faiyum
Egyptian men's footballers
Egyptian expatriate men's footballers
Egypt men's international footballers
Al Ahly SC players
Al Wehda FC players
2000 African Cup of Nations players
Egyptian Premier League players
Saudi Pro League players
Expatriate men's footballers in Saudi Arabia
Egyptian expatriate sportspeople in Saudi Arabia
Men's association football midfielders |
https://en.wikipedia.org/wiki/Chief%20series | In abstract algebra, a chief series is a maximal normal series for a group.
It is similar to a composition series, though the two concepts are distinct in general: a chief series is a maximal normal series, while a composition series is a maximal subnormal series.
Chief series can be thought of as breaking the group down into less complicated pieces, which may be used to characterize various qualities of the group.
Definition
A chief series is a maximal normal series for a group. Equivalently, a chief series is a composition series of the group G under the action of inner automorphisms.
In detail, if G is a group, then a chief series of G is a finite collection of normal subgroups Ni ⊆ G,
such that each quotient group Ni+1/Ni, for i = 1, 2,..., n − 1, is a minimal normal subgroup of G/Ni. Equivalently, there does not exist any subgroup A normal in G such that Ni < A < Ni+1 for any i. In other words, a chief series may be thought of as "full" in the sense that no normal subgroup of G may be added to it.
The factor groups Ni+1/Ni in a chief series are called the chief factors of the series. Unlike composition factors, chief factors are not necessarily simple. That is, there may exist a subgroup A normal in Ni+1 with Ni < A < Ni+1, but A is not normal in G. However, the chief factors are always characteristically simple, that is, they have no proper nontrivial characteristic subgroups. In particular, a finite chief factor is a direct product of isomorphic simple groups.
Properties
Existence
Finite groups always have a chief series, though infinite groups need not have a chief series. For example, the group of integers Z with addition as the operation does not have a chief series. To see this, note Z is cyclic and abelian, and so all of its subgroups are normal and cyclic as well. Supposing there exists a chief series Ni leads to an immediate contradiction: N1 is cyclic and thus is generated by some integer a, however the subgroup generated by 2a is a nontrivial normal subgroup properly contained in N1, contradicting the definition of a chief series.
Uniqueness
When a chief series for a group exists, it is generally not unique. However, a form of the Jordan–Hölder theorem states that the chief factors of a group are unique up to isomorphism, independent of the particular chief series they are constructed from In particular, the number of chief factors is an invariant of the group G, as well as the isomorphism classes of the chief factors and their multiplicities.
Other properties
In abelian groups, chief series and composition series are identical, as all subgroups are normal.
Given any normal subgroup N ⊆ G, one can always find a chief series in which N is one of the elements (assuming a chief series for G exists in the first place.) Also, if G has a chief series and N is normal in G, then both N and G/N have chief series. The converse also holds: if N is normal in G and both N and G/N have chief series, G has a chief series as well.
Re |
https://en.wikipedia.org/wiki/Bailar%20twist | The Bailar twist is a mechanism proposed for the racemization of octahedral complexes containing three bidentate chelate rings. Such complexes typically adopt an octahedral molecular geometry, in which case they possess helical chirality. One pathway by which these compounds can racemize is via the formation of a trigonal prismatic intermediate with D3h point group symmetry. This pathway is named in honor of John C. Bailar, Jr., an inorganic chemist who investigated this process. An alternative pathway is called the Ray–Dutt twist.
See also
Pseudorotation
Bartell mechanism
Berry mechanism
Ray–Dutt twist
Fluxional molecule
References
Molecular geometry
Stereochemistry
Coordination chemistry |
https://en.wikipedia.org/wiki/Ray%E2%80%93Dutt%20twist | The Ray–Dutt twist is a mechanism proposed for the racemization of octahedral complexes containing three bidentate chelate rings. Such complexes typically adopt an octahedral molecular geometry in their ground states, in which case they possess helical chirality. The pathway entails formation of an intermediate of C2v point group symmetry. An alternative pathway that also does not break any metal-ligand bonds is called the Bailar twist. Both of these mechanism product complexes wherein the ligating atoms (X in the scheme) are arranged in an approximate trigonal prism.
This pathway is called the Ray–Dutt twist in honor of Priyadaranjan Ray (not Prafulla Chandra Ray) and N. K. Dutt, inorganic chemists at the Indian Association for the Cultivation of Science abbr. IACS who proposed this process.
See also
Pseudorotation
Bailar twist
Bartell mechanism
Berry mechanism
Fluxional molecule
Indian Association for the Cultivation of Science (IACS)
References
Molecular geometry
Stereochemistry
Coordination chemistry |
https://en.wikipedia.org/wiki/Blade%20geometry | The term blade geometry refers to the physical properties of a sword blade: cross-section (or grind) and taper.
Cross-section
The cross-section of a blade is the primary way of determining its function and place in history.
Early Middle Ages
Early Viking and medieval European blades tended to have a lenticular cross-section. This type of design lacks a strong central ridge in the middle of the blade. The flexibility these blades have illustrates the purpose that they served, as primarily cutting weapons, that could also be used with the thrust.
Late Middle Ages
With the improvement in the defensive capabilities of armor in the High and Late Middle Ages, the cross-section of the sword blade adapted to suit the needs of warriors. Swords began to favour rigidity over flexibility as more rigid blades allowed for the stronger thrusts that were used to pierce armour. These blades were made with a diamond cross-section, which could be more or less acute, depending on the purpose of the blade. Weapons such as the Estoc, for example, would have little to no cutting edge, but they would be very rigid and strong on the thrust. This is opposed by the Longsword which was usually a multi-purpose weapon used for both thrusting and cutting.
The diamond cross-section could also be hollow ground for greater edge sharpness and thrust efficiency, while retaining strong central ridges.
Taper
There are two types of physical blade taper: distal and profile.
Distal tapering refers to a blade's cross-section thinning from its base to its tip. This is used to create the handling characteristics of individual blades and the amount of distal taper varies depending upon the intended purpose of the blade. Many modern replica blades are not made with any distal taper, resulting in a blade that, when wielded, will feel unresponsive and heavy.
Profile taper refers to narrowing upon the edges of the flat of the blade. Blades with a more gradual taper are meant for cutting, whereas blades with an acute taper are usually meant for thrusting.
References
http://www.myarmoury.com/feature_properties.html
Swords
Blade weapons |
https://en.wikipedia.org/wiki/Lincoln%20%28footballer%2C%20born%201983%29 | Abraão Lincoln Martins, or simply Lincoln (born 14 June 1983), is a Brazilian striker who currently plays for Brasiliense.
Club statistics
References
External links
1983 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Bolivia
Expatriate men's footballers in Japan
Avispa Fukuoka players
Shonan Bellmare players
Thespakusatsu Gunma players
Oriente Petrolero players
Paulista Futebol Clube players
Brasiliense FC players
J2 League players
Men's association football forwards
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Dolor | Dolor may refer to:
Pain
Suffering
The unit of measurement in utilitarianism, see Felicific calculus#Hedons and dolors
Dolor (sculpture), a work by Clemente Islas Allende in Mexico City
See also
Dolors, a given name |
https://en.wikipedia.org/wiki/Bruno%20Quadros | Bruno Everton Quadros, or simply Bruno Quadros (born February 3, 1977), is a Brazilian manager and former defender who currently works as the assistant head coach of Cerezo Osaka.
Club statistics
Honors
FC Tokyo
J.League Cup : 2009
References
External links
1977 births
Living people
Brazilian men's footballers
Brazilian football managers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Expatriate men's footballers in Turkey
Expatriate men's footballers in Cyprus
Campeonato Brasileiro Série A players
Süper Lig players
J1 League players
J2 League players
Cypriot First Division players
CR Flamengo footballers
Botafogo de Futebol e Regatas players
Galatasaray S.K. footballers
Sport Club do Recife players
Associação Desportiva São Caetano players
Guarani FC players
Cruzeiro Esporte Clube players
Cerezo Osaka players
Hokkaido Consadole Sapporo players
FC Tokyo players
Alki Larnaca FC players
Clube Atlético Linense players
Clube Atlético Linense managers
Duque de Caxias Futebol Clube managers
Marília Atlético Clube managers
Men's association football defenders
Footballers from Rio de Janeiro (city) |
https://en.wikipedia.org/wiki/Edinaldo%20%28footballer%2C%20born%201987%29 | Edinaldo Batista dos Santos, or simply Edinaldo (born April 2, 1987), is a Brazilian midfielder. He last played for Mito HollyHock in the J2 League.
Club statistics
References
External links
1987 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
J2 League players
Mito HollyHock players
Expatriate men's footballers in Japan
Men's association football forwards |
https://en.wikipedia.org/wiki/Hermite%20constant | In mathematics, the Hermite constant, named after Charles Hermite, determines how long a shortest element of a lattice in Euclidean space can be.
The constant γn for integers n > 0 is defined as follows. For a lattice L in Euclidean space Rn with unit covolume, i.e. vol(Rn/L) = 1, let λ1(L) denote the least length of a nonzero element of L. Then is the maximum of λ1(L) over all such lattices L.
The square root in the definition of the Hermite constant is a matter of historical convention.
Alternatively, the Hermite constant γn can be defined as the square of the maximal systole of a flat n-dimensional torus of unit volume.
Example
The Hermite constant is known in dimensions 1–8 and 24.
For n = 2, one has γ2 = . This value is attained by the hexagonal lattice of the Eisenstein integers.
Estimates
It is known that
A stronger estimate due to Hans Frederick Blichfeldt is
where is the gamma function.
See also
Loewner's torus inequality
References
Systolic geometry
Geometry of numbers
Mathematical constants |
https://en.wikipedia.org/wiki/Autoregressive%20fractionally%20integrated%20moving%20average | In statistics, autoregressive fractionally integrated moving average models are time series models that generalize ARIMA (autoregressive integrated moving average) models by allowing non-integer values of the differencing parameter. These models are useful in modeling time series with long memory—that is, in which deviations from the long-run mean decay more slowly than an exponential decay. The acronyms "ARFIMA" or "FARIMA" are often used, although it is also conventional to simply extend the "ARIMA(p, d, q)" notation for models, by simply allowing the order of differencing, d, to take fractional values. Fractional differencing and the ARFIMA model were introduced in the early 1980s by Clive Granger, Roselyne Joyeux, and Jonathan Hosking.
Basics
In an ARIMA model, the integrated part of the model includes the differencing operator (1 − B) (where B is the backshift operator) raised to an integer power. For example,
where
so that
In a fractional model, the power is allowed to be fractional, with the meaning of the term identified using the following formal binomial series expansion
ARFIMA(0, d, 0)
The simplest autoregressive fractionally integrated model, ARFIMA(0, d, 0), is, in standard notation,
where this has the interpretation
ARFIMA(0, d, 0) is similar to fractional Gaussian noise (fGn): with d = H−, their covariances have the same power-law decay. The advantage of fGn over ARFIMA(0,d,0) is that many asymptotic relations hold for finite samples. The advantage of ARFIMA(0,d,0) over fGn is that it has an especially simple spectral density—
—and it is a particular case of ARFIMA(p, d, q), which is a versatile family of models.
General form: ARFIMA(p, d, q)
An ARFIMA model shares the same form of representation as the ARIMA(p, d, q) process, specifically:
In contrast to the ordinary ARIMA process, the "difference parameter", d, is allowed to take non-integer values.
Enhancement to ordinary ARMA models
The enhancement to ordinary ARMA models is as follows:
Take the original data series and high-pass filter it with fractional differencing enough to make the result stationary, and remember the order d of this fractional difference, d usually between 0 and 1 ... possibly up to 2+ in more extreme cases. Fractional difference of 2 is the 2nd derivative or 2nd difference.
note: applying fractional differencing changes the units of the problem. If we started with Prices then take fractional differences, we no longer are in Price units.
determining the order of differencing to make a time series stationary may be an iterative, exploratory process.
Compute plain ARMA terms via the usual methods to fit to this stationary temporary data set which is in ersatz units.
Forecast either to existing data (static forecast) or "ahead" (dynamic forecast, forward in time) with these ARMA terms.
Apply the reverse filter operation (fractional integration to the same level d as in step 1) to the forecasted series, to return the forecast to the o |
https://en.wikipedia.org/wiki/Sums%20of%20powers | In mathematics and statistics, sums of powers occur in a number of contexts:
Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.
Faulhaber's formula expresses as a polynomial in , or alternatively in terms of a Bernoulli polynomial.
Fermat's right triangle theorem states that there is no solution in positive integers for and .
Fermat's Last Theorem states that is impossible in positive integers with .
The equation of a superellipse is . The squircle is the case , .
Euler's sum of powers conjecture (disproved) concerns situations in which the sum of integers, each a th power of an integer, equals another th power.
The Fermat-Catalan conjecture asks whether there are an infinitude of examples in which the sum of two coprime integers, each a power of an integer, with the powers not necessarily equal, can equal another integer that is a power, with the reciprocals of the three powers summing to less than 1.
Beal's conjecture concerns the question of whether the sum of two coprime integers, each a power greater than 2 of an integer, with the powers not necessarily equal, can equal another integer that is a power greater than 2.
The Jacobi–Madden equation is in integers.
The Prouhet–Tarry–Escott problem considers sums of two sets of th powers of integers that are equal for multiple values of .
A taxicab number is the smallest integer that can be expressed as a sum of two positive third powers in distinct ways.
The Riemann zeta function is the sum of the reciprocals of the positive integers each raised to the power , where is a complex number whose real part is greater than 1.
The Lander, Parkin, and Selfridge conjecture concerns the minimal value of in
Waring's problem asks whether for every natural number there exists an associated positive integer such that every natural number is the sum of at most th powers of natural numbers.
The successive powers of the golden ratio φ obey the Fibonacci recurrence:
Newton's identities express the sum of the th powers of all the roots of a polynomial in terms of the coefficients in the polynomial.
The sum of cubes of numbers in arithmetic progression is sometimes another cube.
The Fermat cubic, in which the sum of three cubes equals another cube, has a general solution.
The power sum symmetric polynomial is a building block for symmetric polynomials.
The sum of the reciprocals of all perfect powers including duplicates (but not including 1) equals 1.
The Erdős–Moser equation, where and are positive integers, is conjectured to have no solutions other than .
The sums of three cubes cannot equal 4 or 5 modulo 9, but it is unknown whether all remaining integers can be expressed in this form.
The sums of powers is related to the Bernoulli polynomials |
https://en.wikipedia.org/wiki/Alfred%20Nijhuis | Alfred Nijhuis (, born 23 March 1966) is a Dutch former professional footballer who played as defender.
Career statistics
References
External links
1966 births
Living people
Footballers from Utrecht (city)
Men's association football defenders
Dutch men's footballers
FC Twente players
ASC Schöppingen players
MSV Duisburg players
Urawa Red Diamonds players
Expatriate men's footballers in Japan
Dutch expatriate sportspeople in Japan
J1 League players
Borussia Dortmund players
Bundesliga players
2. Bundesliga players
Expatriate men's footballers in Germany
Dutch expatriate men's footballers
SuS Stadtlohn managers
Heracles Almelo non-playing staff |
https://en.wikipedia.org/wiki/Cut%20point | In topology, a cut-point is a point of a connected space such that its removal causes the resulting space to be disconnected. If removal of a point doesn't result in disconnected spaces, this point is called a non-cut point.
For example, every point of a line is a cut-point, while no point of a circle is a cut-point.
Cut-points are useful to determine whether two connected spaces are homeomorphic by counting the number of cut-points in each space. If two spaces have different number of cut-points, they are not homeomorphic. A classic example is using cut-points to show that lines and circles are not homeomorphic.
Cut-points are also useful in the characterization of topological continua, a class of spaces which combine the properties of compactness and connectedness and include many familiar spaces such as the unit interval, the circle, and the torus.
Definition
Formal definitions
A cut-point of a connected T1 topological space X, is a point p in X such that X - {p} is not connected. A point which is not a cut-point is called a non-cut point.
A non-empty connected topological space X is a cut-point space if every point in X is a cut point of X.
Basic examples
A closed interval [a,b] has infinitely many cut-points. All points except for its end points are cut-points and the end-points {a,b} are non-cut points.
An open interval (a,b) also has infinitely many cut-points like closed intervals. Since open intervals don't have end-points, it has no non-cut points.
A circle has no cut-points and it follows that every point of a circle is a non-cut point.
Notations
A cutting of X is a set {p,U,V} where p is a cut-point of X, U and V form a separation of X-{p}.
Also can be written as X\{p}=U|V.
Theorems
Cut-points and homeomorphisms
Cut-points are not necessarily preserved under continuous functions. For example: f: [0, 2] → R2, given by f(x) = (cos x, sin x). Every point of the interval (except the two endpoints) is a cut-point, but f(x) forms a circle which has no cut-points.
Cut-points are preserved under homeomorphisms. Therefore, cut-point is a topological invariant.
Cut-points and continua
Every continuum (compact connected Hausdorff space) with more than one point, has at least two non-cut points. Specifically, each open set which forms a separation of resulting space contains at least one non-cut point.
Every continuum with exactly two noncut-points is homeomorphic to the unit interval.
If K is a continuum with points a,b and K-{a,b} isn't connected, K is homeomorphic to the unit circle.
Topological properties of cut-point spaces
Let X be a connected space and x be a cut point in X such that X\{x}=A|B. Then {x} is either open or closed. if {x} is open, A and B are closed. If {x} is closed, A and B are open.
Let X be a cut-point space. The set of closed points of X is infinite.
Irreducible cut-point spaces
Definitions
A cut-point space is irreducible if no proper subset of it is a cut-point space.
The Khalimsky line: Le |
https://en.wikipedia.org/wiki/Gilberto%20Garc%C3%ADa%20%28footballer%2C%20born%201987%29 | Gilberto García Olarte (born 27 January 1987), also known as Alcatraz is a Colombian professional footballer who plays for Deportivo Pasto.
Career statistics
Honours
Atlético Nacional
Categoría Primera A (1): 2015 Finalización
Copa Colombia (1): 2016
Copa Libertadores (1): 2016
Superliga Colombiana (1): 2016
Deportivo Pasto
Categoría Primera B (1): 2011
References
External links
BDFA profile
Living people
1987 births
Footballers from Santa Marta
Men's association football defenders
Colombian men's footballers
Colombian expatriate men's footballers
Colombia men's international footballers
Deportes Tolima footballers
Cúcuta Deportivo footballers
Atlético Bucaramanga footballers
Deportivo Pasto footballers
Deportivo Cali footballers
Once Caldas footballers
Real Valladolid players
Independiente Medellín footballers
Atlético Nacional footballers
Águilas Doradas Rionegro players
Deportivo Pereira footballers
Categoría Primera A players
Copa Libertadores-winning players
Categoría Primera B players
La Liga players
Colombian expatriate sportspeople in Spain
Expatriate men's footballers in Spain
20th-century Colombian people
21st-century Colombian people |
https://en.wikipedia.org/wiki/Chika%20%28footballer%29 | Celso Moraes, or simply Chika (born August 4, 1979), is a Brazilian defender. He has played for Thespa Kusatsu.
Celso previously played for Ji-Paraná in the Copa do Brasil.
Club statistics
References
External links
1979 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
J2 League players
Japan Football League players
Thespakusatsu Gunma players
Expatriate men's footballers in Japan
Men's association football defenders |
https://en.wikipedia.org/wiki/Marlon%20%28footballer%2C%20born%201987%29 | Mário Brandão da Silveira, or simply Marlon (born 26 January 1987), is a Brazilian striker.
Marlon previously played for Thespa Kusatsu in the J2 League.
Club statistics
References
External links
1987 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
J2 League players
Thespakusatsu Gunma players
Expatriate men's footballers in Japan
Men's association football forwards |
https://en.wikipedia.org/wiki/Adiel%20%28footballer%29 | Adiel de Oliveira Amorim (born 13 August 1980), known simply as Adiel, is a Brazilian former professional footballer who played as a midfielder.
Club statistics
References
External links
1980 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Santos FC players
Brazilian expatriate sportspeople in Japan
Urawa Red Diamonds players
Brazilian expatriate sportspeople in Kuwait
Botafogo Futebol Clube (SP) players
Shonan Bellmare players
Expatriate men's footballers in Japan
Expatriate men's footballers in China
Brazilian expatriate sportspeople in China
China League One players
J1 League players
J2 League players
Expatriate men's footballers in Kuwait
Men's association football midfielders
Qadsia SC players
Kuwait Premier League players
People from Cubatão
Footballers from São Paulo (state) |
https://en.wikipedia.org/wiki/Edmilson%20Alves | Edmilson Alves (born February 17, 1976), is a Brazilian midfielder.
Club statistics
References
External links
Profile at Oita Trinita
1976 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Londrina Esporte Clube players
Ceará Sporting Club players
Clube Atlético Juventus players
Fortaleza Esporte Clube players
Expatriate men's footballers in Japan
Campeonato Brasileiro Série B players
J1 League players
J2 League players
Oita Trinita players
Vissel Kobe players
Roasso Kumamoto players
Ulsan Hyundai FC players
K League 1 players
Expatriate men's footballers in South Korea
Brazilian expatriate sportspeople in South Korea
Men's association football midfielders |
https://en.wikipedia.org/wiki/Dario%20Dabac | Dario Dabac (; born 23 May 1978) is a Croatian retired footballer and manager.
Club statistics
References
External links
1978 births
Living people
People from Senj
Men's association football fullbacks
Croatian men's footballers
NK Zagreb players
Dynamo Dresden players
1. FC Union Berlin players
SpVgg Greuther Fürth players
SV Ried players
Sanfrecce Hiroshima players
HNK Rijeka players
Al-Arabi SC (Kuwait) players
NK Nehaj players
Chongqing Liangjiang Athletic F.C. players
Shenyang Zhongze F.C. players
Croatian Football League players
Oberliga (football) players
Regionalliga players
2. Bundesliga players
Austrian Football Bundesliga players
J1 League players
J2 League players
Kuwait Premier League players
First Football League (Croatia) players
China League One players
Croatian expatriate men's footballers
Expatriate men's footballers in Germany
Expatriate men's footballers in Austria
Expatriate men's footballers in Japan
Expatriate men's footballers in Kuwait
Expatriate men's footballers in China
Croatian expatriate sportspeople in Germany
Croatian expatriate sportspeople in Austria
Croatian expatriate sportspeople in Japan
Croatian expatriate sportspeople in Kuwait
Croatian expatriate sportspeople in China
Croatian football managers
Sichuan Jiuniu F.C. managers
Croatian expatriate football managers
Expatriate football managers in China
HNK Rijeka non-playing staff |
https://en.wikipedia.org/wiki/Northern%20Ireland%20Statistics%20and%20Research%20Agency | The Northern Ireland Statistics and Research Agency (NISRA, ) is an executive agency within the Department of Finance in Northern Ireland. The organisation is responsible for the collection and publication of statistics related to the economy, population and society of Northern Ireland. It is responsible for conducting the decennial census, with the last Census in Northern Ireland held on 21 March 2021, and incorporates the General Register Office (GRO) for Northern Ireland which is responsible for the registration of births, marriages, civil partnerships and deaths.
See also
Central Statistics Office (Ireland)
Office for National Statistics
UK Statistics Authority
Census in the United Kingdom
External links
Northern Ireland Executive
Demographics of Northern Ireland
National statistical services
Statistical organisations in the United Kingdom |
https://en.wikipedia.org/wiki/Lemoine%27s%20conjecture | In number theory, Lemoine's conjecture, named after Émile Lemoine, also known as Levy's conjecture, after Hyman Levy, states that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime.
History
The conjecture was posed by Émile Lemoine in 1895, but was erroneously attributed by MathWorld to Hyman Levy who pondered it in the 1960s.
A similar conjecture by Sun in 2008 states that all odd integers greater than 3 can be represented as the sum of a prime number and the product of two consecutive positive integers ( p+x(x+1) ).
Formal definition
To put it algebraically, 2n + 1 = p + 2q always has a solution in primes p and q (not necessarily distinct) for n > 2. The Lemoine conjecture is similar to but stronger than Goldbach's weak conjecture.
Example
For example, 47 = 13 + 2 × 17 = 37 + 2 × 5 = 41 + 2 × 3 = 43 + 2 × 2. counts how many different ways 2n + 1 can be represented as p + 2q.
Evidence
According to MathWorld, the conjecture has been verified by Corbitt up to 109. A blog post in June of 2019 additionally claimed to have verified the conjecture up to 1010.
See also
Lemoine's conjecture and extensions
Notes
References
Emile Lemoine, L'intermédiare des mathématiciens, 1 (1894), 179; ibid 3 (1896), 151.
H. Levy, "On Goldbach's Conjecture", Math. Gaz. 47 (1963): 274
L. Hodges, "A lesser-known Goldbach conjecture", Math. Mag., 66 (1993): 45–47. .
John O. Kiltinen and Peter B. Young, "Goldbach, Lemoine, and a Know/Don't Know Problem", Mathematics Magazine, 58(4) (Sep., 1985), pp. 195–203. .
Richard K. Guy, Unsolved Problems in Number Theory New York: Springer-Verlag 2004: C1
External links
Levy's Conjecture by Jay Warendorff, Wolfram Demonstrations Project.
Additive number theory
Conjectures about prime numbers
Unsolved problems in number theory |
https://en.wikipedia.org/wiki/Giuseppe%20Zappella | Giuseppe Zappella (born May 4, 1973, in Milan) is an Italian football player.
Club career statistics
External links
Profile at aic.football.it
1973 births
Living people
Italian men's footballers
Italian expatriate men's footballers
AC Milan players
Como 1907 players
AC Monza players
US Avellino 1912 players
US Catanzaro 1929 players
US Alessandria Calcio 1912 players
Urawa Red Diamonds players
Expatriate men's footballers in Japan
Italian expatriate sportspeople in Japan
J1 League players
ASD Calcio Ivrea players
Serie B players
Vis Pesaro dal 1898 players
Men's association football defenders
Footballers from Milan |
https://en.wikipedia.org/wiki/Tyranny%20of%20averages | The tyranny of averages is a phrase used in applied statistics to describe the often overlooked fact that the mean does not provide any information about the shape of the probability distribution of a data set or skewness, and that decisions or analysis based on only the mean—as opposed to median and standard deviation—may be faulty.
A UN Development Program press release discusses a real-world example:
A new report launched 1 July [2005] warns that in Asia and the Pacific, the rising prosperity and fast growth in populous countries like China and India is hiding widespread extreme poverty in the Least Developed Countries (LDCs). The result is potentially very debilitating to development efforts in the 14 Asia-Pacific LDCs.
This "tyranny of averages" to which the report refers tends to mask the stark contrast between the Asia-Pacific LDCs' sluggish economies and the success of their far more populous neighbours.
See also
Law of large numbers
Law of averages
Trimean
References
Mecklin, J.M. (1918) "The Tyranny of the Average Man", International Journal of Ethics, 28 (2), 240–252
Misuse of statistics
Jargon |
https://en.wikipedia.org/wiki/Relaxation%20%28iterative%20method%29 | In numerical mathematics, relaxation methods are iterative methods for solving systems of equations, including nonlinear systems.
Relaxation methods were developed for solving large sparse linear systems, which arose as finite-difference discretizations of differential equations. They are also used for the solution of linear equations for linear least-squares problems and also for systems of linear inequalities, such as those arising in linear programming. They have also been developed for solving nonlinear systems of equations.
Relaxation methods are important especially in the solution of linear systems used to model elliptic partial differential equations, such as Laplace's equation and its generalization, Poisson's equation. These equations describe boundary-value problems, in which the solution-function's values are specified on boundary of a domain; the problem is to compute a solution also on its interior. Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences.
Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation, it resembles repeated application of a local smoothing filter to the solution vector. These are not to be confused with relaxation methods in mathematical optimization, which approximate a difficult problem by a simpler problem whose "relaxed" solution provides information about the solution of the original problem.
Model problem of potential theory
When φ is a smooth real-valued function on the real numbers, its second derivative can be approximated by:
Using this in both dimensions for a function φ of two arguments at the point (x, y), and solving for φ(x, y), results in:
To approximate the solution of the Poisson equation:
numerically on a two-dimensional grid with grid spacing h, the relaxation method assigns the given values of function φ to the grid points near the boundary and arbitrary values to the interior grid points, and then repeatedly performs the assignment
φ := φ* on the interior points, where φ* is defined by:
until convergence.
The method is easily generalized to other numbers of dimensions.
Convergence and acceleration
While the method converges under general conditions, it typically makes slower progress than competing methods. Nonetheless, the study of relaxation methods remains a core part of linear algebra, because the transformations of relaxation theory provide excellent preconditioners for new methods. Indeed, the choice of preconditioner is often more important than the choice of iterative method.
Multigrid methods may be used to accelerate the methods. One can first compute an approximation on a coarser grid – usually the double spacing 2h – and use that solution with interpolated values for the other grid points as the initial assignment. This can then also be done recursively for the coarser computation.
See also
In li |
https://en.wikipedia.org/wiki/Counternull | In statistics, and especially in the statistical analysis of psychological data, the counternull is a statistic used to aid the understanding and presentation of research results. It revolves around the effect size, which is the mean magnitude of some effect divided by the standard deviation.
The counternull value is the effect size that is just as well supported by the data as the null hypothesis. In particular, when results are drawn from a distribution that is symmetrical about its mean, the counternull value is exactly twice the observed effect size.
The null hypothesis is a hypothesis set up to be tested against an alternative. Thus the counternull is an alternative hypothesis that, when used to replace the null hypothesis, generates the same p-value as had the original null hypothesis of “no difference.”
Some researchers contend that reporting the counternull, in addition to the p-value, serves to counter two common errors of judgment:
assuming that failure to reject the null hypothesis at the chosen level of statistical significance means that the observed size of the "effect" is zero; and
assuming that rejection of the null hypothesis at a particular p-value means that the measured "effect" is not only statistically significant, but also scientifically important.
These arbitrary statistical thresholds create a discontinuity, causing unnecessary confusion and artificial controversy.
Other researchers prefer confidence intervals as a means of countering these common errors.
See also
File drawer problem
Publication bias
References
Further reading
Rosnow, R. L., & Rosenthal, R. (1996). Computing contrasts, effect sizes, and counternulls on other people's published data: General procedures for research consumers. Psychological Methods, 1, 331-340
Psychometrics
Statistical hypothesis testing |
https://en.wikipedia.org/wiki/Run%20average | In baseball statistics, run average (RA) refers to measures of the rate at which runs are allowed or scored. For pitchers, the run average is the number of runs—earned or unearned—allowed per nine innings. It is calculated using this formula:
where
R = Runs
IP = Innings pitched
Run average for pitchers differs from the more commonly used earned run average (ERA) by adding unearned runs to the numerator. This measure is also known as total run average (TRA) or runs allowed average. For batters, the run average is the number of runs scored per at bat.
Run average for pitchers
Although presentations of pitching statistics generally feature the ERA rather than the RA, the latter statistic is notable for both historical and analytical reasons. For early leagues or leagues for which statistics must be calculated from box scores, such as the Negro leagues, data on earned runs may be unavailable and RA may be the only statistic available. The analytical case for RA appeared as early as 1976, when sportswriter Leonard Koppett proposed that RA would be a better measure of pitcher performance than ERA. Subsequently, sabermetrician Bill James wrote, "I think that the distinction between earned runs and unearned runs is silly and artificial, a distinction having no meaning except in the eyes of some guy up in the press box."
In baseball, defense—that is, preventing the opponent from scoring runs—is the joint responsibility of the pitcher and the fielders. ERA attempts to adjust for some of the influence of the fielders on a pitcher's runs allowed by removing runs that are scored because of fielding errors—that is, unearned runs. However, removing unearned runs doesn't adequately adjust for the effects of defensive support, because it makes no adjustment for other important aspects of fielding, such as proficiency at turning double plays, throwing out base stealers, and fielding range. Errors are the only aspect of fielding that ERA adjusts for, and are generally regarded as a small part of fielding in modern baseball.
Another problem with ERA is the inconsistency with which official scorers call plays as errors. The rules give scorers considerable discretion regarding the plays that can be called as errors. Researcher Craig R. Wright found large differences between teams in the rate at which their scorers called errors, and even found some evidence of home team bias—that is, calling errors to favor the statistics of players for the home team.
While ERA doesn't charge the pitcher for the runs that result from errors, it may tend to over correct for the influence of fielding. Even though unearned runs would not have scored without an error, in most cases the pitcher also contributes to the scoring of the unearned run—either by allowing the opposing player to reach base via a walk or hit, or by allowing a subsequent batter a hit that advances and scores the runner. During the early days of baseball history, this over correction for fielding errors |
https://en.wikipedia.org/wiki/Midhinge | In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of location.
Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator.
The midhinge is related to the interquartile range (IQR), the difference of the third and first quartiles (i.e. ), which is a measure of statistical dispersion. The two are complementary in sense that if one knows the midhinge and the IQR, one can find the first and third quartiles.
The use of the term "hinge" for the lower or upper quartiles derives from John Tukey's work on exploratory data analysis in the late 1970s, and "midhinge" is a fairly modern term dating from around that time. The midhinge is slightly simpler to calculate than the trimean (), which originated in the same context and equals the average of the median () and the midhinge.
See also
Interquartile mean
L-estimator
References
External links
H-spread at MathWorld
Means
Exploratory data analysis |
https://en.wikipedia.org/wiki/Simon%20Wheeldon | Simon Wheeldon (born August 30, 1966) is a former ice hockey player. He played for the New York Rangers and Winnipeg Jets.
Career statistics
Regular season and playoffs
International
Awards
WHL West Second All-Star Team – 1985 & 1986
External links
1966 births
Living people
Baltimore Skipjacks players
Canadian ice hockey centres
Colorado Rangers players
Denver Rangers players
Edmonton Oilers draft picks
Flint Spirits players
Ice hockey people from Vancouver
Ice hockey players at the 1998 Winter Olympics
Ice hockey players at the 2002 Winter Olympics
Kelowna Buckaroos players
Moncton Hawks players
München Barons players
New Haven Nighthawks players
New York Rangers players
Nova Scotia Oilers players
Olympic ice hockey players for Austria
VEU Feldkirch players
Victoria Cougars (WHL) players
Winnipeg Jets (1979–1996) players |
https://en.wikipedia.org/wiki/Dwayne%20Pentland | Dwayne Pentland (born February 28, 1953, in Vancouver, British Columbia) is a retired ice hockey player. He played in the World Hockey Association for the Houston Aeros.
Career statistics
External links
1953 births
Living people
Albuquerque Six-Guns players
Brandon Wheat Kings players
Canadian ice hockey defencemen
Edmonton Oilers (WHA) draft picks
Fort Wayne Komets players
Houston Aeros (WHA) players
Ice hockey people from Vancouver
New Haven Nighthawks players
New York Rangers draft picks
Oklahoma City Blazers (1965–1977) players
Providence Reds players
San Diego Mariners (PHL) players
Western International Hockey League players
Canadian expatriate ice hockey players in the United States |
https://en.wikipedia.org/wiki/Caseolus%20calculus | Caseolus calculus (common name: Madeiran land snail) is a species of small air-breathing land snails, terrestrial pulmonate gastropod molluscs in the family Geomitridae, the hairy snails and their allies.
Distribution and conservation status
This species lives in Europe. It is mentioned in annexes II and IV of Habitats Directive.
References
External links
calculus
Habitats Directive Species |
https://en.wikipedia.org/wiki/Jonathan%20Ch%C3%A1vez | Jonathan Daniel Chávez (born 8 January 1989, in La Plata) is an Argentine football midfielder.
External links
Jonathan Chávez – Argentine Primera statistics at Futbol XXI
Jonathan Chávez at BDFA.com.ar
1989 births
Living people
Footballers from La Plata
Argentine men's footballers
Argentine expatriate men's footballers
Men's association football midfielders
Club de Gimnasia y Esgrima La Plata footballers
Defensa y Justicia footballers
C.D. Cobreloa footballers
Boca Unidos footballers
Godoy Cruz Antonio Tomba footballers
Club Atlético Brown footballers
Club y Biblioteca Ramón Santamarina footballers
Club Atlético Atlanta footballers
Deportes Vallenar footballers
Chilean Primera División players
Primera B de Chile players
Argentine Primera División players
Primera B Metropolitana players
Primera Nacional players
Expatriate men's footballers in Chile
Argentine expatriate sportspeople in Chile |
https://en.wikipedia.org/wiki/Stanislav%20Zhmakin | Stanislav Zhmakin (born June 25, 1982) is a Russian ice hockey winger who current plays for Yugra Khanty-Mansiysk in the Kontinental Hockey League.
Career statistics
Personal life
Zhmakin married Russian singer Daria Vodyahina in June 2010.
References
External links
1982 births
Living people
Avtomobilist Yekaterinburg players
HC CSKA Moscow players
HC Lada Togliatti players
HC Spartak Moscow players
Sportspeople from Penza
Russian ice hockey forwards
Salavat Yulaev Ufa players
Severstal Cherepovets players |
https://en.wikipedia.org/wiki/Curtis%20Cooper%20%28mathematician%29 | Curtis Niles Cooper is an American mathematician who is currently a professor at the University of Central Missouri, in the Department of Mathematics and Computer Science.
GIMPS
Using software from the GIMPS project, Cooper and Steven Boone found the 43rd known Mersenne prime on their 700 PC cluster on December 15, 2005. The prime, 230,402,457 − 1, is 9,152,052 digits long and is the ninth Mersenne prime for GIMPS.
Cooper and Boone became the first GIMPS contributors to find two primes when they also found the 44th known Mersenne prime, 232,582,657 − 1 (or M32,582,657), which has 9,808,358 digits . This prime was discovered on September 4, 2006 using a PC cluster of over 850 machines. This is the tenth Mersenne prime for GIMPS.
On January 25, 2013, Cooper found his third Mersenne prime of 257,885,161 − 1.
On September 17, 2015, Cooper's computer reported yet another Mersenne prime, 274,207,281 - 1, which was the largest known prime number at 22,338,618 decimal digits. The report was, however, unnoticed until January 7, 2016.
Areas of research
Cooper's own work has mainly been in elementary number theory, especially work related to digital representations of numbers. He collaborated extensively with Robert E. Kennedy. They have worked with Niven numbers, among other results, showing that no 21 consecutive integers can all be Niven numbers, and introduced the notion of tau numbers, numbers whose total number of divisors are itself a divisor of the number. Independent of Kennedy, Cooper has also done work about generalizations of geometric series, and their application to probability.
Cooper is also the editor of the publication Fibonacci Quarterly.
Notes
External links
Curtis Cooper's homepage
Living people
Iowa State University alumni
University of Central Missouri faculty
21st-century American mathematicians
Number theorists
Year of birth missing (living people)
Place of birth missing (living people)
Mathematicians from Missouri |
https://en.wikipedia.org/wiki/Curtis%20Cooper | Curtis Cooper may refer to:
Curtis Cooper (activist) (1932–2000), in Savannah, Georgia
Curtis Cooper (mathematician), professor at the University of Central Missouri's Department of Mathematics and Computer Science
Curtis Cooper (Casualty), a character from British soap opera Casualty |
https://en.wikipedia.org/wiki/Masatoshi%20G%C3%BCnd%C3%BCz%20Ikeda | Masatoşi Gündüz İkeda (25 February 1926 – 9 February 2003), was a Japanese-born Turkish mathematician known for his contributions to the field of algebraic number theory.
Early years
Ikeda was born on 25 February 1926 in Tokyo, Japan, to Junzo Ikeda, head of the statistics department of an insurance company, and his wife Yaeko Ikeda. He was the youngest child with a brother and two sisters. He grew up reading mathematics books belonging to his father. During his school years, he bought himself used books about mathematics and the life story of mathematicians. He was very impressed by the French mathematician Évariste Galois (1811–1832).
Academic career
Ikeda graduated from the mathematics department of Osaka University in 1948. He received a Ph.D. degree with his thesis "On Absolutely Segregated Algebras", written in 1953 under the direction of Kenjiro Shoda. He was appointed associate professor in 1955. He pursued scientific research at the University of Hamburg in Germany, under the supervision of Helmut Hasse (1898–1979) between 1957 and 1959. On a suggestion from Hasse, he went to Turkey in 1960 and landed at Ege University in İzmir. In 1961, he was appointed a foreigner specialist in the Faculty of Science at the same university.
In 1964, Ikeda married Turkish biochemist Emel Ardor, whom he met and followed to Turkey. He was naturalized, converted to Islam and adopted the Turkish name Gündüz. He became associate professor in 1965 and a full professor in 1966. In 1968, with permission of the university, he went to the Middle East Technical University (METU) in Ankara as a visiting professor for one year. However, following the end of his term, he was offered a permanent post as a full professor, which he accepted upon the proposal of the mathematician Cahit Arf, whom he had known since his early years in Turkey.
From time to time, Ikeda was invited as a visiting professor to various universities such as the University of Hamburg (1966), San Diego State University, California (1971), and Yarmouk University in Irbid, Jordan (1984, 1985–86). In 1976, Ikeda carried out research work at Princeton University. In 1976, Ikeda went to Hacettepe University in Ankara, where he chaired the mathematics department until 1978, before he returned to METU. He retired in 1992 at METU. His scientific devotion was in Galois theory.
Among the research institutions Ikeda served were TÜBİTAK Marmara Research Center and Turkish National Research Institute of Electronics and Cryptology. Finally, he worked at the Feza Gürsey Basic Sciences Research Center in Istanbul.
Ikeda was a member of the Basic Sciences Board at the Scientific and Technological Research Council of Turkey (TÜBİTAK), and served as the head of the Mathematic Research Unit at the METU.
Family life and death
Ikeda died on 9 February 2003, in Ankara. Following a religious funeral service held on 12 February at Kocatepe Mosque, he was laid to rest at the Karşıyaka Cemetery. He was the father of t |
https://en.wikipedia.org/wiki/Bolza%20surface | In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus with the highest possible order of the conformal automorphism group in this genus, namely of order 48 (the general linear group of matrices over the finite field ). The full automorphism group (including reflections) is the semi-direct product of order 96. An affine model for the Bolza surface can be obtained as the locus of the equation
in . The Bolza surface is the smooth completion of the affine curve. Of all genus hyperbolic surfaces, the Bolza surface maximizes the length of the systole . As a hyperelliptic Riemann surface, it arises as the ramified double cover of the Riemann sphere, with ramification locus at the six vertices of a regular octahedron inscribed in the sphere, as can be readily seen from the equation above.
The Bolza surface has attracted the attention of physicists, as it provides a relatively simple model for quantum chaos; in this context, it is usually referred to as the Hadamard–Gutzwiller model. The spectral theory of the Laplace–Beltrami operator acting on functions on the Bolza surface is of interest to both mathematicians and physicists, since the surface is conjectured to maximize the first positive eigenvalue of the Laplacian among all compact, closed Riemann surfaces of genus with constant negative curvature.
Triangle surface
The Bolza surface is a triangle surface – see Schwarz triangle. More specifically, the Fuchsian group defining the Bolza surface is a subgroup of the group generated by reflections in the sides of a hyperbolic triangle with angles . The group of orientation preserving isometries is a subgroup of the index-two subgroup of the group of reflections, which consists of products of an even number of reflections, which has an abstract presentation in terms of generators and relations as well as . The Fuchsian group defining the Bolza surface is also a subgroup of the (3,3,4) triangle group, which is a subgroup of index 2 in the triangle group. The group does not have a realization in terms of a quaternion algebra, but the group does.
Under the action of on the Poincare disk, the fundamental domain of the Bolza surface is a regular octagon with angles and corners at
where . Opposite sides of the octagon are identified under the action of the Fuchsian group. Its generators are the matrices
where and , along with their inverses. The generators satisfy the relation
These generators are connected to the length spectrum, which gives all of the possible lengths of geodesic loops. The shortest such length is called the systole of the surface. The systole of the Bolza surface is
The element of the length spectrum for the Bolza surface is given by
where runs through the positive integers (but omitting 4, 24, 48, 72, 140, and various higher values) and where is the unique odd integer that minimizes
It is possible to obtain an equivalent c |
https://en.wikipedia.org/wiki/Hannan%E2%80%93Quinn%20information%20criterion | In statistics, the Hannan–Quinn information criterion (HQC) is a criterion for model selection. It is an alternative to Akaike information criterion (AIC) and Bayesian information criterion (BIC). It is given as
where is the log-likelihood, k is the number of parameters, and n is the number of observations.
Burnham & Anderson (2002, p. 287) say that HQC, "while often cited, seems to have seen little use in practice". They also note that HQC, like BIC, but unlike AIC, is not an estimator of Kullback–Leibler divergence. Claeskens & Hjort (2008, ch. 4) note that HQC, like BIC, but unlike AIC, is not asymptotically efficient; however, it misses the optimal estimation rate by a very small factor. They further point out that whatever method is being used for fine-tuning the criterion will be more important in practice than the term , since this latter number is small even for very large ; however, the term ensures that, unlike AIC, HQC is strongly consistent. It follows from the law of the iterated logarithm that any strongly consistent method must miss efficiency by at least a factor, so in this sense HQC is asymptotically very well-behaved. Van der Pas and Grünwald prove that model selection based on a modified Bayesian estimator, the so-called switch distribution, in many cases behaves asymptotically like HQC, while retaining the advantages of Bayesian methods such as the use of priors etc.
See also
Akaike information criterion
Bayesian information criterion
Deviance information criterion
Focused information criterion
Shibata information criterion
References
Aznar Grasa, A. (1989). Econometric Model Selection: A New Approach, Springer.
Burnham, K.P. and Anderson, D.R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. Springer-Verlag. .
Claeskens, G. and Hjort, N.L. (2008). Model Selection and Model Averaging, Cambridge.
Hannan, E. J., and B. G. Quinn (1979), "The Determination of the order of an autoregression", Journal of the Royal Statistical Society, Series B, 41: 190–195.
Van der Pas, S.L.; Grünwald, P.D. (2017). "Almost the best of three worlds." To appear in Statistica Sinica, DOI 10.5705/ss.202016.0011, 2017.
Chen, C et al. Order Determination for Autoregressive Processes Using Resampling methods Statistica Sinica 3:1993, http://www3.stat.sinica.edu.tw/statistica/oldpdf/A3n214.pdf
Regression variable selection
Model selection |
https://en.wikipedia.org/wiki/Maths%20%2B%20English | Maths + English is the third studio album by English rapper Dizzee Rascal. The album went gold in the UK after selling over 100,000 copies.
Background
Maths + English entered the UK Albums Chart at number seven, one position higher than his second album, Showtime (2004), which charted at number eight and his debut, Boy in da Corner (2003), which peaked at number 23.
The track "Wanna Be" features guest vocals from English pop singer Lily Allen. It pays tribute to the 1976 musical Bugsy Malone, specifically the song "So You Want to Be a Boxer?" which shares the same musical arrangements for the sections sung by Lily Allen.
Joss Stone was expected to feature on the song "Da Feelin'", but Dizzee Rascal stated that the song sounded too "poppy" with Joss Stone's hook, so she does not appear on the album.
The track "Pussy'ole" is rumoured to be a Wiley diss. Wiley responded to the track in a video circulating on YouTube, in which he also takes jabs at rappers Kano and Lethal Bizzle.
American hip hop duo UGK are featured on the track "Where's da G's". In return, Dizzee was featured on the track "Two Type of Bitches" along with Pimpin' Ken on UGK's 2007 album Underground Kingz.
On 29 April 2008 Definitive Jux released Maths + English on their independent label in the United States. The Def Jux version features new studio tracks "G.H.E.T.T.O." and "Driving" as well as a remix of the UGK-assisted "Where's da G's" by Def Jux label head El-P. It does not however contain the track "Pussyole (Old Skool)", due to sample clearance issues.
Track listing
Notes
"Pussyole (Old Skool)" samples Lyn Collins' "Think (About It)" and Galactic Force Band's "Space Dust".
"Sirens" was omitted from the US version.
"Da Feelin'" was co-mixed by Shy FX.
"Wanna Be" samples "So You Wanna Be a Boxer" from the film musical Bugsy Malone.
Charts
References
2007 albums
Dizzee Rascal albums
XL Recordings albums |
https://en.wikipedia.org/wiki/1996%20Canadian%20census | The 1996 Canadian census was a detailed enumeration of the Canadian population. Census day was May 14, 1996. On that day, Statistics Canada attempted to count every person in Canada. The total population count of Canada was 28,846,761. This was a 5.7% increase over the 1991 census of 27,296,859.
The previous census was the 1991 census and the following census was in 2001 census.
Canada by the numbers
A summary of information about Canada.
Population by province
Demographics
Mother tongue
Population by mother tongue of Canada's official languages:
Aboriginal peoples
Population of Aboriginal peoples in Canada:
Ethnic origin
Population by ethnic origin. Only those origins with more than 250,000 respondents are included here. This is based entirely on self reporting.
Visible minorities
Age
Population by age:
See also
List of population of Canada by years
Demographics of Canada
Ethnic groups in Canada
History of immigration to Canada
References
External links
1996 Census - Statistics Canada's page on the 1996 census.
Census
1996 censuses
Censuses in Canada |
https://en.wikipedia.org/wiki/Parallelotope | In geometry, a parallelotope may refer to:
A generalization of a parallelepiped and parallelogram
A generalization of a parallelohedron and parallelogon, this includes all parallelohedra in the first sense
See also
Zonotope |
https://en.wikipedia.org/wiki/A%20K%20Peters | A K Peters, Ltd. was a publisher of scientific and technical books, specializing in mathematics and in computer graphics, robotics, and other fields of computer science. They published the journals Experimental Mathematics and the Journal of Graphics Tools, as well as mathematics books geared to children.
Background
Klaus Peters wrote a doctoral dissertation on complex manifolds at the University of Erlangen in 1962, supervised by Reinhold Remmert. He then joined Springer Verlag, becoming their first specialist mathematics editor. As a Springer director from 1971, he hired Alice Merker for Springer New York: they were married that year, and moved to Heidelberg. Leaving Springer, they founded Birkhäuser Boston in 1979; Birkhäuser ran into financial difficulties, and was taken over by Springer. Klaus and Alice then spent a period running a Boston office for Harcourt Brace Jovanovich and their imprint Academic Press. With the takeover of Harcourt Brace Jovanovich by General Cinema Corporation, the couple then found funding from Elwyn Berlekamp to start their own company.
Company history
The company was founded in November 1992 by Alice and Klaus Peters, and maintained as a privately held corporation by the Peters. In 2006 William Randolph Hearst III and David Mumford joined the board. According to Robert J. Lang, who published with them a book on origami and mathematics, A K Peters "was a business, but first and foremost [Klaus] really wanted to create books that were works of art." The Encyclopedia of the Consumer Movement noted A K Peters as "a small publisher who enjoys a fine reputation in Mathematics".
In 2010, A K Peters was acquired by CRC Press, which is owned by Taylor & Francis. In January 2012, Taylor & Francis terminated the employment of Alice and Klaus Peters. On July 7, 2014, Klaus Peters died.
Topics
Experimental mathematics
In 1992 David Epstein, Klaus Peters and Silvio Levy set up the journal Experimental Mathematics, with scope the use of computers in pure mathematics. At the time the Notices of the American Mathematical Society was running a "Computers and Mathematics" section, launched in 1988. The particular focus of the "experimental mathematics" included in the journal was the computer-assisted development of mathematical conjectures.
The traditional context in pure mathematics was that "journals only publish theorems"; in this area A K Peters innovated. Klaus Peters had a particular interest in visualization for experimentation in low-dimensional geometry. The Journal of Graphics Tools was published by A K Peters from 1996, after an approach from Andrew Glassner, then with Microsoft Research. They also published the journal Internet Mathematics from its 2003 founding by Fan Chung until the acquisition of the publisher by Taylor & Francis.
A K Peters, with the participation of Jonathan Borwein, published as books three collective works on experimental mathematics: Mathematics by Experiment and Experimentation in Mathe |
https://en.wikipedia.org/wiki/Tychonoff%20plank | In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures. It is defined as the topological product of the two ordinal spaces and , where is the first infinite ordinal and the first uncountable ordinal. The deleted Tychonoff plank is obtained by deleting the point .
Properties
The Tychonoff plank is a compact Hausdorff space and is therefore a normal space. However, the deleted Tychonoff plank is non-normal. Therefore the Tychonoff plank is not completely normal. This shows that a subspace of a normal space need not be normal. The Tychonoff plank is not perfectly normal because it is not a Gδ space: the singleton is closed but not a Gδ set.
The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank.
Notes
See also
List of topologies
References
Topological spaces |
https://en.wikipedia.org/wiki/Lehmer%20sieve | Lehmer sieves are mechanical devices that implement sieves in number theory. Lehmer sieves are named for Derrick Norman Lehmer and his son Derrick Henry Lehmer. The father was a professor of mathematics at the University of California, Berkeley at the time, and his son followed in his footsteps as a number theorist and professor at Berkeley.
A sieve in general is intended to find the numbers which are remainders when a set of numbers are divided by a second set. Generally, they are used in finding solutions of Diophantine equations or to factor numbers. A Lehmer sieve will signal that such solutions are found in a variety of ways depending on the particular construction.
Construction
The first Lehmer sieve in 1926 was made using bicycle chains of varying length, with rods at appropriate points in the chains. As the chains turned, the rods would close electrical switches, and when all the switches were closed simultaneously, creating a complete electrical circuit, a solution had been found. Lehmer sieves were very fast, in one particular case factoring
in 3 seconds.
Built in 1932, a device using gears was shown at the Century of Progress Exposition in Chicago. These had gears representing numbers, just as the chains had before, with holes. Holes left open were the remainders sought. When the holes lined up, a light at one end of the device shone on a photocell at the other, which could stop the machine allowing for the observation of a solution. This incarnation allowed checking of five thousand combinations a second.
In 1936, a version was built using 16 mm film instead of chains, with holes in the film instead of rods. Brushes against the rollers would make electrical contact when the hole reached the top. Again, a full sequence of holes created a complete circuit, indicating a solution.
Several Lehmer sieves are on display at the Computer History Museum. Since then, the same basic idea has been used to design sieves in integrated circuits or software.
See also
Sieve of Eratosthenes
References
Further reading
.
. Also online at the Antique Computer home page.
, chap.XX,XXI.
.
External links
Lehmer sieves, by Dr. Michael R. Williams, Head Curator of The Computer History Museum
Lehmer sieves at Computer History Museum (at the bottom of the page)
History of computing hardware
History of computing hardware
Perforation-based computational tools |
https://en.wikipedia.org/wiki/Tor%20Arne%20Andreassen | Tor Arne Andreassen (born 16 March 1983) is a Norwegian former footballer who played in defence and midfield for Haugesund.
Career statistics
External links
Guardian's Stats Centre
1983 births
Living people
Norwegian men's footballers
FK Haugesund players
Sportspeople from Haugesund
Footballers from Rogaland
SK Vard Haugesund players
Eliteserien players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Compass%20equivalence%20theorem | In geometry, the compass equivalence theorem is an important statement in compass and straightedge constructions. The tool advocated by Plato in these constructions is a divider or collapsing compass, that is, a compass that "collapses" whenever it is lifted from a page, so that it may not be directly used to transfer distances. The modern compass with its fixable aperture can be used to transfer distances directly and so appears to be a more powerful instrument. However, the compass equivalence theorem states that any construction via a "modern compass" may be attained with a collapsing compass. This can be shown by establishing that with a collapsing compass, given a circle in the plane, it is possible to construct another circle of equal radius, centered at any given point on the plane. This theorem is Proposition II of Book I of Euclid's Elements. The proof of this theorem has had a chequered history.
Construction
The following construction and proof of correctness are given by Euclid in his Elements. Although there appear to be several cases in Euclid's treatment, depending upon choices made when interpreting ambiguous instructions, they all lead to the same conclusion, and so, specific choices are given below.
Given points , , and , construct a circle centered at with radius the length of (that is, equivalent to the solid green circle, but centered at ).
Draw a circle centered at and passing through and vice versa (the red circles). They will intersect at point and form the equilateral triangle .
Extend past and find the intersection of and the circle , labeled .
Create a circle centered at and passing through (the blue circle).
Extend past and find the intersection of and the circle , labeled .
Construct a circle centered at and passing through (the dotted green circle)
Because is an equilateral triangle, .
Because and are on a circle around , .
Therefore, .
Because is on the circle , .
Therefore, .
Alternative construction without straightedge
It is possible to prove compass equivalence without the use of the straightedge.
This justifies the use of "fixed compass" moves (constructing a circle of a given radius at a different location) in proofs of the Mohr–Mascheroni theorem, which states that any construction possible with straightedge and compass can be accomplished with compass alone.
Given points , , and , construct a circle centered at with the radius , using only a collapsing compass and no straightedge.
Draw a circle centered at and passing through and vice versa (the blue circles). They will intersect at points and .
Draw circles through with centers at and (the red circles). Label their other intersection .
Draw a circle (the green circle) with center passing through . This is the required circle.
There are several proofs of the correctness of this construction and it is often left as an exercise for the reader. Here is a modern one using transformations.
The line is the perpendicular bisecto |
https://en.wikipedia.org/wiki/Daniel%20Tijolo | Daniel Silva dos Santos, also known as Daniel Tijolo (May 30, 1982 - February 10, 2019), was a Brazilian defensive midfielder who played several years in Japan.
Club statistics
Updated to 23 February 2016.
Honours
Paraná State League: 2006
References
External links
Profile at Oita Trinita
Daniel se apresentam no Cruzeiro
Guardian Stats Centre
1982 births
2019 deaths
Brazilian men's footballers
Brazilian expatriate men's footballers
People from Cabo Frio
Associação Desportiva Cabofriense players
Paysandu Sport Club players
Ituano FC players
Cruzeiro Esporte Clube players
Associação Desportiva São Caetano players
Ventforet Kofu players
Nagoya Grampus players
Oita Trinita players
J1 League players
J2 League players
J3 League players
Expatriate men's footballers in Japan
Brazilian expatriate sportspeople in Japan
Men's association football midfielders
Deaths from lung cancer
Deaths from cancer in Rio de Janeiro (state)
Footballers from Rio de Janeiro (state) |
https://en.wikipedia.org/wiki/Burr%20distribution | In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution and is one of a number of different distributions sometimes called the "generalized log-logistic distribution".
Definitions
Probability density function
The Burr (Type XII) distribution has probability density function:
The parameter scales the underlying variate and is a positive real.
Cumulative distribution function
The cumulative distribution function is:
Applications
It is most commonly used to model household income, see for example: Household income in the U.S. and compare to magenta graph at right.
Random variate generation
Given a random variable drawn from the uniform distribution in the interval , the random variable
has a Burr Type XII distribution with parameters , and . This follows from the inverse cumulative distribution function given above.
Related distributions
When c = 1, the Burr distribution becomes the Pareto Type II (Lomax) distribution.
When k = 1, the Burr distribution is a log-logistic distribution sometimes referred to as the Fisk distribution, a special case of the Champernowne distribution.
The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.
The Dagum distribution, also known as the inverse Burr distribution, is the distribution of 1 / X, where X has the Burr distribution
References
Further reading
External links
Continuous distributions
Systems of probability distributions |
https://en.wikipedia.org/wiki/Moser%20spindle | In graph theory, a branch of mathematics, the Moser spindle (also called the Mosers' spindle or Moser graph) is an undirected graph, named after mathematicians Leo Moser and his brother William, with seven vertices and eleven edges. It is a unit distance graph requiring four colors in any graph coloring, and its existence can be used to prove that the chromatic number of the plane is at least four.
The Moser spindle has also been called the Hajós graph after György Hajós, as it can be viewed as an instance of the Hajós construction. However, the name "Hajós graph" has also been applied to a different graph, in the form of a triangle inscribed within a hexagon.
Construction
As a unit distance graph, the Moser spindle is formed by two rhombi with 60 and 120 degree angles, so that the sides and short diagonals of the rhombi form equilateral triangles. The two rhombi are placed in the plane, sharing one of their acute-angled vertices, in such a way that the remaining two acute-angled vertices are a unit distance apart from each other. The eleven edges of the graph are the eight rhombus sides, the two short diagonals of the rhombi, and the edge between the unit-distance pair of acute-angled vertices.
The Moser spindle may also be constructed graph-theoretically, without reference to a geometric embedding, using the Hajós construction starting with two complete graphs on four vertices. This construction removes an edge from each complete graph, merges two of the endpoints of the removed edges into a single vertex shared by both cliques, and adds a new edge connecting the remaining two endpoints of the removed edge.
Another way of constructing the Moser spindle is as the complement graph of the graph formed from the utility graph K3,3 by subdividing one of its edges.
Application to the Hadwiger–Nelson problem
The Hadwiger–Nelson problem asks how many colors are needed to color the points of the Euclidean plane in such a way that each pair of points at unit distance from each other are assigned different colors. That is, it asks for the chromatic number of the infinite graph whose vertices are all the points in the plane and whose edges are all pairs of points at unit distance.
The Moser spindle requires four colors in any graph coloring: in any three-coloring of one of the two rhombi from which it is formed, the two acute-angled vertices of the rhombi would necessarily have the same color as each other. But if the shared vertex of the two rhombi has the same color as the two opposite acute-angled vertices, then these two vertices have the same color as each other, violating the requirement that the edge connecting them have differently-colored endpoints. This contradiction shows that three colors are impossible, so at least four colors are necessary. Four colors are also sufficient to color the Moser spindle, a fact that follows for instance from the fact that its degeneracy is three.
An alternative proof that the Moser spindle requires four col |
https://en.wikipedia.org/wiki/Ideal%20%28set%20theory%29 | In the mathematical field of set theory, an ideal is a partially ordered collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and the union of any two elements of the ideal must also be in the ideal.
More formally, given a set an ideal on is a nonempty subset of the powerset of such that:
if and then and
if then
Some authors add a fourth condition that itself is not in ; ideals with this extra property are called .
Ideals in the set-theoretic sense are exactly ideals in the order-theoretic sense, where the relevant order is set inclusion. Also, they are exactly ideals in the ring-theoretic sense on the Boolean ring formed by the powerset of the underlying set. The dual notion of an ideal is a filter.
Terminology
An element of an ideal is said to be or , or simply or if the ideal is understood from context. If is an ideal on then a subset of is said to be (or just ) if it is an element of The collection of all -positive subsets of is denoted
If is a proper ideal on and for every either or then is a .
Examples of ideals
General examples
For any set and any arbitrarily chosen subset the subsets of form an ideal on For finite all ideals are of this form.
The finite subsets of any set form an ideal on
For any measure space, subsets of sets of measure zero.
For any measure space, sets of finite measure. This encompasses finite subsets (using counting measure) and small sets below.
A bornology on a set is an ideal that covers
A non-empty family of subsets of is a proper ideal on if and only if its in which is denoted and defined by is a proper filter on (a filter is if it is not equal to ). The dual of the power set is itself; that is, Thus a non-empty family is an ideal on if and only if its dual is a dual ideal on (which by definition is either the power set or else a proper filter on ).
Ideals on the natural numbers
The ideal of all finite sets of natural numbers is denoted Fin.
The on the natural numbers, denoted is the collection of all sets of natural numbers such that the sum is finite. See small set.
The on the natural numbers, denoted is the collection of all sets of natural numbers such that the fraction of natural numbers less than that belong to tends to zero as tends to infinity. (That is, the asymptotic density of is zero.)
Ideals on the real numbers
The is the collection of all sets of real numbers such that the Lebesgue measure of is zero.
The is the collection of all meager sets of real numbers.
Ideals on other sets
If is an ordinal number of uncountable cofinality, the on is the collection of all subsets of that are not stationary sets. This ideal has been studied extensively by W. Hugh Woodin.
Operations on ideals
Given ideals and on underlying sets and respectively, one forms the |
https://en.wikipedia.org/wiki/Wet%20Lake%20%28Warmia-Masuria%20Voivodeship%29 | Wet Lake () is a ribbon lake in the Mrągowskie Lakeland of Poland. There are 5 islands in the lake. It is situated in the Mazurski Landscape Park near Zgon.
Statistics
Length: 7.7 km
Width: 1.6 km
Area: 846 ha
Maximum depth: 51 m
Lakes of Poland
Lakes of Warmian-Masurian Voivodeship |
https://en.wikipedia.org/wiki/Katherine%20St.%20John | Katherine St. John is a professor at the CUNY Graduate Center Department of Computer Science and at Lehman College Department of Mathematics and Computer Science. She is a faculty member at the New York Consortium in Evolutionary Primatology. In 2007 she was selected to be an AWM/MAA Falconer Lecturer
where she gave a presentation on "Comparing Evolutionary Trees". She is also a former American Mathematical Society Council member at large.
References
External links
American bioinformaticians
American women computer scientists
American computer scientists
Living people
Year of birth missing (living people)
21st-century American women |
https://en.wikipedia.org/wiki/AWM/MAA%20Falconer%20Lecture | The Etta Z. Falconer Lecture is an award and lecture series which honors "women who have made distinguished contributions to the mathematical sciences or mathematics education". It is sponsored by the Association for Women in Mathematics and the Mathematical Association of America. The lectures began in 1996 and were named after the mathematician Etta Z. Falconer in 2004 "in memory of Falconer's profound vision and accomplishments in enhancing the movement of minorities and women into scientific careers". The recipient presents the lecture at MathFest each summer.
Recipients
The Falconer Lecturers have been:
1996 Karen E. Smith, MIT, "Calculus mod p"
1997 Suzanne M. Lenhart, University of Tennessee, "Applications of Optimal Control to Various Population Models"
1998 Margaret H. Wright, Bell Labs, "The Interior-Point Revolution in Constrained Optimization"
1999 Chuu-Lian Terng, Northeastern University, "Geometry and Visualization of Surfaces"
2000 Audrey Terras, University of California at San Diego, "Finite Quantum Chaos"
2001 Pat Shure, University of Michigan, "The Scholarship of Learning and Teaching: A Look Back and a Look Ahead"
2002 Annie Selden, Tennessee Technological University, "Two Research Traditions Separated by a Common Subject: Mathematics and Mathematics Education"
2003 Katherine Puckett Layton, Beverly Hills High School, "What I Learned in Forty Years in Beverly Hills 90212"
2004 Bozenna Pasik-Duncan, University of Kansas "Mathematics Education of Tomorrow"
2005 Fern Hunt, National Institute of Standards and Technology, "Techniques for Visualizing Frequency Patterns in DNA"
2006 Trachette Jackson, University of Michigan, "Cancer Modeling: From the Classical to the Contemporary"
2007 Katherine St. John, City University of New York, "Comparing Evolutionary Trees"
2008 Rebecca Goldin, George Mason University, "The Use and Abuse of Statistics in the Media"
2009 Kathleen Adebola Okikiolu, "The Sum of Squares of Wavelengths of a Closed Surface"
2010 Ami Radunskaya, Pomona College, "Mathematical Challenges in the Treatment of Cancer"
2011 Dawn Lott, Delaware State University, "Mathematical Interventions for Aneurysm Treatment"
2012 Karen D. King, National Council of Teachers of Mathematics, "Because I Love Mathematics: The Role of Disciplinary Grounding in Mathematics Education"
2013 Patricia Clark Kenschaft, Montclair State University,"Improving Equity and Education: Why and How"
2014 Marie A. Vitulli, University of Oregon, "From Algebraic to Weak Subintegral Extensions in Algebra and Geometry"
2015 Erica N. Walker, Teachers College, Columbia University, "'A Multiplicity All at Once': Mathematics for Everyone, Everywhere"
2016 Izabella Laba, University of British Columbia, "Harmonic Analysis and Additive Combinatorics on Fractals"
2017 Talithia Williams, Harvey Mudd College, "Not So Hidden Figures: Unveiling Mathematical Talent"
2018 Pamela Gorkin, Bucknell University, "Finding Ellipses"
2019 Tara Holm, Cornell U |
https://en.wikipedia.org/wiki/Chuu-Lian%20Terng | Chuu-Lian Terng () is a Taiwanese-American mathematician. Her research areas are differential geometry and integrable systems, with particular interests in completely integrable Hamiltonian partial differential equations and their relations to differential geometry, the geometry and topology of submanifolds in symmetric spaces, and the geometry of isometric actions.
Education and career
She received her B.S. from National Taiwan University in 1971 and her Ph.D. from Brandeis University in 1976 under the supervision of Richard Palais, whom she later married. She is currently a professor emerita at the University of California at Irvine.
She was a professor at Northeastern University for many years. Before joining Northeastern, she spent two years at the University of California, Berkeley and four years at Princeton University. She also spent two years at the Institute for Advanced Study (IAS) in Princeton and two years at the Max-Planck Institute in Bonn, Germany.
Terng has been an active member of the Association for Women in Mathematics (AWM). She served as AWM President from 1995 to 1997, chaired the Julia Robinson Celebration of Women in Math Conference, which was held July 1–3, 1996, and chaired the Michler Prize and Travel/Mentoring Grant Committees.
Terng has served on the editorial boards of the Transactions of the AMS, the Taiwanese Journal of Mathematics, Communications of Analysis and Geometry, the Proceedings of the AMS, and the Journal of Fixed Point Theory and its Applications.
In 1999, she was selected as the AWM/MAA Falconer Lecturer. Her citation reads:
Her early research concerned the classification of natural vector bundles and natural differential operators between them. She then became interested in submanifold geometry. Her main contributions are developing a structure theory for isoparametric submanifolds in and constructing soliton equations from special submanifolds. Recently, Terng and Karen Uhlenbeck (University of Texas at Austin) have developed a general approach to integrable PDEs that explains their hidden symmetries in terms of loop group actions. She is co-author of the book Submanifold Geometry and Critical Point Theory and an editor of the Journal of Differential Geometry survey volume 4 on "Integrable systems".
Professor Terng served as president of the Association for Women in Mathematics (AWM) from 1995 to 1997 and as Member-at-Large of the Council of the American Mathematical Society (AMS) from 1989 to 1992. She is currently on the Advisory Board of the National Center for Theoretical Sciences in Taiwan, the Steering Committee of the Institute for Advanced Study Park City Summer Institute, and the Editorial Board of the Transactions of the AMS.
Honors
Sloan Fellowship in 1980.
Humboldt Senior Scientist Award in 1997.
AWM/MAA Falconer Lecturer in 1999
Fellow of the American Mathematical Society, 2012.
Fellow of the Association for Women in Mathematics, 2018 (inaugural class).
Recognition
With
|
https://en.wikipedia.org/wiki/Approximation%20to%20the%20identity | In mathematics, an approximation to the identity refers to a sequence or net that converges to the identity in some algebra. Specifically, it can mean:
Nascent delta function, most commonly
Mollifier, more narrowly
Approximate identity, more abstractly |
https://en.wikipedia.org/wiki/Image%20space | Image space may refer to:
Image space - the optical space coordinatizing the visual representation or component of a scene
Image (mathematics) - the set of results of a function, the output object of a morphism |
https://en.wikipedia.org/wiki/Filter%20%28set%20theory%29 | In mathematics, a filter on a set is a family of subsets such that:
and
if and , then
If , and , then
A filter on a set may be thought of as representing a "collection of large subsets", one intuitive example being the neighborhood filter. Filters appear in order theory, model theory, and set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal.
Filters were introduced by Henri Cartan in 1937 and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book Topologie Générale as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of filters from sets to arbitrary partially ordered sets. Specifically, a filter on a set is just a proper order filter in the special case where the partially ordered set consists of the power set ordered by set inclusion.
Preliminaries, notation, and basic notions
In this article, upper case Roman letters like denote sets (but not families unless indicated otherwise) and will denote the power set of A subset of a power set is called (or simply, ) where it is if it is a subset of Families of sets will be denoted by upper case calligraphy letters such as
Whenever these assumptions are needed, then it should be assumed that is non–empty and that etc. are families of sets over
The terms "prefilter" and "filter base" are synonyms and will be used interchangeably.
Warning about competing definitions and notation
There are unfortunately several terms in the theory of filters that are defined differently by different authors.
These include some of the most important terms such as "filter".
While different definitions of the same term usually have significant overlap, due to the very technical nature of filters (and point–set topology), these differences in definitions nevertheless often have important consequences.
When reading mathematical literature, it is recommended that readers check how the terminology related to filters is defined by the author.
For this reason, this article will clearly state all definitions as they are used.
Unfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (for example, the notation for the set of all prefilters on a set) so in such cases this article uses whatever notation is most self describing or easily remembered.
The theory of filters and prefilters is well developed and has a plethora of definitions and notations, many of which are now unceremoniously listed to prevent this article from becoming prolix and to allow for the easy look up of notation and definitions.
Their important properties are described later.
Sets operations
The or in of a family of sets is
and similarly the of is
Throughout, is a map and is a set.
Nets and their tails
A is a set together with a preo |
https://en.wikipedia.org/wiki/Heinrich%20Behmann | Heinrich Behmann (10 January 1891, in Bremen-Aumund – 3 February 1970, in Bremen-Aumund) was a German mathematician. He performed research in the field of set theory and predicate logic.
Behmann studied mathematics in Tübingen, Leipzig and Göttingen. During World War I, he was wounded and received the Iron Cross 2nd Class. David Hilbert supervised the preparation of his doctoral thesis, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead. In 1922 Behmann proved that the monadic predicate calculus is decidable. In 1938 he obtained a professorial chair in mathematics at Halle (Saale). In 1945 he was dismissed for having been a member of the Nazi Party.
External links
Biography (in German)
1891 births
1970 deaths
20th-century German mathematicians
Scientists from Bremen (city)
Recipients of the Iron Cross (1914), 2nd class
German logicians
German philosophers
University of Tübingen alumni
Leipzig University alumni
University of Göttingen alumni
Academic staff of the Martin Luther University of Halle-Wittenberg
Nazi Party members |
https://en.wikipedia.org/wiki/Neutral%20vector | In statistics, and specifically in the study of the Dirichlet distribution, a neutral vector of random variables is one that exhibits a particular type of statistical independence amongst its elements. In particular, when elements of the random vector must add up to certain sum, then an element in the vector is neutral with respect to the others if the distribution of the vector created by expressing the remaining elements as proportions of their total is independent of the element that was omitted.
Definition
A single element of a random vector is neutral if the relative proportions of all the other elements are independent of .
Formally, consider the vector of random variables
where
The values are interpreted as lengths whose sum is unity. In a variety of contexts, it is often desirable to eliminate a proportion, say , and consider the distribution of the remaining intervals within the remaining length. The first element of , viz is defined as neutral if is statistically independent of the vector
Variable is neutral if is independent of the remaining interval: that is, being independent of
Thus , viewed as the first element of , is neutral.
In general, variable is neutral if is independent of
Complete neutrality
A vector for which each element is neutral is completely neutral.
If is drawn from a Dirichlet distribution, then is completely neutral. In 1980, James and Mosimann showed that the Dirichlet distribution is characterised by neutrality.
See also
Generalized Dirichlet distribution
References
Theory of probability distributions
Independence (probability theory) |
https://en.wikipedia.org/wiki/Generalized%20Dirichlet%20distribution | In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and almost twice the number of parameters. Random vectors with a GD distribution are completely neutral.
The density function of is
where we define . Here denotes the Beta function. This reduces to the standard Dirichlet distribution if for ( is arbitrary).
For example, if k=4, then the density function of is
where and .
Connor and Mosimann define the PDF as they did for the following reason. Define random variables with . Then have the generalized Dirichlet distribution as parametrized above, if the are independent beta with parameters , .
Alternative form given by Wong
Wong gives the slightly more concise form for
where for and . Note that Wong defines a distribution over a dimensional space (implicitly defining ) while Connor and Mosiman use a dimensional space with .
General moment function
If , then
where for and . Thus
Reduction to standard Dirichlet distribution
As stated above, if for then the distribution reduces to a standard Dirichlet. This condition is different from the usual case, in which setting the additional parameters of the generalized distribution to zero results in the original distribution. However, in the case of the GDD, this results in a very complicated density function.
Bayesian analysis
Suppose is generalized Dirichlet, and that is multinomial with trials (here ). Writing for and the joint posterior of is a generalized Dirichlet distribution with
where and for
Sampling experiment
Wong gives the following system as an example of how the Dirichlet and generalized Dirichlet distributions differ. He posits that a large urn contains balls of different colours. The proportion of each colour is unknown. Write for the proportion of the balls with colour in the urn.
Experiment 1. Analyst 1 believes that (ie, is Dirichlet with parameters ). The analyst then makes glass boxes and puts marbles of colour in box (it is assumed that the are integers ). Then analyst 1 draws a ball from the urn, observes its colour (say colour ) and puts it in box . He can identify the correct box because they are transparent and the colours of the marbles within are visible. The process continues until balls have been drawn. The posterior distribution is then Dirichlet with parameters being the number of marbles in each box.
Experiment 2. Analyst 2 believes that follows a generalized Dirichlet distribution: . All parameters are again assumed to be positive integers. The analyst makes wooden boxes. The boxes have two areas: one for balls and one for marbles. The balls are coloured but the marbles are not coloured. Then for , he puts balls of colour , and marbles, in to box . He then puts a ball of colour in box . The analyst then draws a ball from the urn. Because the boxes are wood, the analyst cannot tell whic |
https://en.wikipedia.org/wiki/Ulyqbek%20Asanbayev | Ulyqbek Asanbayev (, Ūlyqbek Asanbaev) is retired Kazakh footballer of Uzbek descent.
Career statistics
Club
Last update: 2012
International
Statistics accurate as of match played 10 September 2008
Honours
MHSK Tashkent
Uzbek League (1): 1997
Dustlik
Uzbek League (1): 1999
Pakhtakor Tashkent
Uzbekistan Cup (1): 2001
Zhenis Astana
Kazakhstan Premier League (1): 2001
Kazakhstan Cup (1): 2000-01
Kairat
Kazakhstan Premier League (1): 2004
Kazakhstan Cup (1): 2003
Aktobe
Kazakhstan Premier League (3): 2007, 2008, 2009
Kazakhstan Cup (1): 2008
Ordabasy
Kazakhstan Cup (1): 2011
References
External links
Profile at National Team Website
1979 births
Living people
Sportspeople from Shymkent
Kazakhstani men's footballers
Kazakhstan men's international footballers
Kazakhstan Premier League players
FC Zhenis players
FC Kairat players
Pakhtakor Tashkent FK players
FC Aktobe players
FC Kyzylzhar players
FC Ordabasy players
FC Irtysh Pavlodar players
FC Okzhetpes players
Uzbekistani expatriate sportspeople in Kazakhstan
Men's association football midfielders |
https://en.wikipedia.org/wiki/Iranians%20in%20Japan | Iranians in Japan (, , ) are a minority group, with official statistics recording about 5,000 Iranian migrants in the country. Part of the Iranian diaspora, most live in the Greater Tokyo Area.
Migration history
Ancient history
According to Akihiro Watanabe of the Nara National Research Institute for Cultural Properties, a mokkan (wooden tablet) dating back to the 7th century CE which was found in Nara Prefecture during the 1960s mentions a Persian official who lived and worked in Japan. Watanabe said that the official may have taught mathematics, citing Iran's expertise in the subject. The mokkan was deciphered in 2016 with the help of technology which allowed researchers to read characters not previously visible. Around the time the mokkan was inscribed, Nara would have been an ethnically diverse metropolitan area associated with the Silk Road and about to become Japan's capital city. Before the mokkan was discovered, the first written account of Persians in Japan was in the Nihon Shoki (The Chronicles of Japan, which was finished in 720). The book describes the arrival in Japan in 634 of several people from a place known in Japanese as Tokhārā (believed to be Tokharistan, which would have been part of the Sasanian Empire) and Dārā, a Persian man who worked for the emperor and returned to his homeland in 660. Another example of interaction between the Persians and the Japanese is the oldest known example of Persian writing in Japan, a one-page document with lines from the Shahnameh and the Vis and Rāmin and Jami' al-tawarikh.the Persian manuscript in Japan Discovered during the 20th century,the paper that was given by several Persians to the Japanese priest Kyōsei during the priest's 1217 trip to the port Quanzhou in China.
Modern history
After the end of the Iran–Iraq War in 1988, a number of Iranians (primarily men with lower-class, military, or criminal backgrounds) traveled to Japan to find work because the war and the Iranian Revolution had had a devastating effect on the Iranian economy. This coincided with an economic boom in Japan which created a need for unskilled laborers, allowing migrant workers without the money to travel to the Western world to secure high-paying jobs and support their families in Iran. Airfare subsidized by Iran Air and a bilateral visa-exemption agreement which had been in place for decades allowed for relatively easy and affordable travel between the countries. The workers arrived in Japan legally and received work permits which allowed them time (typically three months) to find work in Japan. A number of workers had trouble finding work while their permits were in effect, however, and reported that Japanese employers intentionally waited until a worker's permit expired to offer them a job at a fraction of the prevailing wage; deportation was a possibility if they complained about unfair wages. Some Iranians then became low-level yakuza (members of the Japanese mafia), selling illegal drugs and cell phones |
https://en.wikipedia.org/wiki/Infinite%20monkey%20theorem%20in%20popular%20culture | The infinite monkey theorem and its associated imagery is considered a popular and proverbial illustration of the mathematics of probability, widely known to the general public because of its transmission through popular culture rather than because of its transmission via the classroom.
However, this popularity as either presented to or taken in the public's mind often oversimplifies or confuses important aspects of the different scales of the concepts involved: infinity, probability, and time—all of these are in measures beyond average human experience and practical comprehension or comparison.
Popularity
The history of the imagery of "typing monkeys" dates back at least as far as Émile Borel's use of the metaphor in his essay in 1913, and this imagery has recurred many times since in a variety of media.
The Hoffmann and Hofmann paper (2001) referenced a collection compiled by Jim Reeds, titled "The Parable of the Monkeys – a.k.a. The Topos of the Monkeys and the Typewriters".
The enduring, widespread and popular nature of the knowledge of the theorem was noted in a 2001 paper, "Monkeys, Typewriters and Networks – the Internet in the Light of the Theory of Accidental Excellence". In their introduction to that paper, Hoffmann and Hofmann stated: "The Internet is home to a vast assortment of quotations and experimental designs concerning monkeys and typewriters. They all expand on the theory […] that if an infinite number of monkeys were left to bang on an infinite number of typewriters, sooner or later they would accidentally reproduce the complete works of William Shakespeare (or even just one of his sonnets)."
In 2002, a Washington Post article said: "Plenty of people have had fun with the famous notion that an infinite number of monkeys with an infinite number of typewriters and an infinite amount of time could eventually write the works of Shakespeare".
In 2003, an Arts Council funded experiment involving real monkeys and a computer keyboard received widespread press coverage.
In 2007, the theorem was listed by Wired magazine in a list of eight classic thought experiments.
Another study of the history was published in the introduction to a study published in 2007 by Terry Butler, "Monkeying Around with Text".
Today, popular interest in the typing monkeys is sustained by numerous appearances in literature, television and radio, music, and the Internet, as well as graphic novels and stand-up comedy routines. Several collections of cultural references to the theorem have been published.
The following thematic timelines are based on these existing collections. The timelines are not comprehensive – instead, they document notable examples of references to the theorem appearing in various media. The initial timeline starts with some of the early history following Borel, and the later timelines record examples of the history, from the stories by Maloney and Borges in the 1940s, up to the present day.
Early history
1913 – Émile Borel’s essay – “ |
https://en.wikipedia.org/wiki/%282%2C3%2C7%29%20triangle%20group | In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important for its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largest possible order, 84(g − 1), of its automorphism group.
The term "(2,3,7) triangle group" most often refers not to the full triangle group Δ(2,3,7) (the Coxeter group with Schwarz triangle (2,3,7) or a realization as a hyperbolic reflection group), but rather to the ordinary triangle group (the von Dyck group) D(2,3,7) of orientation-preserving maps (the rotation group), which is index 2.
Torsion-free normal subgroups of the (2,3,7) triangle group are Fuchsian groups associated with Hurwitz surfaces, such as the Klein quartic, Macbeath surface and First Hurwitz triplet.
Constructions
Hyperbolic construction
To construct the triangle group, start with a hyperbolic triangle with angles π/2, π/3, and π/7. This triangle, the smallest hyperbolic Schwarz triangle, tiles the plane by reflections in its sides. Consider then the group generated by reflections in the sides of the triangle, which (since the triangle tiles) is a non-Euclidean crystallographic group (a discrete subgroup of hyperbolic isometries) with this triangle for fundamental domain; the associated tiling is the order-3 bisected heptagonal tiling. The (2,3,7) triangle group is defined as the index 2 subgroups consisting of the orientation-preserving isometries, which is a Fuchsian group (orientation-preserving NEC group).
Group presentation
It has a presentation in terms of a pair of generators, g2, g3, modulo the following relations:
Geometrically, these correspond to rotations by , and about the vertices of the Schwarz triangle.
Quaternion algebra
The (2,3,7) triangle group admits a presentation in terms of the group of quaternions of norm 1 in a suitable order in a quaternion algebra. More specifically, the triangle group is the quotient of the group of quaternions by its center ±1.
Let η = 2cos(2π/7). Then from the identity
we see that Q(η) is a totally real cubic extension of Q. The (2,3,7) hyperbolic triangle group is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative algebra by the pair of generators i,j and relations i2 = j2 = η, ij = −ji. One chooses a suitable Hurwitz quaternion order in the quaternion algebra. Here the order is generated by elements
In fact, the order is a free Z[η]-module over the basis . Here the generators satisfy the relations
which descend to the appropriate relations in the triangle group, after quotienting by the center.
Relation to SL(2,R)
Extending the scalars from Q(η) to R (via the standard imbedding), one obtains an isomorphism between the quaternion algebra and the algebra M(2,R) of real 2 by 2 matrices. Choosing a concrete isomorphism allows one to exhibit the (2,3,7) triangle group as a specific Fuchsian group in SL(2,R), specifically as a quotient of the modular group. This can b |
https://en.wikipedia.org/wiki/Laplace%20limit | In mathematics, the Laplace limit is the maximum value of the eccentricity for which a solution to Kepler's equation, in terms of a power series in the eccentricity, converges. It is approximately
0.66274 34193 49181 58097 47420 97109 25290.
Kepler's equation M = E − ε sin E relates the mean anomaly M with the eccentric anomaly E for a body moving in an ellipse with eccentricity ε. This equation cannot be solved for E in terms of elementary functions, but the Lagrange reversion theorem gives the solution as a power series in ε:
or in general
Laplace realized that this series converges for small values of the eccentricity, but diverges for any value of M other than a multiple of π if the eccentricity exceeds a certain value that does not depend on M. The Laplace limit is this value. It is the radius of convergence of the power series.
It is given by the solution to the transcendental equation
No closed-form expression or infinite series is known for the Laplace limit.
History
Laplace calculated the value 0.66195 in 1827. The Italian astronomer Francesco Carlini found the limit 0.66 five years before Laplace. Cauchy in the 1829 gave the precise value 0.66274.
See also
Orbital eccentricity
References
.
.
External links
Orbits
Mathematical constants
Mathematical series |
https://en.wikipedia.org/wiki/VNSO | VNSO may stand for:
Vanuatu National Statistics Office
Vietnam National Symphony Orchestra
Vserossiyskaya Natsionalnaya Skautskaya Organizatsiya, a Russian scouting organization |
https://en.wikipedia.org/wiki/Partogram | A partogram or partograph is a composite graphical record of key data (maternal and fetal) during labour entered against time on a single sheet of paper. Relevant measurements might include statistics such as cervical dilation, fetal heart rate, duration of labour and vital signs.
In, 1954 Friedman prepared the cervicography. In 1972 Philpott and Castle developed the first partograph, by utilizing friedmar’s cervicograph, and adding the relationship of the presenting part to the maternal pelvis.
It is intended to provide an accurate record of the progress in labour, so that any delay or deviation from normal may be detected quickly and treated accordingly. However, a Cochrane review came to the conclusion that there is insufficient evidence to recommend partographs in standard labour management and care.
Components
Patient identification
Time: It is recorded at an interval of one hour. Zero time for spontaneous labour is time of admission in the labour ward and for induced labour is time of induction.
Fetal heart rate: It is recorded at an interval of thirty minutes.
State of membranes and colour of liquor: "I" designates intact membranes, "C" designates clear and "M" designates meconium stained liquor.
Cervical dilatation and descent of head
Uterine contractions: Squares in vertical columns are shaded according to duration and intensity.
Drugs and fluids
Blood pressure: It is recorded in vertical lines at an interval of 2 hours.
Pulse rate: It is also recorded in vertical lines at an interval of 30 minutes.
Oxytocin: Concentration is noted down in upper box; while dose is noted in lower box.
Urine analysis
Temperature record
Advantages
Provides information on single sheet of paper at a glance
Early prediction of deviation from normal progress of labour
Improvement in maternal morbidity, perinatal morbidity and mortality
Limitations
It requires a skilled healthcare worker who can fill and interpret the partograph.
Recent studies have shown there is no evidence that partograph use is detrimental to outcomes.
Often paper-partograph and the equipment required to complete it are unavailable in low resource settings.
Despite decades of training and investment, implementation rates and capacity to correctly use the partograph are very low.
According to some recent literature, cervical dilatation over time is a poor predictor of severe adverse birth outcomes. This raises questions around the validity of a partograph alert line.
Usage
A partograph is contained in the Perinatal Institute's "Birth notes".
Use of a partograph in established labour is recommended by the National Institute for Clinical Excellence (NICE) in the "Intrapartum Care" guideline.
Digital partograph
A digital partograph is an electronic implementation of the standard paper-based partograph/partogram that can work on a mobile or tablet PC. Partograph is a paper-based tool developed by the W.H.O. to monitor labour during pregnancy. The use of the partogra |
https://en.wikipedia.org/wiki/RECSAM | The Regional Centre for Education in Science and Mathematics (RECSAM) is a multinational educational corporation headquartered in Penang, Malaysia. It is one of the founding sister centres of the Southeast Asian Ministers of Education Organisation (SEAMEO), established on 30 November 1965 to promote co-operation in education, science and culture in the Southeast Asian region.
Since its inception in 1967, RECSAM has assisted in the training of educators in science and mathematics at the primary and secondary school levels in Brunei Darussalam, Cambodia, Indonesia, Laos, Malaysia, Myanmar, the Philippines, Singapore, Thailand, Timor Leste and Vietnam.
References
External links
Official website
Penang
Penang |
https://en.wikipedia.org/wiki/Golomb%E2%80%93Dickman%20constant | In mathematics, the Golomb–Dickman constant arises in the theory of random permutations and in number theory. Its value is
It is not known whether this constant is rational or irrational.
Definitions
Let an be the average — taken over all permutations of a set of size n — of the length of the longest cycle in each permutation. Then the Golomb–Dickman constant is
In the language of probability theory, is asymptotically the expected length of the longest cycle in a uniformly distributed random permutation of a set of size n.
In number theory, the Golomb–Dickman constant appears in connection with the average size of the largest prime factor of an integer. More precisely,
where is the largest prime factor of k . So if k is a d digit integer, then is the asymptotic average number of digits of the largest prime factor of k.
The Golomb–Dickman constant appears in number theory in a different way. What is the
probability that second largest prime factor of n is smaller than the square root of the largest prime factor of n? Asymptotically, this probability is .
More precisely,
where is the second largest prime factor n.
The Golomb-Dickman constant also arises when we consider the average length of the largest cycle of any function from a finite set to itself. If X is a finite set, if we repeatedly apply a function f: X → X to any element x of this set, it eventually enters a cycle, meaning that for some k we have for sufficiently large n; the smallest k with this property is the length of the cycle. Let bn be the average, taken over all functions from a set of size n to itself, of the length of the largest cycle. Then Purdom and Williams proved that
Formulae
There are several expressions for . These include:
where is the logarithmic integral,
where is the exponential integral, and
and
where is the Dickman function.
See also
Random permutation
Random permutation statistics
External links
References
Mathematical constants
Permutations |
https://en.wikipedia.org/wiki/Geodesic%20convexity | In mathematics — specifically, in Riemannian geometry — geodesic convexity is a natural generalization of convexity for sets and functions to Riemannian manifolds. It is common to drop the prefix "geodesic" and refer simply to "convexity" of a set or function.
Definitions
Let (M, g) be a Riemannian manifold.
A subset C of M is said to be a geodesically convex set if, given any two points in C, there is a unique minimizing geodesic contained within C that joins those two points.
Let C be a geodesically convex subset of M. A function is said to be a (strictly) geodesically convex function if the composition
is a (strictly) convex function in the usual sense for every unit speed geodesic arc γ : [0, T] → M contained within C.
Properties
A geodesically convex (subset of a) Riemannian manifold is also a convex metric space with respect to the geodesic distance.
Examples
A subset of n-dimensional Euclidean space En with its usual flat metric is geodesically convex if and only if it is convex in the usual sense, and similarly for functions.
The "northern hemisphere" of the 2-dimensional sphere S2 with its usual metric is geodesically convex. However, the subset A of S2 consisting of those points with latitude further north than 45° south is not geodesically convex, since the minimizing geodesic (great circle) arc joining two distinct points on the southern boundary of A leaves A (e.g. in the case of two points 180° apart in longitude, the geodesic arc passes over the south pole).
References
Convex optimization
Riemannian manifolds
Geodesic (mathematics) |
https://en.wikipedia.org/wiki/Chow%E2%80%93Liu%20tree | In probability theory and statistics Chow–Liu tree is an efficient method for constructing a second-order product approximation of a joint probability distribution, first described in a paper by . The goals of such a decomposition, as with such Bayesian networks in general, may be either data compression or inference.
The Chow–Liu representation
The Chow–Liu method describes a joint probability distribution as a product of second-order conditional and marginal distributions. For example, the six-dimensional distribution might be approximated as
where each new term in the product introduces just one new variable, and the product can be represented as a first-order dependency tree, as shown in the figure. The Chow–Liu algorithm (below) determines which conditional probabilities are to be used in the product approximation. In general, unless there are no third-order or higher-order interactions, the Chow–Liu approximation is indeed an approximation, and cannot capture the complete structure of the original distribution. provides a modern analysis of the Chow–Liu tree as a Bayesian network.
The Chow–Liu algorithm
Chow and Liu show how to select second-order terms for the product approximation so that, among all such second-order approximations (first-order dependency trees), the constructed approximation has the minimum Kullback–Leibler divergence to the actual distribution , and is thus the closest approximation in the classical information-theoretic sense. The Kullback–Leibler divergence between a second-order product approximation and the actual distribution is shown to be
where is the mutual information between variable and its parent and is the joint entropy of variable set . Since the terms and are independent of the dependency ordering in the tree, only the sum of the pairwise mutual informations, , determines the quality of the approximation. Thus, if every branch (edge) on the tree is given a weight corresponding to the mutual information between the variables at its vertices, then the tree which provides the optimal second-order approximation to the target distribution is just the maximum-weight tree. The equation above also highlights the role of the dependencies in the approximation: When no dependencies exist, and the first term in the equation is absent, we have only an approximation based on first-order marginals, and the distance between the approximation and the true distribution is due to the redundancies that are not accounted for when the variables are treated as independent. As we specify second-order dependencies, we begin to capture some of that structure and reduce the distance between the two distributions.
Chow and Liu provide a simple algorithm for constructing the optimal tree; at each stage of the procedure the algorithm simply adds the maximum mutual information pair to the tree. See the original paper, , for full details. A more efficient tree construction algorithm for the common case of sparse data was |
https://en.wikipedia.org/wiki/Georgian%20Wikipedia | The Georgian Wikipedia () is a Georgian language edition of free online encyclopedia Wikipedia. Founded in November 2003, it has articles as of .
Statistics
Currently it has 6 administrators and more than 150,000 registered users.
History
In 2014, the Georgian Wikipedia changed its logo to reflect the blue and gold coloring of Ukraine's flag in response to the Russian occupation of Crimea.
In early 2022, the Georgian Wikipedia again changed its logo to reflect the blue and gold coloring of Ukraine's flag in response to the 2022 Russian invasion of Ukraine.
See also
Georgian language
List of Wikipedias
References
External links
Georgian Language Wikipedia
Georgian Wikipedia mobile version
The embassy of the Georgian-language Wikipedia
Wikipedias by language
Internet properties established in 2003
Georgian-language encyclopedias
Georgian-language websites |
https://en.wikipedia.org/wiki/Marina%20Ratner | Marina Evseevna Ratner (; October 30, 1938 – July 7, 2017) was a professor of mathematics at the University of California, Berkeley who worked in ergodic theory. Around 1990, she proved a group of major theorems concerning unipotent flows on homogeneous spaces, known as Ratner's theorems. Ratner was elected to the American Academy of Arts and Sciences in 1992, awarded the Ostrowski Prize in 1993 and elected to the National Academy of Sciences the same year. In 1994, she was awarded the John J. Carty Award from the National Academy of Sciences.
Biographical information
Ratner was born in Moscow, Russian SFSR to a Jewish family, where her father was a plant physiologist and her mother a chemist. Ratner's mother was fired from work in the 1940s for writing to her mother in Israel, then considered an enemy of the Soviet state. Ratner gained an interest in mathematics in her fifth grade. From 1956 to 1961, she studied mathematics and physics at Moscow State University. Here, she became interested in probability theory, inspired by A.N. Kolmogorov and his group. After graduation, she spent four years working in Kolmogorov's applied statistics group. Following this, she returned to Moscow State university for graduate studies were under Yakov G. Sinai, also a student of Kolmogorov. She completed her PhD thesis, titled "Geodesic Flows on Unit Tangent Bundles of Compact Surfaces of Negative Curvature", in 1969. In 1971 she emigrated from the Soviet Union to Israel and she taught at the Hebrew University from 1971 until 1975. She began to work with Rufus Bowen at Berkeley and later emigrated to the United States and became a professor of mathematics at Berkeley. Her work included proofs of conjectures dealing with unipotent flows on quotients of Lie groups made by S. G. Dani and M. S. Raghunathan. For this and other work, she won the John J. Carty Award for the Advancement of Science in 1994. she became only the third woman plenary speaker at International Congress of Mathematicians in 1994.
Marina Ratner died July 7, 2017, at the age of 78.
Selected publications
References
1938 births
2017 deaths
American women mathematicians
Fellows of the American Academy of Arts and Sciences
Members of the United States National Academy of Sciences
20th-century American mathematicians
21st-century American mathematicians
Jewish Russian scientists
Jewish American scientists
University of California, Berkeley faculty
Dynamical systems theorists
20th-century women mathematicians
21st-century women mathematicians
Russian women scientists
20th-century Russian mathematicians
20th-century American women
21st-century American women
20th-century Russian women
21st-century American Jews |
https://en.wikipedia.org/wiki/Subnet%20%28mathematics%29 | In topology and related areas of mathematics, a subnet is a generalization of the concept of subsequence to the case of nets. The analogue of "subsequence" for nets is the notion of a "subnet". The definition is not completely straightforward, but is designed to allow as many theorems about subsequences to generalize to nets as possible.
There are three non-equivalent definitions of "subnet".
The first definition of a subnet was introduced by John L. Kelley in 1955 and later, Stephen Willard introduced his own (non-equivalent) variant of Kelley's definition in 1970.
Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet" but they are each equivalent to the concept of "subordinate filter", which is the analog of "subsequence" for filters (they are not equivalent in the sense that there exist subordinate filters on whose filter/subordinate–filter relationship cannot be described in terms of the corresponding net/subnet relationship).
A third definition of "subnet" (not equivalent to those given by Kelley or Willard) that equivalent to the concept of "subordinate filter" was introduced independently by Smiley (1957), Aarnes and Andenaes (1972), Murdeshwar (1983), and possibly others, although it is not often used.
This article discusses the definition due to Willard (the other definitions are described in the article Filters in topology#Subnets).
Definitions
There are several different non-equivalent definitions of "subnet" and this article will use the definition introduced in 1970 by Stephen Willard, which is as follows:
If and are nets in a set from directed sets and respectively, then is said to be a of ( or a ) if there exists a monotone final function
such that
A function is , , and an if whenever then and it is called if its image is cofinal in
The set being in means that for every there exists some such that that is, for every there exists an such that
Since the net is the function and the net is the function the defining condition may be written more succinctly and cleanly as either or where denotes function composition and is just notation for the function
Subnets versus subsequences
Importantly, a subnet is not merely the restriction of a net to a directed subset of its domain
In contrast, by definition, a of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. Explicitly, a sequence is said to be a of if there exists a strictly increasing sequence of positive integers such that for every (that is to say, such that ). The sequence can be canonically identified with the function defined by Thus a sequence is a subsequence of if and only if there exists a strictly increasing function such that
Subsequences are subnets
Every subsequence is a subnet because if is a subsequence of then the map de |
https://en.wikipedia.org/wiki/Analytic%20function%20of%20a%20matrix | In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size.
This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations.
Extending scalar function to matrix functions
There are several techniques for lifting a real function to a square matrix function such that interesting properties are maintained. All of the following techniques yield the same matrix function, but the domains on which the function is defined may differ.
Power series
If the analytic function has the Taylor expansion
then a matrix function can be defined by substituting by a square matrix: powers become matrix powers, additions become matrix sums and multiplications by coefficients become scalar multiplications. If the series converges for , then the corresponding matrix series converges for matrices such that for some matrix norm that satisfies .
Diagonalizable matrices
A square matrix is diagonalizable, if there is an invertible matrix such that is a diagonal matrix, that is, has the shape
As it is natural to set
It can be verified that the matrix does not depend on a particular choice of .
For example, suppose one is seeking for
One has
for
Application of the formula then simply yields
Likewise,
Jordan decomposition
All complex matrices, whether they are diagonalizable or not, have a Jordan normal form , where the matrix J consists of Jordan blocks. Consider these blocks separately and apply the power series to a Jordan block:
This definition can be used to extend the domain of the matrix function
beyond the set of matrices with spectral radius smaller than the radius of convergence of the power series.
Note that there is also a connection to divided differences.
A related notion is the Jordan–Chevalley decomposition which expresses a matrix as a sum of a diagonalizable and a nilpotent part.
Hermitian matrices
A Hermitian matrix has all real eigenvalues and can always be diagonalized by a unitary matrix P, according to the spectral theorem.
In this case, the Jordan definition is natural. Moreover, this definition allows one to extend standard inequalities for
real functions:
If for all eigenvalues of , then .
(As a convention, is a positive-semidefinite matrix.)
The proof follows directly from the definition.
Cauchy integral
Cauchy's integral formula from complex analysis can also be used to generalize scalar functions to matrix functions. Cauchy's integral formula states that for any analytic function defined on a set , one has
where is a closed simple curve inside the domain enclosing .
Now, replace by a matrix and consider a path inside that encloses all eigenvalues of . One possibility to achieve this is to let be a circle around the origin with radius larger than for an arbitrary matrix norm . Then, is definable by
This integral can |
https://en.wikipedia.org/wiki/Embedding%20problem | In Galois theory, a branch of mathematics, the embedding problem is a generalization of the inverse Galois problem. Roughly speaking, it asks whether a given Galois extension can be embedded into a Galois extension in such a way that the restriction map between the corresponding Galois groups is given.
Definition
Given a field K and a finite group H, one may pose the following question (the so called inverse Galois problem). Is there a Galois extension F/K with Galois group isomorphic to H. The embedding problem is a generalization of this problem:
Let L/K be a Galois extension with Galois group G and let f : H → G be an epimorphism. Is there a Galois extension F/K with Galois group H and an embedding α : L → F fixing K under which the restriction map from the Galois group of F/K to the Galois group of L/K coincides with f?
Analogously, an embedding problem for a profinite group F consists of the following data: Two profinite groups H and G and two continuous epimorphisms φ : F → G and
f : H → G. The embedding problem is said to be finite if the group H is.
A solution (sometimes also called weak solution) of such an embedding problem is a continuous homomorphism γ : F → H such that φ = f γ. If the solution is surjective, it is called a proper solution.
Properties
Finite embedding problems characterize profinite groups. The following theorem gives an illustration for this principle.
Theorem. Let F be a countably (topologically) generated profinite group. Then
F is projective if and only if any finite embedding problem for F is solvable.
F is free of countable rank if and only if any finite embedding problem for F is properly solvable.
References
Vahid Shirbisheh, Galois embedding problems with abelian kernels of exponent p VDM Verlag Dr. Müller, , (2009).
Group theory
Galois theory |
https://en.wikipedia.org/wiki/Morass | Morass may refer to:
Marsh, a wetland
Morass (set theory), an infinite combinatorial structure
The Morass, former name of Inundation, Gibraltar
Palais Morass, a historic building in Heidelberg, Germany, which houses the Kurpfälzisches Museum
Morass (film), a 1922 German silent film |
https://en.wikipedia.org/wiki/Philipp%20Heerwagen | Philipp Heerwagen (born 13 April 1983) is a German professional footballer who plays as a goalkeeper.
Career statistics
References
External links
1983 births
Living people
People from Kelheim
Footballers from Lower Bavaria
German men's footballers
Men's association football goalkeepers
Germany men's youth international footballers
Bundesliga players
2. Bundesliga players
SpVgg Unterhaching players
VfL Bochum players
FC St. Pauli players
FC Ingolstadt 04 players
SV Sandhausen players |
https://en.wikipedia.org/wiki/Tits%20alternative | In mathematics, the Tits alternative, named after Jacques Tits, is an important theorem about the structure of finitely generated linear groups.
Statement
The theorem, proven by Tits, is stated as follows.
Consequences
A linear group is not amenable if and only if it contains a non-abelian free group (thus the von Neumann conjecture, while not true in general, holds for linear groups).
The Tits alternative is an important ingredient in the proof of Gromov's theorem on groups of polynomial growth. In fact the alternative essentially establishes the result for linear groups (it reduces it to the case of solvable groups, which can be dealt with by elementary means).
Generalizations
In geometric group theory, a group G is said to satisfy the Tits alternative if for every subgroup H of G either H is virtually solvable or H contains a nonabelian free subgroup (in some versions of the definition this condition is only required to be satisfied for all finitely generated subgroups of G).
Examples of groups satisfying the Tits alternative which are either not linear, or at least not known to be linear, are:
Hyperbolic groups
Mapping class groups;
Out(Fn);
Certain groups of birational transformations of algebraic surfaces.
Examples of groups not satisfying the Tits alternative are:
the Grigorchuk group;
Thompson's group F.
Proof
The proof of the original Tits alternative is by looking at the Zariski closure of in . If it is solvable then the group is solvable. Otherwise one looks at the image of in the Levi component. If it is noncompact then a ping-pong argument finishes the proof. If it is compact then either all eigenvalues of elements in the image of are roots of unity and then the image is finite, or one can find an embedding of in which one can apply the ping-pong strategy.
Note that the proof of all generalisations above also rests on a ping-pong argument.
References
Infinite group theory
Geometric group theory
Theorems in group theory |
https://en.wikipedia.org/wiki/Stanford%20University%20Mathematics%20Camp | Stanford University Mathematics Camp, or SUMaC, is a competitive summer mathematics program for rising high school juniors and seniors around the world. The camp lasts for 4 weeks, usually from mid-July to mid-August. It is based on the campus of Stanford University.
Like the Ross Program at Ohio State and the PROMYS program at Boston University, SUMaC does not put emphasis on competition-math preparations but focuses instead on advanced undergraduate math topics.
History
SUMaC was founded in 1995 by Professors Rafe Mazzeo and Ralph Cohen of the Stanford Mathematics Department and has been directed from the beginning by Prof. Mazzeo, and Dr. Rick Sommer. Dr. Sommer was an assistant professor in the Stanford Mathematics Department and is currently a deputy director of the Education Program for Gifted Youth (EPGY), at Stanford. He designed the Program I course and has been teaching versions of it since the first SUMaC in 1995. The Program II course was designed and has been taught by Prof. Rafe Mazzeo. (In recent years, the course was cotaught by Dr. Pierre Albin, a former Stanford graduate student who currently teaches at MIT, and is currently taught by Dr. Simon Rubinstein-Salzedo, a postdoctoral fellow in statistics at Stanford.)
Programs
Program I investigates non-constructibility in geometry, classification of patterns in two dimensions, error-correcting codes, cryptography, and the analysis of the Rubik's Cube. The mathematics that is central to solving these problems comes from the areas of abstract algebra and number theory.
Program II contains an introduction to selected topics in combinatorial, differential, and algebraic topology. The program emphasizes developing ideas from, and problems in, geometric topology where methods from abstract algebra and calculus have proven to be effective tools.
Other activities
During the camp, there are frequent guest lectures given by internationally renowned mathematicians. These talks are in the areas of current mathematical research. Ravi Vakil, a current Stanford mathematics professor and a 4-time Putnam Fellow, talked to the students in 2007.
Also in 2007, Tyson Mao, one of the best cube solvers in the world, taught SUMaC students how to solve the Rubik's Cube. Other speakers in 2007 included Drs. Kay Kirkpatrick (MIT), Ted Shifrin (University of Georgia), and Pete Storm (Stanford).
In 2014, John Edmark, professor of art and art history at Stanford, spoke about his current work in mathematics-inspired sculptures, and Brian Conrey, executive director of the American Institute of Mathematics, gave a lecture on the Twin Primes Conjecture and the Riemann Hypothesis.
Students at SUMaC also engage in a variety of sports activities during their free time.
Such sports include basketball, tennis, badminton, table tennis, and ultimate.
SUMaC was the backdrop for Justina Chen Headley's book Nothing But the Truth (and a Few White Lies), a teen novel about a half-Taiwanese girl who finally finds her iden |
https://en.wikipedia.org/wiki/Helen%20G.%20Grundman | Helen Giessler Grundman is an American mathematician. She is the Director of Education and Diversity at the American Mathematical Society and Research Professor Emeritus of Mathematics at Bryn Mawr College. Grundman is noted for her research in number theory and efforts to increase diversity in mathematics.
Education
Helen Grundman earned her PhD in 1989 from the University of California, Berkeley, under the supervision of P. Emery Thomas.
Employment
After receiving her PhD, Grundman spent two years as a C. L. E. Moore instructor at the Massachusetts Institute of Technology. She became a professor at Bryn Mawr College in 1991. In 2016, Grundman was named as the inaugural Director of Education and Diversity for the American Mathematical Society.
Research
In 1994, Grundman proved that sequences of more than 2n consecutive Harshad numbers in base n do not exist.
Honors
In 2017, Grundman was selected as a fellow of the Association for Women in Mathematics in the inaugural class.
Selected publications
References
20th-century American mathematicians
21st-century American mathematicians
Number theorists
American women mathematicians
Living people
Year of birth missing (living people)
Fellows of the Association for Women in Mathematics
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
Massachusetts Institute of Technology School of Science faculty
University of California, Berkeley alumni
Bryn Mawr College faculty
21st-century American women |
https://en.wikipedia.org/wiki/Power%20pitcher | Power pitcher is a term in baseball for a pitcher who relies on pitch velocity at the expense of accuracy. Power pitchers usually record a high number of strikeouts, and statistics such as strikeouts per 9 innings pitched are common measures of power. An average pitcher strikes out about 5 batters per nine innings while a power pitcher will often strike out one or more every inning.
The prototypical power pitcher is National Baseball Hall of Fame member, Nolan Ryan, who struck out a Major League Baseball record 5,714 batters in 5,386 innings. Ryan recorded seven no-hitters, appeared in eight Major League Baseball All-Star Games but also holds the record for most walks issued (2,795). Other prominent power pitchers include Hall of Famers Walter Johnson, Sandy Koufax, Pedro Martínez, Randy Johnson, and Bob Feller. Feller himself famously led his league in strikeouts and walks several times.
The traditional school of thought on power pitching was known as "throw till you blow". However, multimillion-dollar contracts have changed mentalities. The number of pitches thrown is now counted by a team's staff, with particular attention paid to young power arms. The care which some of the older power pitchers took with their arms has allowed for long careers and further opportunity after they have stopped playing.
See also
Finesse pitcher
Power hitter
References
Further reading
Baseball pitching
Baseball strategy
Baseball terminology
ja:投手#技巧派投手 |
https://en.wikipedia.org/wiki/Normalisation%20by%20evaluation | In programming language semantics, normalisation by evaluation (NBE) is a method of obtaining the normal form of terms in the λ-calculus by appealing to their denotational semantics. A term is first interpreted into a denotational model of the λ-term structure, and then a canonical (β-normal and η-long) representative is extracted by reifying the denotation. Such an essentially semantic, reduction-free, approach differs from the more traditional syntactic, reduction-based, description of normalisation as reductions in a term rewrite system where β-reductions are allowed deep inside λ-terms.
NBE was first described for the simply typed lambda calculus. It has since been extended both to weaker type systems such as the untyped lambda calculus using a domain theoretic approach, and to richer type systems such as several variants of Martin-Löf type theory.
Outline
Consider the simply typed lambda calculus, where types τ can be basic types (α), function types (→), or products (×), given by the following Backus–Naur form grammar (→ associating to the right, as usual):
(Types) τ ::= α | τ1 → τ2 | τ1 × τ2
These can be implemented as a datatype in the meta-language; for example, for Standard ML, we might use:
datatype ty = Basic of string
| Arrow of ty * ty
| Prod of ty * ty
Terms are defined at two levels. The lower syntactic level (sometimes called the dynamic level) is the representation that one intends to normalise.
(Syntax Terms) s,t,… ::= var x | lam (x, t) | app (s, t) | pair (s, t) | fst t | snd t
Here lam/app (resp. pair/fst,snd) are the intro/elim forms for → (resp. ×), and x are variables. These terms are intended to be implemented as a first-order datatype in the meta-language:
datatype tm = var of string
| lam of string * tm | app of tm * tm
| pair of tm * tm | fst of tm | snd of tm
The denotational semantics of (closed) terms in the meta-language interprets the constructs of the syntax in terms of features of the meta-language; thus, lam is interpreted as abstraction, app as application, etc. The semantic objects constructed are as follows:
(Semantic Terms) S,T,… ::= LAM (λx. S x) | PAIR (S, T) | SYN t
Note that there are no variables or elimination forms in the semantics; they are represented simply as syntax. These semantic objects are represented by the following datatype:
datatype sem = LAM of (sem -> sem)
| PAIR of sem * sem
| SYN of tm
There are a pair of type-indexed functions that move back and forth between the syntactic and semantic layer. The first function, usually written ↑τ, reflects the term syntax into the semantics, while the second reifies the semantics as a syntactic term (written as ↓τ). Their definitions are mutually recursive as follows:
These definitions are easily implemented in the meta-language:
(* fresh_var : unit -> string *)
val variable_ctr = ref ~1
fun fresh_var () =
(variable_ctr := 1 + !variable_ctr; |
https://en.wikipedia.org/wiki/Francisco%20Chaviano | Francisco Chaviano is a Cuban human rights activist and mathematics professor.
In 1994, he was the President of the Cuban National Council for Human Rights when he documented cases of people who disappeared or died while trying to leave Cuba. He was arrested in March 1994 and sentenced to 15 years in prison a year later by a military court.
Amnesty International listed him as a prisoner of conscience and said that his trial fell short of international standards.
He was released in August 2007 on parole after becoming Cuba's longest serving political prisoner.
References
Amnesty International prisoners of conscience held by Cuba
Cuban human rights activists
Living people
Opposition to Fidel Castro
Year of birth missing (living people)
Cuban prisoners and detainees |
https://en.wikipedia.org/wiki/Modulus%20and%20characteristic%20of%20convexity | In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε-δ definition of uniform convexity as the modulus of continuity does to the ε-δ definition of continuity.
Definitions
The modulus of convexity of a Banach space (X, ||·||) is the function defined by
where S denotes the unit sphere of (X, || ||). In the definition of δ(ε), one can as well take the infimum over all vectors x, y in X such that and .
The characteristic of convexity of the space (X, || ||) is the number ε0 defined by
These notions are implicit in the general study of uniform convexity by J. A. Clarkson (; this is the same paper containing the statements of Clarkson's inequalities). The term "modulus of convexity" appears to be due to M. M. Day.
Properties
The modulus of convexity, δ(ε), is a non-decreasing function of ε, and the quotient is also non-decreasing on . The modulus of convexity need not itself be a convex function of ε. However, the modulus of convexity is equivalent to a convex function in the following sense: there exists a convex function δ1(ε) such that
The normed space is uniformly convex if and only if its characteristic of convexity ε0 is equal to 0, i.e., if and only if for every .
The Banach space is a strictly convex space (i.e., the boundary of the unit ball B contains no line segments) if and only if δ(2) = 1, i.e., if only antipodal points (of the form x and y = −x) of the unit sphere can have distance equal to 2.
When X is uniformly convex, it admits an equivalent norm with power type modulus of convexity. Namely, there exists and a constant such that
Modulus of convexity of the LP spaces
The modulus of convexity is known for the LP spaces. If , then it satisfies the following implicit equation:
Knowing that one can suppose that . Substituting this into the above, and expanding the left-hand-side as a Taylor series around , one can calculate the coefficients:
For , one has the explicit expression
Therefore, .
See also
Uniformly smooth space
Notes
References
Fuster, Enrique Llorens. Some moduli and constants related to metric fixed point theory. Handbook of metric fixed point theory, 133–175, Kluwer Acad. Publ., Dordrecht, 2001.
Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.
.
Vitali D. Milman. Geometric theory of Banach spaces II. Geometry of the unit sphere. Uspechi Mat. Nauk, vol. 26, no. 6, 73–149, 1971; Russian Math. Surveys, v. 26 6, 80–159.
Banach spaces
Convex analysis |
https://en.wikipedia.org/wiki/Waldhausen%20category | In mathematics, a Waldhausen category is a category C equipped with some additional data, which makes it possible to construct the K-theory spectrum of C using a so-called S-construction. It's named after Friedhelm Waldhausen, who introduced this notion (under the term category with cofibrations and weak equivalences) to extend the methods of algebraic K-theory to categories not necessarily of algebraic origin, for example the category of topological spaces.
Definition
Let C be a category, co(C) and we(C) two classes of morphisms in C, called cofibrations and weak equivalences respectively. The triple (C, co(C), we(C)) is called a Waldhausen category if it satisfies the following axioms, motivated by the similar properties for the notions of cofibrations and weak homotopy equivalences of topological spaces:
C has a zero object, denoted by 0;
isomorphisms are included in both co(C) and we(C);
co(C) and we(C) are closed under composition;
for each object A ∈ C the unique map 0 → A is a cofibration, i.e. is an element of co(C);
co(C) and we(C) are compatible with pushouts in a certain sense.
For example, if is a cofibration and is any map, then there must exist a pushout , and the natural map should be cofibration:
Relations with other notions
In algebraic K-theory and homotopy theory there are several notions of categories equipped with some specified classes of morphisms. If C has a structure of an exact category, then by defining we(C) to be isomorphisms, co(C) to be admissible monomorphisms, one obtains a structure of a Waldhausen category on C. Both kinds of structure may be used to define K-theory of C, using the Q-construction for an exact structure and S-construction for a Waldhausen structure. An important fact is that the resulting K-theory spaces are homotopy equivalent.
If C is a model category with a zero object, then the full subcategory of cofibrant objects in C may be given a Waldhausen structure.
S-construction
The Waldhausen S-construction produces from a Waldhausen category C a sequence of Kan complexes , which forms a spectrum. Let denote the loop space of the geometric realization of . Then the group
is the n-th K-group of C. Thus, it gives a way to define higher K-groups. Another approach for higher K-theory is Quillen's Q-construction.
The construction is due to Friedhelm Waldhausen.
biWaldhausen categories
A category C is equipped with bifibrations if it has cofibrations and its opposite category COP has so also. In that case, we denote the fibrations of COP by quot(C).
In that case, C is a biWaldhausen category if C has bifibrations and weak equivalences such that both (C, co(C), we) and (COP, quot(C), weOP) are Waldhausen categories.
Waldhausen and biWaldhausen categories are linked with algebraic K-theory. There, many interesting categories are complicial biWaldhausen categories. For example:
The category of bounded chain complexes on an exact category .
The category of functors when is so.
And |
https://en.wikipedia.org/wiki/Bachmann%E2%80%93Howard%20ordinal | In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard-Bachmann ordinal) is a large countable ordinal.
It is the proof-theoretic ordinal of several mathematical theories, such as Kripke–Platek set theory (with the axiom of infinity) and the system CZF of constructive set theory.
It was introduced by and .
Definition
The Bachmann–Howard ordinal is defined using an ordinal collapsing function:
εα enumerates the epsilon numbers, the ordinals ε such that ωε = ε.
Ω = ω1 is the first uncountable ordinal.
εΩ+1 is the first epsilon number after Ω = εΩ.
ψ(α) is defined to be the smallest ordinal that cannot be constructed by starting with 0, 1, ω and Ω, and repeatedly applying ordinal addition, multiplication and exponentiation, and ψ to previously constructed ordinals (except that ψ can only be applied to arguments less than α, to ensure that it is well defined).
The Bachmann–Howard ordinal is ψ(εΩ+1).
The Bachmann–Howard ordinal can also be defined as for an extension of the Veblen functions φα to certain functions α of ordinals; this extension was carried out by Heinz Bachmann and is not completely straightforward.
References
(Slides of a talk given at Fischbachau.)
Citations
Proof theory
Ordinal numbers |
https://en.wikipedia.org/wiki/List%20of%20SS%20Lazio%20records%20and%20statistics | This is a list of records and statistics in relation to the Italian football club Società Sportiva Lazio.
All time
Divisional movements
Total appearances
Statistics accurate as of 28 May 2023.
Total goals
Statistics accurate as of 29 June 2023.
Serie A record
Scudetti: 2
1973–74, 1999–2000
Total Serie A appearances
Statistics accurate as of 28 May 2023.
Total Serie A goals
Statistics accurate as of 28 May 2023.
European record
Statistics in European competitions
UEFA Cup Winners' Cup: 1
1998–99
UEFA Super Cup: 1
1999
Coppa delle Alpi: 1
1971
Total European appearances
Statistics accurate as of 19 February 2023.
Total European goals
Statistics accurate as of 19 February 2023.
National Cup record
Coppa Italia: 7
1958, 1997–98, 1999–2000, 2003–04, 2008–09, 2012–13, 2018–19
Supercoppa Italiana: 5
1998, 2000, 2009, 2017, 2019
Total national cup appearances
Statistics accurate as of 22 May 2022 and include both Coppa Italia and Supercoppa Italiana matches.
Total national cup goals
Statistics accurate as of 22 May 2022 and include both Coppa Italia and Supercoppa Italiana goals.
Capocannonieri
List of Capocannonieri (Serie A top scorers).
Statistics accurate as of 22 May 2022.
Club records
Statistics accurate as of 1 August 2020.
Largest victory:
13–1 v Pro Roma, Prima Categoria, 10 November 1912.
Largest defeat:
1–8 v Internazionale, Serie A, 18 March 1934. 0–7 v Internazionale, Serie A, 5 March 1961.
Most points in a season:
78 (2019–20)
Fewest points in a season:
15 (1984–85)
Most victories in a season:
24 (2019–20)
Fewest victories in a season:
2 (1984–85)
Most defeats in a season:
21 (1960–61)
Fewest defeats in a season:
3 (1972–73)
Most goals scored in a season:
89 (2017–18)
Fewest goals scored in a season:
16 (1984–85)
Most goals conceded in a season:
66 (1933–34)
Fewest goals conceded in a season:
16 (1972–73)
References
Records
Lazio |
https://en.wikipedia.org/wiki/Multiplicative%20digital%20root | In number theory, the multiplicative digital root of a natural number in a given number base is found by multiplying the digits of together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of .
The multiplicative digital root for the first few positive integers are:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0.
Multiplicative digital roots are the multiplicative equivalent of digital roots.
Definition
Let be a natural number. We define the digit product for base to be the following:
where is the number of digits in the number in base , and
is the value of each digit of the number. A natural number is a multiplicative digital root if it is a fixed point for , which occurs if .
For example, in base , 0 is the multiplicative digital root of 9876, as
All natural numbers are preperiodic points for , regardless of the base. This is because if , then
and therefore
If , then trivially
Therefore, the only possible multiplicative digital roots are the natural numbers , and there are no cycles other than the fixed points of .
Multiplicative persistence
The number of iterations needed for to reach a fixed point is the multiplicative persistence of . The multiplicative persistence is undefined if it never reaches a fixed point.
In base 10, it is conjectured that there is no number with a multiplicative persistence : this is known to be true for numbers . The smallest numbers with persistence 0, 1, ... are:
0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899.
The search for these numbers can be sped up by using additional properties of the decimal digits of these record-breaking numbers. These digits must be sorted, and, except for the first two digits, all digits must be 7, 8, or 9. There are also additional restrictions on the first two digits.
Based on these restrictions, the number of candidates for -digit numbers with record-breaking persistence is only proportional to the square of , a tiny fraction of all possible -digit numbers. However, any number that is missing from the sequence above would have multiplicative persistence > 11; such numbers are believed not to exist, and would need to have over 20,000 digits if they do exist.
Extension to negative integers
The multiplicative digital root can be extended to the negative integers by use of a signed-digit representation to represent each integer.
Programming example
The example below implements the digit product described in the definition above to search for multiplicative digital roots and multiplicative persistences in Python.
def digit_product(x: int, b: int) -> int:
if x == 0:
return 0
total = 1
while x > 1:
if x % b == 0:
return 0
if x % b > 1:
total = total * (x % b)
x = x // b
return total
def multiplicative_digit |
https://en.wikipedia.org/wiki/N-category%20number | In mathematics, the category number of a mathematician is a humorous construct invented by Dan Freed,
intended to measure the capacity of that mathematician to stomach the use of higher categories. It is defined as the largest number n such that they can think about n-categories for a half hour without getting a splitting headache.
See also
n-category
Erdős number
2-category
Weak n-category
References
Higher category theory |
https://en.wikipedia.org/wiki/Strictly%20convex%20space | In mathematics, a strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space is one for which, given any two distinct points x and y on the unit sphere ∂B (i.e. the boundary of the unit ball B of X), the segment joining x and y meets ∂B only at x and y. Strict convexity is somewhere between an inner product space (all inner product spaces being strictly convex) and a general normed space in terms of structure. It also guarantees the uniqueness of a best approximation to an element in X (strictly convex) out of a convex subspace Y, provided that such an approximation exists.
If the normed space X is complete and satisfies the slightly stronger property of being uniformly convex (which implies strict convexity), then it is also reflexive by Milman–Pettis theorem.
Properties
The following properties are equivalent to strict convexity.
A normed vector space (X, || ||) is strictly convex if and only if x ≠ y and || x || = || y || = 1 together imply that || x + y || < 2.
A normed vector space (X, || ||) is strictly convex if and only if x ≠ y and || x || = || y || = 1 together imply that || αx + (1 − α)y || < 1 for all 0 < α < 1.
A normed vector space (X, || ||) is strictly convex if and only if x ≠ 0 and y ≠ 0 and || x + y || = || x || + || y || together imply that x = cy for some constant c > 0;
A normed vector space (X, || ||) is strictly convex if and only if the modulus of convexity δ for (X, || ||) satisfies δ(2) = 1.
See also
Uniformly convex space
Modulus and characteristic of convexity
References
Convex analysis
Normed spaces |
https://en.wikipedia.org/wiki/Stallings%E2%80%93Zeeman%20theorem | In mathematics, the Stallings–Zeeman theorem is a result in algebraic topology, used in the proof of the Poincaré conjecture for dimension greater than or equal to five. It is named after the mathematicians John R. Stallings and Christopher Zeeman.
Statement of the theorem
Let M be a finite simplicial complex of dimension dim(M) = m ≥ 5. Suppose that M has the homotopy type of the m-dimensional sphere Sm and that M is locally piecewise linearly homeomorphic to m-dimensional Euclidean space Rm. Then M is homeomorphic to Sm under a map that is piecewise linear except possibly at a single point x. That is, M \ {x} is piecewise linearly homeomorphic to Rm.
References
Theorems in algebraic topology |
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