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https://en.wikipedia.org/wiki/Lituus%20%28mathematics%29 | The lituus spiral () is a spiral in which the angle is inversely proportional to the square of the radius .
This spiral, which has two branches depending on the sign of , is asymptotic to the axis. Its points of inflexion are at
The curve was named for the ancient Roman lituus by Roger Cotes in a collection of papers entitled Harmonia Mensurarum (1722), which was published six years after his death.
Coordinate representations
Polar coordinates
The representations of the lituus spiral in polar coordinates is given by the equation
where and .
Cartesian coordinates
The lituus spiral with the polar coordinates can be converted to Cartesian coordinates like any other spiral with the relationships and . With this conversion we get the parametric representations of the curve:
These equations can in turn be rearranged to an equation in and :
Divide by :
Solve the equation of the lituus spiral in polar coordinates:
Substitute :
Substitute :
Geometrical properties
Curvature
The curvature of the lituus spiral can be determined using the formula
Arc length
In general, the arc length of the lituus spiral cannot be expressed as a closed-form expression, but the arc length of the lituus spiral can be represented as a formula using the Gaussian hypergeometric function:
where the arc length is measured from .
Tangential angle
The tangential angle of the lituus spiral can be determined using the formula
References
External links
.
Interactive example using JSXGraph.
.
https://hsm.stackexchange.com/a/3181 on the history of the lituus curve.
Spirals
Plane curves |
https://en.wikipedia.org/wiki/Ryo%20Kobayashi | is a former Japanese football player. He is the current first-team coach J2 League club of Thespakusatsu Gunma. His brother is Yoshiyuki Kobayashi.
Club statistics
References
External links
1982 births
Living people
Komazawa University alumni
Association football people from Saitama Prefecture
Japanese actors
Japanese men's footballers
J1 League players
J2 League players
Kashiwa Reysol players
Oita Trinita players
Montedio Yamagata players
Thespakusatsu Gunma players
Men's association football defenders |
https://en.wikipedia.org/wiki/Yuki%20Fukaya | is a former Japanese footballer. He last played for Ehime F.C.
Honors and awards
Oita Trinita
J. League Cup (1) - 2008
Career statistics
Updated to 2 February 2018.
References
External links
1982 births
Living people
Hannan University alumni
Association football people from Aichi Prefecture
Japanese men's footballers
J1 League players
J2 League players
Oita Trinita players
Omiya Ardija players
FC Gifu players
Ehime FC players
Men's association football defenders
People from Okazaki, Aichi |
https://en.wikipedia.org/wiki/Hiroshi%20Ichihara | is a Japanese football player. He plays for Honda Lock.
Club statistics
References
External links
1987 births
Living people
Association football people from Kumamoto Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Oita Trinita players
Sagan Tosu players
Renofa Yamaguchi FC players
Kamatamare Sanuki players
Minebea Mitsumi FC players
Men's association football defenders |
https://en.wikipedia.org/wiki/Daisuke%20Takahashi%20%28footballer%29 | is a former Japanese football player and is the current assistant head coach of J1 League club Cerezo Osaka . His brother is Yutaro Takahashi.
Club statistics
Honors
J.League Cup : 2008
References
External links
Profile at Cerezo Osaka
1983 births
Living people
Fukuoka University alumni
People from Yame, Fukuoka
Association football people from Fukuoka Prefecture
Japanese men's footballers
J1 League players
Oita Trinita players
Cerezo Osaka players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Masaru%20Matsuhashi | is a former Japanese football player who mostly played for Ventforet Kofu.
His older brother Shota is also a professional football player.
Club statistics
Updated to 23 February 2020.
References
External links
Profile at Ventforet Kofu
1985 births
Living people
Waseda University alumni
Association football people from Nagasaki Prefecture
Japanese men's footballers
J1 League players
J2 League players
Oita Trinita players
Ventforet Kofu players
Men's association football defenders |
https://en.wikipedia.org/wiki/Koki%20Kotegawa | is a Japanese football player currently playing for Oita Trinita.
Club Team career statistics
Updated to 7 December 2019.
1Includes Suruga Bank Championship.
References
External links
Profile at Oita Trinita
1989 births
Living people
Association football people from Ōita Prefecture
Japanese men's footballers
J1 League players
J2 League players
Oita Trinita players
Giravanz Kitakyushu players
Men's association football midfielders
Sportspeople from Ōita (city) |
https://en.wikipedia.org/wiki/Yoshiaki%20Fujita | is a Japanese retired footballer of Taiwanese descent.
Club career
Jubilo Iwata
After ten seasons playing for Jubilo Iwata, Fujita retired in December 2020.
Club statistics
Updated to 19 February 2019.
References
External links
Profile at Júbilo Iwata
Profile at Oita Trinita
Fujita's story revealing. In 中文-English version
1983 births
Living people
Juntendo University alumni
Association football people from Tochigi Prefecture
Japanese men's footballers
J1 League players
J2 League players
JEF United Chiba players
Oita Trinita players
Júbilo Iwata players
Men's association football defenders |
https://en.wikipedia.org/wiki/Higher-order%20singular%20value%20decomposition | In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one type of generalization of the matrix singular value decomposition. It has applications in computer vision, computer graphics, machine learning, scientific computing, and signal processing. Some aspects can be traced as far back as F. L. Hitchcock in 1928, but it was L. R. Tucker who developed for third-order tensors the general Tucker decomposition in the 1960s, further advocated by L. De Lathauwer et al. in their Multilinear SVD work that employs the power method, or advocated by Vasilescu and Terzopoulos that developed M-mode SVD a parallel algorithm that employs the matrix SVD.
The term higher order singular value decomposition (HOSVD) was coined be DeLathauwer, but the algorithm referred to commonly in the literature as the HOSVD and attributed to either Tucker or DeLathauwer was developed by Vasilescu and Terzopoulos. Robust and L1-norm-based variants of HOSVD have also been proposed.
Definition
For the purpose of this article, the abstract tensor is assumed to be given in coordinates with respect to some basis as a M-way array, also denoted by , where M is the number of modes and the order of the tensor. is the complex numbers and it includes both the real numbers and the pure imaginary numbers.
Let be a unitary matrix containing a basis of the left singular vectors of the standard mode-m flattening of such that the jth column of corresponds to the jth largest singular value of . Observe that the mode/factor matrix does not depend on the particular on the specific definition of the mode m flattening. By the properties of the multilinear multiplication, we havewhere denotes the conjugate transpose. The second equality is because the 's are unitary matrices. Define now the core tensorThen, the HOSVD of is the decomposition The above construction shows that every tensor has a HOSVD.
Compact HOSVD
As in the case of the compact singular value decomposition of a matrix, it is also possible to consider a compact HOSVD, which is very useful in applications.
Assume that is a matrix with unitary columns containing a basis of the left singular vectors corresponding to the nonzero singular values of the standard factor-m flattening of . Let the columns of be sorted such that the th column of corresponds to the th largest nonzero singular value of . Since the columns of form a basis for the image of , we havewhere the first equality is due to the properties of orthogonal projections (in the Hermitian inner product) and the last equality is due to the properties of multilinear multiplication. As flattenings are bijective maps and the above formula is valid for all , we find as before thatwhere the core tensor is now of size .
Multilinear rank
The multilinear rank of is denoted with rank-. The multilinear rank is a tuple in where . Not all tuples in are multilinea |
https://en.wikipedia.org/wiki/Communications%20in%20Statistics | Communications in Statistics is a peer-reviewed scientific journal that publishes papers related to statistics. It is published by Taylor & Francis in three series, Theory and Methods, Simulation and Computation, and Case Studies, Data Analysis and Applications.
Communications in Statistics – Theory and Methods
This series started publishing in 1970 and publishes papers related to statistical theory and methods. It publishes 20 issues each year. Based on Web of Science, the five most cited papers in the journal are:
Kulldorff M. A spatial scan statistic, 1997, 982 cites.
Holland PW, Welsch RE. Robust regression using iteratively reweighted least-squares, 1977, 526 cites.
Sugiura N. Further analysts of the data by Akaike's information criterion and the finite corrections, 1978, 490 cites.
Hosmer DW, Lemeshow S. Goodness of fit tests for the multiple logistic regression model, 1980, 401 cites.
Iman RL, Conover WJ. Small sample sensitivity analysis techniques for computer models. with an application to risk assessment, 1980, 312 cites.
Abstracting and indexing
Communications in Statistics – Theory and Methods is indexed in the following services:
Current Index to Statistics
Science Citation Index Expanded
Zentralblatt MATH
Communications in Statistics – Simulation and Computation
This series started publishing in 1972 and publishes papers related to computational statistics. It publishes 6 issues each year. Based on Web of Science, the five most cited papers in the journal are:
Iman RL, Conover WJ. A distribution-free approach to inducing rank correlation among input variables, 1982, 519 cites.
Wolfinger R. Covariance structure selection in general mixed models, 1993, 248 cites.
Helland IS, On the structure of partial least squares regression, 1988, 246 cites.
McCulloch JH. Simple consistent estimators of stable distribution parameters, 1986, 191 cites.
Sullivan Pepe M, Anderson GL. A cautionary note on inference for marginal regression models with longitudinal data and general correlated response data, 1994, 162 cites.
Abstracting and indexing
Communications in Statistics – Simulation and Computation is indexed in the following services:
Current Index to Statistics
Science Citation Index Expanded
Zentralblatt MATH
Communications in Statistics: Case Studies, Data Analysis and Applications
This series started publishing in 2015 and publishes case studies and associated data analytic methods in statistics. It publishes 4 online-only issues a year. Based on CrossRef, the three most cited papers in the journal are:
Vandna Jowaheer, Yuvraj Sunecher & Naushad Mamode Khan. A non-stationary BINAR(1) process with negative binomial innovations for modeling the number of goals in the first and second half: The case of Arsenal Football Club, 2016, 2 cites.
Nikolay Kulmatitskiy, Lan Ma Nygren, Kjell Nygren, Jeffrey S. Simonoff & Jing Cao, Survival of Broadway shows: An empirical investigation of recent trends, 2015, 2 cites.
H |
https://en.wikipedia.org/wiki/Ian%20Luder | Ian David Luder (born 13 April 1951) was the 681st Lord Mayor of London, serving from 2008 to 2009.
Biography
Born into a Jewish family as the son of a mathematics teacher, Luder attended The Haberdashers' Aske's Boys' School, Elstree before reading Economics and Economic History at University College London (BA). He then worked as a tax accountant for Arthur Andersen and later Grant Thornton. He regularly comments on tax matters and helped to found the Worshipful Company of Tax Advisers, and is a liveryman of the Coopers' Company. He entered local government as a Labour councillor on Bedford Borough Council, serving for 23 years from 1976 to 1999. Luder also stood for Parliament as the Labour candidate for Yeovil in 1979.
Luder was Aldermanic Sheriff of London for 2007–08 and was elected Lord Mayor on 29 September 2008, taking office in the "Silent Ceremony" on 7 November. He was appointed Commander of the Order of the British Empire (CBE) in the 2010 New Year Honours.
In 2008, Luder and his wife were involved in a dispute with their neighbours over the neighbours' cat. Apparently the Luders had refused their neighbours' request to stop feeding the animal, who was overweight and had a heart condition, and in fact had shut the cat in their home for 36 hours while they were away.
On 28 March 2012, Luder was announced as the new chairman of Basildon and Thurrock University Hospitals NHS Foundation Trust. He took up the post on 1 July that year. Shortly after his selection in January 2015 as a UKIP candidate for the 2015 general election, Luder stood down as Trust chairman.
Politics
In December 2014, Luder was one of five people on the shortlist to become United Kingdom Independence Party (UKIP) candidate for the constituency of South Basildon and East Thurrock at the 2015 general election. At the initial selection meeting he was not chosen as the candidate. Shortly afterwards, the successful candidate, Kerry Smith, resigned as UKIP's nominee for the seat after he was recorded making offensive remarks about fellow party members in a telephone conversation. A new selection was held in January 2015, which Luder won. Luder contested the general election and came second, polling 12,097 votes (26.5% of the total), 7,692 votes behind the incumbent Conservative candidate, Stephen Metcalfe. Smith polled 401 votes and finished in fifth place.
References
External links
Debrett's People of Today
1951 births
Living people
British Jews
Jewish British politicians
People educated at Haberdashers' Boys' School
Alumni of University College London
21st-century lord mayors of London
20th-century British politicians
21st-century British politicians
Sheriffs of the City of London
Commanders of the Order of the British Empire
Labour Party (UK) parliamentary candidates
Councillors in Bedfordshire
Labour Party (UK) councillors
UK Independence Party parliamentary candidates
Masters of the Worshipful Company of Arts Scholars |
https://en.wikipedia.org/wiki/Ba%C4%9Ftala | Bağtala is a village in the municipality of Uzuntala in the Qakh Rayon of Azerbaijan. According to Azerbaijan's State Statistics Committee, only nine people lived in the village as of 2014.
References
Populated places in Qakh District |
https://en.wikipedia.org/wiki/Hiroyuki%20Komoto | is a Japanese football player who plays for Omiya Ardija.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Omiya Ardija
1985 births
Living people
Association football people from Kyoto Prefecture
Japanese men's footballers
J1 League players
J2 League players
Vissel Kobe players
Omiya Ardija players
Men's association football defenders |
https://en.wikipedia.org/wiki/Hideo%20Tanaka%20%28footballer%29 | is a Japanese professional footballer who plays as a midfielder for Japan Football League club Tiamo Hirakata.
Career statistics
Updated to 8 March 2018.
References
External links
Profile at Vissel Kobe
1983 births
Living people
National Institute of Fitness and Sports in Kanoya alumni
People from Uki, Kumamoto
Association football people from Kumamoto Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Vissel Kobe players
Kyoto Sanga FC players
Tegevajaro Miyazaki players
Kamatamare Sanuki players
FC Tiamo Hirakata players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kenji%20Baba | is a retired Japanese footballer.
Club statistics
Updated to 25 February 2019.
References
External links
Profile at Oita Trinita
Profile at Kamatamare Sanuki
"Kenji Baba - Player Profile - Football" - Eurosport Australia
1985 births
Living people
Kindai University alumni
Association football people from Kanagawa Prefecture
People from Hiratsuka, Kanagawa
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Vissel Kobe players
Shonan Bellmare players
Mito HollyHock players
Kamatamare Sanuki players
Oita Trinita players
FC Gifu players
Kagoshima United FC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Masatoshi%20Mihara | is a Japanese football player currently playing for J2 League side Kashiwa Reysol.
Club statistics
Updated to 19 February 2019.
References
External links
Profile at Vissel Kobe
1988 births
Living people
Japanese men's footballers
J1 League players
J2 League players
Sagan Tosu players
Vissel Kobe players
Zweigen Kanazawa players
V-Varen Nagasaki players
Kashiwa Reysol players
Men's association football midfielders
Association football people from Kumamoto |
https://en.wikipedia.org/wiki/Y%C5%8Dsuke%20Ishibitsu | is a former Japanese footballer.
Career statistics
Updated to end of 2020 season.
References
External links
Profile at Kyoto Sanga
1983 births
Living people
Osaka Gakuin University alumni
Association football people from Osaka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Vissel Kobe players
Nagoya Grampus players
Kyoto Sanga FC players
Men's association football defenders
People from Settsu, Osaka |
https://en.wikipedia.org/wiki/Ryosuke%20Matsuoka | is a Japanese football player who plays for Berkeley Goats FC.
Club statistics
Updated to 1 January 2020.
References
External links
Profile at Montedio Yamagata
1984 births
Living people
Hannan University alumni
Association football people from Hyōgo Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Vissel Kobe players
Júbilo Iwata players
Montedio Yamagata players
Fujieda MYFC players
Men's association football midfielders
Sportspeople from Nishinomiya |
https://en.wikipedia.org/wiki/Hiroki%20Kishida | is a former Japanese football player.
Club statistics
References
External links
1981 births
Living people
Kwansei Gakuin University alumni
Association football people from Hyōgo Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Vissel Kobe players
Kataller Toyama players
Fagiano Okayama players
Men's association football forwards |
https://en.wikipedia.org/wiki/Shinichi%20Terada | is a Japanese football player who plays for Ococias Kyoto AC.
Career
On 6 January 2020, Terada joined Ococias Kyoto AC.
Club statistics
Updated to 23 February 2018.
Team honours
AFC Champions League - 2008
J1 League - 2005
Emperor's Cup - 2008
J.League Cup - 2007
References
External links
Profile at Tochigi SC
1985 births
Living people
Association football people from Osaka Prefecture
People from Ibaraki, Osaka
Japanese men's footballers
J1 League players
J2 League players
Gamba Osaka players
Yokohama FC players
Tochigi SC players
Ococias Kyoto AC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Hayato%20Sasaki | is a Japanese football player currently playing for Tochigi SC and currently assistant managers club WE League of MyNavi Sendai.
Club career statistics
Updated to 23 February 2016.
FIFA Club World Cup career statistics
Team honors
AFC Champions League - 2008
Pan-Pacific Championship - 2008
Emperor's Cup - 2008, 2009
References
External links
Profile at Tochigi SC
1982 births
Living people
Osaka Gakuin University alumni
People from Shiogama, Miyagi
Association football people from Miyagi Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Montedio Yamagata players
Gamba Osaka players
Vegalta Sendai players
Kyoto Sanga FC players
Tochigi SC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Takuya%20Takei | is a Japanese football player currently playing for Matsumoto Yamaga FC.
Career statistics
Updated to 23 February 2017.
FIFA Club World Cup Career Statistics
Team honors
Gamba Osaka
AFC Champions League - 2008
Pan-Pacific Championship - 2008
Emperor's Cup - 2008, 2009
J2 League - 2013
References
External links
Profile at Matsumoto Yamaga
1986 births
Living people
Ryutsu Keizai University alumni
Association football people from Tochigi Prefecture
Japanese men's footballers
J1 League players
J2 League players
Gamba Osaka players
Vegalta Sendai players
Matsumoto Yamaga FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Hideya%20Okamoto | is a Japanese footballer currently playing for Tiamo Hirakata.
Club career statistics
Updated to 23 February 2018.
Honours
Gamba Osaka
AFC Champions League (1) : 2008
Kashima Antlers
J. League Cup (1) : 2012
Suruga Bank Championship (1) : 2012
References
External links
Profile at Renofa Yamaguchi
1987 births
Living people
Sportspeople from Sakai, Osaka
Association football people from Osaka Prefecture
Japanese men's footballers
Japan men's youth international footballers
J1 League players
J2 League players
J3 League players
Gamba Osaka players
Avispa Fukuoka players
Kashima Antlers players
Albirex Niigata players
Oita Trinita players
Fagiano Okayama players
Renofa Yamaguchi FC players
AC Nagano Parceiro players
Men's association football forwards |
https://en.wikipedia.org/wiki/Kodai%20Yasuda | is a Japanese football player, currently playing for Ehime FC in the J2 League.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Ehime FC
1989 births
Living people
Association football people from Osaka Prefecture
People from Suita
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Gamba Osaka players
Giravanz Kitakyushu players
Tokyo Verdy players
Gainare Tottori players
Ehime FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Friedhelm%20Waldhausen | Friedhelm Waldhausen (born 1938 in Millich, Hückelhoven, Rhine Province) is a German mathematician known for his work in algebraic topology. He made fundamental contributions in the fields of 3-manifolds and (algebraic) K-theory.
Career
Waldhausen studied mathematics at the universities of Göttingen, Munich and Bonn. He obtained his Ph.D. in 1966 from the University of Bonn; his advisor was Friedrich Hirzebruch and his thesis was entitled "Eine Klasse von 3-dimensionalen Mannigfaltigkeiten" (A class of 3-dimensional manifolds).
After visits to Princeton University, the University of Illinois and the University of Michigan he moved in 1968 to the University of Kiel, where he completed his habilitation (qualified to assume a professorship).
In 1969, he was appointed professor at the Ruhr University Bochum before in 1971 becoming a professor at Bielefeld University, an appointment he held until his retirement in 2004.
Academic work
His early work was mainly on the theory of 3-manifolds. He dealt mainly with Haken manifolds and Heegaard splitting. Among other things, he proved that, roughly speaking, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism, i.e. that closed Haken manifolds are topologically rigid. He put forward the Waldhausen conjecture about Heegaard splitting.
In the mid-seventies, he extended the connection between geometric topology and algebraic K-theory by introducing A-theory, a kind of algebraic K-theory for topological spaces. This led to new foundations for algebraic K-theory (using what are now called Waldhausen categories) and also gave new impetus to the study of highly structured ring spectra.
Recognition
Today, Waldhausen is seen, together with Daniel Quillen, as one of the pioneers of algebraic K-theory. Among others, he was awarded the von Staudt Prize in 2004 along with Günter Harder, and an honorary doctorate from the Universität Osnabrück.
Important publications
Algebraic -theory of spaces, Algebraic and
geometric topology (New Brunswick, N.J., 1983), 318–419, Lecture Notes
in Math., 1126, Springer, Berlin, 1985.
Algebraic -theory of spaces, concordance,
and stable homotopy theory, Algebraic topology and algebraic -theory
(Princeton, N.J., 1983), 392–417, Ann. of Math. Stud., 113, Princeton
Univ. Press, Princeton, NJ, 1987.
(with Marcel Bökstedt) The map ,
Algebraic topology and algebraic -theory (Princeton, N.J., 1983),
418–431, Ann. of Math. Stud., 113, Princeton Univ. Press, Princeton,
NJ, 1987.
See also
Graph manifold
Loop theorem
K-theory of a category
Smith conjecture
Surface subgroup conjecture
Virtually Haken conjecture
History of knot theory
Waldhausen category
Waldhausen S-construction
References
External links
at Bielefeld University
Profile at Zentralblatt MATH:
1938 births
Living people
People from Heinsberg (district)
People from the Rhine Province
20th-century German mathematicians
University of Michigan staff
Topologists
Academic staff of Bielefeld Univer |
https://en.wikipedia.org/wiki/Volcano%20plot | Volcano plot may refer to:
Sabatier principle - a concept in chemical catalysis that relates the optimal concentrations of catalysts and substrates
Volcano plot (statistics) - a type of graph used to relate fold-change to p-value that is commonly used in genomics and other omic experiments involving thousands of data-points |
https://en.wikipedia.org/wiki/Sean%20Whyte%20%28ice%20hockey%29 | Sean Whyte (born May 4, 1970) is a Canadian former professional ice hockey player who briefly played for the Los Angeles Kings in the NHL. Whyte was born in Sudbury, Ontario.
Career statistics
External links
1970 births
Anaheim Bullfrogs players
Canadian ice hockey right wingers
Cornwall Aces players
El Paso Buzzards players
Fort Worth Fire players
Guelph Platers players
Ice hockey people from Greater Sudbury
Living people
Los Angeles Kings draft picks
Los Angeles Kings players
Owen Sound Platers players
Phoenix Cobras players
Phoenix Mustangs players
Phoenix Roadrunners (IHL) players
Tulsa Oilers (1992–present) players
Worcester IceCats players |
https://en.wikipedia.org/wiki/Faithfully%20flat | Faithfully flat may refer to:
Faithfully flat morphism, in the theory of schemes in algebraic geometry
Faithfully flat module, for sequences in algebra |
https://en.wikipedia.org/wiki/Fourier%20sine%20and%20cosine%20series | In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier.
Notation
In this article, denotes a real-valued function on which is periodic with period 2L.
Sine series
If is an odd function with period , then the Fourier Half Range sine series of f is defined to be
which is just a form of complete Fourier series with the only difference that and are zero, and the series is defined for half of the interval.
In the formula we have
Cosine series
If is an even function with a period , then the Fourier cosine series is defined to be
where
Remarks
This notion can be generalized to functions which are not even or odd, but then the above formulas will look different.
See also
Fourier series
Fourier analysis
Least-squares spectral analysis
Bibliography
Fourier series |
https://en.wikipedia.org/wiki/Wrapped%20normal%20distribution | In probability theory and directional statistics, a wrapped normal distribution is a wrapped probability distribution that results from the "wrapping" of the normal distribution around the unit circle. It finds application in the theory of Brownian motion and is a solution to the heat equation for periodic boundary conditions. It is closely approximated by the von Mises distribution, which, due to its mathematical simplicity and tractability, is the most commonly used distribution in directional statistics.
Definition
The probability density function of the wrapped normal distribution is
where μ and σ are the mean and standard deviation of the unwrapped distribution, respectively. Expressing the above density function in terms of the characteristic function of the normal distribution yields:
where is the Jacobi theta function, given by
and
The wrapped normal distribution may also be expressed in terms of the Jacobi triple product:
where and
Moments
In terms of the circular variable the circular moments of the wrapped normal distribution are the characteristic function of the normal distribution evaluated at integer arguments:
where is some interval of length . The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:
The mean angle is
and the length of the mean resultant is
The circular standard deviation, which is a useful measure of dispersion for the wrapped normal distribution and its close relative, the von Mises distribution is given by:
Estimation of parameters
A series of N measurements zn = e iθn drawn from a wrapped normal distribution may be used to estimate certain parameters of the distribution. The average of the series is defined as
and its expectation value will be just the first moment:
In other words, is an unbiased estimator of the first moment. If we assume that the mean μ lies in the interval [−π, π), then Arg will be a (biased) estimator of the mean μ.
Viewing the zn as a set of vectors in the complex plane, the 2 statistic is the square of the length of the averaged vector:
and its expected value is:
In other words, the statistic
will be an unbiased estimator of e−σ2, and ln(1/Re2) will be a (biased) estimator of σ2
Entropy
The information entropy of the wrapped normal distribution is defined as:
where is any interval of length . Defining and , the Jacobi triple product representation for the wrapped normal is:
where is the Euler function. The logarithm of the density of the wrapped normal distribution may be written:
Using the series expansion for the logarithm:
the logarithmic sums may be written as:
so that the logarithm of density of the wrapped normal distribution may be written as:
which is essentially a Fourier series in . Using the characteristic function representation for the wrapped normal distribution in the left side of the integral:
the entropy may be written:
which may be integrated to yield:
See also
Wrapped d |
https://en.wikipedia.org/wiki/Omid%20Khouraj | Omid Khouraj (, born September 20, 1982) is an Iranian footballer who plays for Pas Hamedan F.C. in the IPL.
Club career statistics
Last Update: 3 August 2011
Assist Goals
Honours
Iran's Premier Football League Winner: 1
2003/04 with Pas Tehran
External links
Profile at Persianleague.com
1981 births
Living people
PAS Tehran F.C. players
PAS Hamedan F.C. players
Foolad F.C. players
Persian Gulf Pro League players
Iranian men's footballers
Men's association football defenders
Footballers from Tehran |
https://en.wikipedia.org/wiki/Mehrdad%20Karimian | Mehrdad Karimian (, born September 20, 1982) is a retired Iranian footballer and current coach. Mehrdad is the younger brother of Mehdi Karimian, a former player.
Club career
Club Career Statistics
Last Update 10 May 2013
Assist Goals
External links
Persian League Profile
https://www.youtube.com/watch?v=ydACiq1Y_Ks
1983 births
Living people
Fajr Sepasi Shiraz F.C. players
Bargh Shiraz F.C. players
PAS Hamedan F.C. players
Sanat Mes Kerman F.C. players
Persian Gulf Pro League players
Azadegan League players
Iranian men's footballers
Men's association football midfielders
People from Bushehr
Shahin Bushehr F.C. managers |
https://en.wikipedia.org/wiki/Javad%20Shirzad | Javad Shirzad (, born September 20, 1982) is an Iranian football player. He plays for Malavan. He usually plays in the defender position.
Club career
Club Career Statistics
Last Update 30 April 2013
Assist Goals
Honours
Club
Iran's Premier Football League
Runner up: 1
2010/11 with Esteghlal
Hazfi Cup
Winner: 1
2011/12 with Esteghlal
External links
IPLStats.com
Iranian men's footballers
Iran men's international footballers
Persian Gulf Pro League players
Esteghlal F.C. players
Malavan F.C. players
Foolad F.C. players
PAS Tehran F.C. players
Footballers from Bandar-e Anzali
1982 births
Living people
Men's association football fullbacks |
https://en.wikipedia.org/wiki/Lists%20of%20statistics%20topics | This article itemizes the various lists of statistics topics.
Statistics
Outline of statistics
Outline of regression analysis
Index of statistics articles
List of scientific method topics
List of analyses of categorical data
List of fields of application of statistics
List of graphical methods
List of statistical software
Comparison of statistical packages
List of graphing software
Comparison of Gaussian process software
List of stochastic processes topics
List of matrices used in statistics
Timeline of probability and statistics
List of unsolved problems in statistics
Probability
Topic outline of probability
List of probability topics
Catalog of articles in probability theory
List of probability distributions
List of convolutions of probability distributions
Glossaries and notations
Glossary of experimental design
Glossary of probability and statistics
Notation in probability and statistics
People
List of actuaries
List of statisticians
List of mathematical probabilists
Founders of statistics
Publications
List of important publications in statistics
List of scientific journals in probability
List of scientific journals in statistics
Comparison of statistics journals
Organizations
List of academic statistical associations
List of national and international statistical services
See also
Lists of mathematics topics |
https://en.wikipedia.org/wiki/SPARQL%20Syntax%20Expressions | SPARQL Syntax Expressions (alternatively, SPARQL S-Expressions) is a parse tree (a.k.a. concrete syntax) for representing SPARQL Algebra expressions.
Application
They have been used to apply the BERT language model to create SPARQL queries from natural language questions.
External links
SPARQL Algebra in the W3C SPARQL Query Specification
SPARQL Syntax Expressions in the ARQ query engine
SPARQL Validator that can also print the Algebra expressions
SPARQL Syntax Expressions translations of the DAWG test suite
References
SPARQL
RDF data access |
https://en.wikipedia.org/wiki/Frank%20Morgan%20%28mathematician%29 | Frank Morgan is an American mathematician and the Webster Atwell '21 Professor of Mathematics, Emeritus, at Williams College. He is known for contributions to geometric measure theory, minimal surfaces, and differential geometry, including the resolution of the double bubble conjecture. He was vice-president of the American Mathematical Society and the Mathematical Association of America.
Morgan studied at the Massachusetts Institute of Technology and Princeton University, and received his Ph.D. from Princeton in 1977, under the supervision of Frederick J. Almgren Jr. He taught at MIT for ten years before joining the Williams faculty.
Morgan is the founder of SMALL, one of the largest and best known summer undergraduate Mathematics research programs. In 2012 he became a fellow of the American Mathematical Society.
Frank Morgan is also an avid dancer. He gained eternal fame for his work "Dancing the Parkway".
Mathematical work
He is known for proving, in collaboration with Michael Hutchings, Manuel Ritoré, and Antonio Ros, the Double Bubble conjecture, which states that the minimum-surface-area enclosure of two given volumes is formed by three spherical patches meeting at 120-degree angles at a common circle.
He has also made contributions to the study of manifolds with density, which are Riemannian manifolds together with a measure of volume which is deformed from the standard Riemannian volume form. Such deformed volume measures suggest modifications of the Ricci curvature of the Riemannian manifold, as introduced by Dominique Bakry and Michel Émery. Morgan showed how to modify the classical Heintze-Karcher inequality, which controls the volume of certain cylindrical regions in the space by the Ricci curvature in the region and the mean curvature of the region's cross-section, to hold in the setting of manifolds with density. As a corollary, he was also able to put the Levy-Gromov isoperimetric inequality into this setting. Much of his current work deals with various aspects of isoperimetric inequalities and manifolds with density.
Publications
Textbooks
Calculus Lite. Third edition. A K Peters/CRC Press, Natick, MA, 2001.
Geometric measure theory. A beginner's guide. Fifth edition. Illustrated by James F. Bredt. Elsevier/Academic Press, Amsterdam, 2016. viii+263 pp.
The math chat book. MAA Spectrum. Mathematical Association of America, Washington, DC, 2000. xiv+113 pp.
Real analysis. American Mathematical Society, Providence, RI, 2005. viii+151 pp.
Real analysis and applications. Including Fourier series and the calculus of variations. American Mathematical Society, Providence, RI, 2005. x+197 pp.
Riemannian geometry. A beginner's guide. Second edition. A K Peters, Ltd., Wellesley, MA, 1998. x+156 pp.
Notable articles
Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros. Proof of the double bubble conjecture. Ann. of Math. (2) 155 (2002), no. 2, 459–489. doi:10.2307/3062123
Frank Morgan. Manifolds with density. Notic |
https://en.wikipedia.org/wiki/Colorable | Colorable or colourable may refer to:
Graph coloring in Mathematics
In law, that a legal burden of proof would be met at trial |
https://en.wikipedia.org/wiki/Sebasti%C3%A1n%20Arrieta | Sebastián Alejandro Arrieta (born 21 October 1985 in Añatuya, Santiago del Estero Province) is an Argentine footballer currently playing for Instituto.
External links
Argentine Primera statistics }
Statistics at BDFA
1985 births
Living people
Sportspeople from Santiago del Estero Province
Argentine people of Basque descent
Argentine men's footballers
Men's association football midfielders
Newell's Old Boys footballers
Racing Club de Avellaneda footballers
Atlético de Rafaela footballers
Instituto Atlético Central Córdoba footballers
Unión de Santa Fe footballers
Argentine Primera División players |
https://en.wikipedia.org/wiki/Hiroki%20Nakayama | is a former Japanese football player.
Club statistics
References
External links
1985 births
Living people
Association football people from Kagoshima Prefecture
Japanese men's footballers
J1 League players
J2 League players
Kyoto Sanga FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Jun%20Ando | is a retired Japanese footballer who last played for Kyoto Sanga FC.
Career
After more than a decade in J. League, he opted to retire at the end of the 2020 season.
Career statistics
Updated to January 1st, 2022.
References
External links
Profile at Ehime FC
Profile at Matsumoto Yamaga
1984 births
Living people
Kansai University alumni
Association football people from Shiga Prefecture
Japanese men's footballers
J1 League players
J2 League players
Kyoto Sanga FC players
Cerezo Osaka players
Matsumoto Yamaga FC players
Ehime FC players
Men's association football defenders |
https://en.wikipedia.org/wiki/Morass%20%28set%20theory%29 | In axiomatic set theory, a mathematical discipline, a morass is an infinite combinatorial structure, used to create "large" structures from a "small" number of "small" approximations. They were invented by Ronald Jensen for his proof that cardinal transfer theorems hold under the axiom of constructibility. A far less complex but equivalent variant known as a simplified morass was introduced by Velleman, and the term morass is now often used to mean these simpler structures.
Overview
Whilst it is possible to define so-called gap-n morasses for n > 1, they are so complex that focus is usually restricted to the gap-1 case, except for specific applications. The "gap" is essentially the cardinal difference between the size of the "small approximations" used and the size of the ultimate structure.
A (gap-1) morass on an uncountable regular cardinal κ (also called a (κ,1)-morass) consists of a tree of height κ + 1, with the top level having κ+-many nodes. The nodes are taken to be ordinals, and functions between these ordinals are associated to the edges in the tree order. It is required that the ordinal structure of the top level nodes be "built up" as the direct limit of the ordinals in the branch to that node by the maps π, so the lower level nodes can be thought of as approximations to the (larger) top level node.
A long list of further axioms is imposed to have this happen in a particularly "nice" way.
Variants and equivalents
Velleman and Shelah and Stanley independently developed forcing axioms equivalent to the existence of morasses, to facilitate their use by non-experts. Going further, Velleman showed that the existence of morasses is equivalent to simplified morasses, which are vastly simpler structures. However, the only known construction of a simplified morass in Gödel's constructible universe is by means of morasses, so the original notion retains interest.
Other variants on morasses, generally with added structure, have also appeared over the years. These include universal morasses, whereby every subset of κ is built up through the branches of the morass, mangroves, which are morasses stratified into levels (mangals) at which every branch must have a node, and quagmires.
Simplified morass
Velleman defined gap-1 simplified morasses which are much simpler than gap-1 morasses, and showed that the existence of gap-1 morasses is equivalent to the existence of gap-1 simplified morasses.
Roughly speaking: a (κ,1)-simplified morass M = < φ→, F⇒ > contains a sequence φ→ = < φβ : β ≤ κ > of ordinals such that φβ < κ for β < κ and φκ = κ+, and a double sequence F⇒ = < Fα,β : α < β ≤ κ > where Fα,β are collections of monotone mappings from φα to φβ for α < β ≤ κ with specific (easy but important) conditions.
Velleman's clear definition can be found in, where he also constructed (ω0,1) simplified morasses in ZFC. In he gave similar simple definitions for gap-2 simplified morasses, and in he constructed (ω0,2) simplified morass |
https://en.wikipedia.org/wiki/Graph%20canonization | In graph theory, a branch of mathematics, graph canonization is the problem of finding a canonical form of a given graph G. A canonical form is a labeled graph Canon(G) that is isomorphic to G, such that every graph that is isomorphic to G has the same canonical form as G. Thus, from a solution to the graph canonization problem, one could also solve the problem of graph isomorphism: to test whether two graphs G and H are isomorphic, compute their canonical forms Canon(G) and Canon(H), and test whether these two canonical forms are identical.
The canonical form of a graph is an example of a complete graph invariant: every two isomorphic graphs have the same canonical form, and every two non-isomorphic graphs have different canonical forms. Conversely, every complete invariant of graphs may be used to construct a canonical form. The vertex set of an n-vertex graph may be identified with the integers from 1 to n, and using such an identification a canonical form of a graph may also be described as a permutation of its vertices. Canonical forms of a graph are also called canonical labelings, and graph canonization is also sometimes known as graph canonicalization.
Computational complexity
Clearly, the graph canonization problem is at least as computationally hard as the graph isomorphism problem. In fact, graph isomorphism is even AC0-reducible to graph canonization. However, it is still an open question whether the two problems are polynomial time equivalent.
While the existence of (deterministic) polynomial algorithms for graph isomorphism is still an open problem in computational complexity theory, in 1977 László Babai reported that with probability at least 1 − exp(−O(n)), a simple vertex classification algorithm produces a canonical labeling of a graph chosen uniformly at random from the set of all n-vertex graphs after only two refinement steps. Small modifications and an added depth-first search step produce canonical labeling of such uniformly-chosen random graphs in linear expected time. This result sheds some light on the fact why many reported graph isomorphism algorithms behave well in practice. This was an important breakthrough in probabilistic complexity theory which became widely known in its manuscript form and which was still cited as an "unpublished manuscript" long after it was reported at a symposium.
A commonly known canonical form is the lexicographically smallest graph within the isomorphism class, which is the graph of the class with lexicographically smallest adjacency matrix considered as a linear string.
However, the computation of the lexicographically smallest graph is NP-hard.
For trees, a concise polynomial canonization algorithm requiring O(n) space is presented by . Begin by labeling each vertex with the string 01. Iteratively for each non-leaf x remove the leading 0 and trailing 1 from x's label; then sort x's label along with the labels of all adjacent leaves in lexicographic order. Concatenate these sorted |
https://en.wikipedia.org/wiki/Dan%20Popescu | Daniel Popescu (born 20 February 1988) is a Romanian footballer who plays as a left back for Oțelul Galați.
Career statistics
Club
Honours
Club
Steaua București
League Cup: 2015–16
References
External links
1988 births
Living people
People from Tulcea
Romanian men's footballers
Men's association football defenders
Liga I players
Liga II players
FCM Dunărea Galați players
ASC Oțelul Galați players
ACS Poli Timișoara players
FC Steaua București players
CS Concordia Chiajna players
FCSB II players
ASC Daco-Getica București players
CSM Slatina (football) players
Sportspeople from Tulcea County |
https://en.wikipedia.org/wiki/Luk%C3%A1%C5%A1%20D%C5%BEogan | Lukáš Džogan (pronounced Djogan) (born 1 January 1987) is a professional Slovak football defender who currently plays for FK TATRA Sokoľany.
Career statistics
Last updated: 21 May 2010
External links
Player profile at official club website
http://www.goal.com/en-ng/people/slovakia/33823/lukas-dzogan
1987 births
Living people
Footballers from Košice
Men's association football central defenders
Slovak men's footballers
FC Steel Trans Ličartovce players
FC Lokomotíva Košice players
FC VSS Košice players
Slovak First Football League players |
https://en.wikipedia.org/wiki/Landsberg%E2%80%93Schaar%20relation | In number theory and harmonic analysis, the Landsberg–Schaar relation (or identity) is the following equation, which is valid for arbitrary positive integers p and q:
The standard way to prove it is to put = + ε, where ε > 0 in this identity due to Jacobi (which is essentially just a special case of the Poisson summation formula in classical harmonic analysis):
and then let ε → 0.
A proof using only finite methods was discovered in 2018 by Ben Moore.
If we let q = 1, the identity reduces to a formula for the quadratic Gauss sum modulo p.
The Landsberg–Schaar identity can be rephrased more symmetrically as
provided that we add the hypothesis that pq is an even number.
References
Theorems in analytic number theory |
https://en.wikipedia.org/wiki/Newlands%20Labor%20Act | The Newlands Labor Act, was a 1913 United States federal law, sponsored by Senator Francis G. Newlands of Nevada and drafted by Bureau of Labor Statistics Commissioner Charles Patrick Neill. It created the Board of Mediation and Conciliation (BMC). The BMC was a precursor to today's National Mediation Board (NMB).
Background
In response to railroad strikes during the 1870s and 1880s, Congress passed the Arbitration Act of 1888, which authorized the creation of arbitration panels with the power to investigate the causes of labor disputes and to issue non-binding arbitration awards. The Act was a complete failure: only one panel was ever convened under the Act, and that one, in the case of the 1894 Pullman Strike, issued its report only after the strike had been crushed by a federal court injunction backed by federal troops.
Congress attempted to correct these shortcomings in the Erdman Act, passed in 1898. This law likewise provided for voluntary arbitration, but made any award issued by the panel binding and enforceable in federal court. It also outlawed discrimination against employees for union activities, prohibited "yellow dog" contracts (employee agrees not to join a union while employed), and required both sides to maintain the status quo during any arbitration proceedings and for three months after an award was issued. The arbitration procedures were rarely used.
Newlands Act and the BMC
President Woodrow Wilson signed the Newlands Act on July 15, 1913. The law created the Board of Mediation and Conciliation, which was administered by U.S. Commerce and District Court Judge Martin Augustine Knapp and assisted by U.S. Alabama District Court Judge and Commissioner William Lea Chambers. The Board adjusted and arbitrated disputes between railroad companies and their operating employees, where those disputes threatened to interrupt operation of the carriers to the "serious detriment of the public interest." Voluntary arbitration was also provided for those disputes that could not be settled by mediation.
The Board was functionally replaced on December 26, 1917, by the creation of the United States Railroad Administration, although it continued to exist with its activities restricted to short-line railroads. In 1920, the Esch–Cummins Act (formally called the Transportation Act) created a new Railroad Labor Board which regulated wages and settled disputes, and permanently replaced the duties of the BMC.
On May 20, 1926, Congress repealed the Newlands Labor Act and Title III of the Esch–Cummins Act (which pertained to labor disputes), with the enactment of the Railway Labor Act.
See also
History of rail transport in the United States
References
1913 in American law
1913 in rail transport
United States labor law
United States railroad regulation
United States federal transportation legislation
United States federal labor legislation |
https://en.wikipedia.org/wiki/Asian%20Club%20Championship%20and%20AFC%20Champions%20League%20records%20and%20statistics | This page details statistics of the Asian Club Championship and AFC Champions League.
General performances
Asian Club Championship and AFC Champions League
Titles by club
A total of 24 clubs have won the tournament since its 1967 inception, with Al-Hilal being the only team to win it four times. Clubs from ten countries have provided tournament winners. South Korean clubs have been the most successful, winning a total of twelve titles.
Titles by nation
Titles by city
AFC Champions League era
Titles by club
Titles by nation
Titles by city
Statistics
All-time top 25 AFC Champions League rankings
This table includes results beyond group stage of the AFC Champions League through 2002/03 season, therefore:
It does not include the old Asian Club Championship
It does not include qualifying rounds
All-time table by leagues
This table includes results beyond group stage of the AFC Champions League through 2002/03 season (2002–03 AFC Champions League); qualifying rounds are not included.
Number of participating clubs of the Champions League era (from 2002–2024)
The following table is a list of clubs that have participated in the AFC Champions League (group stage).
Year(s) in Bold : Team advanced to the knockout stage.
Attendance record
Esteghlal holds the record for the most attendance in Asia.Out of the 20 most watched games in the history of Asian club competitions, 10 most watched games belong to Esteghlal, 9 most watched games belong to Persepolis and 1 most watched game belongs to Tractor.
Clubs
Performance review (from 2002–03)
By semi-final appearances
Asian Club Championship and AFC Champions League
The following table is a list of clubs that have participated in the Asian Club Championship and AFC Champions League. Excluding semifinalists from 1987 to 1989–90 seasons. In these seasons, there were no semi-finals as the finalists qualified via a group stage.
Year(s) in Bold: Team was finalist
AFC Champions League era
Year(s) in Bold: Team was finalist
Unbeaten sides
Four sides have been undefeated in multiple seasons:
Al-Hilal (1991–92 and 1999–2000)
Esteghlal (1970 and 1990–91)
Maccabi Tel Aviv (1969 and 1971)
Ulsan Hyundai: (2012 and 2020)
Ten other teams have been undefeated in a single season:
Al-Ittihad (2005)
Daewoo Royals (1985)
Furukawa Electric (1986–87)
Gamba Osaka (2008)
Hapoel Tel Aviv (1967)
Ilhwa Chunma (1995)
Liaoning (1989–90)
Suwon Samsung Bluewings (2001–02)
Thai Farmers Bank (1993–94)
Urawa Red Diamonds (2007)
Consecutive participations
Al-Hilal have the record number of consecutive participations in the AFC Champions League with 12 Times since 2009 .
Biggest wins
The following teams won a single match with goal difference of 8 or more in the AFC Champions League era:
Biggest two-legged wins
The following teams won two-legged matches with goal difference of 5 or more in the knock-out rounds of AFC Champions League era:
Group stage records
Goalscoring and conceding
Most goals score |
https://en.wikipedia.org/wiki/Quantum%20t-design | A quantum t-design is a probability distribution over either pure quantum states or unitary operators which can duplicate properties of the probability distribution over the Haar measure for polynomials of degree t or less. Specifically, the average of any polynomial function of degree t over the design is exactly the same as the average over Haar measure. Here the Haar measure is a uniform probability distribution over all quantum states or over all unitary operators. Quantum t-designs are so called because they are analogous to t-designs in classical statistics, which arose historically in connection with the problem of design of experiments. Two particularly important types of t-designs in quantum mechanics are projective and unitary t-designs.
A spherical design is a collection of points on the unit sphere for which polynomials of bounded degree can be averaged over to obtain the same value that integrating over surface measure on the sphere gives. Spherical and projective t-designs derive their names from the works of Delsarte, Goethals, and Seidel in the late 1970s, but these objects played earlier roles in several branches of mathematics, including numerical integration and number theory. Particular examples of these objects have found uses in quantum information theory, quantum cryptography, and other related fields.
Unitary t-designs are analogous to spherical designs in that they reproduce the entire unitary group via a finite collection of unitary matrices. The theory of unitary 2-designs was developed in 2006 specifically to achieve a practical means of efficient and scalable randomized benchmarking to assess the errors in quantum computing operations, called gates. Since then unitary t-designs have been found useful in other areas of quantum computing and more broadly in quantum information theory and applied to problems as far reaching as the black hole information paradox. Unitary t-designs are especially relevant to randomization tasks in quantum computing since ideal operations are usually represented by unitary operators.
Motivation
In a d-dimensional Hilbert space when averaging over all quantum pure states the natural group is SU(d), the special unitary group of dimension d. The Haar measure is, by definition, the unique group-invariant measure, so it is used to average properties that are not unitarily invariant over all states, or over all unitaries.
A particularly widely used example of this is the spin system. For this system the relevant group is SU(2) which is the group of all 2x2 unitary operators with determinant 1. Since every operator in SU(2) is a rotation of the Bloch sphere, the Haar measure for spin-1/2 particles is invariant under all rotations of the Bloch sphere. This implies that the Haar measure is the rotationally invariant measure on the Bloch sphere, which can be thought of as a constant density distribution over the surface of the sphere.
An important class of complex projective t-designs, are s |
https://en.wikipedia.org/wiki/Julio%20Bevacqua | Julio Maximiliano Bevacqua (born June 9, 1980 in Córdoba) is an Argentine retired footballer. His last club was Delfín SC.
External links
Julio Bevacqua – Argentine Primera statistics at Fútbol XXI
1980 births
Living people
Footballers from Córdoba, Argentina
Argentine men's footballers
Men's association football forwards
San Lorenzo de Almagro footballers
Club Almagro players
Chacarita Juniors footballers
Comisión de Actividades Infantiles footballers
Club Atlético Belgrano footballers
Atlético de Rafaela footballers
S.C. Braga players
Portimonense S.C. players
Panthrakikos F.C. players
S.D. Quito footballers
FC Vaduz players
Argentine expatriate sportspeople in Liechtenstein
Argentine Primera División players
Super League Greece players
Ecuadorian Serie A players
Primeira Liga players
Argentine expatriate men's footballers
Expatriate men's footballers in Portugal
Argentine expatriate sportspeople in Portugal
Expatriate men's footballers in Greece
Expatriate men's footballers in Venezuela
Expatriate men's footballers in Liechtenstein
Expatriate men's footballers in Ecuador |
https://en.wikipedia.org/wiki/Manuel%20de%20Jesus%20Lopes | Manuel de Jesus Lopes (born August 19, 1982) is a Mozambican footballer.
Career
Career statistics
Last update: 27 June 2010
References
External links
Panthraxstats
1982 births
Living people
Footballers from Maputo
Mozambican men's footballers
Mozambican expatriate men's footballers
Mozambique men's international footballers
Men's association football midfielders
PFC Beroe Stara Zagora players
Panetolikos F.C. players
Panthrakikos F.C. players
APOP Kinyras FC players
Expatriate men's footballers in Bulgaria
Expatriate men's footballers in Greece
Expatriate men's footballers in Cyprus
Expatriate men's footballers in Angola
Mozambican expatriate sportspeople in Bulgaria
Mozambican expatriate sportspeople in Greece
Mozambican expatriate sportspeople in Cyprus
Mozambican expatriate sportspeople in Angola
First Professional Football League (Bulgaria) players |
https://en.wikipedia.org/wiki/Vicsek%20fractal | In mathematics the Vicsek fractal, also known as Vicsek snowflake or box fractal, is a fractal arising from a construction similar to that of the Sierpinski carpet, proposed by Tamás Vicsek. It has applications including as compact antennas, particularly in cellular phones.
Box fractal also refers to various iterated fractals created by a square or rectangular grid with various boxes removed or absent and, at each iteration, those present and/or those absent have the previous image scaled down and drawn within them. The Sierpinski triangle may be approximated by a box fractal with one corner removed. The Sierpinski carpet is a box fractal with the middle square removed.
Construction
The basic square is decomposed into nine smaller squares in the 3-by-3 grid. The four squares at the corners and the middle square are left, the other squares being removed. The process is repeated recursively for each of the five remaining subsquares. The Vicsek fractal is the set obtained at the limit of this procedure. The Hausdorff dimension of this fractal is ≈ 1.46497.
An alternative construction (shown below in the left image) is to remove the four corner squares and leave the middle square and the squares above, below, left and right of it. The two constructions produce identical limiting curves, but one is rotated by 45 degrees with respect to the other.
Properties
The Vicsek fractal has the surprising property that it has zero area yet an infinite perimeter, due to its non-integer dimension. At each iteration, four squares are removed for every five retained, meaning that at iteration n the area is (assuming an initial square of side length 1). When n approached infinity, the area approaches zero. The perimeter however is , because each side is divided into three parts and the center one is replaced with three sides, yielding an increase of three to five. The perimeter approaches infinity as n increases.
The boundary of the Vicsek fractal is the Type 1 quadratic Koch curve.
Analogues in higher dimensions
There is a three-dimensional analogue of the Vicsek fractal. It is constructed by subdividing each cube into 27 smaller ones, and removing all but the "center cross", the central cube and the six cubes touching the center of each face. Its Hausdorff dimension is ≈ 1.7712.
Similarly to the two-dimensional Vicsek fractal, this figure has zero volume. Each iteration retains 7 cubes for every 27, resulting in a volume of at iteration n, which approaches zero as n approaches infinity.
There exist an infinite number of cross sections which yield the two-dimensional Vicsek fractal.
See also
Box-counting dimension
Cross crosslet
List of fractals by Hausdorff dimension
Sierpinski carpet
Sierpinski triangle
n-flake
References
External links
Fractals
L-systems |
https://en.wikipedia.org/wiki/List%20of%20OFI%20Crete%20seasons | OFI Crete seasons.
Seasons
Overall seasons table
External links
Rec.Sport.Soccer Statistics Foundation
OFI Crete |
https://en.wikipedia.org/wiki/Demographic%20economics | Demographic economics or population economics is the application of economic analysis to demography, the study of human populations, including size, growth, density, distribution, and vital statistics.
Aspects
Aspects of the subject include:
marriage and fertility
the family
divorce
morbidity and life expectancy/mortality
dependency ratios
migration
population growth
population size
public policy
the demographic transition from "population explosion" to (dynamic) stability or decline.
Other subfields include measuring value of life and the economics of the elderly and the handicapped and of gender, race, minorities, and non-labor discrimination. In coverage and subfields, it complements labor economics and implicates a variety of other economics subjects.
Subareas
The Journal of Economic Literature classification codes are a way of categorizing subjects in economics. There, demographic economics is paired with labour economics as one of 19 primary classifications at JEL: J. It has eight subareas:
General
Demographic Trends and Forecasts
Marriage; Marital Dissolution; Family Structure
Fertility; Family Planning; Child Care; Children; Youth
Economics of the Elderly; Economics of the Handicapped
Economics of Minorities and Races; Non-labor Discrimination
Economics of Gender; Non-labor Discrimination
Value of life; Foregone Income
Public Policy
See also
Cost of raising a child
Family economics
Generational accounting
Growth economics
Retirement age, international comparison
Related:
Income and fertility
Demographic dividend
Demographic transition
Demographic gift
Demographic window
Demographic trap
Preston curve
Development economics
Notes
References
John Eatwell, Murray Milgate, and Peter Newman, ed. ([1987] 1989. Social Economics: The New Palgrave, pp. v-vi. Arrow-page searchable links to entries for:
"Ageing Populations," pp. 1-3, by Robert L. Clark
"Declining Population," pp. 10-15, by Robin Barlow
"Demographic Transition," pp. 16-23, by Ansley J. Coale
"Extended Family," pp. 58-63, by Oliva Harris
"Family," pp. 65-76, by Gary S. Becker
"Fertility," pp.77-89, by Richard A. Easterlin
"Gender," pp. 95-108, by Francine D. Blau
"Race and Economics," pp. 215-218, by H. Stanback
"Value of Life," pp.289-76, by Thomas C. Schelling
Nathan Keyfitz, 1987. "demography," The New Palgrave: A Dictionary of Economics, v. 1, pp. 796–802.
T. Paul Schultz, 1981. Economics of Population. Addison-Wesley. Book review.
John B. Shoven, ed., 2011. Demography and the Economy, University of Chicago Press. Scroll-down description and preview.
Julian L. Simon, 1977. The Economics of Population Growth. Princeton,
_, [1981] 1996. The Ultimate Resource 2, rev. and expanded. Princeton. Description and preview links.
Dennis A. Ahlburg, 1998. "Julian Simon and the Population Growth Debate," Population and Development Review, 24(2), pp. 317-327.
M. Perlman, 1982. [Untitled review of Simon, 1977 & 1981], |
https://en.wikipedia.org/wiki/Freirina | Freirina is a Chilean commune and town in Huasco Province, Atacama Region. The commune spans an area of .
Demographics
According to the 2002 census by the National Statistics Institute, Freirina had 5,666 inhabitants; of these, 3,469 (61.2%) lived in urban areas and 2,107 (38.8%) in rural areas. At that time, there were 2,800 men and 2,866 women. The population grew by 8.5% (445 persons) between the 1992 and 2002 censuses.
Administration
As a commune, Freirina is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2008-2012 alcalde is Roberto Bruzzone Galeb.
Within the electoral divisions of Chile, Freirina is represented in the Chamber of Deputies by Mr. Alberto Robles (PRSD) and Mr. Giovanni Calderón (UDI) as part of the 6th electoral district, (together with Caldera, Tierra Amarilla, Vallenar, Huasco and Alto del Carmen). The commune is represented in the Senate by Isabel Allende Bussi (PS) and Baldo Prokurica Prokurica (RN) as part of the 3rd senatorial constituency (Atacama Region).
Notable people
Nicolasa Montt (1857-1924), poet
References
Communes of Chile
Populated places in Huasco Province
Atacama Desert |
https://en.wikipedia.org/wiki/Alto%20del%20Carmen | Alto del Carmen is a Chilean commune and village in Huasco Province, Atacama Region. The commune spans an area of .
Demographics
According to the 2002 census of the National Statistics Institute, Alto del Carmen had 4,840 inhabitants (2,629 men and 2,211 women), making the commune an entirely rural area. The population grew by 2% (95 persons) between the 1992 and 2002 censuses.
Administration
As a commune, Alto del Carmen is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years.
Within the electoral divisions of Chile, Alto del Carmen is represented in the Chamber of Deputies by Mr. Alberto Robles (PRSD) and Mr. Giovanni Calderón (UDI) as part of the 6th electoral district, (together with Caldera, Tierra Amarilla, Vallenar, Freirina and Huasco). The commune is represented in the Senate by Isabel Allende Bussi (PS) and Baldo Prokurica Prokurica (RN) as part of the 3rd senatorial constituency (Atacama Region).
References
External links
Municipality of Alto del Carmen
Communes of Chile
Populated places in Huasco Province |
https://en.wikipedia.org/wiki/Rocket%20City%20Math%20League | Rocket City Math League (RCML) is a student-run mathematics competition in the United States. Run by students at Virgil I. Grissom High School in Huntsville, Alabama, RCML gets its name from Huntsville's nickname as the "Rocket City". RCML was started in 2001 and has been annually sponsored by the Mu Alpha Theta Math Honor Society. The competition consists of three individual rounds and a team round that was added in 2008. It is divided into five divisions named for NASA programs: Explorer (pre-algebra), Mercury (algebra I), Gemini (geometry), Apollo (algebra II), and Discovery (comprehensive).
Individual rounds
Each of the 3 individual rounds consists of a 10 question test with a 30-minute time limit. Out of the 10 questions, there are four 1-point questions, three 2-point questions, two 3-point questions, and one 4-point question, with the more difficult questions having larger point values. The maximum score on an individual test is 20, and individual tests often contain many interesting space-themed questions.
Team round
The team round is divided into a senior division and a junior division that take separate tests for the team round. It consists of a 15 question test with a 30-minute time limit, in which team members work together to get as many correct answers as possible. Out of the 15 questions, there are five 1-point questions, four 2-point questions, three 3-point questions, two 4-point questions, and one 5-point question, making the maximum score on the team test a 35.
Sources
http://www.rocketcitymath.org
Notes
External links
http://www.mualphatheta.org/Contests/RocketCity.aspx
http://www.artofproblemsolving.com/Wiki/index.php/Rocket_City_Math_League
Culture of Huntsville, Alabama
Mathematics competitions |
https://en.wikipedia.org/wiki/Shinta%20Fukushima | is a Japanese football player. He plays for Verspah Oita.
Club statistics
References
External links
1989 births
Living people
People from Nagakute, Aichi
Association football people from Aichi Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Nagoya Grampus players
Tokushima Vortis players
Verspah Oita players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Oribe%20Niikawa | is a former Japanese football player.
Club statistics
References
External links
1988 births
Living people
Association football people from Gifu Prefecture
Japanese men's footballers
J1 League players
Japan Football League players
Nagoya Grampus players
FC Ryukyu players
Men's association football forwards |
https://en.wikipedia.org/wiki/Toru%20Hasegawa | is a Japanese footballer who plays for Tokushima Vortis.
Club statistics
Updated to end of 2018 season.
References
External links
Profile at Tokushima Vortis
1988 births
Living people
People from Seto, Aichi
Association football people from Aichi Prefecture
Japanese men's footballers
J1 League players
J2 League players
Nagoya Grampus players
Tokushima Vortis players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Wataru%20Inoue%20%28footballer%29 | is a Japanese footballer who plays for Kagoshima United FC.
Club statistics
Updated to 23 February 2016.
References
External links
Profile at Kagoshima United FC
1986 births
Living people
Association football people from Shizuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Nagoya Grampus players
Tokushima Vortis players
Zweigen Kanazawa players
Kagoshima United FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Masaya%20Sato | is a former Japanese football player.
Club statistics
References
External links
1990 births
Living people
Association football people from Shizuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Nagoya Grampus players
Thespakusatsu Gunma players
FC Ryukyu players
Fujieda MYFC players
Men's association football defenders |
https://en.wikipedia.org/wiki/Akira%20Takeuchi%20%28footballer%29 | is a former Japanese football player.
Club career statistics
Updated to 23 February 2018.
References
External links
Profile at Oita Trinita
Profile at Nagoya Grampus
1983 births
Living people
Kokushikan University alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Nagoya Grampus players
JEF United Chiba players
Oita Trinita players
Kamatamare Sanuki players
Men's association football defenders |
https://en.wikipedia.org/wiki/Keiji%20Watanabe | is a former Japanese football player.
Club statistics
References
External links
1985 births
Living people
Association football people from Shizuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Nagoya Grampus players
Japan Soccer College players
JEF United Chiba players
Men's association football defenders |
https://en.wikipedia.org/wiki/Shosuke%20Katayama | is a Japanese former footballer who last played for Roasso Kumamoto.
Career
Katayama retired at the end of the 2019 season.
Club statistics
Updated to 23 February 2017.
References
External links
Profile at Roasso Kumamoto
1983 births
Living people
Kokushikan University alumni
Association football people from Nara Prefecture
Japanese men's footballers
J1 League players
J2 League players
Nagoya Grampus players
Yokohama FC players
Roasso Kumamoto players
Men's association football defenders |
https://en.wikipedia.org/wiki/Tomohiro%20Tsuda | is a Japanese football player currently playing for FC Maruyasu Okazaki.
Club statistics
Updated to 1 January 2020.
References
External links
Profile at Yokohama FC
1986 births
Living people
Association football people from Gifu Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Nagoya Grampus players
Tokushima Vortis players
Yokohama FC players
AC Nagano Parceiro players
FC Maruyasu Okazaki players
Men's association football forwards |
https://en.wikipedia.org/wiki/836%20%28number%29 | 836 (eight hundred [and] thirty-six) is the natural number following 835 and preceding 837.
In mathematics
The factorization of 836 is , so its proper factors are 1, 2, 4, 11, 19, 22, 38, 44, 76, 209, and 418. They sum to 844. As this is greater than 836, it is an abundant number, but no subset sums to 836, so it is not a semiperfect number; therefore it is a weird number. Besides, 836 is the smallest weird number that is also an untouchable number, i.e. there is no n such that the sum of proper factors of n equals 836. (The only smaller weird number 70 is not untouchable, since σ(134) − 134 = 70)
See also
836 (year)
836 Jole (asteroid)
836th Air Division, an inactive United States Air Force organization
836 Naval Air Squadron, a World War II organization in the British Navy
Pi Arietis, designated as object 836 in the Bright Star catalogue
References
Integers |
https://en.wikipedia.org/wiki/Shuto%20Yamamoto | is a Japanese football player currently playing for Shonan Bellmare.
Club statistics
Updated to 5 November 2022.
1Includes Suruga Bank Championship, J. League Championship and FIFA Club World Cup.
National team statistics
Honors
Júbilo Iwata
J.League Cup (1): 2010
Suruga Bank Championship (1): 2011
Kashima Antlers
J1 League (1): 2016
Emperor's Cup (1): 2016
J.League Cup (1): 2015
Japanese Super Cup (1): 2017
AFC Champions League (1): 2018
References
External links
Profile at Kashima Antlers
1985 births
Living people
Waseda University alumni
Association football people from Iwate Prefecture
Japanese men's footballers
J1 League players
Júbilo Iwata players
Kashima Antlers players
Shonan Bellmare players
Footballers at the 2006 Asian Games
Men's association football defenders
Japan men's international footballers
Asian Games competitors for Japan |
https://en.wikipedia.org/wiki/Kyohei%20Suzaki | is a former Japanese football player.
Suzaki previously played for Júbilo Iwata in the J1 League.
Club statistics
National team statistics
Appearances in major competitions
References
External links
1989 births
Living people
Association football people from Mie Prefecture
Japanese men's footballers
J1 League players
J2 League players
Júbilo Iwata players
FC Gifu players
Men's association football defenders |
https://en.wikipedia.org/wiki/Naoki%20Hatta | is a Japanese footballer who plays for Júbilo Iwata.
Career statistics
Updated as of end of 2022 season.
References
External links
Profile at Júbilo Iwata
1986 births
Living people
Association football people from Mie Prefecture
Japanese men's footballers
J1 League players
J2 League players
Júbilo Iwata players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Takashi%20Fujii%20%28footballer%29 | is a former Japanese football player. Fujii previously played for Júbilo Iwata and Ehime FC
Club statistics
References
External links
football news
1986 births
Living people
Association football people from Aichi Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Júbilo Iwata players
Ehime FC players
AC Nagano Parceiro players
Blaublitz Akita players
FC Ryukyu players
Hong Kong First Division League players
Shatin SA players
Sun Hei SC players
Japanese expatriate men's footballers
Japanese expatriate sportspeople in Hong Kong
Expatriate men's footballers in Hong Kong
Men's association football forwards |
https://en.wikipedia.org/wiki/Kota%20Ueda | is a Japanese football player currently playing for Fagiano Okayama.
Club statistics
Updated to end of 2018 season.
References
External links
Profile at Jubilo Iwata
Player profile at Goal.com
1986 births
Living people
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
Júbilo Iwata players
Omiya Ardija players
Fagiano Okayama players
Men's association football midfielders
People from Ōme, Tokyo |
https://en.wikipedia.org/wiki/Ryu%20Okada | is a Japanese footballer, who currently plays for Júbilo Iwata on loan from Jubilo Iwata.
Career statistics
Updated to 23 February 2016.
References
External links
1984 births
Living people
University of Tsukuba alumni
Association football people from Shizuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Júbilo Iwata players
Avispa Fukuoka players
Men's association football midfielders
People from Fujieda, Shizuoka |
https://en.wikipedia.org/wiki/Yusuke%20Inuzuka | is a Japanese football player.
Club statistics
References
External links
1983 births
Living people
Shizuoka Sangyo University alumni
Japanese men's footballers
J1 League players
Japan Football League players
Júbilo Iwata players
Ventforet Kofu players
Sagan Tosu players
Azul Claro Numazu players
Men's association football defenders
Association football people from Hamamatsu |
https://en.wikipedia.org/wiki/International%20Association%20for%20Mathematics%20and%20Computers%20in%20Simulation | The International Association for Mathematics and Computers in Simulation (IMACS) has the goal to establish means of communication between researchers on simulation. It is incorporated in the United States and Belgium, with affiliates in other countries. IMACS organizes conferences, and publishes scientific journals and books in affiliation with commercial publishers.
IMACS journals
Applied Numerical Mathematics (Elsevier)
Mathematics and Computers in Simulation (Elsevier)
Journal of Theoretical and Computational Acoustics (World Scientific)
References
External links
Mathematical societies
Mathematics and Computers
Scientific organizations established in 1956 |
https://en.wikipedia.org/wiki/Von%20Neumann%20paradox | In mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area-preserving affine transformation such that the result is two planar figures of the same size as the original. This was proved in 1929 by John von Neumann, assuming the axiom of choice. It is based on the earlier Banach–Tarski paradox, which is in turn based on the Hausdorff paradox.
Banach and Tarski had proved that, using isometric transformations, the result of taking apart and reassembling a two-dimensional figure would necessarily have the same area as the original. This would make creating two unit squares out of one impossible. But von Neumann realized that the trick of such so-called paradoxical decompositions was the use of a group of transformations that include as a subgroup a free group with two generators. The group of area-preserving transformations (whether the special linear group or the special affine group) contains such subgroups, and this opens the possibility of performing paradoxical decompositions using them.
Sketch of the method
The following is an informal description of the method found by von Neumann. Assume that we have a free group H of area-preserving linear transformations generated by two transformations, σ and τ, which are not far from the identity element. Being a free group means that all its elements can be expressed uniquely in the form for some n, where the s and s are all non-zero integers, except possibly the first and the last . We can divide this group into two parts: those that start on the left with σ to some non-zero power (we call this set A) and those that start with τ to some power (that is, is zero—we call this set B, and it includes the identity).
If we operate on any point in Euclidean 2-space by the various elements of H we get what is called the orbit of that point. All the points in the plane can thus be classed into orbits, of which there are an infinite number with the cardinality of the continuum. Using the axiom of choice, we can choose one point from each orbit and call the set of these points M. We exclude the origin, which is a fixed point in H. If we then operate on M by all the elements of H, we generate each point of the plane (except the origin) exactly once. If we operate on M by all the elements of A or of B, we get two disjoint sets whose union is all points but the origin.
Now we take some figure such as the unit square or the unit disk. We then choose another figure totally inside it, such as a smaller square, centred at the origin. We can cover the big figure with several copies of the small figure, albeit with some points covered by two or more copies. We can then assign each point of the big figure to one of the copies of the small figure. Let us call the sets corresponding to each copy . We shall now make a one-to-one mapping of each point in the big figure to a point in its i |
https://en.wikipedia.org/wiki/Shun%20Morishita | is a Japanese football player currently playing for Iwate Grulla Morioka.
Career statistics
Updated to 19 February 2019.
References
External links
Profile at Jubilo Iwata
1986 births
Living people
Association football people from Mie Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Júbilo Iwata players
Kyoto Sanga FC players
Kawasaki Frontale players
Yokohama FC players
Iwate Grulla Morioka players
Men's association football defenders |
https://en.wikipedia.org/wiki/Baum%E2%80%93Sweet%20sequence | In mathematics the Baum–Sweet sequence is an infinite automatic sequence of 0s and 1s defined by the rule:
bn = 1 if the binary representation of n contains no block of consecutive 0s of odd length;
bn = 0 otherwise;
for n ≥ 0.
For example, b4 = 1 because the binary representation of 4 is 100, which only contains one block of consecutive 0s of length 2; whereas b5 = 0 because the binary representation of 5 is 101, which contains a block of consecutive 0s of length 1.
Starting at n = 0, the first few terms of the Baum–Sweet sequence are:
1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1 ...
Historical motivation
The properties of the sequence were first studied by Leonard E. Baum and Melvin M. Sweet in 1976. In 1949, Khinchin conjectured that there does not exist a non-quadratic algebraic real number having bounded partial quotients in its continued fraction expansion. A counterexample to this conjecture is still not known. Baum and Sweet's paper showed that the same expectation is not met for algebraic power series. They gave an example of cubic power series in whose partial quotients are bounded. (The degree of the power series in Baum and Sweet's result is analogous to the degree of the field extension associated with the algebraic real in Khinchin's conjecture.)
One of the series considered in Baum and Sweet's paper is a root of
The authors show that by Hensel's lemma, there is a unique such root in because reducing the defining equation of modulo gives , which factors as
They go on to prove that this unique root has partial quotients of degree . Before doing so, they state (in the remark following Theorem 2, p 598) that the root can be written in the form
where and for if and only if the binary expansion of contains only even length blocks of 's. This is the origin of the Baum–Sweet sequence.
Mkaouar and Yao proved that the partial quotients of the continued fraction for above do not form an automatic sequence. However, the sequence of partial quotients can be generated by a non-uniform morphism.
Properties
The Baum–Sweet sequence can be generated by a 3-state automaton.
The value of term bn in the Baum–Sweet sequence can be found recursively as follows. If n = m·4k, where m is not divisible by 4 (or is 0), then
Thus b76 = b9 = b4 = b0 = 1, which can be verified by observing that the binary representation of 76, which is 1001100, contains no consecutive blocks of 0s with odd length.
The Baum–Sweet word 1101100101001001..., which is created by concatenating the terms of the Baum–Sweet sequence, is a fixed point of the morphism or string substitution rules
00 → 0000
01 → 1001
10 → 0100
11 → 1101
as follows:
11 → 1101 → 11011001 → 1101100101001001 → 11011001010010011001000001001001 ...
From the morphism rules it can be seen that the Baum–Sweet word contains blocks of consecutive 0s of any length (bn = 0 for all 2k integers in the range 5.2k ≤ n < 6.2k), but it contains no block of three consecutive 1s.
More succinctly |
https://en.wikipedia.org/wiki/The%20Princeton%20Companion%20to%20Mathematics | The Princeton Companion to Mathematics is a book providing an extensive overview of mathematics that was published in 2008 by Princeton University Press. Edited by Timothy Gowers with associate editors June Barrow-Green and Imre Leader, it has been noted for the high caliber of its contributors. The book was the 2011 winner of the Euler Book Prize of the Mathematical Association of America, given annually to "an outstanding book about mathematics".
Topics and organization
The book concentrates primarily on modern pure mathematics rather than applied mathematics, although it does also cover both applications of mathematics and the mathematics that relates to those applications;
it provides a broad overview of the significant ideas and developments in research mathematics. It is organized into eight parts:
An introduction to mathematics, outlining the major areas of study, key definitions, and the goals and purposes of mathematical research.
An overview of the history of mathematics, in seven chapters including the development of important concepts such as number, geometry, mathematical proof, and the axiomatic approach to the foundations of mathematics. A chronology of significant events in mathematical history is also provided later in the book.
Three core sections, totalling approximately 600 pages. The first of these sections provides an alphabetized set of articles on 99 specific mathematical concepts such as the axiom of choice, expander graphs, and Hilbert space. The second core section includes long surveys of 26 branches of research mathematics such as algebraic geometry and combinatorial group theory. The third describes 38 important mathematical problems and theorems such as the four color theorem, the Birch and Swinnerton-Dyer conjecture, and the Halting problem.
A collection of biographies of nearly 100 famous deceased mathematicians, arranged chronologically, also including a history of Nicolas Bourbaki's pseudonymous collaboration.
Essays describing the influences and applications of mathematics in the sciences, technology, business, medicine, and the fine arts.
A section of perspectives on the future of mathematics, problem solving techniques, the ubiquity of mathematics, and advice to young mathematicians.
Despite its length, the range of topics included is selective rather than comprehensive: some important established topics such as diophantine approximation are omitted, transcendental number theory, differential geometry, and cohomology get short shrift, and the most recent frontiers of research are also generally not included.
Target audience
The book's authors have attempted to keep their work accessible by forgoing abstraction and technical nomenclature as much as possible and by making heavy use of concrete examples and illustrations. Compared to the concise and factual coverage of mathematics in sources such as Wikipedia and MathWorld, the articles in the Princeton Companion are intended to be more reflective and discursi |
https://en.wikipedia.org/wiki/Garside%20element | In mathematics, a Garside element is an element of an algebraic structure such as a monoid that has several desirable properties.
Formally, if M is a monoid, then an element Δ of M is said to be a Garside element if the set of all right divisors of Δ,
is the same set as the set of all left divisors of Δ,
and this set generates M.
A Garside element is in general not unique: any power of a Garside element is again a Garside element.
Garside monoid and Garside group
A Garside monoid is a monoid with the following properties:
Finitely generated and atomic;
Cancellative;
The partial order relations of divisibility are lattices;
There exists a Garside element.
A Garside monoid satisfies the Ore condition for multiplicative sets and hence embeds in its group of fractions: such a group is a Garside group. A Garside group is biautomatic and hence has soluble word problem and conjugacy problem. Examples of such groups include braid groups and, more generally, Artin groups of finite Coxeter type.
The name was coined by Patrick Dehornoy and Luis Paris to mark the work on the conjugacy problem for braid groups of Frank Arnold Garside (1915–1988), a teacher at Magdalen College School, Oxford who served as Lord Mayor of Oxford in 1984–1985.
References
Benson Farb, Problems on mapping class groups and related topics (Volume 74 of Proceedings of symposia in pure mathematics) AMS Bookstore, 2006, , p. 357
Patrick Dehornoy, Groupes de Garside, Annales Scientifiques de l'École Normale Supérieure (4) 35 (2002) 267-306. .
Matthieu Picantin, "Garside monoids vs divisibility monoids", Math. Structures Comput. Sci. 15 (2005) 231-242. .
Abstract algebra
Semigroup theory |
https://en.wikipedia.org/wiki/Asymmetric%20graph | In graph theory, a branch of mathematics, an undirected graph is called an asymmetric graph if it has no nontrivial symmetries.
Formally, an automorphism of a graph is a permutation of its vertices with the property that any two vertices and are adjacent if and only if and are adjacent.
The identity mapping of a graph onto itself is always an automorphism, and is called the trivial automorphism of the graph. An asymmetric graph is a graph for which there are no other automorphisms.
Note that the term "asymmetric graph" is not a negation of the term "symmetric graph," as the latter refers to a stronger condition than possessing nontrivial symmetries.
Examples
The smallest asymmetric non-trivial graphs have 6 vertices. The smallest asymmetric regular graphs have ten vertices; there exist ten-vertex asymmetric graphs that are 4-regular and 5-regular. One of the five smallest asymmetric cubic graphs is the twelve-vertex Frucht graph discovered in 1939. According to a strengthened version of Frucht's theorem, there are infinitely many asymmetric cubic graphs.
Properties
The class of asymmetric graphs is closed under complements: a graph G is asymmetric if and only if its complement is. Any n-vertex asymmetric graph can be made symmetric by adding and removing a total of at most n/2 + o(n) edges.
Random graphs
The proportion of graphs on n vertices with nontrivial automorphism tends to zero as n grows, which is informally expressed as "almost all finite graphs are asymmetric". In contrast, again informally, "almost all infinite graphs have nontrivial symmetries." More specifically, countable infinite random graphs in the Erdős–Rényi model are, with probability 1, isomorphic to the highly symmetric Rado graph.
Trees
The smallest asymmetric tree has seven vertices: it consists of three paths of lengths 1, 2, and 3, linked at a common endpoint. In contrast to the situation for graphs, almost all trees are symmetric. In particular, if a tree is chosen uniformly at random among all trees on n labeled nodes, then with probability tending to 1 as n increases, the tree will contain some two leaves adjacent to the same node and will have symmetries exchanging these two leaves.
References
Graph families
Graph |
https://en.wikipedia.org/wiki/AIDS%20orphan | An AIDS orphan is a child who became an orphan because one or both parents died from AIDS.
In statistics from the Joint United Nations Programme on HIV/AIDS (UNAIDS), the World Health Organization (WHO) and the United Nations Children's Fund (UNICEF), the term is used for a child whose mother has died due to AIDS before the child's 15th birthday, regardless of whether the father is still alive. As a result of this definition, one study estimated that 80% of all AIDS orphans still have one living parent.
There are 70,000 new AIDS orphans a year (as of 2001).
Because AIDS affects mainly those who are sexually active, AIDS-related deaths are often people who are their family's primary wage earners. The resulting AIDS orphans frequently depend on the state for care and financial support, particularly in Africa.
The highest number of orphans due to AIDS alive in 2007 was in South Africa (although the definition of AIDS orphan in South African statistics includes children up to the age of 18 who have lost either biological parent). In 2005 the highest number of AIDS orphans as a percentage of all orphans was in Zimbabwe.
See also
List of AIDS-related topics
References
External links
AIDS Orphan Resources Around the Globe
!Nam Child Wiki (Namibian Wiki on Children)
HIV/AIDS
Effects of death on children
Adoption, fostering, orphan care and displacement |
https://en.wikipedia.org/wiki/Pakistanis%20in%20Japan | form the country's third-largest community of immigrants from a Muslim-majority country, trailing only the Indonesian community and Bangladeshi community. As of June 2023, official statistics showed 23,417 registered foreigners of Pakistani origin living in the country. There were a further estimated 3,414 illegal immigrants from Pakistan in Japan as of 2000. The average increase in the Pakistani population is about 2-3 persons per day.
Migration history
As early as 1950, only three years after the independence of Pakistan in 1947 which created the Pakistani state, there were recorded to be four Pakistanis living in Japan. However, Pakistani migration to Japan would not grow to a large scale until the 1980s. The later Pakistani migrants in Japan largely come from a muhajir background; their family history of migration made them consider working overseas as a "natural choice" when they found opportunities at home to be too limited. While Pakistanis saw North America as a good destination to settle down and start a business, Japanese employment agencies commonly advertised in Karachi newspapers in the 1980s, when Japan offered some of the highest wages in the world for unskilled labour; it came to be preferred as a destination by single male migrants, who came without their families. The wages they earned could reach as high as twenty times what they made in Pakistan.
Pakistani citizens once enjoyed the privilege of short-term visa-free entry to Japan, but when controversy arose in Japanese society over illegal foreign workers, the Japanese government revoked this privilege. With little chance of acquiring a work visa or even permission to enter the country, Pakistanis paid as much as ¥300,000 to people smugglers in the late 1980s and early 1990s to enter the country. According to Japanese government statistics, the number of Pakistanis illegally residing in Japan peaked in 1992 at 8,056 individuals and declined after that. However, Pakistani sources suggest that as late as 1999, the total population of Pakistanis was 25,000 and still included a significant amount of illegal immigrants. Some Pakistanis were able to obtain legal resident status by finding Japanese spouses.
Demographics
According to 2008 Japanese government figures, 19.9% of registered Pakistanis lived in Saitama, 17.8% in Tokyo, 12.3% in Kanagawa, 10.4% in Aichi, 8.98% in Chiba, 7.59% in Gunma, 6.02% in Ibaraki, 4.44% in Tochigi, 4.21% in Toyama, 3.27% in Shizuoka and the remaining 4.98% in other prefectures of Japan. Only an estimated 200 Pakistanis hold Japanese citizenship.
Business and employment
Many Pakistanis in Japan run used car export businesses. This trend was believed to have begun in the late 1970s, when one Pakistani working in Japan sent a car back to his homeland. The potential for doing business in used cars also attracted more Pakistanis to come to Japan in the 1990s.
Though many migrants come from a middle-class family background in Pakistan, because they of |
https://en.wikipedia.org/wiki/Rudin%E2%80%93Shapiro%20sequence | In mathematics, the Rudin–Shapiro sequence, also known as the Golay–Rudin–Shapiro sequence, is an infinite 2-automatic sequence named after Marcel Golay, Walter Rudin, and Harold S. Shapiro, who independently investigated its properties.
Definition
Each term of the Rudin–Shapiro sequence is either or . If the binary expansion of is given by
then let
(So is the number of times the block 11 appears in the binary expansion of .)
The Rudin–Shapiro sequence is then defined by
Thus if is even and if is odd.
The sequence is known as the complete Rudin–Shapiro sequence, and starting at , its first few terms are:
0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, ...
and the corresponding terms of the Rudin–Shapiro sequence are:
+1, +1, +1, −1, +1, +1, −1, +1, +1, +1, +1, −1, −1, −1, +1, −1, ...
For example, and because the binary representation of 6 is 110, which contains one occurrence of 11; whereas and because the binary representation of 7 is 111, which contains two (overlapping) occurrences of 11.
Historical motivation
The Rudin–Shapiro sequence was introduced independently by Golay, Rudin, and Shapiro. The following is a description of Rudin's motivation. In Fourier analysis, one is often concerned with the norm of a measurable function . This norm is defined by
One can prove that for any sequence with each in ,
Moreover, for almost every sequence with each is in ,
However, the Rudin–Shapiro sequence satisfies a tighter bound: there exists a constant such that
It is conjectured that one can take , but while it is known that , the best published upper bound is currently . Let be the n-th Shapiro polynomial. Then, when , the above inequality gives a bound on . More recently, bounds have also been given for the magnitude of the coefficients of where .
Shapiro arrived at the sequence because the polynomials
where is the Rudin–Shapiro sequence, have absolute value bounded on the complex unit circle by . This is discussed in more detail in the article on Shapiro polynomials. Golay's motivation was similar, although he was concerned with applications to spectroscopy and published in an optics journal.
Properties
The Rudin–Shapiro sequence can be generated by a 4-state automaton accepting binary representations of non-negative integers as input. The sequence is therefore 2-automatic, so by Cobham's little theorem there exists a 2-uniform morphism with fixed point and a coding such that , where is the Rudin–Shapiro sequence. However, the Rudin–Shapiro sequence cannot be expressed as the fixed point of some uniform morphism alone.
There is a recursive definition
The values of the terms rn and un in the Rudin–Shapiro sequence can be found recursively as follows. If n = m·2k where m is odd then
Thus u108 = u13 + 1 = u3 + 1 = u1 + 2 = u0 + 2 = 2, which can be verified by observing that the binary representation of 108, which is 1101100, contains two sub-strings 11. And so r108 = (−1)2 = +1.
A 2-uniform morphism |
https://en.wikipedia.org/wiki/Makoto%20Oda%20%28footballer%29 | is a former Japanese football player.
Club statistics
References
External links
1989 births
Living people
Japanese men's footballers
J2 League players
Roasso Kumamoto players
AC Nagano Parceiro players
Men's association football defenders
Association football people from Kumamoto |
https://en.wikipedia.org/wiki/Kazuya%20Kawabata | is a former Japanese football player.
Club statistics
References
External links
1981 births
Living people
Sapporo University alumni
Association football people from Hokkaido
People from Tomakomai, Hokkaido
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Hokkaido Consadole Sapporo players
Roasso Kumamoto players
Giravanz Kitakyushu players
V-Varen Nagasaki players
FC Ryukyu players
ReinMeer Aomori players
Men's association football defenders
FISU World University Games gold medalists for Japan
Universiade medalists in football |
https://en.wikipedia.org/wiki/Yusuke%20Suzuki%20%28footballer%29 | is a former Japanese football player.
Suzuki previously played for Roasso Kumamoto in the J2 League.
Club statistics
References
External links
1982 births
Living people
Komazawa University alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Roasso Kumamoto players
AC Nagano Parceiro players
FC Machida Zelvia players
Kamatamare Sanuki players
SC Sagamihara players
Men's association football defenders |
https://en.wikipedia.org/wiki/Yasunobu%20Matsuoka | is a former Japanese football player.
Club statistics
References
External links
Official blog
1986 births
Living people
Association football people from Osaka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Gamba Osaka players
Roasso Kumamoto players
V-Varen Nagasaki players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kiyoshi%20Saito%20%28footballer%29 | is a former Japanese football player.
Saito made five appearances for Roasso Kumamoto in the J2 League Division 2.
Club statistics
References
External links
1982 births
Living people
Tokyo University of Agriculture alumni
Association football people from Miyagi Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Arte Takasaki players
Roasso Kumamoto players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Trinomial%20tree | The trinomial tree is a lattice-based computational model used in financial mathematics to price options. It was developed by Phelim Boyle in 1986. It is an extension of the binomial options pricing model, and is conceptually similar. It can also be shown that the approach is equivalent to the explicit finite difference method for option pricing. For fixed income and interest rate derivatives see Lattice model (finance)#Interest rate derivatives.
Formula
Under the trinomial method, the underlying stock price is modeled as a recombining tree, where, at each node the price has three possible paths: an up, down and stable or middle path. These values are found by multiplying the value at the current node by the appropriate factor , or where
(the structure is recombining)
and the corresponding probabilities are:
.
In the above formulae: is the length of time per step in the tree and is simply time to maturity divided by the number of time steps; is the risk-free interest rate over this maturity; is the corresponding volatility of the underlying; is its corresponding dividend yield.
As with the binomial model, these factors and probabilities are specified so as to ensure that the price of the underlying evolves as a martingale, while the moments considering node spacing and probabilities are matched to those of the log-normal distribution (and with increasing accuracy for smaller time-steps). Note that for , , and to be in the interval the following condition on has to be satisfied .
Once the tree of prices has been calculated, the option price is found at each node largely as for the binomial model, by working backwards from the final nodes to the present node (). The difference being that the option value at each non-final node is determined based on the threeas opposed to two later nodes and their corresponding probabilities.
If the length of time-steps is taken as an exponentially distributed random variable and interpreted as the waiting time between two movements of the stock price then the resulting stochastic process is a birth–death process. The resulting model is soluble and there exist analytic pricing and hedging formulae for various options.
Application
The trinomial model is considered to produce more accurate results than the binomial model when fewer time steps are modelled, and is therefore used when computational speed or resources may be an issue. For vanilla options, as the number of steps increases, the results rapidly converge, and the binomial model is then preferred due to its simpler implementation. For exotic options the trinomial model (or adaptations) is sometimes more stable and accurate, regardless of step-size.
See also
Binomial options pricing model
Valuation of options
Option: Model implementation
Korn–Kreer–Lenssen model
Implied trinomial tree
References
External links
Phelim Boyle, 1986. "Option Valuation Using a Three-Jump Process", International Options Journal 3, 7–12.
Paul Cliffo |
https://en.wikipedia.org/wiki/Tamon%20Machida | is a Japanese former footballer.
Machida previously played for Roasso Kumamoto in the J2 League.
Club statistics
References
External links
1982 births
Living people
University of Tsukuba alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Arte Takasaki players
Roasso Kumamoto players
Sony Sendai FC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Kenichi%20Kawano | is a former Japanese football player.
Club statistics
References
External links
1982 births
Living people
Nippon Bunri University alumni
Association football people from Mie Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Roasso Kumamoto players
Reilac Shiga FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Atsushi%20Ichimura | is a Japanese former football player who last played for Kamatamare Sanuki.
Career
Ichimura retired at the end of the 2019 season.
Club statistics
Updated to 23 February 2020.
References
External links
1984 births
Living people
Association football people from Hokkaido
People from Eniwa, Hokkaido
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Hokkaido Consadole Sapporo players
Roasso Kumamoto players
Yokohama FC players
Kamatamare Sanuki players
Men's association football defenders |
https://en.wikipedia.org/wiki/Daisuke%20Yano | is a former Japanese football player.
Club statistics
References
External links
1984 births
Living people
Association football people from Kumamoto Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Gamba Osaka players
Sagan Tosu players
Roasso Kumamoto players
Men's association football defenders |
https://en.wikipedia.org/wiki/Kosuke%20Yoshii | is a Japanese retired football player.
Club statistics
Updated to 23 February 2019.
References
External links
1986 births
Living people
Association football people from Kagoshima Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Shonan Bellmare players
Roasso Kumamoto players
Kagoshima United FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Masaaki%20Nishimori | is a Japanese former football player.
Club statistics
References
External links
1985 births
Living people
Ryutsu Keizai University alumni
Japanese men's footballers
J2 League players
Japan Football League players
Roasso Kumamoto players
V-Varen Nagasaki players
Renofa Yamaguchi FC players
Men's association football midfielders
Association football people from Kumamoto |
https://en.wikipedia.org/wiki/Daishiro%20Miyazaki | is a former Japanese football player.
Club statistics
References
External links
1983 births
Living people
Komazawa University alumni
Japanese men's footballers
J2 League players
Japan Football League players
Roasso Kumamoto players
Men's association football midfielders
Association football people from Kumamoto |
https://en.wikipedia.org/wiki/Simion%20Stoilow%20Prize | The Simion Stoilow Prize () is the prize offered by the Romanian Academy for achievements in mathematics. It is named in honor of Simion Stoilow.
The prize is awarded either for a mathematical work or for a cycle of works.
The award consists of 30,000 lei and a diploma. The prize was established in 1963 and is awarded annually. Prizes of the Romanian Academy for a particular year are awarded two years later.
Honorees
Honorees of the Simion Stoilow Prize have included:
2020: Victor-Daniel Lie
2019: Marius Ghergu; Bogdan Teodor Udrea
2018: Iulian Cîmpean
2017: Aurel Mihai Fulger
2016: Arghir Dani Zărnescu
2015: No award
2014: Florin Ambro
2013: Petru Jebelean
2012: George Marinescu
2011: Dan Timotin
2010: Laurențiu Leuștean; Mihai Mihăilescu
2009: Miodrag Iovanov; Sebastian Burciu
2008: Nicolae Bonciocat; Călin Ambrozie
2007: Cezar Joița; Bebe Prunaru; Liviu Ignat
2006: Radu Pantilie
2005: Eugen Mihăilescu, for the work "Estimates for the stable dimension for holomorphic maps"; Radu Păltânea, for the cycle of works "Approximation theory using positive linear operators"
2000: Liliana Pavel, for the book Hipergrupuri ("Hypergroups")
1999: Vicențiu Rădulescu for the work "Boundary value problems for nonlinear elliptic equations and hemivariational inequalities"
1995: No award
1994: No award
1993: No award
1992: Florin Rădulescu
1991: Ovidiu Cârjă
1990: Ștefan Mirică
1989: Gelu Popescu
1988: Cornel Pasnicu
1987: Călin-Ioan Gheorghiu; Titus Petrila
1986: Vlad Bally; Paltin Ionescu
1985: Vasile Brânzănescu; Paul Flondor; Dan Polisevschi; Mihai Putinar
1984: Toma Albu; ; Dan Vuza
1983: Mircea Puta; Ion Chițescu; Eugen Popa
1982: Mircea Craioveanu; Mircea Puta
1981: Lucian Bădescu
1980: Dumitru Gașpar; Costel Peligrad; Mihai Pimsner; Sorin T. Popa
1979: Dumitru Motreanu; Dorin Popescu; Ilie Valusescu
1978: Aurel Bejancu; Gheorghe Micula
1977: Alexandru Brezuleanu; Nicolae Radu;
1976: Zoia Ceaușescu; Ion Cuculescu; Nicolae Popa
1975: Șerban Strătilă; Elena Stroescu;
1974: Ioana Ciorănescu; Dan Pascali; Constantin Vârsan
1973: Vasile Istrătescu; Ioan Marusciac; ; Veniamin Urseanu
1972: Bernard Bereanu; Nicolae Pavel; Gustav Peeters; Elena Moldovan Popoviciu
1971: Nicolae Popescu
1970: Viorel Barbu;
1969: Ion Suciu
1968:
1967: Constantin Apostol
1966: Dan Burghelea; Cabiria Andreian Cazacu;
1965: ; Alexandru Lascu
1964: ;
1963: ;
See also
List of mathematics awards
References
Prizes of the Romanian Academy
Mathematics awards |
https://en.wikipedia.org/wiki/Free%20convolution | Free convolution is the free probability analog of the classical notion of convolution of probability measures. Due to the non-commutative nature of free probability theory, one has to talk separately about additive and multiplicative free convolution, which arise from addition and multiplication of free random variables (see below; in the classical case, what would be the analog of free multiplicative convolution can be reduced to additive convolution by passing to logarithms of random variables). These operations have some interpretations in terms of empirical spectral measures of random matrices.
The notion of free convolution was introduced by Dan-Virgil Voiculescu.
Free additive convolution
Let and be two probability measures on the real line, and assume that is a random variable in a non commutative probability space with law and is a random variable in the same non commutative probability space with law . Assume finally that and are freely independent. Then the free additive convolution is the law of . Random matrices interpretation: if and are some independent by Hermitian (resp. real symmetric) random matrices such that at least one of them is invariant, in law, under conjugation by any unitary (resp. orthogonal) matrix and such that the empirical spectral measures of and tend respectively to and as tends to infinity, then the empirical spectral measure of tends to .
In many cases, it is possible to compute the probability measure explicitly by using complex-analytic techniques and the R-transform of the measures and .
Rectangular free additive convolution
The rectangular free additive convolution (with ratio ) has also been defined in the non commutative probability framework by Benaych-Georges and admits the following random matrices interpretation. For , for and are some independent by complex (resp. real) random matrices such that at least one of them is invariant, in law, under multiplication on the left and on the right by any unitary (resp. orthogonal) matrix and such that the empirical singular values distribution of and tend respectively to and as and tend to infinity in such a way that tends to , then the empirical singular values distribution of tends to .
In many cases, it is possible to compute the probability measure explicitly by using complex-analytic techniques and the rectangular R-transform with ratio of the measures and .
Free multiplicative convolution
Let and be two probability measures on the interval , and assume that is a random variable in a non commutative probability space with law and is a random variable in the same non commutative probability space with law . Assume finally that and are freely independent. Then the free multiplicative convolution is the law of (or, equivalently, the law of . Random matrices interpretation: if and are some independent by non negative Hermitian (resp. real symmetric) random matrices such that at least one of t |
https://en.wikipedia.org/wiki/Yoichi%20Futori | is a former Japanese football player.
Club statistics
References
External links
1982 births
Living people
People from Noda, Chiba
Komazawa University alumni
Association football people from Chiba Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Roasso Kumamoto players
Gamba Osaka players
Tokyo Verdy players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Yuichi%20Yamauchi | is a former Japanese football player.
Club statistics
References
External links
1984 births
Living people
Fukuoka University alumni
Japanese men's footballers
J2 League players
Japan Football League players
Roasso Kumamoto players
V-Varen Nagasaki players
Men's association football forwards
Expatriate men's footballers in Thailand
Expatriate men's soccer players in Australia
Sydney United 58 FC players
Blacktown City FC players
Yuichi Yamauchi
Association football people from Kumamoto |
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