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https://en.wikipedia.org/wiki/Fixed-point%20combinator | In mathematics and computer science in general, a fixed point of a function is a value that is mapped to itself by the function.
In combinatory logic for computer science, a fixed-point combinator (or fixpoint combinator) is a higher-order function that returns some fixed point of its argument function, if one exists.
Formally, if the function f has one or more fixed points, then
and hence, by repeated application,
Y combinator
In the classical untyped lambda calculus, every function has a fixed point.
A particular implementation of fix is Curry's paradoxical combinator Y, represented by
In functional programming, the Y combinator can be used to formally define recursive functions in a programming language that does not support recursion.
This combinator may be used in implementing Curry's paradox. The heart of Curry's paradox is that untyped lambda calculus is unsound as a deductive system, and the Y combinator demonstrates this by allowing an anonymous expression to represent zero, or even many values. This is inconsistent in mathematical logic.
Applied to a function with one variable, the Y combinator usually does not terminate. More interesting results are obtained by applying the Y combinator to functions of two or more variables. The additional variables may be used as a counter, or index. The resulting function behaves like a while or a for loop in an imperative language.
Used in this way, the Y combinator implements simple recursion. In the lambda calculus, it is not possible to refer to the definition of a function inside its own body by name. Recursion though may be achieved by obtaining the same function passed in as an argument, and then using that argument to make the recursive call, instead of using the function's own name, as is done in languages which do support recursion natively. The Y combinator demonstrates this style of programming.
An example implementation of Y combinator in two languages is presented below.
# Y Combinator in Python
Y=lambda f: (lambda x: f(x(x)))(lambda x: f(x(x)))
Y(Y)
// Y Combinator in C++
int main() {
auto Y = [](auto f) {
auto f1 = [f](auto x) -> decltype(f) {
// A print statement may be inserted here,
// to observe that we get an infinite loop
// (at least until the stack overflows)
return f(x(x));
};
return f1(f1);
};
Y(Y);
}
Fixed-point combinator
The Y combinator is an implementation of a fixed-point combinator in lambda calculus. Fixed-point combinators may also be easily defined in other functional and imperative languages. The implementation in lambda calculus is more difficult due to limitations in lambda calculus.
The fixed-point combinator may be used in a number of different areas:
General mathematics
Untyped lambda calculus
Typed lambda calculus
Functional programming
Imperative programming
Fixed-point combinators may be applied to a range of different functions, but n |
https://en.wikipedia.org/wiki/Ideal%20class%20group | In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of . The order of the group, which is finite, is called the class number of .
The theory extends to Dedekind domains and their fields of fractions, for which the multiplicative properties are intimately tied to the structure of the class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain.
History and origin of the ideal class group
Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an ideal was formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as put into something like a final form by Carl Friedrich Gauss, a composition law was defined on certain equivalence classes of forms. This gave a finite abelian group, as was recognised at the time.
Later Ernst Kummer was working towards a theory of cyclotomic fields. It had been realised (probably by several people) that failure to complete proofs in the general case of Fermat's Last Theorem by factorisation using the roots of unity was for a very good reason: a failure of unique factorization – i.e., the fundamental theorem of arithmetic – to hold in the rings generated by those roots of unity was a major obstacle. Out of Kummer's work for the first time came a study of the obstruction to the factorisation. We now recognise this as part of the ideal class group: in fact Kummer had isolated the p-torsion in that group for the field of p-roots of unity, for any prime number p, as the reason for the failure of the standard method of attack on the Fermat problem (see regular prime).
Somewhat later again Richard Dedekind formulated the concept of an ideal, Kummer having worked in a different way. At this point the existing examples could be unified. It was shown that while rings of algebraic integers do not always have unique factorization into primes (because they need not be principal ideal domains), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals (that is, every ring of algebraic integers is a Dedekind domain). The size of the ideal class group can be considered as a measure for the deviation of a ring from being a principal ideal domain; a ring is a principal ideal domain if and only if it has a trivial ideal class group.
Definition
If R is an integral domain, define a relation ~ on nonzero fractional ideals of R by I ~ J whenever there exist nonzero elements a and b of R such that (a)I = (b)J. (Here the notation (a) means the principal ideal of R consisting of all the multiples of a.) It is easily shown that this |
https://en.wikipedia.org/wiki/Toeplitz%20matrix | In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix:
Any matrix of the form
is a Toeplitz matrix. If the element of is denoted then we have
A Toeplitz matrix is not necessarily square.
Solving a Toeplitz system
A matrix equation of the form
is called a Toeplitz system if is a Toeplitz matrix. If is an Toeplitz matrix, then the system has at-most only
unique values, rather than . We might therefore expect that the solution of a Toeplitz system would be easier, and indeed that is the case.
Toeplitz systems can be solved by the Levinson algorithm in time. Variants of this algorithm have been shown to be weakly stable (i.e. they exhibit numerical stability for well-conditioned linear systems). The algorithm can also be used to find the determinant of a Toeplitz matrix in time.
A Toeplitz matrix can also be decomposed (i.e. factored) in time. The Bareiss algorithm for an LU decomposition is stable. An LU decomposition gives a quick method for solving a Toeplitz system, and also for computing the determinant.
Algorithms that are asymptotically faster than those of Bareiss and Levinson have been described in the literature, but their accuracy cannot be relied upon.
General properties
An Toeplitz matrix may be defined as a matrix where , for constants . The set of Toeplitz matrices is a subspace of the vector space of matrices (under matrix addition and scalar multiplication).
Two Toeplitz matrices may be added in time (by storing only one value of each diagonal) and multiplied in time.
Toeplitz matrices are persymmetric. Symmetric Toeplitz matrices are both centrosymmetric and bisymmetric.
Toeplitz matrices are also closely connected with Fourier series, because the multiplication operator by a trigonometric polynomial, compressed to a finite-dimensional space, can be represented by such a matrix. Similarly, one can represent linear convolution as multiplication by a Toeplitz matrix.
Toeplitz matrices commute asymptotically. This means they diagonalize in the same basis when the row and column dimension tends to infinity.
For symmetric Toeplitz matrices, there is the decomposition
where is the lower triangular part of .
The inverse of a nonsingular symmetric Toeplitz matrix has the representation
where and are lower triangular Toeplitz matrices and is a strictly lower triangular matrix.
Discrete convolution
The convolution operation can be constructed as a matrix multiplication, where one of the inputs is converted into a Toeplitz matrix. For example, the convolution of and can be formulated as:
This approach can be extended to compute autocorrelation, cross-correlation, moving average etc.
Infinite Toeplitz matrix
A bi-infinite Toeplitz matrix (i.e. entries indexed by ) induces a linear operator on .
The induced operator |
https://en.wikipedia.org/wiki/Runcinated%205-orthoplexes | In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.
There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantellations. Four are more simply constructed relative to the 5-cube.
Runcinated 5-orthoplex
Alternate names
Runcinated pentacross
Small prismated triacontiditeron (Acronym: spat) (Jonathan Bowers)
Coordinates
The vertices of the can be made in 5-space, as permutations and sign combinations of:
(0,1,1,1,2)
Images
Runcitruncated 5-orthoplex
Alternate names
Runcitruncated pentacross
Prismatotruncated triacontiditeron (Acronym: pattit) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a runcitruncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of
(±3,±2,±1,±1,0)
Images
Runcicantellated 5-orthoplex
Alternate names
Runcicantellated pentacross
Prismatorhombated triacontiditeron (Acronym: pirt) (Jonathan Bowers)
Coordinates
The vertices of the runcicantellated 5-orthoplex can be made in 5-space, as permutations and sign combinations of:
(0,1,2,2,3)
Images
Runcicantitruncated 5-orthoplex
Alternate names
Runcicantitruncated pentacross
Great prismated triacontiditeron (gippit) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a runcicantitruncated 5-orthoplex having an edge length of are given by all permutations of coordinates and sign of:
Images
Snub 5-demicube
The snub 5-demicube defined as an alternation of the omnitruncated 5-demicube is not uniform, but it can be given Coxeter diagram or and symmetry [32,1,1]+ or [4,(3,3,3)+], and constructed from 10 snub 24-cells, 32 snub 5-cells, 40 snub tetrahedral antiprisms, 80 2-3 duoantiprisms, and 960 irregular 5-cells filling the gaps at the deleted vertices.
Related polytopes
This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
x3o3o3x4o - spat, x3x3o3x4o - pattit, x3o3x3x4o - pirt, x3x3x3x4o - gippit
External links
Polytopes of Various Dimensions, Jonathan Bowers
Runcinated uniform polytera (spid), Jonathan Bowers
Multi-dimensional Glossary
5-polytopes |
https://en.wikipedia.org/wiki/Charles%20Keene%20%28racing%20driver%29 | Charles Keene (born in Beaver Falls, Pennsylvania) was an American racecar driver active during the formative years of the auto racing.
Career statistics
By season
Indy 500 results
References
External links
Indianapolis 500 drivers
People from Beaver Falls, Pennsylvania
Sportspeople from Beaver County, Pennsylvania
Racing drivers from Pennsylvania
Year of birth missing
Year of death missing |
https://en.wikipedia.org/wiki/M.%20taiwanensis | M. taiwanensis may refer to:
Maackia taiwanensis, a legume species found only in Taiwan
Macrothele taiwanensis, a mygalomorph spider species in the genus Macrothele
Methanocalculus taiwanensis, an Archaea species in the genus Methanocalculus
Myotis taiwanensis, a bat species |
https://en.wikipedia.org/wiki/Quasisymmetric | In mathematics, quasisymmetric may refer to
Quasisymmetric functions in algebraic combinatorics
Quasisymmetric maps in complex analysis or metric spaces.
Quasi-symmetric designs in combinatorial design theory. |
https://en.wikipedia.org/wiki/Lunocet | The Lunocet is a biomimetic monofin intended to reduce drag and augment human swimming ability underwater. It is modeled after a dolphin's tail, by replicating the geometry, scale, and morphology dynamics of what the manufacturer calls the "lunate tail propulsor," as the shape is similar to a crescent moon. During the COVID-19 pandemic, the manufacturer stopped producing the Lunocet due to supply issues.
Description
The Lunocet is a 5.3 lb (2.4 kg) monofin designed/invented by Ted Ciamillo. It is built from aluminum and rubber with a span of 30 inches. It is attached to a diver using road cycling or triathlon shoes screwed to an aluminum plate oriented at a 30-degree angle to the resting plane of the flukes, to compensate for the angle of the ankle when extended.
Features
The manufacturer claims an advantage over traditional fins because of its hydrodynamic shape. It has a high aspect ratio and large surface area, which can propel an experienced diver to a maximum speed of eight mph (13 km/h). That later claimed result is exactly the world record for 50 m apnea with traditional monofin.
A spring-loaded mechanism adjusts the angle of attack of the flukes using the water pressure from swimming. The rubber spring can be set on three tension settings from flexible for slow relatively efficient swimming to more rigid for sprinting.
A cambered foil can be more efficient than a symmetrical foil, but a rigid cambered foil is efficient for producing lift in just one direction. The Lunocet uses the water pressure to induce camber in the flexible flukes which are symmetric at rest. The camber is dynamic and changes sides with up and down stroke. The flexibility is mostly at the tips and trailing edge.
Road cycling or triathlon shoes are used to attach the fin to the diver. Power transfer is through the rigid soles into the peduncle (the part which transitions the feet to the flukes). All road cycling and triathlon shoes have a standard bolt-hole pattern which mates with the bolt-hole pattern on the foot deck and can be attached using six stainless steel screws.
See also
Diving equipment
References
External links
Archive of Lunocet Blog
2014 Archive of LUNOCET
Diver propulsion equipment |
https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood%20zeta-function%20conjectures | In mathematics, the Hardy–Littlewood zeta-function conjectures, named after Godfrey Harold Hardy and John Edensor Littlewood, are two conjectures concerning the distances between zeros and the density of zeros of the Riemann zeta function.
Conjectures
In 1914, Godfrey Harold Hardy proved that the Riemann zeta function has infinitely many real zeros.
Let be the total number of real zeros, be the total number of zeros of odd order of the function , lying on the interval
.
Hardy and Littlewood claimed two conjectures. These conjectures – on the distance between real zeros of and on the density of zeros of on intervals for sufficiently great , and with as less as possible value of , where is an arbitrarily small number – open two new directions in the investigation of the Riemann zeta function.
1. For any there exists such that for and the interval contains a zero of odd order of the function .
2. For any there exist and , such that for and the inequality is true.
Status
In 1942, Atle Selberg studied the problem 2 and proved that for any there exists such and , such that for and the inequality is true.
In his turn, Selberg made his conjecture that it's possible to decrease the value of the exponent for which was proved 42 years later by A.A. Karatsuba.
References
Conjectures
Zeta and L-functions |
https://en.wikipedia.org/wiki/Granville%20number | In mathematics, specifically number theory, Granville numbers, also known as -perfect numbers, are an extension of the perfect numbers.
The Granville set
In 1996, Andrew Granville proposed the following construction of a set :
Let , and for any integer larger than 1, let if
A Granville number is an element of for which equality holds, that is, is a Granville number if it is equal to the sum of its proper divisors that are also in . Granville numbers are also called -perfect numbers.
General properties
The elements of can be -deficient, -perfect, or -abundant. In particular, 2-perfect numbers are a proper subset of .
S-deficient numbers
Numbers that fulfill the strict form of the inequality in the above definition are known as -deficient numbers. That is, the -deficient numbers are the natural numbers for which the sum of their divisors in is strictly less than themselves:
S-perfect numbers
Numbers that fulfill equality in the above definition are known as -perfect numbers. That is, the -perfect numbers are the natural numbers that are equal the sum of their divisors in . The first few -perfect numbers are:
6, 24, 28, 96, 126, 224, 384, 496, 1536, 1792, 6144, 8128, 14336, ...
Every perfect number is also -perfect. However, there are numbers such as 24 which are -perfect but not perfect. The only known -perfect number with three distinct prime factors is 126 = 2 · 32 · 7.
S-abundant numbers
Numbers that violate the inequality in the above definition are known as -abundant numbers. That is, the -abundant numbers are the natural numbers for which the sum of their divisors in is strictly greater than themselves:
They belong to the complement of . The first few -abundant numbers are:
12, 18, 20, 30, 42, 48, 56, 66, 70, 72, 78, 80, 84, 88, 90, 102, 104, ...
Examples
Every deficient number and every perfect number is in because the restriction of the divisors sum to members of either decreases the divisors sum or leaves it unchanged. The first natural number that is not in is the smallest abundant number, which is 12. The next two abundant numbers, 18 and 20, are also not in . However, the fourth abundant number, 24, is in because the sum of its proper divisors in is:
1 + 2 + 3 + 4 + 6 + 8 = 24
In other words, 24 is abundant but not -abundant because 12 is not in . In fact, 24 is -perfect - it is the smallest number that is -perfect but not perfect.
The smallest odd abundant number that is in is 2835, and the smallest pair of consecutive numbers that are not in are 5984 and 5985.
References
Number theory |
https://en.wikipedia.org/wiki/Calhoun%20Hall | Calhoun Hall (abbreviated CAL) is a building located on the University of Texas at Austin campus, built in 1968. The building is named after John William Calhoun, a mathematics professor, university comptroller from 1925 to 1937, and university president from 1937 to 1939.
References
1968 establishments in Texas
University and college buildings completed in 1968
University of Texas at Austin campus |
https://en.wikipedia.org/wiki/Ben%20Jones%20%28racing%20driver%29 | Ben Jones (April 2, 1903 Schlater, Mississippi – December 23, 1938 Uniontown, Alabama) was an American racecar driver.
Career statistics
By season
Indy 500 results
References
External links
Indianapolis 500 drivers
1903 births
1938 deaths
People from Leflore County, Mississippi
Racing drivers from Mississippi |
https://en.wikipedia.org/wiki/Probability%20box | A probability box (or p-box) is a characterization of uncertain numbers consisting of both aleatoric and epistemic uncertainties that is often used in risk analysis or quantitative uncertainty modeling where numerical calculations must be performed. Probability bounds analysis is used to make arithmetic and logical calculations with p-boxes.
An example p-box is shown in the figure at right for an uncertain number x consisting of a left (upper) bound and a right (lower) bound on the probability distribution for x. The bounds are coincident for values of x below 0 and above 24. The bounds may have almost any shape, including step functions, so long as they are monotonically increasing and do not cross each other. A p-box is used to express simultaneously incertitude (epistemic uncertainty), which is represented by the breadth between the left and right edges of the p-box, and variability (aleatory uncertainty), which is represented by the overall slant of the p-box.
Interpretation
There are dual interpretations of a p-box. It can be understood as bounds on the cumulative probability associated with any x-value. For instance, in the p-box depicted at right, the probability that the value will be 2.5 or less is between 4% and 36%. A p-box can also be understood as bounds on the x-value at any particular probability level. In the example, the 95th percentile is sure to be between 9 and 16.
If the left and right bounds of a p-box are sure to enclose the unknown distribution, the bounds are said to be rigorous, or absolute. The bounds may also be the tightest possible such bounds on the distribution function given the available information about it, in which case the bounds are therefore said to be best-possible. It may commonly be the case, however, that not every distribution that lies within these bounds is a possible distribution for the uncertain number, even when the bounds are rigorous and best-possible.
Mathematical definition
P-boxes are specified by left and right bounds on the distribution function (or, equivalently, the survival function) of a quantity and, optionally, additional information constraining the quantity's mean and variance to specified intervals, and specified constraints on its distributional shape (family, unimodality, symmetry, etc.). A p-box represents a class of probability distributions consistent with these constraints.
A distribution function on the real numbers , is a function for which whenever , and the limit of at +∞ is 1 and the limit at −∞ is 0. A p-box is a set of distributions functions F satisfying the following constraints, for specified distribution functions F, and specified bounds m1 ≤ m2 on the expected value of the distribution and specified bounds v1 ≤ v2 on the variance of the distribution.
where integrals of the form are Riemann–Stieltjes integrals.
Thus, the constraints are that the distribution function F falls within prescribed bounds, the mean of the distribution is in th |
https://en.wikipedia.org/wiki/Credal%20set | In mathematics, a credal set is a set of probability distributions or, more generally, a set of (possibly only finitely additive) probability measures. A credal set is often assumed or constructed to be a closed convex set. It is intended to express uncertainty or doubt about the probability model that should be used, or to convey the beliefs of a Bayesian agent about the possible states of the world.
If a credal set is closed and convex, then, by the Krein–Milman theorem, it can be equivalently described by its extreme points . In that case, the expectation for a function of with respect to the credal set forms a closed interval , whose lower bound is called the lower prevision of , and whose upper bound is called the upper prevision of :
where denotes a probability measure, and with a similar expression for (just replace by in the above expression).
If is a categorical variable, then the credal set can be considered as a set of probability mass functions over .
If additionally is also closed and convex, then the lower prevision of a function of can be simply evaluated as:
where denotes a probability mass function.
It is easy to see that a credal set over a Boolean variable cannot have more than two extreme points (because the only closed convex sets in are closed intervals), while credal sets over variables that can take three or more values can have any arbitrary number of extreme points.
See also
Imprecise probability
Dempster–Shafer theory
Probability box
Robust Bayes analysis
Upper and lower probabilities
References
Further reading
Bayesian inference
Probability bounds analysis |
https://en.wikipedia.org/wiki/Jim%20Hill%20%28racing%20driver%29 | Jim Hill (1890 in Indianapolis, Indiana – ?) was an American racecar driver.
Career statistics
By season
Indianapolis 500 results
References
External links
1890 births
Year of death missing
Indianapolis 500 drivers
Racing drivers from Indianapolis |
https://en.wikipedia.org/wiki/Torsion%20field | Torsion field can refer to:
A torsion tensor in differential geometry.
The field used in Einstein–Cartan theory and other alternatives to general relativity that involve torsion of spacetime
Torsion field (pseudoscience), a field alleged to make faster-than-light communication and paranormal phenomena possible |
https://en.wikipedia.org/wiki/Barrier%20cone | In mathematics, specifically functional analysis, the barrier cone is a cone associated to any non-empty subset of a Banach space. It is closely related to the notions of support functions and polar sets.
Definition
Let X be a Banach space and let K be a non-empty subset of X. The barrier cone of K is the subset b(K) of X∗, the continuous dual space of X, defined by
Related notions
The function
defined for each continuous linear functional ℓ on X, is known as the support function of the set K; thus, the barrier cone of K is precisely the set of continuous linear functionals ℓ for which σK(ℓ) is finite.
The set of continuous linear functionals ℓ for which σK(ℓ) ≤ 1 is known as the polar set of K. The set of continuous linear functionals ℓ for which σK(ℓ) ≤ 0 is known as the (negative) polar cone of K. Clearly, both the polar set and the negative polar cone are subsets of the barrier cone.
References
Functional analysis |
https://en.wikipedia.org/wiki/Runcinated%205-cubes | In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube.
There are 8 unique degrees of runcinations of the 5-cube, along with permutations of truncations and cantellations. Four are more simply constructed relative to the 5-orthoplex.
Runcinated 5-cube
Alternate names
Small prismated penteract (Acronym: span) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a runcinated 5-cube having edge length 2 are all permutations of:
Images
Runcitruncated 5-cube
Alternate names
Runcitruncated penteract
Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers)
Construction and coordinates
The Cartesian coordinates of the vertices of a runcitruncated 5-cube having edge length 2 are all permutations of:
Images
Runcicantellated 5-cube
Alternate names
Runcicantellated penteract
Prismatorhombated penteract (Acronym: prin) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a runcicantellated 5-cube having edge length 2 are all permutations of:
Images
Runcicantitruncated 5-cube
Alternate names
Runcicantitruncated penteract
Biruncicantitruncated pentacross
great prismated penteract () (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a runcicantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:
Images
Related polytopes
These polytopes are a part of a set of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin
External links
Polytopes of Various Dimensions, Jonathan Bowers
Runcinated uniform polytera (spid), Jonathan Bowers
Multi-dimensional Glossary
5-polytopes |
https://en.wikipedia.org/wiki/Stericated%205-cubes | In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube.
There are eight degrees of sterication for the 5-cube, including permutations of runcination, cantellation, and truncation. The simple stericated 5-cube is also called an expanded 5-cube, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-cube. The highest form, the steriruncicantitruncated 5-cube, is more simply called an omnitruncated 5-cube with all of the nodes ringed.
Stericated 5-cube
Alternate names
Stericated penteract / Stericated 5-orthoplex / Stericated pentacross
Expanded penteract / Expanded 5-orthoplex / Expanded pentacross
Small cellated penteractitriacontaditeron (Acronym: scant) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a stericated 5-cube having edge length 2 are all permutations of:
Images
The stericated 5-cube is constructed by a sterication operation applied to the 5-cube.
Steritruncated 5-cube
Alternate names
Steritruncated penteract
Celliprismated triacontaditeron (Acronym: capt) (Jonathan Bowers)
Construction and coordinates
The Cartesian coordinates of the vertices of a steritruncated 5-cube having edge length 2 are all permutations of:
Images
Stericantellated 5-cube
Alternate names
Stericantellated penteract
Stericantellated 5-orthoplex, stericantellated pentacross
Cellirhombated penteractitriacontiditeron (Acronym: carnit) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a stericantellated 5-cube having edge length 2 are all permutations of:
Images
Stericantitruncated 5-cube
Alternate names
Stericantitruncated penteract
Steriruncicantellated triacontiditeron / Biruncicantitruncated pentacross
Celligreatorhombated penteract (cogrin) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of an stericantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:
Images
Steriruncitruncated 5-cube
Alternate names
Steriruncitruncated penteract / Steriruncitruncated 5-orthoplex / Steriruncitruncated pentacross
Celliprismatotruncated penteractitriacontiditeron (captint) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of an steriruncitruncated penteract having an edge length of 2 are given by all permutations of coordinates and sign of:
Images
Steritruncated 5-orthoplex
Alternate names
Steritruncated pentacross
Celliprismated penteract (Acronym: cappin) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a steritruncated 5-orthoplex, centered at the origin, are all permutations of
Images
Stericantitruncated 5-orthoplex
Alternate names
Stericantitruncated pentacross
Celligreatorhombated triacontaditeron (cogart) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of an stericantitruncat |
https://en.wikipedia.org/wiki/Bockstein%20spectral%20sequence | In mathematics, the Bockstein spectral sequence is a spectral sequence relating the homology with mod p coefficients and the homology reduced mod p. It is named after Meyer Bockstein.
Definition
Let C be a chain complex of torsion-free abelian groups and p a prime number. Then we have the exact sequence:
Taking integral homology H, we get the exact couple of "doubly graded" abelian groups:
where the grading goes: and the same for
This gives the first page of the spectral sequence: we take with the differential . The derived couple of the above exact couple then gives the second page and so forth. Explicitly, we have that fits into the exact couple:
where and (the degrees of i, k are the same as before). Now, taking of
we get:
.
This tells the kernel and cokernel of . Expanding the exact couple into a long exact sequence, we get: for any r,
.
When , this is the same thing as the universal coefficient theorem for homology.
Assume the abelian group is finitely generated; in particular, only finitely many cyclic modules of the form can appear as a direct summand of . Letting we thus see is isomorphic to .
References
J. P. May, A primer on spectral sequences
Spectral sequences |
https://en.wikipedia.org/wiki/Johann%20Sabath | Johann Sabath (born 4 June 1939) is a German former professional footballer who played as a defender.
Career statistics
References
External links
1939 births
Living people
German men's footballers
West German men's footballers
Footballers from Duisburg
Men's association football defenders
Bundesliga players
MSV Duisburg players
VfL Bochum players
1. FC Bocholt players |
https://en.wikipedia.org/wiki/Cantellated%205-orthoplexes | In five-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex.
There are 6 cantellation for the 5-orthoplex, including truncations. Some of them are more easily constructed from the dual 5-cube.
Cantellated 5-orthoplex
Alternate names
Cantellated 5-orthoplex
Bicantellated 5-demicube
Small rhombated triacontiditeron (Acronym: sart) (Jonathan Bowers)
Coordinates
The vertices of the can be made in 5-space, as permutations and sign combinations of:
(0,0,1,1,2)
Images
The cantellated 5-orthoplex is constructed by a cantellation operation applied to the 5-orthoplex.
Cantitruncated 5-orthoplex
Alternate names
Cantitruncated pentacross
Cantitruncated triacontiditeron (Acronym: gart) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a cantitruncated 5-orthoplex, centered at the origin, are all sign and coordinate permutations of
(±3,±2,±1,0,0)
Images
Related polytopes
These polytopes are from a set of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
x3o3x3o4o - sart, x3x3x3o4o - gart
External links
Polytopes of Various Dimensions, Jonathan Bowers
Multi-dimensional Glossary
5-polytopes |
https://en.wikipedia.org/wiki/James%20reduced%20product | In topology, a branch of mathematics, the James reduced product or James construction J(X) of a topological space X with given basepoint e is the quotient of the disjoint union of all powers X, X2, X3, ... obtained by identifying points (x1,...,xk−1,e,xk+1,...,xn) with (x1,...,xk−1, xk+1,...,xn). In other words, its underlying set is the free monoid generated by X (with unit e). It was introduced by .
For a connected CW complex X, the James reduced product J(X) has the same homotopy type as ΩΣX, the loop space of the suspension of X.
The commutative analogue of the James reduced product is called the infinite symmetric product.
References
Algebraic topology |
https://en.wikipedia.org/wiki/Dold%E2%80%93Thom%20theorem | In algebraic topology, the Dold-Thom theorem states that the homotopy groups of the infinite symmetric product of a connected CW complex are the same as its reduced homology groups. The most common version of its proof consists of showing that the composition of the homotopy group functors with the infinite symmetric product defines a reduced homology theory. One of the main tools used in doing so are quasifibrations. The theorem has been generalised in various ways, for example by the Almgren isomorphism theorem.
There are several other theorems constituting relations between homotopy and homology, for example the Hurewicz theorem. Another approach is given by stable homotopy theory. Thanks to the Freudenthal suspension theorem, one can see that the latter actually defines a homology theory. Nevertheless, none of these allow one to directly reduce homology to homotopy. This advantage of the Dold-Thom theorem makes it particularly interesting for algebraic geometry.
The theorem
Dold-Thom theorem. For a connected CW complex X one has πnSP(X) ≅ H̃n(X), where H̃n denotes reduced homology and SP stands for the infinite symmetric product.
It is also very useful that there exists an isomorphism φ : πnSP(X) → H̃n(X) which is compatible with the Hurewicz homomorphism h: πn(X) → H̃n(X), meaning that one has a commutative diagram
where i* is the map induced by the inclusion i: X = SP1(X) → SP(X).
The following example illustrates that the requirement of X being a CW complex cannot be dropped offhand:
Let X = CH ∨ CH be the wedge sum of two copies of the cone over the Hawaiian earring. The common point of the two copies is supposed to be the point 0 ∈ H meeting every circle. On the one hand, H1(X) is an infinite group while H1(CH) is trivial. On the other hand, π1(SP(X)) ≅ π1(SP(CH)) × π1(SP(CH)) holds since φ : SP(X) × SP(Y) → SP(X ∨ Y) defined by φ([x1, ..., xn], [y1, ..., yn]) = ([x1, ..., xn, y1, ..., yn]) is a homeomorphism for compact X and Y.
But this implies that either π1(SP(CH)) ≅ H1(CH) or π1(SP(X)) ≅ H1(X) does not hold.
Sketch of the proof
One wants to show that the family of functors hn = πn ∘ SP defines a homology theory. Dold and Thom chose in their initial proof a slight modification of the Eilenberg-Steenrod axioms, namely calling a family of functors (h̃n)n∈N0 from the category of basepointed, connected CW complexes to the category of abelian groups a reduced homology theory if they satisfy
If f ≃ g: X → Y, then f* = g*: h̃n(X) → h̃n(Y), where ≃ denotes pointed homotopy equivalence.
There are natural boundary homomorphisms ∂ : h̃n(X/A) → h̃n−1(A) for each pair (X, A) with X and A being connected, yielding an exact sequence where i: A → X is the inclusion and q: X → X/A is the quotient map.
h̃n(S1) = 0 for n ≠ 1, where S1 denotes the circle.
Let (Xλ) be the system of compact subspaces of a pointed space X containing the basepoint. Then (Xλ) is a direct system together with the inclusions. Denote by respectively the inclu |
https://en.wikipedia.org/wiki/Albrecht%20Dold | Albrecht Dold (5 August 1928 – 26 September 2011) was a German mathematician specializing in algebraic topology who proved the Dold–Thom theorem, the Dold–Kan correspondence, and introduced Dold manifolds, Dold–Puppe stabilization, and Dold fibrations.
Life
Albrecht Dold was born in Triberg, and studied mathematics and physics at Heidelberg University, earning a Ph.D. degree in 1954 under the direction of Herbert Seifert. He visited the Institute for Advanced Study in Princeton in 1956–58, and taught at Columbia University in 1960–62 and at the University of Zürich in 1962–63. In 1963 he returned to Heidelberg, where he stayed most of his career, till his retirement in 1996.
Dold's work in algebraic topology, in particular, his work on fixed-point theory has made him known in economics as well as mathematics. His book "Lectures on Algebraic Topology" is a standard reference among economists as well as mathematicians. He had 19 doctoral students, including Mónica Clapp, Eberhard Freitag, Volker Puppe, and Carl-Friedrich Bödigheimer, as well as 58 descendants.
Dold was married to Yvonne Dold-Samplonius, a distinguished historian of mathematics.
References
1928 births
2011 deaths
20th-century German mathematicians
21st-century German mathematicians
Topologists
Heidelberg University alumni
Academic staff of Heidelberg University
People from Schwarzwald-Baar-Kreis |
https://en.wikipedia.org/wiki/Dold%20manifold | In mathematics, a Dold manifold is one of the manifolds , where is the involution that acts as −1 on the m-sphere and as complex conjugation on the complex projective space . These manifolds were constructed by , who used them to give explicit generators for René Thom's unoriented cobordism ring. Note that , the real projective space of dimension m, and .
References
Algebraic topology
Manifolds |
https://en.wikipedia.org/wiki/A5%20polytope | {{DISPLAYTITLE:A5 polytope}}
In 5-dimensional geometry, there are 19 uniform polytopes with A5 symmetry. There is one self-dual regular form, the 5-simplex with 6 vertices.
Each can be visualized as symmetric orthographic projections in the Coxeter planes of the A5 Coxeter group and other subgroups.
Graphs
Symmetric orthographic projections of these 19 polytopes can be made in the A5, A4, A3, A2 Coxeter planes. Ak graphs have [k+1] symmetry. For even k and symmetrically nodea_1ed-diagrams, symmetry doubles to [2(k+1)].
These 19 polytopes are each shown in these 4 symmetry planes, with vertices and edges drawn and vertices colored by the number of overlapping vertices in each projective position.
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover, New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, and Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2, 10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
External links
Notes
5-polytopes |
https://en.wikipedia.org/wiki/B5%20polytope | {{DISPLAYTITLE:B5 polytope}}
In 5-dimensional geometry, there are 31 uniform polytopes with B5 symmetry. There are two regular forms, the 5-orthoplex, and 5-cube with 10 and 32 vertices respectively. The 5-demicube is added as an alternation of the 5-cube.
They can be visualized as symmetric orthographic projections in Coxeter planes of the B5 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 32 polytopes can be made in the B5, B4, B3, B2, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry.
These 32 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
External links
Notes
5-polytopes |
https://en.wikipedia.org/wiki/D5%20polytope | {{DISPLAYTITLE:D5 polytope}}
In 5-dimensional geometry, there are 23 uniform polytopes with D5 symmetry, 8 are unique, and 15 are shared with the B5 symmetry. There are two special forms, the 5-orthoplex, and 5-demicube with 10 and 16 vertices respectively.
They can be visualized as symmetric orthographic projections in Coxeter planes of the D6 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 8 polytopes can be made in the D5, D4, D3, A3, Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry. The B5 plane is included, with only half the [10] symmetry displayed.
These 8 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Notes
5-polytopes |
https://en.wikipedia.org/wiki/Symmetric%20product%20%28topology%29 | In algebraic topology, the nth symmetric product of a topological space consists of the unordered n-tuples of its elements. If one fixes a basepoint, there is a canonical way of embedding the lower-dimensional symmetric products into the higher-dimensional ones. That way, one can consider the colimit over the symmetric products, the infinite symmetric product. This construction can easily be extended to give a homotopy functor.
From an algebraic point of view, the infinite symmetric product is the free commutative monoid generated by the space minus the basepoint, the basepoint yielding the identity element. That way, one can view it as the abelian version of the James reduced product.
One of its essential applications is the Dold-Thom theorem, stating that the homotopy groups of the infinite symmetric product of a connected CW complex are the same as the reduced homology groups of that complex. That way, one can give a homotopical definition of homology.
Definition
Let X be a topological space and n ≥ 1 a natural number. Define the nth symmetric product of X or the n-fold symmetric product of X as the space
Here, the symmetric group Sn acts on Xn by permuting the factors. Hence, the elements of SPn(X) are the unordered n-tuples of elements of X. Write [x1, ..., xn] for the point in SPn(X) defined by (x1, ..., xn) ∈ Xn.
Note that one can define the nth symmetric product in any category where products and colimits exist. Namely, one then has canonical isomorphisms φ : X × Y → Y × X for any objects X and Y and can define the action of the transposition on Xn as , thereby inducing an action of the whole Sn on Xn. This means that one can consider symmetric products of objects like simplicial sets as well. Moreover, if the category is cartesian closed, the distributive law X × (Y ∐ Z) ≅ X × Y ∐ X × Z holds and therefore one gets
If (X, e) is a based space, it is common to set SP0(X) = {e}. Further, Xn can then be embedded into Xn+1 by sending (x1, ..., xn) to (x1, ..., xn, e). This clearly induces an embedding of SPn(X) into SPn+1(X). Therefore, the infinite symmetric product can be defined as
A definition avoiding category theoretic notions can be given by taking SP(X) to be the union of the increasing sequence of spaces SPn(X) equipped with the direct limit topology. This means that a subset of SP(X) is open if and only if all its intersections with the SPn(X) are open. We define the basepoint of SP(X) as [e]. That way, SP(X) becomes a based space as well.
One can generalise this definition as well to pointed categories where products and colimits exist. Namely, in this case one has a canonical map Xn → Xn+1, induced by the identity Xn → Xn and the zero map Xn → X. So this results in a direct system of the symmetric products, too and one can therefore define its colimit as the infinite symmetric product.
Examples
SPn(I) is the same as the n-dimensional standard simplex Δn, where I denotes the unit interval.
SPn(S1) can be identifie |
https://en.wikipedia.org/wiki/Boolean-valued | Boolean-valued usually refers to:
in most applied fields: something taking one of two values (example: True or False, On or Off, 1 or 0) referring to two-element Boolean algebra (the Boolean domain), e.g. Boolean-valued function or Boolean data type
in mathematics: something taking values over an arbitrary, abstract Boolean algebra, for example Boolean-valued model
See also
Boolean algebra further explains the distinction
Mathematical concepts
Logic and statistics |
https://en.wikipedia.org/wiki/Ramin%20Takloo-Bighash | Ramin Takloo-Bighash (born 1974) is a mathematician who works in the field of automorphic forms and Diophantine geometry and is a professor at the University of Illinois at Chicago.
Mathematical career
Takloo-Bighash graduated from the Sharif University of Technology, where he enrolled after winning a Silver medal at the 1992 International Mathematical Olympiad. In 2001, Takloo-Bighash graduated under Joseph Shalika from Johns Hopkins University. He spent 2001-2007 at Princeton University, first as an instructor and then as an assistant professor. He is a professor at the University of Illinois at Chicago.
Research
Takloo-Bighash computed the local factors of spinor L-function attached to generic automorphic forms on the symplectic group GSp(4). He has joint works with Joseph Shalika and Yuri Tschinkel on the distribution of rational points on certain group compactifications. He is a co-author, with Steven J. Miller, of An Invitation To Modern Number Theory (Princeton University Press, 2006).
Books
External links
Takloo-Bighash's web page at UIC
1974 births
Living people
21st-century Iranian mathematicians
Iranian Mathematics Competition Medalists
Number theorists
Johns Hopkins University alumni |
https://en.wikipedia.org/wiki/List%20of%20FC%20Seoul%20managers | This article is regarding all FC Seoul managers.
Statistics
Managerial history
Match results
※ Win%, Draw%, Lose%, GFA, GAA: Only K League regular season (included K League Championship) and League Cup matches are counted.
※ Penalty shoot-outs results in 1993, 1998, 1999, 2000 seasons are not counted by K League's principle of official statistics.
Honours
References
FC Seoul Matchday Magazine
External links
FC Seoul Official Website
L2 |
https://en.wikipedia.org/wiki/Steenrod%20homology | In algebraic topology, Steenrod homology is a homology theory for compact metric spaces introduced by , based on regular cycles.
It is similar to the homology theory introduced rather sketchily by Andrey Kolmogorov in 1936.
References
Homology theory |
https://en.wikipedia.org/wiki/2011%20DPR%20Korea%20Football%20League | Statistics of DPR Korea Football League for the 2011 season.
Overview
The championship was played over six rounds, after which the top four teams – April 25, Kigwanch'a, Sobaeksu, and Amrokkang – played a final tournament in P'yŏngyang in November 2011, which was won by April 25.
Final standings
Clubs
4.25 (Namp'o)
Amrokkang (P'yŏngyang)
Ponghwasan
Kyŏnggong'ŏp
Kigwanch'a (Sinŭiju)
Maebong
Man'gyŏngbong (P'yŏngyang)
P'yōngyang City Sports Club (P'yŏngyang)
Rimyŏngsu (Sariwŏn)
Ryongnamsan
Sobaeksu (P'yŏngyang)
References
DPR Korea Football League seasons
1
Korea
Korea |
https://en.wikipedia.org/wiki/Yasha%20Asley | Yasha Asley is a British mathematics child prodigy of Iranian descent.
Life
Raised solely by his father, Moussa, Yasha was gifted at maths from an early age. When he started primary school at the age of four, he was described as having the mathematical ability of a typical twelve-year-old. While attending school Asley passed the A-level exams and became the youngest person to have achieved the 'A' grade in Mathematics at the age of 8 in 2011, surpassing a record previously held by March Tian Boedihardjo.
For three years prior to starting his university education his full-time schooling was split, three days a week at primary school and two days a week he learned mathematics with Professor Alexander Gorban at the University of Leicester.
Although he had met UK university entrance requirements by scoring top grades and was in a position to apply to UK universities, he wanted to continue in the standard education system. In an ITV interview, on 24 July 2018, Yasha said, "... It wasn't our first choice [to go to university]. But then it seemed like the headteacher had told my teachers to slow me down... So we decided that I really had no choice [other than to go to university]." He began his university course at the age of twelve.
In 2017, at the age of 15, he became the university's youngest-ever graduate receiving a first-class degree in mathematics.
References
Living people
People from Leicester
21st-century British mathematicians
2000s births |
https://en.wikipedia.org/wiki/Hilton%27s%20theorem | In algebraic topology, Hilton's theorem, proved by , states that the loop space of a wedge of spheres is homotopy-equivalent to a product of loop spaces of spheres.
showed more generally that the loop space of the suspension of a wedge of spaces can be written as an infinite product of loop spaces of suspensions of smash products.
References
1955 in mathematics
Theorems in algebraic topology |
https://en.wikipedia.org/wiki/Isotypic%20component | The isotypic component of weight of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight .
Definition
A finite-dimensional module of a reductive Lie algebra (or of the corresponding Lie group) can be decomposed into irreducible submodules
.
Each finite-dimensional irreducible representation of is uniquely identified (up to isomorphism) by its highest weight
, where denotes the highest weight module with highest weight .
In the decomposition of , a certain isomorphism class might appear more than once, hence
.
This defines the isotypic component of weight of V:
where is maximal.
See also
Lie algebra representation
Weight (representation theory)
Semisimple representation#Isotypic decomposition
References
Representation theory of Lie algebras |
https://en.wikipedia.org/wiki/Sole%20trader%20insolvency | According to the Office for National Statistics, sole proprietors represented 23.8% of all UK enterprise in 2010. Of that number, more than half a million sole traders were operating via the PAYE or VAT system alone. Sole traders are a distinct legal entity, operating as one type of UK business structure. In the event of financial problems affecting the business, they are subject to different rules to those that govern companies.
Sole trader insolvency occurs when the business cannot meet financial obligations. It may be that bills cannot be paid on time, leading to debts which eventually attract legal action by creditors. Insolvency does not automatically equate to bankruptcy; definitions of insolvency are provided within the Insolvency Act 1986. Cash flow insolvency occurs when a business cannot meet its credit obligations as they fall due. Balance sheet insolvency occurs when the businesses’ liabilities exceed its assets. According to Business Link there are a number of factors that can lead to sole trader insolvency. These can include late invoicing for goods or services, accepting orders that exceed its financial capacity to deliver, failure to recover debts, excess inventory and unsuitable credit arrangements and often personal drawings taken in excess of profit.
UK Insolvency statistics
During 2010, the recorded number of individual insolvencies in England and Wales was 135,089 according to the UK Government’s Insolvency Service (including provisional figures from the final quarter). The figures had fallen by 13.6% during the final quarter compared to the same quarter during 2009. 12,049 individuals declared formal bankruptcy, a drop of almost a third (29.2%) on the previous year. 12,058 entered formal Individual Voluntary Arrangements (IVAs), representing an annual drop of 5.4%. A further 6,172 entered a formal Debt Relief Order (DRO), an increase of 15.4% on the previous year’s figures. The levels of self-employed bankruptcies had fallen slightly by the third quarter of 2010 to make up 11.9% of the total within England and Wales. This represented an improvement on the number of bankrupt sole traders during the previous year and up to June 2010.
UK Insolvency Law and sole traders
Insolvency Act 1986
This legislation provides the legal framework for two key formal insolvency solutions relevant to sole traders: namely bankruptcy and Individual Voluntary Arrangements. It also makes provision for company insolvency
Bankruptcy laws vary somewhat between Scotland, Northern Ireland, Wales and England. In England, Wales & Northern Ireland, the applicable law is the Insolvency Act 1986. Bankruptcy requires the surrender of all valuable assets to the Official Receiver, including any property interests. It is extremely unlikely the business activities would be permitted to continue. Additionally, there are quite a number of other legal restrictions upon the bankrupt individual. Individuals are therefore cautioned by the Insolvency Servi |
https://en.wikipedia.org/wiki/Vincent%20Blondel | Vincent Daniel Blondel (born April 28, 1965) is a Belgian professor of applied mathematics and current rector of the University of Louvain (UCLouvain) and a visiting professor at the Massachusetts Institute of Technology (MIT). Blondel's research lies in the area of mathematical control theory and theoretical computer science. He is mostly known for his contributions in computational complexity in control, multi-agent coordination and complex networks.
Education
Blondel studied philosophy, mathematics, engineering and computer science in Louvain-la-Neuve, Grenoble, London and Oxford. He completed a master thesis in engineering at the Institut National Polytechnique de Grenoble, he holds a MSc in mathematics from Imperial College of Science and Technology and a degree in philosophy, a master's degree in engineering (summa cum laude) and a PhD in applied mathematics from Université catholique de Louvain.
Career
In 1993-1994 he was a Göran Gustafsson Fellow at the Royal Institute of Technology (Stockholm) and in 1994-1995 he was a Research Fellow at the National Institute for Research in Computer Science and Control (INRIA) in Paris. From 1995 to 1999 he was an assistant professor at the Institute of Mathematics of the Université de Liège before joining the Louvain School of Engineering of UCLouvain where he has been since then. He was a research visitor with the Australian National University, the University of California at Berkeley, the Santa Fe Institute, the Mittag-Leffler Institute of the Royal Swedish Academy of Sciences and Harvard University. He was a visiting professor of the Ecole Nationale Supérieure in Lyon in 1998 and at the Université Paris VII - Diderot in 1999, 2000 and 2002. In 2005-2006 he was a visiting professor and a Fulbright scholar with the Department of Electrical Engineering and Computer Science of the Massachusetts Institute of Technology. In 2010-2011 he was a visiting professor with the MIT Laboratory for Information and Decision Systems (LIDS) of the Massachusetts Institute of Technology.
Blondel is a former associate editor of the European Journal of Control (Springer) and of Systems and Control Letters (Elsevier). He is an editor of the journal Mathematics of Control, Signals, and Systems He has published about 100 journal articles and 6 books.
At UCLouvain, Blondel has founded the Group on Large Graphs and Networks. He has supervised 20 doctoral and postdoctoral researchers and 15 visiting professors. He was department head in 2003-2010 and a university president candidate in 2009.
In 2013, Blondel has become the dean of the Louvain School of Engineering.
He was elected the Rector of the University of Louvain for the term 2014-2019, and reelected again in 2019 until 2024.
Honors and awards
Grant from the Trustees of the Mathematics Institute of Oxford University (1992)
Prize Agathon De Potter of the Belgian Royal Academy of Science (1993)
Prize Paul Dubois of the Montefiore Institute (1993)
Trienna |
https://en.wikipedia.org/wiki/List%20of%20South%20Korean%20regions%20by%20GDP | This is a list of South Korean regions by GDP. All data are sourced from the latest regional statistics published by the South Korean Government, the OECD and the International Monetary Fund (IMF). The South Korean won has been converted to the international dollar using the IMF's Purchasing Power Parity conversion rate.
By GDP (PPP, 2016)
By GDP per capita (nominal) (2021)
See also
Administrative divisions of South Korea
Economy of South Korea
List of subnational entities
References
S
GDP
GDP
South Korea |
https://en.wikipedia.org/wiki/Posetal%20category | In mathematics, specifically category theory, a posetal category, or thin category, is a category whose homsets each contain at most one morphism. As such, a posetal category amounts to a preordered class (or a preordered set, if its objects form a set). As suggested by the name, the further requirement that the category be skeletal is often assumed for the definition of "posetal"; in the case of a category that is posetal, being skeletal is equivalent to the requirement that the only isomorphisms are the identity morphisms, equivalently that the preordered class satisfies antisymmetry and hence, if a set, is a poset.
All diagrams commute in a posetal category. When the commutative diagrams of a category are interpreted as a typed equational theory whose objects are the types, a codiscrete posetal category corresponds to an inconsistent theory understood as one satisfying the axiom x = y at all types.
Viewing a 2-category as an enriched category whose hom-objects are categories, the hom-objects of any extension of a posetal category to a 2-category having the same 1-cells are monoids.
Some lattice-theoretic structures are definable as posetal categories of a certain kind, usually with the stronger assumption of being skeletal. For example, under this assumption, a poset may be defined as a small posetal category, a distributive lattice as a small posetal distributive category, a Heyting algebra as a small posetal finitely cocomplete cartesian closed category, and a Boolean algebra as a small posetal finitely cocomplete *-autonomous category. Conversely, categories, distributive categories, finitely cocomplete cartesian closed categories, and finitely cocomplete *-autonomous categories can be considered the respective categorifications of posets, distributive lattices, Heyting algebras, and Boolean algebras.
References
Category theory |
https://en.wikipedia.org/wiki/Beltrami%20equation | In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation
for w a complex distribution of the complex variable z in some open set U, with derivatives that are locally L2, and where μ is a given complex function in L∞(U) of norm less than 1, called the Beltrami coefficient, and where and are Wirtinger derivatives. Classically this differential equation was used by Gauss to prove the existence locally of isothermal coordinates on a surface with analytic Riemannian metric. Various techniques have been developed for solving the equation. The most powerful, developed in the 1950s, provides global solutions of the equation on C and relies on the Lp theory of the Beurling transform, a singular integral operator defined on Lp(C) for all 1 < p < ∞. The same method applies equally well on the unit disk and upper half plane and plays a fundamental role in Teichmüller theory and the theory of quasiconformal mappings. Various uniformization theorems can be proved using the equation, including the measurable Riemann mapping theorem and the simultaneous uniformization theorem. The existence of conformal weldings can also be derived using the Beltrami equation. One of the simplest applications is to the Riemann mapping theorem for simply connected bounded open domains in the complex plane. When the domain has smooth boundary, elliptic regularity for the equation can be used to show that the uniformizing map from the unit disk to the domain extends to a C∞ function from the closed disk to the closure of the domain.
Metrics on planar domains
Consider a 2-dimensional Riemannian manifold, say with an (x, y) coordinate system on it. The curves of constant x on that surface typically don't intersect the curves of constant y orthogonally. A new coordinate system (u, v) is called isothermal when the curves of constant u do intersect the curves of constant v orthogonally and, in addition, the parameter spacing is the same — that is, for small enough h, the little region with and is nearly square, not just nearly rectangular. The Beltrami equation is the equation that has to be solved in order to construct isothermal coordinate systems.
To see how this works, let S be an open set in C and let
be a smooth metric g on S. The first fundamental form of g
is a positive real matrix (E > 0, G > 0, EG − F2 > 0) that varies smoothly with x and y.
The Beltrami coefficient of the metric g is defined to be
This coefficient has modulus strictly less than one since the identity
implies that
Let f(x,y) =(u(x,y),v(x,y)) be a smooth diffeomorphism of S onto another open set T in C. The map f preserves orientation just when its Jacobian is positive:
And using f to pull back to S the standard Euclidean metric ds2 = du2 + dv2 on T induces a metric on S given by
a metric whose first fundamental form is
When f both preserves orientation and induces a metric that differs from the original metric g only by a positive, smoothl |
https://en.wikipedia.org/wiki/Mean%20absolute%20scaled%20error | In statistics, the mean absolute scaled error (MASE) is a measure of the accuracy of forecasts. It is the mean absolute error of the forecast values, divided by the mean absolute error of the in-sample one-step naive forecast. It was proposed in 2005 by statistician Rob J. Hyndman and Professor of Decision Sciences Anne B. Koehler, who described it as a "generally applicable measurement of forecast accuracy without the problems seen in the other measurements." The mean absolute scaled error has favorable properties when compared to other methods for calculating forecast errors, such as root-mean-square-deviation, and is therefore recommended for determining comparative accuracy of forecasts.
Rationale
The mean absolute scaled error has the following desirable properties:
Scale invariance: The mean absolute scaled error is independent of the scale of the data, so can be used to compare forecasts across data sets with different scales.
Predictable behavior as : Percentage forecast accuracy measures such as the Mean absolute percentage error (MAPE) rely on division of , skewing the distribution of the MAPE for values of near or equal to 0. This is especially problematic for data sets whose scales do not have a meaningful 0, such as temperature in Celsius or Fahrenheit, and for intermittent demand data sets, where occurs frequently.
Symmetry: The mean absolute scaled error penalizes positive and negative forecast errors equally, and penalizes errors in large forecasts and small forecasts equally. In contrast, the MAPE and median absolute percentage error (MdAPE) fail both of these criteria, while the "symmetric" sMAPE and sMdAPE fail the second criterion.
Interpretability: The mean absolute scaled error can be easily interpreted, as values greater than one indicate that in-sample one-step forecasts from the naïve method perform better than the forecast values under consideration.
Asymptotic normality of the MASE: The Diebold-Mariano test for one-step forecasts is used to test the statistical significance of the difference between two sets of forecasts. To perform hypothesis testing with the Diebold-Mariano test statistic, it is desirable for , where is the value of the test statistic. The DM statistic for the MASE has been empirically shown to approximate this distribution, while the mean relative absolute error (MRAE), MAPE and sMAPE do not.
Non seasonal time series
For a non-seasonal time series, the mean absolute scaled error is estimated by
where the numerator ej is the forecast error for a given period (with J, the number of forecasts), defined as the actual value (Yj) minus the forecast value (Fj) for that period: ej = Yj − Fj, and the denominator is the mean absolute error of the one-step "naive forecast method" on the training set (here defined as t = 1..T), which uses the actual value from the prior period as the forecast: Ft = Yt−1
Seasonal time series
For a seasonal time series, the mean absolute scaled error is estimate |
https://en.wikipedia.org/wiki/Inflation-restriction%20exact%20sequence | In mathematics, the inflation-restriction exact sequence is an exact sequence occurring in group cohomology and is a special case of the five-term exact sequence arising from the study of spectral sequences.
Specifically, let G be a group, N a normal subgroup, and A an abelian group which is equipped with an action of G, i.e., a homomorphism from G to the automorphism group of A. The quotient group G/N acts on
AN = { a ∈ A : na = a for all n ∈ N}.
Then the inflation-restriction exact sequence is:
0 → H 1(G/N, AN) → H 1(G, A) → H 1(N, A)G/N → H 2(G/N, AN) →H 2(G, A)
In this sequence, there are maps
inflation H 1(G/N, AN) → H 1(G, A)
restriction H 1(G, A) → H 1(N, A)G/N
transgression H 1(N, A)G/N → H 2(G/N, AN)
inflation H 2(G/N, AN) →H 2(G, A)
The inflation and restriction are defined for general n:
inflation Hn(G/N, AN) → Hn(G, A)
restriction Hn(G, A) → Hn(N, A)G/N
The transgression is defined for general n
transgression Hn(N, A)G/N → Hn+1(G/N, AN)
only if Hi(N, A)G/N = 0 for i ≤ n − 1.
The sequence for general n may be deduced from the case n = 1 by dimension-shifting or from the Lyndon–Hochschild–Serre spectral sequence.
References
Homological algebra |
https://en.wikipedia.org/wiki/Beurling%20zeta%20function | In mathematics, a Beurling zeta function is an analogue of the Riemann zeta function where the ordinary primes are replaced by a set of Beurling generalized primes: any sequence of real numbers greater than 1 that tend to infinity. These were introduced by .
A Beurling generalized integer is a number that can be written as a product of Beurling generalized primes. Beurling generalized the usual prime number theorem to Beurling generalized primes. He showed that if the number N(x) of Beurling generalized integers less than x is of the form N(x) = Ax + O(x log−γx) with γ > 3/2 then the number of Beurling generalized primes less than x is asymptotic to x/log x, just as for ordinary primes,
but if γ = 3/2 then this conclusion need not hold.
See also
Abstract analytic number theory
References
Zeta and L-functions |
https://en.wikipedia.org/wiki/Opposite%20group | In group theory, a branch of mathematics, an opposite group is a way to construct a group from another group that allows one to define right action as a special case of left action.
Monoids, groups, rings, and algebras can be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc.
Definition
Let be a group under the operation . The opposite group of , denoted , has the same underlying set as , and its group operation is defined by .
If is abelian, then it is equal to its opposite group. Also, every group (not necessarily abelian) is naturally isomorphic to its opposite group: An isomorphism is given by . More generally, any antiautomorphism gives rise to a corresponding isomorphism via , since
Group action
Let be an object in some category, and be a right action. Then is a left action defined by , or .
See also
Opposite ring
Opposite category
External links
http://planetmath.org/oppositegroup
Group theory
Representation theory |
https://en.wikipedia.org/wiki/Peter%20Orno | Beginning in 1974, the fictitious Peter Orno (alternatively, Peter Ørno, P. Ørno, and P. Orno) appeared as the author of research papers in mathematics. According to Robert Phelps, the name "P. Orno" is a pseudonym that was inspired by "porno", an abbreviation for "pornography". Orno's short papers have been called "elegant" contributions to functional analysis. Orno's theorem on linear operators is important in the theory of Banach spaces. Research mathematicians have written acknowledgments that have thanked Orno for stimulating discussions and for Orno's generosity in allowing others to publish his results. The Mathematical Association of America's journals have also published more than a dozen problems whose solutions were submitted in the name of Orno.
Biography
Peter Orno appears as the author of short papers written by an anonymous mathematician; thus "Peter Orno" is a pseudonym. According to Robert R. Phelps, the name "P. Orno" was inspired by "porno", a shortening of "pornography".
Orno's papers list his affiliation as the Department of Mathematics at Ohio State University. This affiliation is confirmed in the description of Orno as a "special creation" at Ohio State in Pietsch's History of Banach spaces and linear operators.
The publications list of Ohio State mathematician Gerald Edgar includes two items that were published under the name of Orno. Edgar indicates that he published them "as Peter Ørno".
Research
His papers feature "surprisingly simple" proofs and solutions to open problems in functional analysis and approximation theory, according to reviewers from Mathematical Reviews: In one case, Orno's "elegant" approach was contrasted with the previously known "elementary, but masochistic" approach. Peter Orno's "permanent interest and sharp criticism stimulated" the "work" on Lectures on Banach spaces of analytic functions by Aleksander Pełczyński, which includes several of Orno's unpublished results. Tomczak-Jaegermann thanked Peter Orno for his stimulating discussions.
Selected publications
Peter Orno has published in research journals and in collections; his papers have always been short, having lengths between one and three pages. Orno has also established himself as a formidable solver of mathematical problems in peer-reviewed journals published by the Mathematical Association of America.
Research papers
According to Mathematical Reviews (), this paper proves the following theorem, which has come to be known as "Orno's theorem": Suppose that E and F are Banach lattices, where F is an infinite-dimensional vector space that contains no Riesz subspace that is uniformly isomorphic to the sequence space equipped with the supremum norm. If each linear operator in the uniform closure of the finite-rank operators from E to F has a Riesz decomposition as the difference of two positive operators, then E can be renormed so that it is an L-space (in the sense of Kakutani and Birkhoff).
According to Mathematical Reviews (), |
https://en.wikipedia.org/wiki/John%20Quackenbush | John Quackenbush is an American computational biologist and genome scientist. He is a professor of biostatistics and computational biology and a professor of cancer biology at the Dana–Farber Cancer Institute (DFCI), as well as the director of its Center for Cancer Computational Biology (CCCB). Quackenbush also holds an appointment as a professor of computational biology and bioinformatics in the Department of Biostatistics at the Harvard School of Public Health.
Education and early life
A native of Mountain Top, Pennsylvania, Quackenbush attended Bishop Hoban High School in Wilkes Barre, graduating in 1979, after which he attended the California Institute of Technology, where he earned a bachelor's degree in physics. He went on to earn a doctorate in theoretical physics from the University of California, Los Angeles in 1990.
Career and research
After working two years as a postdoctoral fellow in physics, Quackenbush was awarded a Special Emphasis Research Career Award from the National Center for Human Genome Research (the predecessor of the National Human Genome Research Institute), and subsequently spent the next two years at the Salk Institute working on physical maps of human chromosome 11, followed by another two years at Stanford University developing new laboratory and computational strategies for sequencing the human genome.
In 1997, Quackenbush joined the faculty of The Institute for Genomic Research (TIGR) in Rockville, Maryland, where his focus began to shift to post-genomic applications, with an emphasis on microarray analysis. Using a combination of laboratory and computational approaches, Quackenbush and his group developed analytical methods based on the integration of data across domains to derive biological meaning from high-dimensional data.
In 2005, Quackenbush was appointed to his current positions at the Dana–Farber Cancer Institute and the Harvard School of Public Health. Four years later, he launched the DFCI's Center for Cancer Computational Biology, which he directs and which provides broad-based bioinformatics and computational biology support to the research community through a collaborative consulting model, and which also performs and analyzes large-scale second-generation DNA sequencing.
Quackenbush's current research focuses on the analysis of human cancer using systems biology-based approaches to understanding and modeling the biological networks that underlie disease. This has led him and his colleagues to make fundamental discoveries about the role that variation in gene expression plays in defining biological phenotypes.
In 2010, Quackenbush and his colleagues at CCCB, together with investigators at National Jewish Health's Center for Genes, Environment and Health, University of Pittsburgh's Dorothy P. and Richard P. Simmons Center for Interstitial Lung Disease, Boston University's Section for Computational Biomedicine and the Pulmonary Center, and the University of Colorado Denver's Genomics Core Facil |
https://en.wikipedia.org/wiki/Benjamin%20Osgood%20Peirce | Benjamin Osgood Peirce (February 11, 1854 – January 14, 1914) was an American mathematician and a holder of the Hollis Chair of Mathematics and Natural Philosophy at Harvard from 1888 until his death in 1914.
Early life
Benjamin Osgood Peirce was born to M. (née Seccomb) and Benjamin Osgood Peirce on February 11, 1854, in Beverly, Massachusetts. In 1876, he graduated from Harvard College. He then received a PhD from the Leipzig University in 1879. He then studied in Berlin, Germany for another year.
Career
Peirce taught at the Boston Latin School for one year. From 1881 to 1884, he taught mathematics at Harvard University. He then taught mathematics and physics as an assistant professor until 1888. In 1888, he became the Hollis Chair of Mathematics and Natural Philosophy.
Peirce was elected to the Council of the American Mathematical Society, serving from 1896 to 1898. He was a founder of the American Physical Society when it began in 1899 and was elected to the National Academy of Sciences (United States) in 1906. He was honoured with election to foreign academies such as the Mathematical Circle of Palermo and the French Physical Society. In 1910 he was awarded an honorary degree by Harvard University. In 1912 he represented Harvard University at the celebrations for the 250th anniversary of the founding of the Royal Society of London.
Personal life
Peirce married Isabella Turnbull Landreth in 1882. Together, they had two daughters.
Removed by several degrees, he was a cousin of Charles Sanders Peirce, whose father, Benjamin Peirce, worked as the academic advisor to Joseph Lovering, Benjamin Osgood Peirce's predecessor as holder of the Hollis Chair of Mathematics and Natural Philosophy.
Death
Peirce died at his home in Cambridge from angina on January 14, 1914. He is buried at the Central Cemetery in Beverly.
Works
A list of all the publications of Benjamin Osgood Peirce:
(with Edward B Lefavour) "On the effect of armatures on the magnetic state of electromagnets", Proc. Amer. Acad. Arts Sciences 10 (1875), 385–386.
"On the induction spark produced in breaking a galvanic circuit between the poles of a magnet", Proc. Amer. Acad. Arts Sciences 11 (1875), 218–227.
"On a new method of comparing the electromotive forces of two batteries and measuring their internal resistance", Proc. Amer. Acad. Arts Sciences 12 (1877), 137–140.
"On a new method of measuring the resistance of a galvanic battery", Proc. Amer. Acad. Arts Sciences 12 (1877), 140–142.
"Note on the determination of the law of propagation of heat in the interior of a solid body", Proc. Amer. Acad. Arts Sciences 12 (1877), 143–149.
(with Edward B Lefavour) "Preliminary work on the determination of the law of propagation of heat in the interior of solid bodies", Proc. Amer. Acad. Arts Sciences 13 (1877), 128–140.
"Über die Emissionsspectra der Haloid-verbindungen des Quecksilbers", Annalen der Physik und Chemie 242 (4) (1879), 597–599.
Über die Electromotorische Knifte von |
https://en.wikipedia.org/wiki/L.%20M.%20Milne-Thomson | Louis Melville Milne-Thomson CBE FRSE RAS (1 May 1891 – 21 August 1974) was an English applied mathematician who wrote several classic textbooks on applied mathematics, including The Calculus of Finite Differences, Theoretical Hydrodynamics, and Theoretical Aerodynamics. He is also known for developing several mathematical tables such as Jacobian Elliptic Function Tables. The Milne-Thomson circle theorem and the Milne-Thomson method for finding a holomorphic function are named after him. Milne-Thomson was made a Commander of the Order of the British Empire (CBE) in 1952.
Early years and career
Milne-Thomson was born in Ealing, London, England on 1 May 1891 to Colonel Alexander Milne-Thomson, a physician and Eva Mary Milne, the daughter of the Revd J. Milne. He was the eldest of his parents' sons. He studied at Clifton College in Bristol as a classical scholar for three years. After securing a scholarship, Milne-Thomson joined Corpus Christi College, Cambridge in 1909 and received part I of the Mathematical Tripos in 1911. He graduated with distinction as a Wrangler in 1913.
In 1914 Milne-Thomson joined Winchester College in Hampshire as an assistant mathematics master and taught there for next seven years. In 1921 he was appointed professor of mathematics at the Royal Naval College, Greenwich and remained there until retirement at the age of 65.
In 1933 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were Bevan Baker, John Marshall, Edward Thomas Copson, and Herbert Turnbull.
After retiring from the Royal Naval College, Milne-Thomson took up various posts as Visiting Professor at various institutions around the world, including the Brown University at Rhode Island, the US Army Mathematics Research Center at the University of Wisconsin (1958–1960), the University of Arizona (1961–1970), University of Rome (1968), the University of Queensland (1969), the University of Calgary (1970), and finally the University of Otago (1971). At the end of a long career Milne-Thomson quit academia in 1971 and went to live in Sevenoaks, Kent where he died at the age of 83.
Work
During the early stages of his career, he developed and compiled several mathematical tables such as the Standard Four Figure Mathematical Tables jointly constructed with L. J. Comrie and published in 1931, Standard Table of Square Roots (1932), and Jacobian Elliptic Function Tables (1932). Later, Milne-Thomson wrote the chapters on Elliptic Integrals and Jacobian Elliptic Functions in the classic NBS AMS 55 handbook.
In 1933 Milne-Thomson published his first book, The Calculus of Finite Differences which became a classic textbook and the original text was reprinted in 1951. In the mid 1930s, Milne-Thomson developed an interest in hydrodynamics and later in aerodynamics. This led to publication of two popular textbooks titled Theoretical Hydrodynamics and Theoretical Aerodynamics. The Theoretical Hydrodynamics published by Macmillan & Co. Ltd., London, appeare |
https://en.wikipedia.org/wiki/Robert%20Goldblatt | __notoc__
Robert Ian Goldblatt (born 1949) is a mathematical logician who is Emeritus Professor in the School of Mathematics and Statistics at Victoria University, Wellington, New Zealand. His doctoral advisor was Max Cresswell. His most popular books are Logics of Time and Computation and Topoi: the Categorial Analysis of Logic. He has also written a graduate level textbook on hyperreal numbers which is an introduction to nonstandard analysis.
He has been Coordinating Editor of The Journal of Symbolic Logic and a Managing Editor of Studia Logica.
He was elected Fellow and Councillor of the Royal Society of New Zealand, President of the New Zealand Mathematical Society, and represented New Zealand to the International Mathematical Union.
In 2012 he was awarded the
Jones Medal for lifetime achievement in mathematics.
Books and handbook chapters
1979: Topoi: The Categorial Analysis of Logic, North-Holland. Revised edition 1984. Dover Publications edition 2006. Internet edition, Project Euclid.
Benjamin C. Pierce recommends it as an "excellent beginner book", praising it for the use of simple set-theoretic examples and motivating intuitions, but noted that it "is sometimes criticized by category theorists for being misleading on some aspects of the subject, and for presenting long and difficult proofs where simple ones are available." But the preface of the Dover edition observes (p. xv) that "This is a book about logic, rather than category theory per se. It aims to explain, in an introductory way, how certain logical ideas are illuminated by a category-theoretic perspective."
1982: Axiomatising the Logic of Computer Programming, Lecture Notes in Computer Science 130, Springer-Verlag.
1987: Orthogonality and Spacetime Geometry, Universitext Springer-Verlag
1987: Logics of Time and Computation. CSLI Lecture Notes, 7. Stanford University, Center for the Study of Language and Information . Second edition 1992.
1993: Mathematics of Modality, CSLI Publications,
1998: Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. Graduate Texts in Mathematics, 188. Springer-Verlag.
Reviewer Perry Smith for MathSciNet wrote: "The author's ideas on how to achieve both intelligibility and rigor, explained in the preface, will be useful reading for anyone intending to teach nonstandard analysis."
2006: "Mathematical Modal Logic: a View of its Evolution" in Modalities in the Twentieth Century, Volume 7 of the Handbook of the History of Logic, edited by Dov M. Gabbay and John Woods, Elsevier, pp. 1–98.
2011: Quantifiers, Propositions and Identity: Admissible Semantics for Quantified Modal and Substructural Logics, Cambridge University Press and the Association for Symbolic Logic.
See also
Influence of non-standard analysis
References
External links
Home page
Living people
New Zealand mathematicians
New Zealand logicians
Mathematical logicians
Victoria University of Wellington alumni
Academic staff of Victoria University of Wellin |
https://en.wikipedia.org/wiki/Ivar%20Ekeland | Ivar I. Ekeland (born 2 July 1944, Paris) is a French mathematician of Norwegian descent. Ekeland has written influential monographs and textbooks on nonlinear functional analysis, the calculus of variations, and mathematical economics, as well as popular books on mathematics, which have been published in French, English, and other languages. Ekeland is known as the author of Ekeland's variational principle and for his use of the Shapley–Folkman lemma in optimization theory. He has contributed to the periodic solutions of Hamiltonian systems and particularly to the theory of Kreĭn indices for linear systems (Floquet theory). Ekeland helped to inspire the discussion of chaos theory in Michael Crichton's 1990 novel Jurassic Park.
Biography
Ekeland studied at the École Normale Supérieure (1963–1967). He is a senior research fellow at the French National Centre for Scientific Research (CNRS). He obtained his doctorate in 1970. He teaches mathematics and economics at the Paris Dauphine University, the École Polytechnique, the École Spéciale Militaire de Saint-Cyr, and the University of British Columbia in Vancouver. He was the chairman of Paris-Dauphine University from 1989 to 1994.
Ekeland is a recipient of the D'Alembert Prize and the Jean Rostand prize. He is also a member of the Norwegian Academy of Science and Letters.
Popular science: Jurassic Park by Crichton and Spielberg
Ekeland has written several books on popular science, in which he has explained parts of dynamical systems, chaos theory, and probability theory. These books were first written in French and then translated into English and other languages, where they received praise for their mathematical accuracy as well as their value as literature and as entertainment.
Through these writings, Ekeland had an influence on Jurassic Park, on both the novel and film. Ekeland's Mathematics and the unexpected and James Gleick's Chaos inspired the discussions of chaos theory in the novel Jurassic Park by Michael Crichton. When the novel was adapted for the film Jurassic Park by Steven Spielberg, Ekeland and Gleick were consulted by the actor Jeff Goldblum as he prepared to play the mathematician specializing in chaos theory.
Research
Ekeland has contributed to mathematical analysis, particularly to variational calculus and mathematical optimization.
Variational principle
In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exists a nearly optimal solution to a class of optimization problems.
Ekeland's variational principle can be used when the lower level set of a minimization problem is not compact, so that the Bolzano–Weierstrass theorem can not be applied. Ekeland's principle relies on the completeness of the metric space.
Ekeland's principle leads to a quick proof of the Caristi fixed point theorem.
Ekeland was associated with the University of Paris when he proposed this theorem.
Variational theory of Hamilt |
https://en.wikipedia.org/wiki/ITL%201%20statistical%20regions%20of%20England | International Territorial Level (ITL) is a geocode standard for referencing the subdivisions of the United Kingdom for statistical purposes, used by the Office for National Statistics (ONS). Between 2003 and 2021, as part of the European Union and European Statistical System, the geocode standard used for the United Kingdom were Nomenclature of Territorial Units for Statistics or NUTS. The NUTS code for the UK was UK and the NUTS standard had a hierarchy of three levels, with 12 first level regions, which are currently mirrored by the ITL classification, of which 9 regions are in England. The sub-structure corresponds to administrative divisions within the country. Formerly, the further NUTS divisions IV and V existed; these have now been replaced by Local Administrative Units (LAU-1 and LAU-2 respectively).
Between 1994 and 2011, the nine regions had an administrative role in the implementation of UK Government policy, and as the areas covered by (mostly indirectly) elected bodies.
List of regions
The ITL 1 statistical regions correspond with the regions of England as used by the UK's Office for National Statistics. Prior to 2021, all codes had "UK" instead of "TL" for Territorial Level.
TLC. North East (used in NUTS as UKC)
TLD. North West (used in NUTS as UKD)
TLE. Yorkshire and the Humber (used in NUTS as UKE)
TLF. East Midlands (used in NUTS as UKF)
TLG. West Midlands (used in NUTS as UKG)
TLH. East of England (used in NUTS as UKH)
TLI. London (used in NUTS as UKI)
TLJ. South East (used in NUTS as UKJ)
TLK. South West (used in NUTS as UKK)
Greater London has a directly elected Mayor and Assembly. The other eight regions have Local authority leaders' boards, which have a role in coordinating local government on a regional level, with members appointed by local government bodies. These boards replaced indirectly elected regional assemblies, which were established in 1994 and undertook a range of co-ordinating, lobbying, scrutiny and strategic planning functions until their abolition.
Sub-structure of the regions
Each region of England is divided into a range of metropolitan and non-metropolitan counties. For ITL purposes, these subdivisions are formally known as ITL levels 2 and 3.
London region is divided into
London boroughs (ITL 3, usual grouped)
All other regions are divided into
metropolitan counties (ITL 2)
shire counties (ITL 2 or 3 depending on the region) and
unitary authorities (usually ITL 3).
See also
Regions of England
Historical and alternative regions of England
International Territorial Level
List of articles about local government in the United Kingdom
Nomenclature of Territorial Units for Statistics
References
External links
Local Government Boundary Commission for England
Dept of Communities and Local Government
England
Regionalism (politics) in the United Kingdom
Types of subdivision in the United Kingdom |
https://en.wikipedia.org/wiki/Bratteli%E2%80%93Vershik%20diagram | In mathematics, a Bratteli–Veršik diagram is an ordered, essentially simple Bratteli diagram (V, E) with a homeomorphism on the set of all infinite paths called the Veršhik transformation. It is named after Ola Bratteli and Anatoly Vershik.
Definition
Let X = {(e1, e2, ...) | ei ∈ Ei and r(ei) = s(ei+1)} be the set of all paths in the essentially simple Bratteli diagram (V, E). Let Emin be the set of all minimal edges in E, similarly let Emax be the set of all maximal edges. Let y be the unique infinite path in Emax. (Diagrams which possess a unique infinite path are called "essentially simple".)
The Veršhik transformation is a homeomorphism φ : X → X defined such that φ(x) is the unique minimal path if x = y. Otherwise x = (e1, e2,...) | ei ∈ Ei where at least one ei ∉ Emax. Let k be the smallest such integer. Then φ(x) = (f1, f2, ..., fk−1, ek + 1, ek+1, ... ), where ek + 1 is the successor of ek in the total ordering of edges incident on r(ek) and (f1, f2, ..., fk−1) is the unique minimal path to ek + 1.
The Veršhik transformation allows us to construct a pointed topological system (X, φ, y) out of any given ordered, essentially simple Bratteli diagram. The reverse construction is also defined.
Equivalence
The notion of graph minor can be promoted from a well-quasi-ordering to an equivalence relation if we assume the relation is symmetric. This is the notion of equivalence used for Bratteli diagrams.
The major result in this field is that equivalent essentially simple ordered Bratteli diagrams correspond to topologically conjugate pointed dynamical systems. This allows us apply results from the former field into the latter and vice versa.
See also
Markov odometer
Notes
Further reading
Application-specific graphs
Operator algebras |
https://en.wikipedia.org/wiki/Alpha%20value | Alpha value (designated α value) may refer to:
Significance level in statistics
Alpha compositing |
https://en.wikipedia.org/wiki/McConnell%20equation | In physical chemistry, the McConnell equation gives the probability of an unpaired electron in an in aromatic radical compound (such as benzene radical anion ) being on a particular atom. It relates this probability, known as the "spin density", to its proportional dependence on the hyperfine splitting constant.
The equation is
where is the hyperfine splitting constant, is the spin density, and is an empirical constant that can range from 2.0 to 2.5 mT.
History
The equation is named after Harden M. McConnell of Stanford University, who first presented it in 1956.
References
Chemical physics |
https://en.wikipedia.org/wiki/Mikl%C3%B3s%20Simonovits | Miklós Simonovits (4 September 1943 in Budapest) is a Hungarian mathematician who currently works at the Rényi Institute of Mathematics in Budapest and is a member of the Hungarian Academy of Sciences. He is on the advisory board of the journal Combinatorica. He is best known for his work in extremal graph theory and was awarded Széchenyi Prize in 2014. Among other things, he discovered the method of progressive induction which he used to describe graphs which do not contain a predetermined graph and the number of edges is close to maximal. With Lovász, he gave a randomized algorithm using O(n7 log2 n)
separation calls to approximate the volume of a convex body within a fixed relative error.
Simonovits was also one of the most frequent collaborators with Paul Erdős, co-authoring 21 papers with him.
Career
He began his university studies at the Mathematics department of Eötvös Loránd University in 1962, after winning a silver and bronze medal at the International Mathematics Olympiad in 1961 and 1962 respectively. He got his diploma in mathematics from the university in 1967 and defended his PhD under Vera T. Sós in 1971. He taught as an assistant professor and then associate professor at Eötvös Loránd, from 1971 to 1979, mainly combinatorics and analysis. He joined Alfréd Rényi Institute of Mathematics in 1979. In the coming years, he was appointed as the professor in Discrete mathematics. He was also a visiting professor at a number of foreign institutions in US and Canada. He was also a visiting researcher at Moscow State University, Charles University, Prague, Warsaw University, Denmark and various institutions in India. He was elected as a corresponding member at the Hungarian Academy of Sciences in 2001 and full membership was awarded in 2008.
Academic work
His main research interests are Combinatorics, Extremal Graph Theory, Theoretical Computer Science and Random Graphs.
He discovered the method of progressive induction which he used to describe graphs which do not contain a predetermined graph and the number of edges is close to maximal. With Laszlo Lovász, he gave a randomized algorithm using O(n7 log2 n)
separation calls to approximate the volume of a convex body within a fixed relative error.
He is a long-time collaborator of Endre Szemeredi and worked with him closely.
Simonovits was also one of the most frequent collaborators with Paul Erdős, co-authoring 21 papers with him.
Family
His father Simonovits István (1907–1985) was a doctor and a hematologist. He was a member of the Hungarian Academy of Sciences. Beke Anna, his mother, was a mathematics and physics teacher, who also worked in a book publishing company.
Awards
Tibor Szele-Medal (1989)
Academy Award (1993)
Széchenyi-Prize (2014)
Key publications
A limit theorem in graph theory (with Erdős Pál, 1966)
Anti-Ramsey theorems (coauthor, 1973)
On the Structure of Edge Graphs-2 (coauthor, 1976)
Spanning Retracts of a Partially Ordered Set (coauthor, 1980)
Compa |
https://en.wikipedia.org/wiki/Formal%20semantics%20%28natural%20language%29 | Formal semantics is the study of grammatical meaning in natural languages using formal tools from logic, mathematics and theoretical computer science. It is an interdisciplinary field, sometimes regarded as a subfield of both linguistics and philosophy of language. It provides accounts of what linguistic expressions mean and how their meanings are composed from the meanings of their parts. The enterprise of formal semantics can be thought of as that of reverse-engineering the semantic components of natural languages' grammars.
Overview
Formal semantics studies the denotations of natural language expressions. High-level concerns include compositionality, reference, and the nature of meaning. Key topic areas include scope, modality, binding, tense, and aspect. Semantics is distinct from pragmatics, which encompasses aspects of meaning which arise from interaction and communicative intent.
Formal semantics is an interdisciplinary field, often viewed as a subfield of both linguistics and philosophy, while also incorporating work from computer science, mathematical logic, and cognitive psychology. Within philosophy, formal semanticists typically adopt a Platonistic ontology and an externalist view of meaning. Within linguistics, it is more common to view formal semantics as part of the study of linguistic cognition. As a result, philosophers put more of an emphasis on conceptual issues while linguists are more likely to focus on the syntax–semantics interface and crosslinguistic variation.
Central concepts
Truth conditions
The fundamental question of formal semantics is what you know when you know how to interpret expressions of a language. A common assumption is that knowing the meaning of a sentence requires knowing its truth conditions, or in other words knowing what the world would have to be like for the sentence to be true. For instance, to know the meaning of the English sentence "Nancy smokes" one has to know that it is true when the person Nancy performs the action of smoking.
However, many current approaches to formal semantics posit that there is more to meaning than truth-conditions. In the formal semantic framework of inquisitive semantics, knowing the meaning of a sentence also requires knowing what issues (i.e. questions) it raises. For instance "Nancy smokes, but does she drink?" conveys the same truth-conditional information as the previous example but also raises an issue of whether Nancy drinks. Other approaches generalize the concept of truth conditionality or treat it as epiphenomenal. For instance in dynamic semantics, knowing the meaning of a sentence amounts to knowing how it updates a context.
Pietroski treats meanings as instructions to build concepts.
Compositionality
The Principle of Compositionality is the fundamental assumption in formal semantics. This principle states that the denotation of a complex expression is determined by the denotations of its parts along with their mode of composition. For instance, |
https://en.wikipedia.org/wiki/Reinhard%20Majgl | Reinhard Majgl (born 4 December 1949) is a retired German football forward.
Career
Statistics
References
External links
1949 births
Living people
German men's footballers
Bundesliga players
2. Bundesliga players
VfL Bochum players
K.A.S. Eupen players
SC Fortuna Köln players
1. FC Bocholt players
Place of birth missing (living people)
Men's association football forwards |
https://en.wikipedia.org/wiki/Robert%20Tibshirani | Robert Tibshirani (born July 10, 1956) is a professor in the Departments of Statistics and Biomedical Data Science at Stanford University. He was a professor at the University of Toronto from 1985 to 1998. In his work, he develops statistical tools for the analysis of complex datasets, most recently in genomics and proteomics.
His most well-known contributions are the
Lasso method, which proposed the use of L1 penalization in regression and related problems, and Significance Analysis of Microarrays.
Education and early life
Tibshirani was born on 10 July 1956 in Niagara Falls, Ontario, Canada. He received his B. Math. in statistics and computer science from the University of Waterloo in 1979 and a Master's degree in Statistics from University of Toronto in 1980. Tibshirani joined the doctoral program at Stanford University in 1981 and received his Ph.D. in 1984 under the supervision of Bradley Efron. His dissertation was entitled "Local likelihood estimation".
Honors and awards
Tibshirani received the COPSS Presidents' Award in 1996. Given jointly by the world's leading
statistical societies, the award recognizes outstanding contributions to statistics by a statistician under the age of 40. He is a fellow of
the Institute of Mathematical Statistics and the American Statistical Association. He won an E.W.R. Steacie Memorial Fellowship from the Natural Sciences and Engineering Research Council of Canada in 1997. He was elected a Fellow of the Royal Society of Canada in 2001 and a member of the National Academy of Sciences in 2012.
Tibshirani was made the 2012 Statistical Society of Canada's Gold Medalist at their yearly meeting in Guelph, Ontario for "exceptional contributions to methodology and theory for the analysis of complex data sets, smoothing and regression methodology, statistical learning, and classification, and application areas that include public health, genomics, and proteomics". He gave his Gold Medal Address at the 2013 meeting in Edmonton. He was elected to the Royal Society in 2019. Tibshirani was named as the 2021 recipient of the ISI Founders of Statistics Prize for his 1996 paper Regression Shrinkage and Selection via the Lasso.
Personal life
His son, Ryan Tibshirani, with whom he occasionally publishes scientific papers, is a professor at UC Berkeley in the Department of Statistics.
Publications
Tibshirani is a prolific author of scientific works on various topics in applied statistics, including statistical learning, data mining, statistical computing, and bioinformatics. He along with his collaborators has authored about 250 scientific articles. Many of Tibshirani's scientific articles were coauthored by his longtime collaborator, Trevor Hastie. Tibshirani is one of the most ISI Highly Cited Authors in Mathematics by the ISI Web of Knowledge. He has coauthored the following books:
T. Hastie and R. Tibshirani, Generalized Additive Models, Chapman and Hall, 1990.
B. Efron and R. Tibshirani, An Introduction to the |
https://en.wikipedia.org/wiki/Discoid | Discoid may refer to:
Disk (mathematics), the region in a plane enclosed by a circle
Medicine
Furosemide, a medication sold under the trade name Discoid
Discoid meniscus, a human anatomical variant
Discoid lupus erythematosus, a chronic skin condition in humans
Canine discoid lupus erythematosus, the equivalent condition in dogs
Nummular dermatitis, also known as discoid eczema
Biology
Discoid head, a type of floret arrangement in Asteraceae flower heads
Discoidal cleavage, a type of partial cleavage in embryos
Discoidin domain, a protein domain of many blood coagulation factors
Blaberus discoidalis, the discoid cockroach
Dictyostelium discoideum, a slime mold
Other
Discoidal stele, a funerary stele type found in Basque country
See also
Disc (disambiguation) |
https://en.wikipedia.org/wiki/Dieter%20Schwemmle | Dieter Schwemmle (born 28 July 1949) is a German former footballer who played as a forward.
Career
Statistics
References
External links
1949 births
Living people
German men's footballers
Bundesliga players
VfB Stuttgart players
VfB Stuttgart II players
FC Twente players
Kickers Offenbach players
FC Biel-Bienne players
AC Bellinzona players
VfL Bochum players
Bulova SA players
Men's association football forwards
Footballers from Stuttgart |
https://en.wikipedia.org/wiki/Heinz%20Kn%C3%BCwe | Heinz Knüwe (born 16 January 1956) is a German retired professional footballer who played as a defender.
Career statistics
References
External links
1956 births
Living people
German men's footballers
Men's association football defenders
Bundesliga players
2. Bundesliga players
TSV 1860 Munich players
SV Lippstadt 08 players
SC Herford players
VfL Bochum players
Hannover 96 players
SC Verl players |
https://en.wikipedia.org/wiki/Induced%20character | In mathematics, an induced character is the character of the representation V of a finite group G induced from a representation W of a subgroup H ≤ G. More generally, there is also a notion of induction of a class function f on H given by the formula
If f is a character of the representation W of H, then this formula for calculates the character of the induced representation V of G.
The basic result on induced characters is Brauer's theorem on induced characters. It states that every irreducible character on G is a linear combination with integer coefficients of characters induced from elementary subgroups.
References
Group theory |
https://en.wikipedia.org/wiki/Numerical%20semigroup | In mathematics, a numerical semigroup is a special kind of a semigroup. Its underlying set is the set of all nonnegative integers except a finite number and the binary operation is the operation of addition of integers. Also, the integer 0 must be an element of the semigroup. For example, while the set {0, 2, 3, 4, 5, 6, ...} is a numerical semigroup, the set {0, 1, 3, 5, 6, ...} is not because 1 is in the set and 1 + 1 = 2 is not in the set. Numerical semigroups are commutative monoids and are also known as numerical monoids.
The definition of numerical semigroup is intimately related to the problem of determining nonnegative integers that can be expressed in the form x1n1 + x2 n2 + ... + xr nr for a given set {n1, n2, ..., nr} of positive integers and for arbitrary nonnegative integers x1, x2, ..., xr. This problem had been considered by several mathematicians like Frobenius (1849–1917) and Sylvester (1814–1897) at the end of the 19th century. During the second half of the twentieth century, interest in the study of numerical semigroups resurfaced because of their applications in algebraic geometry.
Definition and examples
Definition
Let N be the set of nonnegative integers. A subset S of N is called a numerical semigroup if the following conditions are satisfied.
0 is an element of S
N − S, the complement of S in N, is finite.
If x and y are in S then x + y is also in S.
There is a simple method to construct numerical semigroups. Let A = {n1, n2, ..., nr} be a nonempty set of positive integers. The set of all integers of the form x1 n1 + x2 n2 + ... + xr nr is the subset of N generated by A and is denoted by 〈 A 〉. The following theorem fully characterizes numerical semigroups.
Theorem
Let S be the subsemigroup of N generated by A. Then S is a numerical semigroup if and only if gcd (A) = 1. Moreover, every numerical semigroup arises in this way.
Examples
The following subsets of N are numerical semigroups.
〈 1 〉 = {0, 1, 2, 3, ...}
〈 1, 2 〉 = {0, 1, 2, 3, ...}
〈 2, 3 〉 = {0, 2, 3, 4, 5, 6, ...}
Let a be a positive integer. 〈 a, a + 1, a + 2, ... , 2a – 1 〉 = {0, a, a + 1, a + 2, a + 3, ...}.
Let b be an odd integer greater than 1. Then 〈 2, b 〉 = {0, 2, 4, . . . , b − 3 , b − 1, b, b + 1, b + 2, b + 3 , ...}.
Well-tempered harmonic semigroup H={0,12,19,24,28,31,34,36,38,40,42,43,45,46,47,48,...}
Embedding dimension, multiplicity
The set A is a set of generators of the numerical semigroup 〈 A 〉. A set of generators of a numerical semigroup is a minimal system
of generators if none of its proper subsets generates the numerical semigroup. It is known that
every numerical semigroup S has a unique minimal system of generators and also that this minimal system of generators is finite. The cardinality of the minimal set of generators is called the embedding dimension of the numerical semigroup S and is denoted by e(S). The smallest member in the minimal system of generators is called the multiplicity of the numerical semigroup S an |
https://en.wikipedia.org/wiki/Akira%20Takabe | is a former Japanese football player.
Club statistics
References
External links
1982 births
Living people
Toyo University alumni
Association football people from Yamanashi Prefecture
Japanese men's footballers
J1 League players
Japan Football League players
Tokyo Verdy players
Roasso Kumamoto players
Reilac Shiga FC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Sandro%20da%20Silva | Sandro André da Silva (born March 5, 1974) is a former Brazilian football player.
Club statistics
References
External links
J. League
1974 births
Living people
Brazilian men's footballers
J1 League players
Kashima Antlers players
Royal Antwerp F.C. players
Sociedade Esportiva Palmeiras players
C.D. Guadalajara footballers
São José Esporte Clube players
Club Olimpia footballers
Avaí FC players
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Expatriate men's footballers in Mexico
Expatriate men's footballers in Paraguay
Expatriate men's footballers in Belgium
Expatriate men's footballers in Venezuela
Expatriate men's footballers in Chile
Men's association football midfielders |
https://en.wikipedia.org/wiki/Eiichiro%20Ozaki | is a Japanese football player who plays for Fukui United.
Club statistics
Updated to 23 February 2018.
References
External links
1984 births
Living people
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Singapore Premier League players
Albirex Niigata players
Albirex Niigata Singapore FC players
Gainare Tottori players
Azul Claro Numazu players
Men's association football defenders
Association football people from Shizuoka (city) |
https://en.wikipedia.org/wiki/Alexandre%20Bortolato | Alexandre Jose Bortolato (born November 10, 1973) is a former Brazilian football player.
Club statistics
References
External links
J. League
1973 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
J2 League players
Montedio Yamagata players
Expatriate men's footballers in Japan
Men's association football forwards |
https://en.wikipedia.org/wiki/Santos%20%28footballer%2C%20born%201983%29 | Rafael dos Santos Franciscatti (born April 9, 1983) is a former Brazilian football player.
Club statistics
References
External links
J. League
1983 births
Living people
Brazilian men's footballers
J2 League players
Shonan Bellmare players
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Men's association football midfielders |
https://en.wikipedia.org/wiki/Atsushi%20Terui | is a former Japanese football player.
Club statistics
References
External links
1980 births
Living people
Kokushikan University alumni
Association football people from Iwate Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Shonan Bellmare players
Arte Takasaki players
Tochigi SC players
Vanraure Hachinohe players
Men's association football defenders |
https://en.wikipedia.org/wiki/Jefferson%20%28footballer%2C%20born%201981%29 | Jefferson Vieira da Cruz (born July 3, 1981) is a Brazilian football player. He was active from 1999 to 2009, and retired on January 1, 2010.
Club statistics
References
External links
1981 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
J2 League players
CR Flamengo footballers
Bangu Atlético Clube players
Sagan Tosu players
Yokohama FC players
Fagiano Okayama players
Expatriate men's footballers in Japan
Men's association football forwards |
https://en.wikipedia.org/wiki/Hidehito%20Shirao | is a former Japanese football player.
Club statistics
References
External links
1980 births
Living people
Kokushikan University alumni
Association football people from Kagoshima Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Ventforet Kofu players
Matsumoto Yamaga FC players
V-Varen Nagasaki players
FC Ryukyu players
Men's association football forwards |
https://en.wikipedia.org/wiki/Tomohisa%20Yoshida | is a former Japanese football player.
Club statistics
References
External links
1984 births
Living people
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Oita Trinita players
Ehime FC players
Mito HollyHock players
Zweigen Kanazawa players
FC Machida Zelvia players
Fukushima United FC players
Nara Club players
Men's association football midfielders |
https://en.wikipedia.org/wiki/George%20Box%20Medal | The George Box Medal is an insignia of an award named after the statistician George Box. It is awarded annually by the European Network for Business and Industrial Statistics (ENBIS) in recognition of outstanding work in the development and the application of statistical methods in European business and industry.
Past Recipients
Source: ENBIS
2003 George Box
2004 Søren Bisgaard
2005 Sir David Cox
2006 Gerry Hahn
2007 Poul Thyregod
2008 Doug Montgomery
2009 Tony Greenfield
2010 David Stewardson
2011 Henry Philip Wynn
2012 Bill Woodall
2013 David Steinberg
2014 John F. MacGregor
2015 Geoffrey Vining
2016 David J. Hand
2017 C. F. Jeff Wu
2018 Ron S. Kenett
2019 Ronald J.M.M. Does
2020 William Q. Meeker
2021 Christine Anderson-Cook
2022 Jianjun Shi
See also
List of mathematics awards
References
External links
. ENBIS website.
Mathematics awards
Statistical awards |
https://en.wikipedia.org/wiki/Koichi%20Hirono | is a former Japanese football player.
Club statistics
References
External links
library.footballjapan.jp
1980 births
Living people
Aichi Gakuin University alumni
Association football people from Nara Prefecture
Japanese men's footballers
J1 League players
J2 League players
Nagoya Grampus players
Yokohama FC players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Pontryagin%20cohomology%20operation | In mathematics, a Pontryagin cohomology operation is a cohomology operation taking cohomology classes in H2n(X,Z/prZ) to H2pn(X,Z/pr+1Z) for some prime number p. When p=2 these operations were introduced by and were named Pontrjagin squares by (with the term "Pontryagin square" also being used). They were generalized to arbitrary primes by .
See also
Steenrod operation
References
Algebraic topology |
https://en.wikipedia.org/wiki/Focused%20information%20criterion | In statistics, the focused information criterion (FIC) is a method for selecting the most appropriate model among a set of competitors for a given data set. Unlike most other model selection strategies, like the Akaike information criterion (AIC), the Bayesian information criterion (BIC) and the deviance information criterion (DIC), the FIC does not attempt to assess the overall fit of candidate models but focuses attention directly on the parameter of primary interest with the statistical analysis, say , for which competing models lead to different estimates, say for model . The FIC method consists in first developing an exact or approximate expression for the precision or quality of each estimator, say for , and then use data to estimate these precision measures, say . In the end the model with best estimated precision is selected. The FIC methodology was developed by Gerda Claeskens and Nils Lid Hjort, first in two 2003 discussion articles in Journal of the American Statistical Association and later on in other papers and in their 2008 book.
The concrete formulae and implementation for FIC depend firstly on the particular parameter of interest, the choice of which does not depend on mathematics but on the scientific and statistical context. Thus the FIC apparatus may be selecting one model as most appropriate for estimating a quantile of a distribution but preferring another model as best for estimating the mean value. Secondly, the FIC formulae depend on the specifics of the models used for the observed data and also on how precision is to be measured. The clearest case is where precision is taken to be mean squared error, say in terms of squared bias and variance for the estimator associated with model . FIC formulae are then available in a variety of situations, both for handling parametric, semiparametric and nonparametric situations, involving separate estimation of squared bias and variance, leading to estimated precision . In the end the FIC selects the model with smallest estimated mean squared error.
Associated with the use of the FIC for selecting a good model is the FIC plot, designed to give a clear and informative picture of all estimates, across all candidate models, and their merit. It displays estimates on the axis along with FIC scores on the axis; thus estimates found to the left in the plot are associated with the better models and those found in the middle and to the right stem from models less or not adequate for the purpose of estimating the focus parameter in question.
Generally speaking, complex models (with many parameters relative to sample size) tend to lead to estimators with small bias but high variance; more parsimonious models (with fewer parameters) typically yield estimators with larger bias but smaller variance. The FIC method balances the two desired data of having small bias and small variance in an optimal fashion. The main difficulty lies with the bias , as it involves the distance from the expecte |
https://en.wikipedia.org/wiki/Robust%20Bayesian%20analysis | In statistics, robust Bayesian analysis, also called Bayesian sensitivity analysis, is a type of sensitivity analysis applied to the outcome from Bayesian inference or Bayesian optimal decisions.
Sensitivity analysis
Robust Bayesian analysis, also called Bayesian sensitivity analysis, investigates the robustness of answers from a Bayesian analysis to uncertainty about the precise details of the analysis. An answer is robust if it does not depend sensitively on the assumptions and calculation inputs on which it is based. Robust Bayes methods acknowledge that it is sometimes very difficult to come up with precise distributions to be used as priors. Likewise the appropriate likelihood function that should be used for a particular problem may also be in doubt. In a robust Bayes approach, a standard Bayesian analysis is applied to all possible combinations of prior distributions and likelihood functions selected from classes of priors and likelihoods considered empirically plausible by the analyst. In this approach, a class of priors and a class of likelihoods together imply a class of posteriors by pairwise combination through Bayes' rule. Robust Bayes also uses a similar strategy to combine a class of probability models with a class of utility functions to infer a class of decisions, any of which might be the answer given the uncertainty about best probability model and utility function. In both cases, the result is said to be robust if it is approximately the same for each such pair. If the answers differ substantially, then their range is taken as an expression of how much (or how little) can be confidently inferred from the analysis.
Although robust Bayes methods are clearly inconsistent with the Bayesian idea that uncertainty should be measured by a single additive probability measure and that personal attitudes and values should always be measured by a precise utility function, they are often accepted as a matter of convenience (e.g., because the cost or schedule do not allow the more painstaking effort needed to get a precise measure and function). Some analysts also suggest that robust methods extend the traditional Bayesian approach by recognizing incertitude as of a different kind of uncertainty. Analysts in the latter category suggest that the set of distributions in the prior class is not a class of reasonable priors, but that it is rather a reasonable class of priors. The idea is that no single distribution is reasonable as a model of ignorance, but considered as a whole, the class is a reasonable model for ignorance.
Robust Bayes methods are related to important and seminal ideas in other areas of statistics such as robust statistics and resistance estimators. The arguments in favor of a robust approach are often applicable to Bayesian analyses. For example, some criticize methods that must assume the analyst is "omniscient" about certain facts such as model structure, distribution shapes and parameters. Because such facts are t |
https://en.wikipedia.org/wiki/Postnikov%20square | In algebraic topology, a Postnikov square is a certain cohomology operation from a first cohomology group H1 to a third cohomology group H3, introduced by . described a generalization taking classes in Ht to H2t+1.
References
PDF
Algebraic topology |
https://en.wikipedia.org/wiki/Rectified%20Gaussian%20distribution | In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (constant 0) and a continuous distribution (a truncated Gaussian distribution with interval ) as a result of censoring.
Density function
The probability density function of a rectified Gaussian distribution, for which random variables X having this distribution, derived from the normal distribution are displayed as , is given by
Here, is the cumulative distribution function (cdf) of the standard normal distribution:
is the Dirac delta function
and, is the unit step function:
Mean and variance
Since the unrectified normal distribution has mean and since in transforming it to the rectified distribution some probability mass has been shifted to a higher value (from negative values to 0), the mean of the rectified distribution is greater than
Since the rectified distribution is formed by moving some of the probability mass toward the rest of the probability mass, the rectification is a mean-preserving contraction combined with a mean-changing rigid shift of the distribution, and thus the variance is decreased; therefore the variance of the rectified distribution is less than
Generating values
To generate values computationally, one can use
and then
Application
A rectified Gaussian distribution is semi-conjugate to the Gaussian likelihood, and it has been recently applied to factor analysis, or particularly, (non-negative) rectified factor analysis.
Harva proposed a variational learning algorithm for the rectified factor model, where the factors follow a mixture of rectified Gaussian; and later Meng proposed an infinite rectified factor model coupled with its Gibbs sampling solution, where the factors follow a Dirichlet process mixture of rectified Gaussian distribution, and applied it in computational biology for reconstruction of gene regulatory networks.
Extension to general bounds
An extension to the rectified Gaussian distribution was proposed by Palmer et al., allowing rectification between arbitrary lower and upper bounds. For lower and upper bounds and respectively, the cdf, is given by:
where is the cdf of a normal distribution with mean and variance . The mean and variance of the rectified distribution is calculated by first transforming the constraints to be acting on a standard normal distribution:
Using the transformed constraints, the mean and variance, and respectively, are then given by:
where is the error function. This distribution was used by Palmer et al. for modelling physical resource levels, such as the quantity of liquid in a vessel, which is bounded by both 0 and the capacity of the vessel.
See also
Folded normal distribution
Half-normal distribution
Half-t distribution
Modified half-normal distribution
Truncated normal distribution
Reference |
https://en.wikipedia.org/wiki/Zenon%20Ivanovich%20Borevich | Zenon Ivanovich Borevich Зенон Иванович Боревич (7 November 1922 – 26 February 1995) was a Russian mathematician who worked on homological algebra, algebraic number theory, integral representations, and linear groups.
Biography
Zenon Borevich completed his master's thesis titled "Regarding the theory of local fields" in 1951 and his doctoral dissertation titled "Regarding the multiplicative groups of normal R-extensions of local fields" in 1967.
In 1968, Borevich became Professor of Mathematics at Saint Petersburg State University (then named Leningrad State University) and was put in charge of the Department of Higher Algebra and Number Theory. He became Dean of the university's entire Department of Mathematics and Mechanics in 1973 and remained in that position until 1984. He continued to head up Higher Algebra and Number Theory until his retirement in 1992.
Borevich authored more than 100 publications and works, including the textbook "Determinants and Matrices" and the monograph "Number Theory" (together with Shafarevich).
Publications
References
External links
International Algebraic Conference dedicated to the memory of Z. I. Borevich
1922 births
1995 deaths
20th-century Russian mathematicians |
https://en.wikipedia.org/wiki/Zoghman%20Mebkhout | Zoghman Mebkhout (born 1949 ) (زغمان مبخوت) is a French-Algerian mathematician. He is known for his work in algebraic analysis, geometry and representation theory, more precisely on the theory of D-modules.
Career
Mebkhout is currently a research director at the French National Centre for Scientific Research and in 2002 Zoghman received the Servant Medal from the CNRS a prize given every two years with an amount of €10,000.
Notable works
In September 1979 Mebkhout presented the Riemann–Hilbert correspondence, which is a generalization of Hilbert's twenty-first problem to higher dimensions. The original setting was for Riemann surfaces, where it was about the existence of regular differential equations with prescribed monodromy groups.
In higher dimensions, Riemann surfaces are replaced by complex manifolds of dimension > 1. Certain systems of partial differential equations (linear and having very special properties for their solutions) and possible monodromies of their solutions correspond. An independent proof of this result was presented by Masaki Kashiwara in April 1980.
Zoghman is now largely known as a specialist in D-modules theory.
Recognition
Zoghman is one of the first modern international-caliber North-African mathematicians. A symposium in Spain was held on his sixtieth birthday. He was invited to the Institute for Advanced Study and gave a recent talk at Institut Fourier.
Alexander Grothendieck said that Mebkhout's name was hidden and his role neglected in an operation headed by Pierre Deligne in the Luminy congress in June 1981. He calls it "a great disgrace of the mathematical world of this century" and is one of the reasons of Grothendieck's departure from mathematics.
References
Differential equations
Representation theory
1949 births
Algerian mathematicians
Living people
People from Naâma Province
21st-century Algerian people |
https://en.wikipedia.org/wiki/Bonnie%20Stewart | Bonnie Madison Stewart (July 10, 1914 – April 15, 1994) was a professor of mathematics at Michigan State University from 1940 to 1980. He earned his Ph.D. from the University of Wisconsin–Madison in 1941, under the supervision of Cyrus Colton MacDuffee.
Contributions
Number theory
In 1952, the first edition of his book, Theory of Numbers, was published.
Stewart's contributions to number theory also include a complete characterization of the practical numbers in terms of their factorizations, which he published in 1954, a year before Wacław Sierpiński's independent discovery of the same result.
Geometry
In 1970 he published a book, Adventures among the toroids. A study of orientable polyhedra with regular faces, in which he discussed what are now called Stewart toroids. The book was handwritten in calligraphy with many formulas and illustrations. Like the Platonic solids, Archimedean solids, and Johnson solids, the Stewart polyhedra have regular polygons as faces. The first three categories are all convex, whereas Stewart toroids have polygonal-faced tunnels.
Selected publications
B. M. Stewart, Theory of Numbers, Macmillan, 1952. 2nd ed., Macmillan, 1964.
.
B. M. Stewart, Adventures Among the Toroids (1970) This 5"x13" edition was self-published by the author (printed by The John Henry Company) and has no ISBN.
B. M. Stewart, Adventures Among the Toroids, Revised Second Edition (1980) 11"x8.5". , converted much later to
References
External links
Stewart Toroids (Toroidal Solids with Regular Polygon Faces)
Stewart's polyhedra
Stewart toroids
1914 births
1994 deaths
20th-century American mathematicians
University of Wisconsin–Madison alumni
Michigan State University faculty |
https://en.wikipedia.org/wiki/Makoto%20Watanabe%20%28footballer%29 | is a former Japanese football player.
Club statistics
References
External links
1980 births
Living people
Kokushikan University alumni
Association football people from Shizuoka Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Ventforet Kofu players
Kataller Toyama players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Ernst%20Hairer | Ernst Hairer (born 19 June 1949) is a professor of mathematics at the University of Geneva known for his work in numerical analysis.
His PhD was completed at the University of Innsbruck.
He is the father of the mathematician Martin Hairer, who won the Fields medal in 2014.
Hairer is a member of the editorial boards for the journals Mathematics of Computation and Journal of Scientific Computing.
He wrote, with others, Solving Ordinary Differential Equations (Hairer, Nørsett, Wanner) and L'analyse au fil de l'histoire (Hairer, Wanner).
Awards and honors
Hairer holds a doctor honoris causa from the University of Lund.
In 2003 he was awarded, jointly with Gerhard Wanner, the Peter Henrici Prize by the Society for Industrial and Applied Mathematics.
In 2009 a conference on scientific computing and differential equations was held in honor of his 60th birthday, at the University of Geneva, and in the same year the 7th International Conference in Numerical Analysis and Applied Mathematics in Crete was dedicated in his honor.
In 2009–2010 he was John-von-Neumann guest professor at the Technical University of Munich.
See also
Bulirsch–Stoer algorithm
References
1949 births
Living people
20th-century Austrian mathematicians
21st-century Austrian mathematicians
Austrian educators
Academic staff of the University of Geneva |
https://en.wikipedia.org/wiki/K.%20David%20Elworthy | Kenneth David Elworthy is a Professor Emeritus of Mathematics at the University of Warwick. He works on stochastic analysis, stochastic differential equations and geometric analysis.
Life and career
Elworthy was born on 21 December 1940. He was educated at Bristol Grammar School, and in 1959 went up to Merton College, Oxford to read Mathematics as an undergraduate. In 1964 he married Susan Margaret Anderson. His DPhil was completed in 1967 under the supervision of Michael Francis Atiyah at the University of Oxford. Before his retirement from Warwick, he held a personal chair there. He also serves on the advisory board of the Center for Integrative Mathematical Sciences at Keio University in Tokyo, Japan.
Together with Xue-Mei Li and Yves Le Jan he wrote the books "The Geometry of Filtering" and "On the Geometry of Diffusion Operators and Stochastic Flows".
References
External links
Website at the University of Warwick
Academics of the University of Warwick
English mathematicians
Living people
1940 births
Alumni of Merton College, Oxford |
https://en.wikipedia.org/wiki/Bo%C3%A1z%20Klartag | Boáz Klartag (; born 25 April 1978) is an Israeli mathematician. He currently is a professor at the Weizmann Institute, and prior to that he was a professor at the Department of Pure Mathematics of Tel Aviv University, where he earned his doctorate under the supervision of Vitali Milman. Klartag made contributions in asymptotic geometric analysis and won the 2008 EMS Prize, as well as the 2010 Erdős Prize. He is an editor of the Journal d'Analyse Mathématique.
Selected works
References
External links
1978 births
Living people
21st-century Israeli mathematicians
Tel Aviv University alumni
International Mathematical Olympiad participants
Erdős Prize recipients |
https://en.wikipedia.org/wiki/Chandan%20Roy%20Sanyal | Chandan Roy Sanyal (born 30 January 1980) is an Indian actor who is known for his work in the Hindi and Bengali language films of India. After graduating with a degree in mathematics, he made his acting debut in the 2006 film Rang de Basanti, in a minor role. He then received critical acclaim for his supporting roles in the 2009 action film Kaminey and in his Bengali debut, the 2010 film Mahanagar@Kolkata. Sanyal then went on to receive recognition for his performances in the 2012 Bengali drama film Aparajita Tumi and as an insomniac artist in the 2013 romantic drama Prague.
Early life
Born and brought up in Delhi, Sanyal hails from a Bengali family. Chandan studied at Raisina Bengali School and then Zakir Husain College, where he received an honors degree in Mathematics.
Career
Sanyal made his debut as Batukeshwar Dutt in 2006's Rakeysh Omprakash Mehra film Rang De Basanti but his big break came with Vishal Bhardwaj's 2009 movie Kaminey where he got a lot of appreciation from the critics as well as audience.
Chandan also did an excellent job in the latest 2012 film "Aparajita Tumi". He has been signed on as one of the two leads in the film Kaanchi.
Filmography
Films
Web series
References
External links
1979 births
Living people
Male actors from Delhi
Indian male models
Bengali people
Indian male film actors
Male actors in Bengali cinema
Male actors in Hindi cinema |
https://en.wikipedia.org/wiki/2003%20Vegalta%20Sendai%20season | 2003 Vegalta Sendai season.
Competitions
Domestic results
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Vegalta Sendai
Vegalta Sendai seasons |
https://en.wikipedia.org/wiki/2003%20Kashima%20Antlers%20season | 2003 Kashima Antlers season
Competitions
Domestic results
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Kashima Antlers
Kashima Antlers seasons |
https://en.wikipedia.org/wiki/2003%20Urawa%20Red%20Diamonds%20season | 2003 Urawa Red Diamonds season
Competitions
Domestic results
J.League 1
Emperor's Cup
J.League Cup
Player statistics
Other pages
J. League official site
Urawa Red Diamonds
Urawa Red Diamonds seasons |
https://en.wikipedia.org/wiki/2003%20JEF%20United%20Ichihara%20season | 2003 JEF United Ichihara season
Competitions
Domestic results
J.League 1
Emperor's Cup
J.League Cup
Player statistics
Other pages
J. League official site
JEF United Ichihara
JEF United Chiba seasons |
https://en.wikipedia.org/wiki/2003%20Kashiwa%20Reysol%20season | 2003 Kashiwa Reysol season
Competitions
Domestic results
J.League 1
Emperor's Cup
J.League Cup
Player statistics
Other pages
J. League official site
Kashiwa Reysol
Kashiwa Reysol seasons |
https://en.wikipedia.org/wiki/2003%20FC%20Tokyo%20season | 2003 FC Tokyo season
Competitions
Domestic results
J.League 1
Emperor's Cup
J.League Cup
International results
Player statistics
Other pages
J. League official site
Tokyo
2003 |
https://en.wikipedia.org/wiki/2003%20Tokyo%20Verdy%201969%20season | 2003 Tokyo Verdy 1969 season
Competitions
Domestic results
J.League 1
Emperor's Cup
J.League Cup
Player statistics
Other pages
J. League official site
Tokyo Verdy 1969
Tokyo Verdy seasons |
https://en.wikipedia.org/wiki/2003%20Yokohama%20F.%20Marinos%20season | 2003 Yokohama F. Marinos season
Competitions
Domestic results
J.League 1
Emperor's Cup
J.League Cup
Player statistics
Kits
Other pages
J.League official site
Yokohama F. Marinos
Yokohama F. Marinos seasons |
https://en.wikipedia.org/wiki/2003%20J%C3%BAbilo%20Iwata%20season | 2003 Júbilo Iwata season
Competitions
Domestic results
J.League 1
Emperor's Cup
J.League Cup
Player statistics
Other pages
J. League official site
Jubilo Iwata
Júbilo Iwata seasons |
https://en.wikipedia.org/wiki/2003%20Nagoya%20Grampus%20Eight%20season | 2003 Nagoya Grampus Eight season
Competitions
Domestic results
J.League 1
Emperor's Cup
J.League Cup
Player statistics
Other pages
J. League official site
Nagoya Grampus Eight
Nagoya Grampus seasons |
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