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https://en.wikipedia.org/wiki/Fernando%20Moner
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Fernando Daniel Moner (born 30 December 1967) is a retired Argentine football player.
Club statistics
External links
1967 births
Living people
Argentine men's footballers
Argentine expatriate men's footballers
Footballers from Buenos Aires Province
San Lorenzo de Almagro footballers
La Liga players
Atlético Madrid footballers
Expatriate men's footballers in Spain
Argentine expatriate sportspeople in Spain
Expatriate men's footballers in Japan
Japan Soccer League players
J1 League players
J2 League players
Yokohama Flügels players
Yokohama FC players
Atlético Tucumán footballers
Club Atlético Platense footballers
Unión de Santa Fe footballers
Club Atlético Huracán footballers
Men's association football defenders
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https://en.wikipedia.org/wiki/Wilks%27s%20lambda%20distribution
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In statistics, Wilks' lambda distribution (named for Samuel S. Wilks), is a probability distribution used in multivariate hypothesis testing, especially with regard to the likelihood-ratio test and multivariate analysis of variance (MANOVA).
Definition
Wilks' lambda distribution is defined from two independent Wishart distributed variables as the ratio distribution of their determinants,
given
independent and with
where p is the number of dimensions. In the context of likelihood-ratio tests m is typically the error degrees of freedom, and n is the hypothesis degrees of freedom, so that is the total degrees of freedom.
Approximations
Computations or tables of the Wilks' distribution for higher dimensions are not readily available and one usually resorts to approximations.
One approximation is attributed to M. S. Bartlett and works for large m allows Wilks' lambda to be approximated with a chi-squared distribution
Another approximation is attributed to C. R. Rao.
Properties
There is a symmetry among the parameters of the Wilks distribution,
Related distributions
The distribution can be related to a product of independent beta-distributed random variables
As such it can be regarded as a multivariate generalization of the beta distribution.
It follows directly that for a one-dimension problem, when the Wishart distributions are one-dimensional with (i.e., chi-squared-distributed), then the Wilks' distribution equals the beta-distribution with a certain parameter set,
From the relations between a beta and an F-distribution, Wilks' lambda can be related to the F-distribution when one of the parameters of the Wilks lambda distribution is either 1 or 2, e.g.,
and
See also
Chi-squared distribution
Dirichlet distribution
F-distribution
Gamma distribution
Hotelling's T-squared distribution
Student's t-distribution
Wishart distribution
References
Continuous distributions
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https://en.wikipedia.org/wiki/Symmetrization
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In mathematics, symmetrization is a process that converts any function in variables to a symmetric function in variables.
Similarly, antisymmetrization converts any function in variables into an antisymmetric function.
Two variables
Let be a set and be an additive abelian group. A map is called a if
It is called an if instead
The of a map is the map
Similarly, the or of a map is the map
The sum of the symmetrization and the antisymmetrization of a map is
Thus, away from 2, meaning if 2 is invertible, such as for the real numbers, one can divide by 2 and express every function as a sum of a symmetric function and an anti-symmetric function.
The symmetrization of a symmetric map is its double, while the symmetrization of an alternating map is zero; similarly, the antisymmetrization of a symmetric map is zero, while the antisymmetrization of an anti-symmetric map is its double.
Bilinear forms
The symmetrization and antisymmetrization of a bilinear map are bilinear; thus away from 2, every bilinear form is a sum of a symmetric form and a skew-symmetric form, and there is no difference between a symmetric form and a quadratic form.
At 2, not every form can be decomposed into a symmetric form and a skew-symmetric form. For instance, over the integers, the associated symmetric form (over the rationals) may take half-integer values, while over a function is skew-symmetric if and only if it is symmetric (as ).
This leads to the notion of ε-quadratic forms and ε-symmetric forms.
Representation theory
In terms of representation theory:
exchanging variables gives a representation of the symmetric group on the space of functions in two variables,
the symmetric and antisymmetric functions are the subrepresentations corresponding to the trivial representation and the sign representation, and
symmetrization and antisymmetrization map a function into these subrepresentations – if one divides by 2, these yield projection maps.
As the symmetric group of order two equals the cyclic group of order two (), this corresponds to the discrete Fourier transform of order two.
n variables
More generally, given a function in variables, one can symmetrize by taking the sum over all permutations of the variables, or antisymmetrize by taking the sum over all even permutations and subtracting the sum over all odd permutations (except that when the only permutation is even).
Here symmetrizing a symmetric function multiplies by – thus if is invertible, such as when working over a field of characteristic or then these yield projections when divided by
In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for there are others – see representation theory of the symmetric group and symmetric polynomials.
Bootstrapping
Given a function in variables, one can obtain a symmetric function in variables by taking the sum over -element subsets of the variables.
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https://en.wikipedia.org/wiki/National%20Board%20for%20Higher%20Mathematics
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The National Board for Higher Mathematics (NBHM), founded in 1983 by the Indian Government, is a board in India intended to foster the development of higher mathematics, help in the establishment and development of mathematics centres, and give financial assistance to research projects and to doctoral and post-doctoral scholars. It is funded by the Department of Atomic Energy and is an autonomous body. NBHM functions autonomously preparing its budget based on the funds made available by Dept. of Atomic Energy.
Programs
The NBHM gives direct financial support to the following (among others):
The Mathematical Olympiads in India: NBHM finances the MO Cell and also arranges money for the IMO Training Camp and India's participation in the International Mathematical Olympiad.
Undergraduate and masters' scholarships.
Research Scholarship
For M.A. (Master of Arts) and M.Sc. (Master of Science) in Mathematics, NBHM conducts the Nationwide test.
The Institution
One of the most popular movements in the spreading Higher Mathematics in India taken up by NBHM is its funding to various university and institute libraries across India for books and journals in mathematics. NBHM manages its whole Library Movement through its National Library Committee. The business of the committee is divided into five major zones of India, viz. North, East, West, South and North Central zones. Besides extending the financial support to host of libraries in India, NBHM has recognized 12 libraries across India as the Regional Libraries. The Regional Libraries are expected to cater to the needs of the mathematicians and allied scientists in terms of library services. People from every walk of life can become a member of the Regional Libraries if they are doing research in mathematics of allied topics or if they are studying higher mathematics.
Nurture Program
National Board for Higher Mathematics in India, conducts a program (called Nurture Program) for 20 students of class XII, from the senior batch of the IMOTC who have shown a special interest in mathematics. Any of these students formally pursuing mathematics at the undergraduate level (B.A./B. Sc or an integrated M.A./M.S. course) is entitled to a scholarship of Rs. 2500/- (Rs. Two Thousand Five Hundred only) per month from NBHM. Selected students may participate in the nurture program even if they choose not to pursue a formal program in mathematics at the undergraduate level.
The nurture program is of four-year duration. Each batch of students is assigned to a faculty of active research mathematicians. The faculty devises a syllabus for the students for each academic year and also guides them with proper references. The faculty also keeps in touch with the students through post. At the end of a year the students are called for a contact program with the faculty for three to four weeks. During this period the faculty arranges lectures on diverse topics and clears specific difficulties of the students. In addition, the
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https://en.wikipedia.org/wiki/1994%E2%80%9395%20First%20League%20of%20FR%20Yugoslavia
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Statistics of First League of FR Yugoslavia () for the 1994–95 season.
Overview
Just as the previous season, the league consisted of 2 groups, A and B, each containing 10 clubs. Both groups were played in league system. By winter break all clubs in each group had met each other twice, home and away, with the bottom four from A group moving to group B, and being replaced by the top four from the B group. At the end of the season the same situation happened with four teams being replaced from A and B groups, and in addition, the bottom three clubs from the B group were relegated into the Second League of FR Yugoslavia for the next season and replaced by the top three from that league.
At the end of the season Red Star Belgrade became champions.
FK Partizan striker Savo Milošević become the league's top-scorer for second consecutive time, this time with 30 goals.
The relegated clubs were FK Spartak Subotica, FK Sutjeska Nikšić, FK Rudar Pljevlja.
That was the first season when Yugoslav clubs again qualified to the UEFA competitions after three years of ban due to UN embargo.
Teams
Autumn
IA league
Table
Results
IB league
Table
Results
Spring
IA league
Table
Results
IB league
Table
Results
UEFA Cup Playoff
Vojvodina was qualified to the 1995–96 UEFA Cup but they are not admitted, along with Partizan, because the country coefficient of Yugoslavia has been recalculated due to the split up. Budućnost Podgorica was qualified to the 1995 UEFA Intertoto Cup.
Winning squad
Champions: Red Star Belgrade (coach: Ljupko Petrović)
Players (league matches/league goals)
Dejan Petković (36/8)
Marko Perović (33/7)
Ivan Adžić (33/6)
Darko Kovačević (31/24)
Zvonko Milojević (31/0) -goalkeeper-
Nebojša Krupniković (30/23)
Dejan Stefanović (30/9)
Nenad Sakić (29/0)
Mitko Stojkovski (28/1)
Goran Đorović (28/0)
Bratislav Živković (25/2)
Nikola Radmanović (24/0)
Goran Stojiljković (17/7)
Srđan Bajčetić (14/1) sold to Celta de Vigo after 1st half of the season
Predrag Stanković (13/2)
Jovan Stanković (13/0)
Darko Pivaljević (7/2)
Zoran Riznić (7/2)
Aleksandar Kristić (7/0)
Dejan Stanković (7/0)
Zoran Mašić (6/3)
Milan Simeunović (4/0) -goalkeeper-
Perica Ognjenović (3/0)
Zoran Đorović (2/0)
Vinko Marinović (2/0)
Božidar Bandović (1/0)
Miodrag Božović (1/0)
Žarko Dragaš (1/0)
Darko Ljubojević (1/0)
Rade Mojović (1/0) -goalkeeper-
Top goalscorers
References
External links
Table at RSSSF
Yugoslav First League seasons
Yugo
1994–95 in Yugoslav football
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https://en.wikipedia.org/wiki/1995%E2%80%9396%20First%20League%20of%20FR%20Yugoslavia
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Statistics of First League of FR Yugoslavia () for the 1995–96 season.
Overview
Just as in the two previous seasons, the league was divided into 2 groups, A and B, consisting each one of 10 clubs. Both groups were played in league system. By winter break all clubs in each group meet each other twice, home and away, with the bottom four classified from A group moving to the group B, and being replaced by the top four from the B group. At the end of the season the same situation happened with four teams being replaced from A and B groups, adding the fact that the bottom two clubs from the B group were relegated into the Second League of FR Yugoslavia for the next season and replaced by the top two from that league.
At the end of the season FK Partizan were the champions.
The league top-scorer was FK Čukarički striker Vojislav Budimirović with 23 goals.
The relegated clubs were FK Napredak Kruševac and FK Radnički Beograd.
Autumn
IA league
Table
Results
IB league
Table
Results
Spring
IA league
Table
Results
IB league
Table
Results
IA Playoff
Relegation playoff
Winning squad
Champions: Partizan Belgrade (Coach: Ljubiša Tumbaković)
Players (league matches/league goals)
Ivica Kralj
Nikola Damjanac
Viktor Trenevski
Bratislav Mijalković
Darko Tešović
Gjorgji Hristov
Ivan Tomić
Dražen Bolić
Mladen Krstajić
Predrag Pažin
Đorđe Svetličić
Dejan Peković
Dejan Vukićević
Damir Čakar
Dragan Ćirić
Zoran Mirković
Niša Saveljić
Zoran Đurić
Igor Taševski
Albert Nađ
Marko Marković
Rahim Beširović
Source:
Top goalscorers
References
External links
Tables and results at RSSSF
Yugoslav First League seasons
Yugo
1995–96 in Yugoslav football
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https://en.wikipedia.org/wiki/Loomis%E2%80%93Whitney%20inequality
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In mathematics, the Loomis–Whitney inequality is a result in geometry, which in its simplest form, allows one to estimate the "size" of a -dimensional set by the sizes of its -dimensional projections. The inequality has applications in incidence geometry, the study of so-called "lattice animals", and other areas.
The result is named after the American mathematicians Lynn Harold Loomis and Hassler Whitney, and was published in 1949.
Statement of the inequality
Fix a dimension and consider the projections
For each 1 ≤ j ≤ d, let
Then the Loomis–Whitney inequality holds:
Equivalently, taking we have
implying
A special case
The Loomis–Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space to its "average widths" in the coordinate directions. This is in fact the original version published by Loomis and Whitney in 1949 (the above is a generalization).
Let E be some measurable subset of and let
be the indicator function of the projection of E onto the jth coordinate hyperplane. It follows that for any point x in E,
Hence, by the Loomis–Whitney inequality,
and hence
The quantity
can be thought of as the average width of in the th coordinate direction. This interpretation of the Loomis–Whitney inequality also holds if we consider a finite subset of Euclidean space and replace Lebesgue measure by counting measure.
The following proof is the original one
Corollary. Since , we get a loose isoperimetric inequality:
Iterating the theorem yields and more generallywhere enumerates over all projections of to its dimensional subspaces.
Generalizations
The Loomis–Whitney inequality is a special case of the Brascamp–Lieb inequality, in which the projections πj above are replaced by more general linear maps, not necessarily all mapping onto spaces of the same dimension.
References
Sources
Incidence geometry
Geometric inequalities
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https://en.wikipedia.org/wiki/Uarini
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Uarini is a municipality located in the Brazilian state of Amazonas. According to estimates of the Brazilian Institute of Geography and Statistics (IBGE), its population was 13,690 inhabitants in 2020. Its area is 10,246 km2.
The municipality contains 38% of the Baixo Juruá Extractive Reserve, created in 2001.
History
It has its history linked to the history of Tefé , which goes back to the village founded at the end of century XVII by the Jesuit Samuel Fritz . Until the end of the seventeenth century, disputes between the Spaniards and the Portuguese overlapped in the territory, only consolidating under the military occupation of Portugal in 1790. As a municipality, Tefé came to possess an area of 500,000 km2. From the middle of the 19th century onwards, dismemberment of its territory began, giving rise to the new municipalities of São Paulo de Olivença , Coari , Fonte Boa, São Felipe (now Eirunepé), Xibauá (now Carauari) Japurá and Maraã.
At the end of 1981 Tefé had an administrative structure in which five sub-districts were planned: Tefé, Caiambé, Alvarães, Jarauá and Uarini.
Economy
Primary Sector
Agriculture: is the most productive economic activity, with special emphasis for the culture of the cassava , from which the flour of Uarini is made. The brown-nut is in 2nd place in the economy. It has crops of rice, beans, jute, mallow, corn and sugarcane between temporary crops and, mango, avocado, banana, orange and lemon among permanent crops.
Livestock : in economic terms livestock has insignificant role.
Poultry: practiced in essentially domestic molds, aimed at subsistence and local consumption, not generating income for families.
Plant Extractivism: it reaches its greatest expression in the exploitation of native rubber, Brazil nut and wood .
Secondary Sector
Industries: pottery, baking, carpentry, furniture and metallurgy.
Tertiary sector
Retail: retailer.
Of the 5 counties in the country that had decreases in the HDI between 1991 and 2000, three are from Amazonas : Uarini, whose HDI increased from 0.611 to 0.599; Silves, from 0.684 to 0.675; And São Sebastião do Uatumã , from 0.661 to 0.659. This occurred solely because of decreases registered in the dimension of income , which were not offset by the positive increments observed in the dimensions longevity and education .
References
Municipalities in Amazonas (Brazilian state)
Populated places on the Amazon
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https://en.wikipedia.org/wiki/Scherk%20surface
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In mathematics, a Scherk surface (named after Heinrich Scherk) is an example of a minimal surface. Scherk described two complete embedded minimal surfaces in 1834; his first surface is a doubly periodic surface, his second surface is singly periodic. They were the third non-trivial examples of minimal surfaces (the first two were the catenoid and helicoid). The two surfaces are conjugates of each other.
Scherk surfaces arise in the study of certain limiting minimal surface problems and in the study of harmonic diffeomorphisms of hyperbolic space.
Scherk's first surface
Scherk's first surface is asymptotic to two infinite families of parallel planes, orthogonal to each other, that meet near z = 0 in a checkerboard pattern of bridging arches. It contains an infinite number of straight vertical lines.
Construction of a simple Scherk surface
Consider the following minimal surface problem on a square in the Euclidean plane: for a natural number n, find a minimal surface Σn as the graph of some function
such that
That is, un satisfies the minimal surface equation
and
What, if anything, is the limiting surface as n tends to infinity? The answer was given by H. Scherk in 1834: the limiting surface Σ is the graph of
That is, the Scherk surface over the square is
More general Scherk surfaces
One can consider similar minimal surface problems on other quadrilaterals in the Euclidean plane. One can also consider the same problem on quadrilaterals in the hyperbolic plane. In 2006, Harold Rosenberg and Pascal Collin used hyperbolic Scherk surfaces to construct a harmonic diffeomorphism from the complex plane onto the hyperbolic plane (the unit disc with the hyperbolic metric), thereby disproving the Schoen–Yau conjecture.
Scherk's second surface
Scherk's second surface looks globally like two orthogonal planes whose intersection consists of a sequence of tunnels in alternating directions. Its intersections with horizontal planes consists of alternating hyperbolas.
It has implicit equation:
It has the Weierstrass–Enneper parameterization
,
and can be parametrized as:
for and . This gives one period of the surface, which can then be extended in the z-direction by symmetry.
The surface has been generalised by H. Karcher into the saddle tower family of periodic minimal surfaces.
Somewhat confusingly, this surface is occasionally called Scherk's fifth surface in the literature. To minimize confusion it is useful to refer to it as Scherk's singly periodic surface or the Scherk-tower.
External links
Scherk's first surface in MSRI Geometry
Scherk's second surface in MSRI Geometry
Scherk's minimal surfaces in Mathworld
References
Minimal surfaces
Differential geometry
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https://en.wikipedia.org/wiki/Cardinality%20%28disambiguation%29
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Cardinality may refer to:
Cardinality of a set, a measure of the "number of elements" of a set in mathematics
Cardinality of a musical set, the number of pitch classes
Cardinality (data modeling), a term in database design, e.g. many-to-many or one-to-many relationships
Cardinality (SQL statements), a term used in SQL statements which describes the "uniqueness" of the data in a given column
Cardinal utility, in contrast with ordinal utility, in economics
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https://en.wikipedia.org/wiki/Dean%20Williams%20%28basketball%29
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Dean Williams (born 17 February 1977 in England) is a British former professional basketball player. He used to play for Reading Rockets and he is now teaching maths in Jumeirah College in Dubai.
The 6 ft 3 in tall Guard was educated at Marjon's and has played internationally for the England national team. Williams started his professional basketball career for the London Towers, having previously played for the amateur club Brixton Topcats.
He left the Towers in 2000 to join National Basketball League team the Plymouth Raiders, where he spent four successful seasons before moving back into the top-flight with teammate Roderick Wellington, both of whom joined British Basketball League team the Thames Valley Tigers in 2004, the same year that the Raiders also made the jump to the BBL. After one season with the Tigers, the club folded, and through the efforts of local fans a new club was set up to replace them, the Guildford Heat, of which Dean was a part of the first roster of the new club. In 2009, Williams moved to newly created Essex Pirates to become part of their first roster, just like at Guildford 4 years earlier. He played for Guildford Heat until the end of the 2011/12 season. He joined Reading Rockets at the start of the 2012/13 season. During the 2012/2013 season, Dean suffered a knee injury, a partially torn ACL. He had surgery, which was successful. Following the surgery, Dean decided to focus on his career as a Maths teacher. He is currently a successful full-time mathematics teacher at Jumeirah College in Dubai, UAE, teaching Ks3, GCSE and A-Levels.
References
Living people
1977 births
English men's basketball players
Plymouth Raiders players
Surrey Scorchers players
Black British sportsmen
English schoolteachers
English expatriates in the United Arab Emirates
Alumni of Plymouth Marjon University
Guards (basketball)
British Basketball League players
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https://en.wikipedia.org/wiki/Schoen%E2%80%93Yau%20conjecture
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In mathematics, the Schoen–Yau conjecture is a disproved conjecture in hyperbolic geometry, named after the mathematicians Richard Schoen and Shing-Tung Yau.
It was inspired by a theorem of Erhard Heinz (1952). One method of disproof is the use of Scherk surfaces, as used by Harold Rosenberg and Pascal Collin (2006).
Setting and statement of the conjecture
Let be the complex plane considered as a Riemannian manifold with its usual (flat) Riemannian metric. Let denote the hyperbolic plane, i.e. the unit disc
endowed with the hyperbolic metric
E. Heinz proved in 1952 that there can exist no harmonic diffeomorphism
In light of this theorem, Schoen conjectured that there exists no harmonic diffeomorphism
(It is not clear how Yau's name became associated with the conjecture: in unpublished correspondence with Harold Rosenberg, both Schoen and Yau identify Schoen as having postulated the conjecture). The Schoen(-Yau) conjecture has since been disproved.
Comments
The emphasis is on the existence or non-existence of an harmonic diffeomorphism, and that this property is a "one-way" property. In more detail: suppose that we consider two Riemannian manifolds M and N (with their respective metrics), and write
if there exists a diffeomorphism from M onto N (in the usual terminology, M and N are diffeomorphic). Write
if there exists an harmonic diffeomorphism from M onto N. It is not difficult to show that (being diffeomorphic) is an equivalence relation on the objects of the category of Riemannian manifolds. In particular, is a symmetric relation:
It can be shown that the hyperbolic plane and (flat) complex plane are indeed diffeomorphic:
so the question is whether or not they are "harmonically diffeomorphic". However, as the truth of Heinz's theorem and the falsity of the Schoen–Yau conjecture demonstrate, is not a symmetric relation:
Thus, being "harmonically diffeomorphic" is a much stronger property than simply being diffeomorphic, and can be a "one-way" relation.
References
Disproved conjectures
Hyperbolic geometry
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https://en.wikipedia.org/wiki/Decomposition%20matrix
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In mathematics, and in particular modular representation theory, a decomposition matrix is a matrix that results from writing the irreducible ordinary characters in terms of the irreducible modular characters, where the entries of the two sets of characters are taken to be over all conjugacy classes of elements of order coprime to the characteristic of the field. All such entries in the matrix are non-negative integers. The decomposition matrix, multiplied by its transpose, forms the Cartan matrix, listing the composition factors of the projective modules.
References
See also
Matrix decomposition
Representation theory of groups
Matrices
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https://en.wikipedia.org/wiki/Brauer%E2%80%93Suzuki%20theorem
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In mathematics, the Brauer–Suzuki theorem, proved by , , , states that if a finite group has a generalized quaternion Sylow 2-subgroup and no non-trivial normal subgroups of odd order, then the group has a center of order 2. In particular, such a group cannot be simple.
A generalization of the Brauer–Suzuki theorem is given by Glauberman's Z* theorem.
References
gives a detailed proof of the Brauer–Suzuki theorem.
Theorems about finite groups
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https://en.wikipedia.org/wiki/Line%E2%80%93line%20intersection
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In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or another line. Distinguishing these cases and finding the intersection have uses, for example, in computer graphics, motion planning, and collision detection.
In three-dimensional Euclidean geometry, if two lines are not in the same plane, they have no point of intersection and are called skew lines. If they are in the same plane, however, there are three possibilities: if they coincide (are not distinct lines), they have an infinitude of points in common (namely all of the points on either of them); if they are distinct but have the same slope, they are said to be parallel and have no points in common; otherwise, they have a single point of intersection.
The distinguishing features of non-Euclidean geometry are the number and locations of possible intersections between two lines and the number of possible lines with no intersections (parallel lines) with a given line.
Formulas
A necessary condition for two lines to intersect is that they are in the same plane—that is, are not skew lines. Satisfaction of this condition is equivalent to the tetrahedron with vertices at two of the points on one line and two of the points on the other line being degenerate in the sense of having zero volume. For the algebraic form of this condition, see .
Given two points on each line
First we consider the intersection of two lines and in two-dimensional space, with line being defined by two distinct points and , and line being defined by two distinct points and .
The intersection of line and can be defined using determinants.
The determinants can be written out as:
When the two lines are parallel or coincident, the denominator is zero.
Given two points on each line segment
The intersection point above is for the infinitely long lines defined by the points, rather than the line segments between the points, and can produce an intersection point not contained in either of the two line segments. In order to find the position of the intersection in respect to the line segments, we can define lines and in terms of first degree Bézier parameters:
(where and are real numbers). The intersection point of the lines is found with one of the following values of or , where
and
with
There will be an intersection if and . The intersection point falls within the first line segment if , and it falls within the second line segment if . These inequalities can be tested without the need for division, allowing rapid determination of the existence of any line segment intersection before calculating its exact point.
Given two line equations
The and coordinates of the point of intersection of two non-vertical lines can easily be found using the following substitutions and rearrangements.
Suppose that two lines have the equations and where and are the slopes (gradients) of the lines and where and are the -intercepts of the lines. At the point whe
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https://en.wikipedia.org/wiki/Harvey%20Butchart
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John Harvey Butchart (May 10, 1907 – May 29, 2002) was a mathematics professor who was well known for his hiking exploits in and around the Grand Canyon in Arizona, United States. Beginning in 1945, Butchart explored the Grand Canyon's backcountry on foot. He wrote extensively about his adventures and influenced generations of canyoneers.
Sparse human communities have lived, worked, and traveled in the harsh and beautiful Canyon terrain at least since the Ancestral Puebloans. Native Americans occupied parts of the canyon in the mid 1800s when American explorers first arrived, followed over time by prospectors, miners, researchers, and the outdoor tourists who now dominate the community. Most of the millions of visitors to Grand Canyon National Park remain in the developed South and North Rim areas. A smaller number hike into the Canyon along a few well-maintained trails. A very small number venture beyond, into the true wilderness of the Park. Harvey Butchart led the way, using extraordinary physical exertion and individual skill to travel where few others can.
Upon moving to Flagstaff in 1945, he began exploring the Grand Canyon. After hiking most of the main routes, he began to explore unofficial routes, old Native American trails, and even animal trails. His mentors included Merrel Clubb and Emery Kolb. He sometimes hiked alone but most often traveled with friends and students. That community of enthusiasts persists to the present, focused around the Backcountry Information Center. A 2007 biography tells his story and that of foot exploration in the Grand Canyon.
In contrast to his predecessors, Butchart kept a detailed log of his explorations, which would eventually reach more than 1,000 pages. He recorded 1,024 days spent in the Canyon, and over walked. He climbed 83 summits within the Canyon, and scaled the walls at 164 places, claiming 25 first ascents. He was credited with discovering over 100 rim-to-river routes within the Canyon.
By 1963 Butchart was the acknowledged expert on backcountry hiking. In multiple trips over several years, he had completed the very first route from one end of the national park to the other, except for about four miles below Great Thumb and Tahuta Points.
Colin Fletcher relied heavily on Butchart's knowledge to plan his own hike through the whole park in a single journey later that year.
Fletcher wrote:
Today, Harvey Butchart is a compact, coiled-spring fifty-five – and a happy and devoted schizophrenic. Teaching mathematics is only one of his worlds. At intervals he lives in a quite different reality. His three-year-old grandson, a young man of perception, recently heard someone use the words “Grand Canyon.” “Where Grandpa lives?“ he asked, just to make sure.
Beginning in 1970, Butchart published three slim volumes of trail notes from his exploration records.
He continued hiking until 1987. In 1998, the books were republished in one volume with additional new material. Butchart's famously cryptic t
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https://en.wikipedia.org/wiki/Mohamed%20Hamdoud
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Mohamed "Moha" Hamdoud (born June 9, 1976 in El Biar, Alger, Algeria) is an Algerian former football player.
Club career
International career
Career statistics
Club
Honours
Won the Algerian League four times with USM Alger in 1996, 2002, 2003 and 2005
Won the Algerian Cup five times with USM Alger in 1997, 1999, 2001, 2003 and 2004
Semi-finalist in the African Champions League twice with USM Alger in 1997 and 2003
Runner-up in the Algerian League three times with USM Alger in 1998, 2001 and 2004
Finalist in the Algerian Cup two times with USM Alger in 2006 and 2007
Has 5 caps for the Algerian National Team
References
External links
1976 births
Living people
Algerian men's footballers
Algeria men's international footballers
USM Alger players
Paradou AC players
JS El Biar players
People from El Biar
Footballers from Algiers
Algeria men's under-23 international footballers
Men's association football defenders
Competitors at the 1997 Mediterranean Games
Mediterranean Games competitors for Algeria
21st-century Algerian people
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https://en.wikipedia.org/wiki/Johann%20Schweigger
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Johann Salomo Christoph Schweigger (8 April 1779 – 6 September 1857) was a German chemist, physicist, and professor of mathematics born in Erlangen.
J.S.C.Schweigger was the son of Friedrich Christian Lorenz Schweigger, professor of theologie in Erlangen (1786 until his death in 1802). He studied philosophy in Erlangen. His PhD involved the Homeric Question revived at that time by Friedrich August Wolf. Johann Tobias Mayer, Georg Friedrich Hildebrandt and Karl Christian von Langsdorf convinced him to switch to physics and chemistry and he lectured on this subjects in Erlangen until 1803 before taking a position as schoolteacher in Bayreuth and in 1811 in Nuremberg. During 1816-1819 he was appointed professor of philosophy in Erlangen teaching physics and chemistry. 1816 he was elected member of the Leopoldina. 1819 he moved on to the university of Halle.
In 1820 he built the first sensitive galvanometer, naming it after Luigi Galvani. He created this instrument, acceptable for actual measurement as well as detection of small amounts of electric current, by wrapping a coil of wire around a graduated compass. The instrument was initially called a multiplier.
He is the father of Karl Ernst Theodor Schweigger and adopted one of his students Franz Wilhelm Schweigger-Seidel as his own son.
Written works
Einleitung in die Mythologie auf dem Standpunkte der Naturwissenschaft, Halle (1836) - Introduction to mythology, from the standpoint of natural science.
Über naturwissenschaftliche Mysterien in ihrem Verhältnis zur Litteratur des Altertums, Halle (1843) - Involving scientific mysteries in their relation to the literature of antiquity.
Über das Elektron der Alten, Greifswald (1848) - On the electron of the past.
Über die stöchiometrischen Reihen, Halle (1853) - On the stoichiometry series.
References
J. S. C. Schweigger: His Romanticism and His Crystal Electrical Theory of Matter by H. A. M. Snelders (1971)
External links
(German)
1779 births
1857 deaths
19th-century German chemists
19th-century German inventors
19th-century German physicists
People from Erlangen
People from the Principality of Bayreuth
University of Erlangen-Nuremberg alumni
Academic staff of the Martin Luther University of Halle-Wittenberg
19th-century German mathematicians
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https://en.wikipedia.org/wiki/Dendroid%20%28topology%29
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In mathematics, a dendroid is a type of topological space, satisfying the properties that it is hereditarily unicoherent (meaning that every subcontinuum of X is unicoherent), arcwise connected, and forms a continuum. The term dendroid was introduced by Bronisław Knaster lecturing at the University of Wrocław, although these spaces were studied earlier by Karol Borsuk and others.
proved that dendroids have the fixed-point property: Every continuous function from a dendroid to itself has a fixed point. proved that every dendroid is tree-like, meaning that it has arbitrarily fine open covers whose nerve is a tree. The more general question of whether every tree-like continuum has the fixed-point property, posed by ,
was solved in the negative by David P. Bellamy, who gave an example of a tree-like continuum without the fixed-point property.
In Knaster's original publication on dendroids, in 1961, he posed the problem of characterizing the dendroids which can be embedded into the Euclidean plane. This problem remains open. Another problem posed in the same year by Knaster, on the existence of an uncountable collection of dendroids with the property that no dendroid in the collection has a continuous surjection onto any other dendroid in the collection, was solved by and , who gave an example of such a family.
A locally connected dendroid is called a dendrite. A cone over the Cantor set (called a Cantor fan) is an example of a dendroid that is not a dendrite.
References
Continuum theory
Trees (topology)
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https://en.wikipedia.org/wiki/Dendrite%20%28mathematics%29
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In mathematics, a dendrite is a certain type of topological space that may be characterized either as a locally connected dendroid or equivalently as a locally connected continuum that contains no simple closed curves.
Importance
Dendrites may be used to model certain types of Julia set. For example, if 0 is pre-periodic, but not periodic, under the function , then the Julia set of is a dendrite: connected, without interior.
References
See also
Misiurewicz point
Real tree, a related concept defined using metric spaces instead of topological spaces
Dendroid (topology) and unicoherent space, two more general types of tree-like topological space
Continuum theory
Trees (topology)
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https://en.wikipedia.org/wiki/Mark%20Embree
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Mark Embree is professor of computational and applied mathematics at Virginia Tech in Blacksburg, Virginia. Until 2013, he was a professor of computational and applied mathematics at Rice University in Houston, Texas.
Mark Embree was awarded Man of the Year and Outstanding Student in the College of Arts and Sciences at Virginia Tech in 1996. He was also a Rhodes Scholar at the University of Oxford, where he completed his doctorate.
Early life
Mark Embree attended Thomas Jefferson High School for Science and Technology.
Research
His main research interests are Krylov subspace methods, non-normal operators and spectral perturbation theory, Toeplitz matrices, random matrices, and damped wave operators.
Books
Dr Mark Embree wrote a book with Lloyd N. Trefethen titled Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators.
See also
Embree–Trefethen constant
External links
Dr. Embree's Virginia Tech Homepage
Dr. Embree's Rice Homepage
Dr. Embree's Mathematical Genealogy
Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators
References
20th-century American mathematicians
American Rhodes Scholars
Rice University faculty
Virginia Tech alumni
Year of birth missing (living people)
Living people
Place of birth missing (living people)
21st-century American mathematicians
Thomas Jefferson High School for Science and Technology alumni
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https://en.wikipedia.org/wiki/Prior%20information
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Prior information may refer to
Prior probability
A prior information notice (PIN) issued in advance of procurement actions for the purposes of government procurement in the European Union
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https://en.wikipedia.org/wiki/Group%20technology
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Group technology or TZ is a manufacturing technique in which parts having similarities in geometry, manufacturing process and/or functions are manufactured in one location using a small number of machines or processes. Group technology is based on a general principle that many problems are similar and by grouping similar problems, a single solution can be found to a set of problems, thus saving time and effort.
The group of similar parts is known as part family and the group of machineries used to process an individual part family is known as machine cell. It is not necessary for each part of a part family to be processed by every machine of corresponding machine cell. This type of manufacturing in which a part family is produced by a machine cell is known as cellular manufacturing.
The manufacturing efficiencies are generally increased by employing GT because the required operations may be confined to only a small cell and thus avoiding the need for transportation of in-process parts.
Group technology is an approach in which similar parts are identified and grouped together in order to take advantage of the similarities in design and production. Similarities among parts permit them to be classified into part families.
The advantage of GT can be divided into three groups:
Engineering
Manufacturing
Process Planning
Disadvantages of GT Manufacturing :
Involves less manufacturing flexibility
Increases the machine down time as machines are grouped as cells which may not be functional throughout the production process.
References
Secondary sector of the economy
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https://en.wikipedia.org/wiki/List%20of%20United%20States%20men%27s%20international%20soccer%20players
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The following tables include various statistics for players on the United States men's national soccer team (featuring all caps, goals, assists and goalkeeper wins and shutouts) from the team's inception in 1916 through the October 17, 2023 game with Ghana.
Appearances
Players capped since 2022 are shown in Bold.
Goals
Active players are shown in Bold.
Assists
Active players are shown in Bold.
USSF did not begin tracking assists until the 1970s. The top twenty are most likely accurate as no players before the mid-1980s amassed more than twenty or thirty caps. For example, Boris Bandov, the player active before 1980 with the highest number of caps, played 33 times between 1976 and 1983, while Perry Van der Beck played 23 times between 1979 and 1985. With the typical low scores of the times, it was unlikely any players before the 1980s assisted on more than a handful of goals.
Wins
Active players are shown in Bold.
Shutouts
Active players are shown in Bold.
See also
List of United States men's national soccer team hat-tricks
List of United States women's international soccer players
Association football player non-biographical articles
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https://en.wikipedia.org/wiki/CHAMPS%20Project
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The CHAMPS Project (Creating High Achievements in Mathematics, Problem-solving, and Science) is a combined effort of the Mississippi School for Mathematics and Science and the Mississippi University for Women aimed at improving various aspects of education in Mississippi. The goal of the CHAMPS Project is to improve student achievement and teacher quality in mathematics through a sustained program of professional development for K-8th grade teachers based on College and Career Readiness mathematics content, teaching strategies, and utilizing formative assessment to inform and guide instruction.
References
External links
MSMS CHAMPS Project
MSMS homepage
Education in Mississippi
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https://en.wikipedia.org/wiki/Palmer%20High%20School%20%28Alaska%29
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Palmer High School is a high school located in the Matanuska-Susitna Borough in the city of Palmer, Alaska. It offers classes in fine arts, mathematics, world languages, physical education and health, science, English, social sciences, and career and technical education. Student support services are available for students.
Sports
Palmer High School's sports include, baseball, wrestling, swimming, diving, cross country running, cross country skiing, track and field, football, ice hockey, volleyball, e-sports, and soccer.
Machetanz Field is located on campus.
History
The school was established in 1936.
Curriculum
The foreign languages offered are French and Japanese. IB classes (see below) are offered in Humanities/Literature, Math, Biology, Chemistry, History, Music, Art, Agriculture, and Foreign Languages. In 2012 the school began offering the APEX online education program.
International Baccalaureate
Palmer High School has been an International Baccalaureate World School since 1999. It is one of two International Baccalaureate schools in Alaska.
Notable alumni
Talis Colberg (1976), former Alaska State Attorney General
References
External links
Palmer High School homepage
Public high schools in Alaska
Schools in Matanuska-Susitna Borough, Alaska
International Baccalaureate schools in Alaska
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https://en.wikipedia.org/wiki/EDGE%20Foundation
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The EDGE Foundation (Enhancing Diversity in Graduate Education) is an organization which helps women get advanced degrees in mathematics.
History
The EDGE program was launched in 1998 by Sylvia Bozeman and Rhonda Hughes to support female students pursuing graduate degrees in the mathematical sciences. The first EDGE summer session was held at Bryn Mawr College in 1998 and the location alternated between Bryn Mawr and Spelman College until 2003. Since 2003 the summer program has been hosted by Pomona College, Florida A&M University, Harvey Mudd College, Mills College, New College of Florida, North Carolina A&T State University, North Carolina State University, Purdue University, and Texas Tech.
In 2013, in response to an overwhelming push from former EDGE participants, the Sylvia Bozeman and Rhonda Hughes EDGE Foundation was established. The mission of this 501(c)(3) non-profit organization is to support and oversee all EDGE programming.
Purpose
The EDGE program is designed to offer comprehensive mentoring for women pursuing careers in the mathematical sciences. Activities are designed to provide ongoing support toward the academic development and research productivity at several critical stages, including entering graduate students, advanced graduate students, postdocs and early career mathematicians. Along with its signature summer session, the Foundation supports an annual conference, mini-sabbaticals for research collaborations, regional research symposia, regional mentoring clusters, travel support for research talks, and other open-ended mentoring activities. The EDGE Summer Session is a four-week residential program for women entering graduate programs in the mathematical sciences. The workshops are immersion experiences that simulate the fast pace of studying graduate level mathematics.
Sponsors
EDGE receives support from The National Science Foundation. Other sponsors include:
American Mathematical Society
Bryn Mawr College
California State Polytechnic University, Pomona
Cornell University
Henry Luce Foundation
National Security Agency
Pomona College
Spelman College
Springer Publishing
Texas Tech University
University of Michigan
University of Nebraska
University of Washington
Worcester Polytechnic Institute
Impact and awards
In 2015 EDGE received the Presidential Award for Excellence in Science, Mathematics and Engineering Mentoring (PAESMEM) The citation for the award commented on the phenomenal success of this organization, and noted that at the time of the award over 200 women had participated in 16 EDGE summer sessions. Fifty-six women (of whom 46 percent are minorities) had completed Ph.D. programs, and over 65 were still working toward their Ph.D. Remarkably, in 2009, EDGE participants accounted for over 35 percent of all Ph.D.s granted to African-American women.
In 2019, Karen Uhlenbeck became the first woman to be awarded the prestigious Abel Prize, which is considered the Nobel Prize of mathematics. On May 21, 2019
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https://en.wikipedia.org/wiki/Kurepa%20tree
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In set theory, a Kurepa tree is a tree (T, <) of height ω1, each of whose levels is at most countable, and has at least ℵ2 many branches. This concept was introduced by . The existence of a Kurepa tree (known as the Kurepa hypothesis, though Kurepa originally conjectured that this was false) is consistent with the axioms of ZFC: Solovay showed in unpublished work that there are Kurepa trees in Gödel's constructible universe . More precisely, the existence of Kurepa trees follows from the diamond plus principle, which holds in the constructible universe. On the other hand, showed that if a strongly inaccessible cardinal is Lévy collapsed to ω2 then, in the resulting model, there are no Kurepa trees. The existence of an inaccessible cardinal is in fact equiconsistent with the failure of the Kurepa hypothesis, because if the Kurepa hypothesis is false then the cardinal ω2 is inaccessible in the constructible universe.
A Kurepa tree with fewer than 2ℵ1 branches is known as a Jech–Kunen tree.
More generally if κ is an infinite cardinal, then a κ-Kurepa tree is a tree of height κ with more than κ branches but at most |α| elements of each infinite level α<κ, and the Kurepa hypothesis for κ is the statement that there is a κ-Kurepa tree. Sometimes the tree is also assumed to be binary. The existence of a binary κ-Kurepa tree is equivalent to the existence of a Kurepa family: a set of more than κ subsets of κ such that their intersections with any infinite ordinal α<κ form a set of cardinality at most α. The Kurepa hypothesis is false if κ is an ineffable cardinal, and conversely Jensen showed that in the constructible universe for any uncountable regular cardinal κ there is a κ-Kurepa tree unless κ is ineffable.
Specializing a Kurepa tree
A Kurepa tree can be "killed" by forcing the existence of a function whose value on any non-root node is an ordinal less than the rank of the node, such that whenever three nodes, one of which is a lower bound for the other two, are mapped to the same ordinal, then the three nodes are comparable. This can be done without collapsing ℵ1, and results in a tree with exactly ℵ1 branches.
See also
Aronszajn tree
Suslin tree
References
Trees (set theory)
Independence results
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https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Srinivasa%20Ramanujan
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Srinivasa Ramanujan (1887 – 1920) is the eponym of all of the topics listed below.
Mathematics
Brocard–Ramanujan Diophantine equation
Dougall–Ramanujan identity
Landau–Ramanujan constant
Ramanujan's congruences
Hardy–Ramanujan number
Hardy–Ramanujan theorem
Hardy–Ramanujan asymptotic formula
Ramanujan identity
Ramanujan machine
Ramanujan–Nagell equation
Ramanujan–Peterssen conjecture
Ramanujan–Soldner constant
Ramanujan summation
Ramanujan theta function
Ramanujan graph
Ramanujan's tau function
Ramanujan's ternary quadratic form
Ramanujan prime
Ramanujan's constant
Ramanujan's lost notebook
Ramanujan's master theorem
Ramanujan's sum
Rogers–Ramanujan identities
Rogers–Ramanujan continued fraction
Ramanujan–Sato series
Ramanujan magic square
Journals
Hardy–Ramanujan Journal
Journal of the Ramanujan Mathematical Society
Ramanujan Journal
Institutions and societies
Ramanujan College, University of Delhi
Ramanujan Institute for Advanced Study in Mathematics
Srinivasa Ramanujan Institute of Technology
Ramanujan Mathematical Society
Srinivasa Ramanujan Centre at Sastra University https://sas.sastra.edu/src/
Srinivasa Ramanujan Concept School
Ramanujan Hostel, Indian Institute of Management, Calcutta
Ramanujan computer centre, Department of Mathematics, Rajdhani College, University of Delhi
Srinivisa Ramanujan Library, Indian Institute of Science Education and Research, Pune
Prizes and awards
Srinivasa Ramanujan Medal
SASTRA Ramanujan Prize
DST-ICTP-IMU Ramanujan Prize
Ramanujan Prize, University of Madras
Places
Ramanujan IT City, Chennai
Ramanujan Math Park, Chittoor, Andhra Pradesh, India
References
Ramanujan
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https://en.wikipedia.org/wiki/F%20%28disambiguation%29
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F is the sixth letter of the Latin alphabet.
F may also refer to:
Science and technology
Mathematics
F or f, the number 15 in hexadecimal and higher positional systems
pFq, the hypergeometric function
F-distribution, a continuous probability distribution
F-test, a statistical test
f, SI prefix femto, factor 10−15
, Fibonacci number
Computing and engineering
F (programming language), a subset of Fortran 95
F Sharp (programming language), a functional and object-oriented language for the .NET platform.
F* (programming language), a dependently typed functional language for the .NET platform.
F-measure, the harmonic mean of precision and recall
f, in programming languages often used to represent the floating point
F connector, used for inlet in cable modems
F crimp, a type of solderless electrical connection
F band (NATO), a radio frequency band from 3 to 4 GHz
F band (waveguide), a millimetre wave band from 90 to 140 GHz
Physics
°F, degree on the Fahrenheit temperature scale
F, for farad, a unit for electric capacitance
or ℱ, the Faraday constant
, focal length of a lens
f-number (sometimes called f-ratio, f-stop, or written f/), the focal length divided by the aperture diameter
, , , force
, frequency
F (or A), Helmholtz Free Energy
F, the electromagnetic field tensor in electromagnetism
Often generalized to include the field tensors of other interactions, as in i.e. the gluon field strength tensor.
F region, part of the ionosphere
Chemistry
Fluorine, symbol F, a chemical element
f-block, a block of elements in the periodic table of elements
F (or Phe), abbreviation for compound phenylalanine
Biology
Form (botany)
Form (zoology)
Bioavailability, which is expressed as the letter "F" in medical chemistry equations
Haplogroup F (Y-DNA), a patrilineal haplogroup
Haplogroup F (mtDNA), a matrilineal haplogroup
Music
F (musical note)
F major, a scale
F minor, a scale
F major chord, Chord names and symbols (popular music)
f, Forte (music)
F (album), an album by Japanese singer Masaharu Fukuyama
F Album, an album by Japanese duo KinKi Kids
"F", song on the double A-side single "Tsume Tsume Tsume/F" by Japanese metal band Maximum the Hormone
Transport
F Sixth Avenue Local, a rapid transit service of the New York City Subway
F (S-train), trains on the ring line of Copenhagen's commuter train network
F Market & Wharves, a heritage streetcar line operated by the San Francisco Municipal Railway
Tokyo Metro Fukutoshin Line, a subway service operated by the Tokyo Metro, labeled
The NYSE ticker symbol of the Ford Motor Company
, the official West Japan Railway Company service symbol for the Osaka Higashi Line.
Ford F-Series, a series of full-size pickup trucks from Ford Motor Company
Military
F, a class of designation in the United States military aircraft designation systems which stands for "Fighter"
F, a common designation for fighter aircraft
Foxtrot, the military time zone code for UTC+06:00
F band (NATO),
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https://en.wikipedia.org/wiki/U%20%28disambiguation%29
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U is the twenty-first letter of the Latin alphabet.
U may also refer to:
Science
Mathematics
, union (set theory)
U-set, a set of uniqueness
U, the unitary group
Chemistry
Uranium, symbol U, a chemical element
u, the Dalton (unit), a unified atomic mass unit to express atomic and molecular masses
Astronomy
U, ultraviolet magnitude, in a UBV photometric system
U, an October 16 through 31 discovery in the provisional designation of a comet
U (trans-Neptunian object), a possible extremely distant trans-Neptunian object
Computing
<u>, a now-deprecated HTML element denoting underlined text
U, representing 1.75" as the minimal height of a rack unit
U, representing a 10x10x10cm CubeSat
U, universal Turing machine
Biology
U, abbreviation for uracil
U, mitochondrial haplogroup U
Other scientific uses
U, a common notation for potential energy
U, the middle of an edge joining a hexagonal and a square face of the Brillouin zone of a face-centered cubic lattice, in solid-state physics
U, one of the two subcarrier-modulated color-difference channels in the YUV colorspace, in video
U, the recommended symbol for a system's internal energy, in thermodynamics
U, units, any of various standard units of biological activity, such as insulin units, enzyme units, or penicillin units
U, the enzyme unit
International units, for which the symbol U is sometimes used
U, the abbreviation for selenocysteine, an uncommon amino acid containing selenium
u, the alternative abbreviation for the SI prefix "micro-" when the Greek letter mu (µ) is not available
U or U-value, the symbol for the overall heat transfer coefficient or thermal transmittance
Transportation
U Line, also known as the Uijeongbu Line, a light metro line in Seoul
Transilien Line U, a line of the Paris transport network
U (Los Angeles Railway), a former streetcar line in Los Angeles
Yurikamome, an automated guideway line in Tokyo, labeled
The official West Japan Railway Company service symbol for:
Sakurai Line.
Kibi Line.
Music
’u’, the first opera in the Klingon language
"U" (Kendrick Lamar song), 2015
"U" (Super Junior song), 2006
U (Incredible String Band album), 1970
U (NiziU album), 2021
U (Tourist album), 2016 album by Tourist
"U", a song by KNK from Remain
"U", a song by Pearl Jam on their album Lost Dogs
"U", a song by S.E.S., released in 2002
"U", an album by The Enid, released in 2019
"U", a song by Treasure (band), released in 2022
People
An honorific used in Burmese names
An alternative spelling of Woo (Korean surname), including a list of people with that family name
An alternative spelling of Woo (Korean given name), including a list of people with that given name
U of Goryeo (1363–1389), king of Goryeo (Korea), often called King Woo
Places
U, Federated States of Micronesia
Ü-Tsang, one of three historical provinces of Tibet
Ü (region), a particular region of Ü-Tsang
Language
/u/, the close back rounded vowel in the Internation
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https://en.wikipedia.org/wiki/Central%20Bureau%20of%20Investigation%20and%20Statistics
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The Central Bureau of Investigation and Statistics (CBIS) (, commonly known as Zhongtong, ), was an intelligence unit under the Organisation Department of the Central Executive Committee of the Kuomintang. It was one of Chiang Kai-shek's two police and military intelligence agencies, the other being the Military Bureau of Investigation and Statistics headed by Dai Li from 1929 until his death in 1946. The CBIS focused on civilian intelligence, while the MBIS targeted military activities.
The CBIS bureau was largely superseded by the Ministry of Justice Investigation Bureau in Taiwan after 1949.
History
The previous body of CBIS had its origin in the CC Clique, which was founded in 1927 as a secret spying agency.
In 1931, Chen Lifu was appointed the head of the Kuomintang's Organization Department and he set up the intelligence unit.
In 1935, this intelligence body was re-organized as the Central Bureau of Investigation and Statistics.
See also
Ministry of Justice Investigation Bureau
National Security Bureau (Taiwan)
Bureau of Investigation and Statistics
Republic of China (1912–1949)
Kuomintang
References
Further reading
Warlord cliques in Republican China
Taiwanese intelligence agencies
Defunct intelligence agencies
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https://en.wikipedia.org/wiki/Dehn%E2%80%93Sommerville%20equations
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In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their general form was established by Duncan Sommerville in 1927. The Dehn–Sommerville equations can be restated as a symmetry condition for the [[h-vector|h-vector]] of the simplicial polytope and this has become the standard formulation in recent combinatorics literature. By duality, analogous equations hold for simple polytopes.
Statement
Let P be a d-dimensional simplicial polytope. For i = 0, 1, ..., d − 1, let fi denote the number of i-dimensional faces of P. The sequence
is called the 'f-vector of the polytope P. Additionally, set
Then for any k = −1, 0, ..., d − 2, the following Dehn–Sommerville equation' holds:
When k = −1, it expresses the fact that Euler characteristic of a (d − 1)-dimensional simplicial sphere is equal to 1 + (−1)d − 1.
Dehn–Sommerville equations with different k are not independent. There are several ways to choose a maximal independent subset consisting of equations. If d is even then the equations with k = 0, 2, 4, ..., d − 2 are independent. Another independent set consists of the equations with k = −1, 1, 3, ..., d − 3. If d is odd then the equations with k = −1, 1, 3, ..., d − 2 form one independent set and the equations with k = −1, 0, 2, 4, ..., d − 3 form another.
Equivalent formulations
Sommerville found a different way to state these equations:
where 0 ≤ k ≤ (d−1). This can be further facilitated introducing the notion of h-vector of P. For k = 0, 1, ..., d, let
The sequence
is called the h-vector of P. The f-vector and the h-vector uniquely determine each other through the relation
Then the Dehn–Sommerville equations can be restated simply as
The equations with 0 ≤ k ≤ (d−1) are independent, and the others are manifestly equivalent to them.
Richard Stanley gave an interpretation of the components of the h-vector of a simplicial convex polytope P in terms of the projective toric variety X associated with (the dual of) P. Namely, they are the dimensions of the even intersection cohomology groups of X:
(the odd intersection cohomology groups of X are all zero). In this language, the last form of the Dehn–Sommerville equations, the symmetry of the h-vector, is a manifestation of the Poincaré duality in the intersection cohomology of X.
References
Branko Grünbaum, Convex Polytopes. Second edition. Graduate Texts in Mathematics, Vol. 221, Springer, 2003
Richard P. Stanley, Combinatorics and Commutative Algebra. Second edition. Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1996.
D. M. Y. Sommerville (1927) The relations connecting the angle sums and volume of a polytope in space of n dimensions. Proceedings of the Royal Society Series A, 115:103–19, weblink from JSTOR.
Günter M. Ziegler, Lectures on Polytopes
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https://en.wikipedia.org/wiki/Duncan%20Sommerville
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Duncan MacLaren Young Sommerville (1879–1934) was a Scottish mathematician and astronomer. He compiled a bibliography on non-Euclidean geometry and also wrote a leading textbook in that field. He also wrote Introduction to the Geometry of N Dimensions, advancing the study of polytopes. He was a co-founder and the first secretary of the New Zealand Astronomical Society.
Sommerville was also an accomplished watercolourist, producing a series New Zealand landscapes.
The middle name 'MacLaren' is spelt using the old orthography M'Laren in some sources, for example the records of the Royal Society of Edinburgh.
Early life
Sommerville was born on 24 November 1879 in Beawar in India, where his father the Rev Dr James Sommerville, was employed as a missionary by the United Presbyterian Church of Scotland. His father had been responsible for establishing the hospital at Jodhpur, Rajputana.
The family returned home to Perth, Scotland, where Duncan spent 4 years at a private school, before completing his education at Perth Academy. His father died in his youth. He lived with his mother at 12 Rose Terrace. Despite his father's death, he won a scholarship, allowing him to continue his studies to university level.
He then studied mathematics at the University of St Andrews in Fife, graduating MA in 1902. He then began as an assistant lecturer at the university. In 1905 he gained his doctorate (DSc) for his thesis, Networks of the Plane in Absolute Geometry and was promoted to lecturer. He continued teaching mathematics at St Andrews until 1915.
In projective geometry the method of Cayley–Klein metrics had been used in the 19th century to model non-euclidean geometry. In 1910 Duncan wrote "Classification of geometries with projective metrics". The classification is described by Daniel Corey as follows:
He classifies them into 9 types of plane geometries, 27 in dimension 3, and more generally 3n in dimension n. A number of these geometries have found applications, for instance in physics.
In 1910 Sommerville reported to the British Association on the need for a bibliography on non-euclidean geometry, noting that the field had no International Association like the Quaternion Society to sponsor it.
In 1911 Sommerville published his compiled bibliography of works on non-euclidean geometry, and it received favorable reviews. In 1970 Chelsea Publishing issued a second edition which referred to collected works then available of some of the cited authors.
Sommerville was elected a Fellow of the Royal Society of Edinburgh in 1911. His proposers were Peter Redford Scott Lang, Robert Alexander Robertson, William Peddie and George Chrystal.
Family
In 1912 he married Louisa Agnes Beveridge.
Work in New Zealand
In 1915 Sommerville went to New Zealand to take up the Chair of Pure and Applied Mathematics at the Victoria College of Wellington.
Duncan became interested in honeycombs and wrote "Division of space by congruent triangles and tetrahedra" in 1923. The fo
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https://en.wikipedia.org/wiki/Ban%20number
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In recreational mathematics, a ban number is a number that does not contain a particular letter when spelled out in English; in other words, the letter is "banned." Ban numbers are not precisely defined, since some large numbers do not follow the standards of number names (such as googol and googolplex).
There are several published sequences of ban numbers:
The aban numbers do not contain the letter A. The first few aban numbers are 1 through 999, 1,000,000 through 1,000,999, 2,000,000 through 2,000,999, ... The word "and" is not counted.
The eban numbers do not contain the letter E. The first few eban numbers are 2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, 2000, 2002, 2004, ... . The sequence was coined in 1990 by Neil Sloane. Coincidentally, all the numbers in the sequence are even.
The iban numbers do not contain the letter I. The first few iban numbers are 1, 2, 3, 4, 7, 10, 11, 12, 14, 17, 20, 21, 22, 23, 24, 27, 40, ... . Since all -illion numbers contain the letter I, there are exactly 30,275 iban numbers, the largest being 777,777.
The oban numbers do not contain the letter O. The first few oban numbers are 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 23, 25, 26, ... . Since "thousand" and all the -illion numbers contain the letter O, there are exactly 454 oban numbers, the largest being 999.
The tban numbers do not contain the letter T. The first few tban numbers are 1, 4, 5, 6, 7, 9, 11, 100, 101, 104, 105, 106, 107, 109, 111, 400, 401, 404, 405, 406, ... .
The uban numbers do not contain the letter U. The first few uban numbers are 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, ... .
Basic properties
Aban numbers
For 1<N<109, aban numbers are numbers which the integer part of N/1000 is divisible by 1000.
Eban numbers
Eban numbers are even, due to "one", "three", "five", "seven", "nine", "eleven" and the suffix -teen all containing 'e's.
Further reading
External links
Integer sequences
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https://en.wikipedia.org/wiki/Mallows%27s%20Cp
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In statistics, Mallows's , named for Colin Lingwood Mallows, is used to assess the fit of a regression model that has been estimated using ordinary least squares. It is applied in the context of model selection, where a number of predictor variables are available for predicting some outcome, and the goal is to find the best model involving a subset of these predictors. A small value of means that the model is relatively precise.
Mallows's Cp has been shown to be equivalent to Akaike information criterion in the special case of Gaussian linear regression.
Definition and properties
Mallows's Cp addresses the issue of overfitting, in which model selection statistics such as the residual sum of squares always get smaller as more variables are added to a model. Thus, if we aim to select the model giving the smallest residual sum of squares, the model including all variables would always be selected. Instead, the Cp statistic calculated on a sample of data estimates the sum squared prediction error (SSPE) as its population target
where is the fitted value from the regression model for the ith case, E(Yi | Xi) is the expected value for the ith case, and σ2 is the error variance (assumed constant across the cases). The mean squared prediction error (MSPE) will not automatically get smaller as more variables are added. The optimum model under this criterion is a compromise influenced by the sample size, the effect sizes of the different predictors, and the degree of collinearity between them.
If P regressors are selected from a set of K > P, the Cp statistic for that particular set of regressors is defined as:
where
is the error sum of squares for the model with P regressors,
Ypi is the predicted value of the ith observation of Y from the P regressors,
S2 is the estimation of residuals variance after regression on the complete set of K regressors and can be estimated by ,
and N is the sample size.
Alternative definition
Given a linear model such as:
where:
are coefficients for predictor variables
represents error
An alternate version of Cp can also be defined as:
where
RSS is the residual sum of squares on a training set of data
is the number of predictors
and refers to an estimate of the variance associated with each response in the linear model (estimated on a model containing all predictors)
Note that this version of the Cp does not give equivalent values to the earlier version, but the model with the smallest Cp from this definition will also be the same model with the smallest Cp from the earlier definition.
Limitations
The Cp criterion suffers from two main limitations
the Cp approximation is only valid for large sample size;
the Cp cannot handle complex collections of models as in the variable selection (or feature selection) problem.
Practical use
The Cp statistic is often used as a stopping rule for various forms of stepwise regression. Mallows proposed the statistic as a criterion for selecting among many altern
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https://en.wikipedia.org/wiki/Brian%20Blank
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Brian Blank (September 3, 1953 – December 9, 2018) was a Canadian/American associate professor of mathematics at Washington University in St. Louis.
Blank was born in Montreal, Quebec, and earned an undergraduate degree in mathematics from McGill University. He received his Masters and Ph.D. in 1980 from Cornell University, with Anthony Knapp as advisor. His 20th century work involved harmonic analysis. He also co-authored a pair of calculus textbooks with his Washington University colleague, Steven Krantz. Titled Calculus: Single Variable and Calculus: Multivariable, the textbooks and workbooks used to be used in calculus classes at Washington University.
Blank died on December 9, 2018, due to complications of acute congestive heart failure.
References
External links
1953 births
2018 deaths
Anglophone Quebec people
Academics from Montreal
McGill University Faculty of Science alumni
Cornell University alumni
Washington University in St. Louis faculty
Washington University in St. Louis mathematicians
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https://en.wikipedia.org/wiki/Terre%20Haute%20North%20Vigo%20High%20School
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Terre Haute North Vigo High School, also known as Terre Haute North (THN), is a public high school located in Terre Haute, Indiana.
Academics
Advanced Placement classes include Calculus AB, Physics (Mechanics), Physics (Electricity and Magnetism), U.S. History, World History, U.S. Government, Biology, Chemistry, English, Spanish, and more.
Athletics
There are 21 varsity teams at THN: boys' and girls' cross country, soccer, tennis, golf, basketball, swimming & diving and track & field; boys' football, wrestling, baseball; girls' volleyball, softball, and dance team. Terre Haute North was a part of the Metropolitan Interscholastic Conference (MIC) from 1997 until 2013 when they joined Conference Indiana.
Notable alumni
Brian Dorsett (1979) is a retired professional baseball player who played eight seasons for the Cleveland Indians, Los Angeles Angels of Anaheim, New York Yankees, San Diego Padres, Cincinnati Reds and Chicago Cubs.
Josh Phegley (2006) is an MLB catcher, currently playing for the Chicago Cubs. Phegley formerly played for the Chicago White Sox and the Oakland Athletics.
Anthony Thompson (1986) is a former Indiana University football standout (1986–1989) and NFL running back (1990–92). He was inducted into the College Football Hall of Fame in 2007.
Steve Weatherford (2001) is a National Football League punter. He played for the New Orleans Saints, Kansas City Chiefs, Jacksonville Jaguars, New York Jets and New York Giants from 2006 to 2015, becoming a Super Bowl XLVI champion with the Giants.
Clyde Lovellette (Terre Haute Garfield (1948) before consolidation) former professional basketball player; 3x NBA Champion, 3x All-American, NCAA MOP (1952), 1952 Olympic Gold Medalist.
Terry Dischinger (Terre Haute Garfield (1959) before consolidation) former professional basketball player; 3x All-American, NBA Rookie of the Year, 1960 Olympic Gold Medalist.
Frank Hamblen (Terre Haute Garfield (1965) before consolidation) was a professional basketball coach and 7-time NBA champion, Phil Jackson's assistant with Chicago Bulls and Los Angeles Lakers and interim head coach. He is a member of the Indiana Basketball Hall of Fame.
Ron Greene (Terre Haute Gerstmeyer (1957) before consolidation) is a former college basketball head coach.
Tommy John (Terre Haute Gerstmeyer (1961) before consolidation) is a former Major League Baseball pitcher. He played for seven different teams from 1963 through 1989 and had 288 MLB victories. He is also known for his recovery and comeback from a revolutionary ulnar collateral ligament surgery, that has since become universally known as Tommy John surgery.
Steve Newton (Terre Haute Gerstmeyer (1959) before consolidation) is a former college basketball head coach.
See also
List of high schools in Indiana
Terre Haute South Vigo High School
References
External links
Terre Haute North Homepage
Public high schools in Indiana
Former Southern Indiana Athletic Conference members
Educational institutions es
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https://en.wikipedia.org/wiki/Countably%20compact%20space
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In mathematics a topological space is called countably compact if every countable open cover has a finite subcover.
Equivalent definitions
A topological space X is called countably compact if it satisfies any of the following equivalent conditions:
(1) Every countable open cover of X has a finite subcover.
(2) Every infinite set A in X has an ω-accumulation point in X.
(3) Every sequence in X has an accumulation point in X.
(4) Every countable family of closed subsets of X with an empty intersection has a finite subfamily with an empty intersection.
(1) (2): Suppose (1) holds and A is an infinite subset of X without -accumulation point. By taking a subset of A if necessary, we can assume that A is countable.
Every has an open neighbourhood such that is finite (possibly empty), since x is not an ω-accumulation point. For every finite subset F of A define . Every is a subset of one of the , so the cover X. Since there are countably many of them, the form a countable open cover of X. But every intersect A in a finite subset (namely F), so finitely many of them cannot cover A, let alone X. This contradiction proves (2).
(2) (3): Suppose (2) holds, and let be a sequence in X. If the sequence has a value x that occurs infinitely many times, that value is an accumulation point of the sequence. Otherwise, every value in the sequence occurs only finitely many times and the set is infinite and so has an ω-accumulation point x. That x is then an accumulation point of the sequence, as is easily checked.
(3) (1): Suppose (3) holds and is a countable open cover without a finite subcover. Then for each we can choose a point that is not in . The sequence has an accumulation point x and that x is in some . But then is a neighborhood of x that does not contain any of the with , so x is not an accumulation point of the sequence after all. This contradiction proves (1).
(4) (1): Conditions (1) and (4) are easily seen to be equivalent by taking complements.
Examples
The first uncountable ordinal (with the order topology) is an example of a countably compact space that is not compact.
Properties
Every compact space is countably compact.
A countably compact space is compact if and only if it is Lindelöf.
Every countably compact space is limit point compact.
For T1 spaces, countable compactness and limit point compactness are equivalent.
Every sequentially compact space is countably compact. The converse does not hold. For example, the product of continuum-many closed intervals with the product topology is compact and hence countably compact; but it is not sequentially compact.
For first-countable spaces, countable compactness and sequential compactness are equivalent.
For metrizable spaces, countable compactness, sequential compactness, limit point compactness and compactness are all equivalent.
The example of the set of all real numbers with the standard topology shows that neither local compactness nor σ-compactness nor pa
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https://en.wikipedia.org/wiki/Sequentially%20compact%20space
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In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in .
Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (if one assumes countable choice). However, there exist sequentially compact topological spaces that are not compact, and compact topological spaces that are not sequentially compact.
Examples and properties
The space of all real numbers with the standard topology is not sequentially compact; the sequence given by for all natural numbers is a sequence that has no convergent subsequence.
If a space is a metric space, then it is sequentially compact if and only if it is compact. The first uncountable ordinal with the order topology is an example of a sequentially compact topological space that is not compact. The product of copies of the closed unit interval is an example of a compact space that is not sequentially compact.
Related notions
A topological space is said to be limit point compact if every infinite subset of has a limit point in , and countably compact if every countable open cover has a finite subcover. In a metric space, the notions of sequential compactness, limit point compactness, countable compactness and compactness are all equivalent (if one assumes the axiom of choice).
In a sequential (Hausdorff) space sequential compactness is equivalent to countable compactness.
There is also a notion of a one-point sequential compactification—the idea is that the non convergent sequences should all converge to the extra point.
See also
Notes
References
Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970). .
Compactness (mathematics)
Properties of topological spaces
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https://en.wikipedia.org/wiki/Smale%27s%20problems
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Smale's problems are a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998 and republished in 1999. Smale composed this list in reply to a request from Vladimir Arnold, then vice-president of the International Mathematical Union, who asked several mathematicians to propose a list of problems for the 21st century. Arnold's inspiration came from the list of Hilbert's problems that had been published at the beginning of the 20th century.
Table of problems
In later versions, Smale also listed three additional problems, "that don't seem important enough to merit a place on our main list, but it would still be nice to solve them:"
Mean value problem
Is the three-sphere a minimal set (Gottschalk's conjecture)?
Is an Anosov diffeomorphism of a compact manifold topologically the same as the Lie group model of John Franks?
See also
Millennium Prize Problems
Simon problems
References
Unsolved problems in mathematics
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https://en.wikipedia.org/wiki/Ricardinho%20%28footballer%2C%20born%20May%201984%29
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José Ricardo dos Santos Oliveira or simply Ricardinho (born May 19, 1984 in João Pessoa, Brazil) is a Brazilian striker, who plays for Santa Cruz Futebol Clube.
Club statistics
References
External links
CBF
Living people
1984 births
Brazilian men's footballers
Brazilian expatriate men's footballers
Santa Cruz Futebol Clube players
Sociedade Esportiva Palmeiras players
Grêmio Foot-Ball Porto Alegrense players
Figueirense FC players
Botafogo de Futebol e Regatas players
Botafogo Futebol Clube (SP) players
Mogi Mirim Esporte Clube players
Kashiwa Reysol players
Paulista Futebol Clube players
Associação Desportiva São Caetano players
Guaratinguetá Futebol players
J1 League players
Expatriate men's footballers in South Korea
Jeju United FC players
Expatriate men's footballers in Japan
Brazilian expatriate sportspeople in South Korea
Men's association football forwards
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https://en.wikipedia.org/wiki/Tse%20Man%20Wing
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Tse Man Wing (, born 5 January 1983 in Hong Kong) is a former Hong Kong professional footballer who currently plays as an amateur player for Hong Kong First Division club Tai Po.
Career statistics
Club career
As of 4 April 2008
International career
As of 6 December 2006
References
External links
Tse Man Wing at HKFA
1983 births
Living people
Men's association football defenders
Hong Kong men's footballers
South China AA players
Hong Kong Rangers FC players
Sun Hei SC players
Southern District FC players
Eastern Sports Club footballers
Hong Kong Sapling players
Hong Kong First Division League players
Hong Kong Premier League players
Hong Kong men's international footballers
Footballers at the 2006 Asian Games
Asian Games competitors for Hong Kong
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https://en.wikipedia.org/wiki/British%20flag%20theorem
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In Euclidean geometry, the British flag theorem says that if a point P is chosen inside a rectangle ABCD then the sum of the squares of the Euclidean distances from P to two opposite corners of the rectangle equals the sum to the other two opposite corners.
As an equation:
The theorem also applies to points outside the rectangle, and more generally to the distances from a point in Euclidean space to the corners of a rectangle embedded into the space. Even more generally, if the sums of squares of distances from a point P to the two pairs of opposite corners of a parallelogram are compared, the two sums will not in general be equal, but the difference between the two sums will depend only on the shape of the parallelogram and not on the choice of P.
The theorem can also be thought of as a generalisation of the Pythagorean theorem. Placing the point P on any of the four vertices of the rectangle yields the square of the diagonal of the rectangle being equal to the sum of the squares of the width and length of the rectangle, which is the Pythagorean theorem.
Proof
Drop perpendicular lines from the point P to the sides of the rectangle, meeting sides AB, BC, CD, and AD at points W, X, Y and Z respectively, as shown in the figure. These four points WXYZ form the vertices of an orthodiagonal quadrilateral.
By applying the Pythagorean theorem to the right triangle AWP, and observing that WP = AZ, it follows that
and by a similar argument the squares of the lengths of the distances from P to the other three corners can be calculated as
and
Therefore:
Isosceles trapezoid
The British flag theorem can be generalized into a statement about (convex) isosceles trapezoids. More precisely for a trapezoid with parallel sides and and interior point the following equation holds:
In the case of a rectangle the fraction evaluates to 1 and hence yields the original theorem.
Naming
This theorem takes its name from the fact that, when the line segments from P to the corners of the rectangle are drawn, together with the perpendicular lines used in the proof, the completed figure resembles a Union Flag.
See also
Pizza theorem
References
Further reading
Nguyen Minh Ha, Dao Thanh Oai: An interesting application of the British flag theorem. Global Journal of Advanced Research on Classical and Modern Geometries, Volume 4 (2015), issue 1, pp. 31–34.
Martin Gardner, Dana S. Richards (ed.): The Colossal Book of Short Puzzles and Problems. W. W. Norton, 2006, , pp. 147, 159 (problem 6.16)
External links
British Flag Theorem at artofproblemsolving.com
Can You Solve Microsoft's Rectangle Corners Interview Question? (video, 5:41 mins)
interacive illustration of the British flag theorem for rectangles und for isosceles trapezoids
Euclidean geometry
Theorems about quadrilaterals
Pythagorean theorem
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https://en.wikipedia.org/wiki/Shrinkage%20%28statistics%29
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In statistics, shrinkage is the reduction in the effects of sampling variation. In regression analysis, a fitted relationship appears to perform less well on a new data set than on the data set used for fitting. In particular the value of the coefficient of determination 'shrinks'. This idea is complementary to overfitting and, separately, to the standard adjustment made in the coefficient of determination to compensate for the subjunctive effects of further sampling, like controlling for the potential of new explanatory terms improving the model by chance: that is, the adjustment formula itself provides "shrinkage." But the adjustment formula yields an artificial shrinkage.
A shrinkage estimator is an estimator that, either explicitly or implicitly, incorporates the effects of shrinkage. In loose terms this means that a naive or raw estimate is improved by combining it with other information. The term relates to the notion that the improved estimate is made closer to the value supplied by the 'other information' than the raw estimate. In this sense, shrinkage is used to regularize ill-posed inference problems.
Shrinkage is implicit in Bayesian inference and penalized likelihood inference, and explicit in James–Stein-type inference. In contrast, simple types of maximum-likelihood and least-squares estimation procedures do not include shrinkage effects, although they can be used within shrinkage estimation schemes.
Description
Many standard estimators can be improved, in terms of mean squared error (MSE), by shrinking them towards zero (or any other finite constant value). In other words, the improvement in the estimate from the corresponding reduction in the width of the confidence interval can outweigh the worsening of the estimate introduced by biasing the estimate towards zero (see bias-variance tradeoff).
Assume that the expected value of the raw estimate is not zero and consider other estimators obtained by multiplying the raw estimate by a certain parameter. A value for this parameter can be specified so as to minimize the MSE of the new estimate. For this value of the parameter, the new estimate will have a smaller MSE than the raw one. Thus it has been improved. An effect here may be to convert an unbiased raw estimate to an improved biased one.
Examples
A well-known example arises in the estimation of the population variance by sample variance. For a sample size of n, the use of a divisor n − 1 in the usual formula (Bessel's correction) gives an unbiased estimator, while other divisors have lower MSE, at the expense of bias. The optimal choice of divisor (weighting of shrinkage) depends on the excess kurtosis of the population, as discussed at mean squared error: variance, but one can always do better (in terms of MSE) than the unbiased estimator; for the normal distribution a divisor of n + 1 gives one which has the minimum mean squared error.
Methods
Types of regression that involve shrinkage estimates include ridge regression
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https://en.wikipedia.org/wiki/Contraharmonic%20mean
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In mathematics, a contraharmonic mean is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, , where p = 2.
Definition
The contraharmonic mean of a set of positive numbers is defined as the arithmetic mean of the squares of the numbers divided by the arithmetic mean of the numbers:
Properties
It is easy to show that this satisfies the characteristic properties of a mean of some list of values :
The first property implies the fixed point property, that for all k > 0,
The contraharmonic mean is higher in value than the arithmetic mean and also higher than the root mean square:
where x is a list of values, H is the harmonic mean, G is geometric mean, L is the logarithmic mean, A is the arithmetic mean, R is the root mean square and C is the contraharmonic mean. Unless all values of x are the same, the ≤ signs above can be replaced by <.
The name contraharmonic may be due to the fact that when taking the mean of only two variables, the contraharmonic mean is as high above the arithmetic mean as the arithmetic mean is above the harmonic mean (i.e., the arithmetic mean of the two variables is equal to the arithmetic mean of their harmonic and contraharmonic means).
Two-variable formulae
From the formulas for the arithmetic mean and harmonic mean of two variables we have:
Notice that for two variables the average of the harmonic and contraharmonic means is exactly equal to the arithmetic mean:
As a gets closer to 0 then H(a, b) also gets closer to 0. The harmonic mean is very sensitive to low values. On the other hand, the contraharmonic mean is sensitive to larger values, so as a approaches 0 then C(a, b) approaches b (so their average remains A(a, b)).
There are two other notable relationships between 2-variable means. First, the geometric mean of the arithmetic and harmonic means is equal to the geometric mean of the two values:
The second relationship is that the geometric mean of the arithmetic and contraharmonic means is the root mean square:
The contraharmonic mean of two variables can be constructed geometrically using a trapezoid (see ).
Additional constructions
The contraharmonic mean can be constructed on a circle similar to the way the Pythagorean means of two variables are constructed. The contraharmonic mean is the remainder of the diameter on which the harmonic mean lies.
Properties
The contraharmonic mean of a random variable is equal to the sum of the arithmetic mean and the variance divided by the arithmetic mean. Since the variance is always ≥0 the contraharmonic mean is always greater than or equal to the arithmetic mean.
The ratio of the variance and the mean was proposed as a test statistic by Clapham. This statistic is the contraharmonic mean less one.
It is also related to Katz's statistic
where m is the mean, s2 the variance and n is the sample size.
Jn is asymptotically normally distributed with a mean of zero and variance of 1.
Uses
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https://en.wikipedia.org/wiki/Cristian%20Leiva
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Cristian Gustavo Leiva (born 26 September 1977) is an Argentine football midfielder who plays for Americo Tesorieri.
External links
Statistics at Guardian Stats Centre
Cristian Leiva at Football Lineups
Argentine Primera statistics
Argentine men's footballers
Men's association football midfielders
Club Atlético Banfield footballers
Cruz Azul footballers
Club de Gimnasia y Esgrima La Plata footballers
R.S.C. Anderlecht players
R. Charleroi S.C. players
Club Olimpia footballers
Godoy Cruz Antonio Tomba footballers
San Lorenzo de Almagro footballers
Arsenal de Sarandí footballers
Expatriate men's footballers in Paraguay
Sportspeople from La Rioja Province, Argentina
1977 births
Living people
Argentine Primera División players
Belgian Pro League players
Argentine expatriate men's footballers
Expatriate men's footballers in Belgium
Expatriate men's footballers in Mexico
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https://en.wikipedia.org/wiki/Defective
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Defective may refer to::
Defective matrix, in algebra
Defective verb, in linguistics
Defective, or haser, in Hebrew orthography, a spelling variant that does not include mater lectionis
Something presenting an anomaly, such as a product defect, making it nonfunctional
See also
Defect (disambiguation)
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https://en.wikipedia.org/wiki/Elias%20Loomis
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Elias Loomis (August 7, 1811 – August 15, 1889) was an American mathematician. He served as a professor of mathematics and natural philosophy at Western Reserve College (now Case Western Reserve University), the University of the City of New York and Yale University. During his tenure at Western Reserve College in 1838, he established the Loomis Observatory, currently the second oldest observatory in the United States.
Life and work
Loomis was born in Willington, Connecticut in 1811. He graduated at Yale College in 1830, was a tutor there for three years (1833–36), and then spent the next year in scientific investigation in Paris. On his return, Loomis served as professor of mathematics and natural philosophy for eight years (1836–44) at Western Reserve College in Hudson, Ohio, now Case Western Reserve University. During his tenure, he opened up the Loomis Observatory in 1838, currently the second oldest observatory in the United States. From 1844 to 1860 he held the professorship of natural philosophy and mathematics in the University of the City of New York, and in the latter year became professor of natural philosophy in Yale. Professor Loomis published (besides many papers in the American Journal of Science and in the Transactions of the American Philosophical Society) many textbooks on mathematics, including Analytical Geometry and of the Differential and Integral Calculus, published in 1835.
In 1859 Alexander Wylie, assistant director of London Missionary Press in Shanghai, in cooperation with fellow Chinese scholar Li Shanlan, translated Elias Loomis's book on Geometry, Differential and Integral Calculus into Chinese. The Chinese text was subsequently translated twice by Japanese scholars into Japanese and published in Japan. Loomis's writings thus played an important role in the transfer of analytical mathematical knowledge to the Far East.
Great Auroral Exhibition of 1859
In his memoir of Loomis, Hubert Anson Newton summarized Loomis's work on the historical Geomagnetic Storm of 1859.
Closely connected with terrestrial magnetism, and to be considered with it, is the Aurora Borealis. In the week that covered the end of August and the beginning of September, 1859, there occurred an exceedingly brilliant display of the Northern Lights. Believing that an exhaustive discussion of a single aurora promised to do more for the promotion of science than an imperfect study of an indefinite number of them, Professor Loomis undertook at once to collect and to collate accounts of this display. A large number of such accounts were secured from North America, from Europe, from Asia, and from places in the Southern Hemisphere; especially all the reports from the Smithsonian observers and correspondents, were placed in his hands by the Secretary, Professor Henry.
These observations and the discussions of them were given to the public during the following two years, in a series of nine papers in the American Journal of Science.
Few, if any, d
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https://en.wikipedia.org/wiki/Trichotomy
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A trichotomy can refer to:
Law of trichotomy, a mathematical law that every real number is either positive, negative, or zero
Trichotomy theorem, in finite group theory
Trichotomy (jazz trio), Australian jazz band, collaborators with Danny Widdicombe on a 2019 album
Trichotomy (philosophy), series of three terms used by various thinkers
Trichotomy (speciation), three groups from a common ancestor, where it is unclear or unknown in what chronological order the three groups split
Trichotomous or 3-forked branching in botany
See also
Tripartite (disambiguation)
Triune (disambiguation)
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https://en.wikipedia.org/wiki/Cauchy%27s%20theorem%20%28geometry%29
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Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states that
convex polytopes in three dimensions with congruent corresponding faces must be congruent to each other. That is, any polyhedral net formed by unfolding the faces of the polyhedron onto a flat surface, together with gluing instructions describing which faces should be connected to each other, uniquely determines the shape of the original polyhedron. For instance, if six squares are connected in the pattern of a cube, then they must form a cube: there is no convex polyhedron with six square faces connected in the same way that does not have the same shape.
This is a fundamental result in rigidity theory: one consequence of the theorem is that, if one makes a physical model of a convex polyhedron by connecting together rigid plates for each of the polyhedron faces with flexible hinges along the polyhedron edges, then this ensemble of plates and hinges will necessarily form a rigid structure.
Statement
Let P and Q be combinatorially equivalent 3-dimensional convex polytopes; that is, they are convex polytopes with isomorphic face lattices. Suppose further that each pair of corresponding faces from P and Q are congruent to each other, i.e. equal up to a rigid motion. Then P and Q are themselves congruent.
To see that convexity is necessary, consider a regular icosahedron. One can "push in" a vertex to create a nonconvex polyhedron that is still combinatorially equivalent to the regular icosahedron. Another way to see it, is to take the pentagonal pyramid around a vertex, and reflect it with respect to its base.
History
The result originated in Euclid's Elements, where solids are called equal if the same holds for their faces. This version of the result was proved by Cauchy in 1813 based on earlier work by Lagrange. An error in Cauchy's proof of the main lemma was corrected by Ernst Steinitz, Isaac Jacob Schoenberg, and Aleksandr Danilovich Aleksandrov. The corrected proof of Cauchy is so short and elegant, that it is considered to be one of the Proofs from THE BOOK.
Generalizations and related results
The result does not hold on a plane or for non-convex polyhedra in : there exist non-convex flexible polyhedra that have one or more degrees of freedom of movement that preserve the shapes of their faces. In particular, the Bricard octahedra are self-intersecting flexible surfaces discovered by a French mathematician Raoul Bricard in 1897. The Connelly sphere, a flexible non-convex polyhedron homeomorphic to a 2-sphere, was discovered by Robert Connelly in 1977.
Although originally proven by Cauchy in three dimensions, the theorem was extended to dimensions higher than 3 by Alexandrov (1950).
Cauchy's rigidity theorem is a corollary from Cauchy's theorem stating that a convex polytope cannot be deformed so that its faces remain rigid.
In 1974 Herman Gluck showed that in a certain precise sense almost all simply connected closed surfaces are rigid.
Dehn'
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https://en.wikipedia.org/wiki/Southwest%20Virginia%20Governor%27s%20School%20for%20Science%2C%20Mathematics%2C%20and%20Technology
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The Southwest Virginia Governor's School for Science, Mathematics, and Technology is one of Virginia's 18 state-initiated magnet Governor's Schools. It is a part-time school where 11th and 12th grade students take advanced classes in the morning (receiving their remaining classes from their home high school.)
The school utilizes the combined resources of the participating school divisions to provide programs which facilitate the acquisition of scientific and technical knowledge through laboratory investigation and research. Students attend the Governor's School for half a day to take science, math, and research classes before returning to their neighborhood high schools. While all classes in the program satisfy high school requirements, they all count for college credit too.
The program's approach combines traditional classroom and laboratory training with specialized experiences such as visits from successful scientists and universities, unique internships, and various field research.
Faculty
Participating school systems
Carroll County
Floyd County
Galax City
Giles County
Montgomery County
Pulaski County
Radford City
Smyth County
Wythe County
Offered courses
College Chemistry
College Physics
Pre-Calculus
AP Statistics/Research
College Biology
University Physics
Applied Calculus
Engineering Calculus
Advanced Calculus
Vector Calculus
Differential Equations
Linear Algebra
Analytic Geometry
Introduction to Microcomputer Software
AP Environmental Science
Anatomy/Physiology
Organic Chemistry
Astronomy
References
External links
Southwest Virginia Governor's School
Public high schools in Virginia
Magnet schools in Virginia
Schools in Pulaski County, Virginia
Educational institutions established in 1990
1990 establishments in Virginia
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https://en.wikipedia.org/wiki/Chan%27s%20algorithm
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In computational geometry, Chan's algorithm, named after Timothy M. Chan, is an optimal output-sensitive algorithm to compute the convex hull of a set of points, in 2- or 3-dimensional space.
The algorithm takes time, where is the number of vertices of the output (the convex hull). In the planar case, the algorithm combines an algorithm (Graham scan, for example) with Jarvis march (), in order to obtain an optimal time. Chan's algorithm is notable because it is much simpler than the Kirkpatrick–Seidel algorithm, and it naturally extends to 3-dimensional space. This paradigm has been independently developed by Frank Nielsen in his Ph.D. thesis.
Algorithm
Overview
A single pass of the algorithm requires a parameter which is between 0 and (number of points of our set ). Ideally, but , the number of vertices in the output convex hull, is not known at the start. Multiple passes with increasing values of are done which then terminates when (see below on choosing parameter ).
The algorithm starts by arbitrarily partitioning the set of points into subsets with at most points each; notice that .
For each subset , it computes the convex hull, , using an algorithm (for example, Graham scan), where is the number of points in the subset. As there are subsets of points each, this phase takes time.
During the second phase, Jarvis's march is executed, making use of the precomputed (mini) convex hulls, . At each step in this Jarvis's march algorithm, we have a point in the convex hull (at the beginning, may be the point in with the lowest y coordinate, which is guaranteed to be in the convex hull of ), and need to find a point such that all other points of are to the right of the line , where the notation simply means that the next point, that is , is determined as a function of and . The convex hull of the set , , is known and contains at most points (listed in a clockwise or counter-clockwise order), which allows to compute in time by binary search. Hence, the computation of for all the subsets can be done in time. Then, we can determine using the same technique as normally used in Jarvis's march, but only considering the points (i.e. the points in the mini convex hulls) instead of the whole set . For those points, one iteration of Jarvis's march is which is negligible compared to the computation for all subsets. Jarvis's march completes when the process has been repeated times (because, in the way Jarvis march works, after at most iterations of its outermost loop, where is the number of points in the convex hull of , we must have found the convex hull), hence the second phase takes time, equivalent to time if is close to (see below the description of a strategy to choose such that this is the case).
By running the two phases described above, the convex hull of points is computed in time.
Choosing the parameter
If an arbitrary value is chosen for , it may happen that . In that case, after steps in the secon
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https://en.wikipedia.org/wiki/Consensus%20theorem
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In Boolean algebra, the consensus theorem or rule of consensus is the identity:
The consensus or resolvent of the terms and is . It is the conjunction of all the unique literals of the terms, excluding the literal that appears unnegated in one term and negated in the other. If includes a term which is negated in (or vice versa), the consensus term is false; in other words, there is no consensus term.
The conjunctive dual of this equation is:
Proof
Consensus
The consensus or consensus term of two conjunctive terms of a disjunction is defined when one term contains the literal and the other the literal , an opposition. The consensus is the conjunction of the two terms, omitting both and , and repeated literals. For example, the consensus of and is . The consensus is undefined if there is more than one opposition.
For the conjunctive dual of the rule, the consensus can be derived from and through the resolution inference rule. This shows that the LHS is derivable from the RHS (if A → B then A → AB; replacing A with RHS and B with (y ∨ z) ). The RHS can be derived from the LHS simply through the conjunction elimination inference rule. Since RHS → LHS and LHS → RHS (in propositional calculus), then LHS = RHS (in Boolean algebra).
Applications
In Boolean algebra, repeated consensus is the core of one algorithm for calculating the Blake canonical form of a formula.
In digital logic, including the consensus term in a circuit can eliminate race hazards.
History
The concept of consensus was introduced by Archie Blake in 1937, related to the Blake canonical form. It was rediscovered by Samson and Mills in 1954 and by Quine in 1955. Quine coined the term 'consensus'. Robinson used it for clauses in 1965 as the basis of his "resolution principle".
References
Further reading
Roth, Charles H. Jr. and Kinney, Larry L. (2004, 2010). "Fundamentals of Logic Design", 6th Ed., p. 66ff.
Boolean algebra
Theorems in lattice theory
Theorems in propositional logic
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https://en.wikipedia.org/wiki/Dense-in-itself
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In general topology, a subset of a topological space is said to be dense-in-itself or crowded
if has no isolated point.
Equivalently, is dense-in-itself if every point of is a limit point of .
Thus is dense-in-itself if and only if , where is the derived set of .
A dense-in-itself closed set is called a perfect set. (In other words, a perfect set is a closed set without isolated point.)
The notion of dense set is distinct from dense-in-itself. This can sometimes be confusing, as "X is dense in X" (always true) is not the same as "X is dense-in-itself" (no isolated point).
Examples
A simple example of a set that is dense-in-itself but not closed (and hence not a perfect set) is the set of irrational numbers (considered as a subset of the real numbers). This set is dense-in-itself because every neighborhood of an irrational number contains at least one other irrational number . On the other hand, the set of irrationals is not closed because every rational number lies in its closure. Similarly, the set of rational numbers is also dense-in-itself but not closed in the space of real numbers.
The above examples, the irrationals and the rationals, are also dense sets in their topological space, namely . As an example that is dense-in-itself but not dense in its topological space, consider . This set is not dense in but is dense-in-itself.
Properties
A singleton subset of a space can never be dense-in-itself, because its unique point is isolated in it.
The dense-in-itself subsets of any space are closed under unions. In a dense-in-itself space, they include all open sets. In a dense-in-itself T1 space they include all dense sets. However, spaces that are not T1 may have dense subsets that are not dense-in-itself: for example in the space with the indiscrete topology, the set is dense, but is not dense-in-itself.
The closure of any dense-in-itself set is a perfect set.
In general, the intersection of two dense-in-itself sets is not dense-in-itself. But the intersection of a dense-in-itself set and an open set is dense-in-itself.
See also
Nowhere dense set
Glossary of topology
Dense order
Notes
References
Topology
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https://en.wikipedia.org/wiki/Dirichlet%20process
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In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. It is often used in Bayesian inference to describe the prior knowledge about the distribution of random variables—how likely it is that the random variables are distributed according to one or another particular distribution.
As an example, a bag of 100 real-world dice is a random probability mass function (random pmf)—to sample this random pmf you put your hand in the bag and draw out a die, that is, you draw a pmf. A bag of dice manufactured using a crude process 100 years ago will likely have probabilities that deviate wildly from the uniform pmf, whereas a bag of state-of-the-art dice used by Las Vegas casinos may have barely perceptible imperfections. We can model the randomness of pmfs with the Dirichlet distribution.
The Dirichlet process is specified by a base distribution and a positive real number called the concentration parameter (also known as scaling parameter). The base distribution is the expected value of the process, i.e., the Dirichlet process draws distributions "around" the base distribution the way a normal distribution draws real numbers around its mean. However, even if the base distribution is continuous, the distributions drawn from the Dirichlet process are almost surely discrete. The scaling parameter specifies how strong this discretization is: in the limit of , the realizations are all concentrated at a single value, while in the limit of the realizations become continuous. Between the two extremes the realizations are discrete distributions with less and less concentration as increases.
The Dirichlet process can also be seen as the infinite-dimensional generalization of the Dirichlet distribution. In the same way as the Dirichlet distribution is the conjugate prior for the categorical distribution, the Dirichlet process is the conjugate prior for infinite, nonparametric discrete distributions. A particularly important application of Dirichlet processes is as a prior probability distribution in infinite mixture models.
The Dirichlet process was formally introduced by Thomas Ferguson in 1973.
It has since been applied in data mining and machine learning, among others for natural language processing, computer vision and bioinformatics.
Introduction
Dirichlet processes are usually used when modelling data that tends to repeat previous values in a so-called "rich get richer" fashion. Specifically, suppose that the generation of values can be simulated by the following algorithm.
Input: (a probability distribution called base distribution), (a positive real number called scaling parameter)
For :
a) With probability draw from .
b) With probability set , where is the number of pr
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https://en.wikipedia.org/wiki/Frank%20Kelly%20%28mathematician%29
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Francis Patrick Kelly, CBE, FRS (born 28 December 1950) is Professor of the Mathematics of Systems at the Statistical Laboratory, University of Cambridge. He served as Master of Christ's College, Cambridge from 2006 to 2016.
Kelly's research interests are in random processes, networks and optimisation, especially in very large-scale systems such as telecommunication or transportation networks. In the 1980s, he worked with colleagues in Cambridge and at British Telecom's Research Labs on Dynamic Alternative Routing in telephone networks, which was implemented in BT's main digital telephone network. He has also worked on the economic theory of pricing to congestion control and fair resource allocation in the internet. From 2003 to 2006 he served as Chief Scientific Advisor to the United Kingdom Department for Transport.
Kelly was elected a Fellow of the Royal Society in 1989. In December 2006 he was elected 37th Master of Christ's College, Cambridge. He was appointed Commander of the Order of the British Empire (CBE) in the 2013 New Year Honours for services to mathematical science.
Awards
1979 Davidson Prize of the University of Cambridge
1989 Guy Medal in Silver of the Royal Statistical Society
1989 Fellow of the Royal Society
1992 Lanchester Prize of the Institute for Operations Research and the Management Sciences
1997 Naylor Prize of the London Mathematical Society
2001 Honorary D.Sc. from Heriot-Watt University
2005 IEEE Koji Kobayashi Computers and Communications Award
2006 Companionship of OR by the Operational Research Society
2008 John von Neumann Theory Prize of the Institute for Operations Research and the Management Sciences
2009 SIGMETRICS Achievement Award
2009 EURO Gold Medal from European Operational Research Society
2013 Commander of the Order of the British Empire in the New Years Honours List for "services to mathematical sciences"
2015 IEEE Alexander Graham Bell Medal, for "Creating principled mathematical foundations for the design and analysis of congestion control, routing, and blocking in modern communication networks"
2015 David Crighton Medal of the London Mathematical Society and Institute of Mathematics and its Applications
Works
References
External links
Frank Kelly's homepage. Retrieved 8 December 2006
Biography from Frank Kelly's website. Retrieved 8 December 2006
Who's Who
1950 births
Alumni of Van Mildert College, Durham
British operations researchers
Fellows of the Royal Society
English mathematicians
John von Neumann Theory Prize winners
Living people
Cambridge mathematicians
Masters of Christ's College, Cambridge
Probability theorists
Queueing theorists
Commanders of the Order of the British Empire
David Crighton medalists
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https://en.wikipedia.org/wiki/Average%20path%20length
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Average path length, or average shortest path length is a concept in network topology that is defined as the average number of steps along the shortest paths for all possible pairs of network nodes. It is a measure of the efficiency of information or mass transport on a network.
Concept
Average path length is one of the three most robust measures of network topology, along with its clustering coefficient and its degree distribution. Some examples are: the average number of clicks which will lead you from one website to another, or the number of people you will have to communicate through, on an average, to contact a complete stranger. It should not be confused with the diameter of the network, which is defined as the longest geodesic, i.e., the longest shortest path between any two nodes in the network (see Distance (graph theory)).
The average path length distinguishes an easily negotiable network from one, which is complicated and inefficient, with a shorter average path length being more desirable. However, the average path length is simply what the path length will most likely be. The network itself might have some very remotely connected nodes and many nodes, which are neighbors of each other.
Definition
Consider an unweighted directed graph with the set of vertices . Let , where denote the shortest distance between and .
Assume that if cannot be reached from . Then, the average path length is:
where is the number of vertices in .
Applications
In a real network like the Internet, a short average path length facilitates the quick transfer of information and reduces costs. The efficiency of mass transfer in a metabolic network can be judged by studying its average path length. A power grid network will have fewer losses if its average path length is minimized.
Most real networks have a very short average path length leading to the concept of a small world where everyone is connected to everyone else through a very short path.
As a result, most models of real networks are created with this condition in mind. One of the first models which tried to explain real networks was the random network model. It was later followed by the Watts and Strogatz model, and even later there were the scale-free networks starting with the BA model. All these models had one thing in common: they all predicted very short average path length. The average path lengths of some networks are listed in Table.
The average path length depends on the system size but does not change drastically with it. Small world network theory predicts that the average path length changes proportionally to log n, where n is the number of nodes in the network.
References
Network theory
Graph invariants
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https://en.wikipedia.org/wiki/Mathematica%20%28disambiguation%29
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Wolfram Mathematica is a computer algebra system and programming language.
Mathematica may also refer to:
Mathematica Inc. (1968–1986), a defunct research and software company
Mathematica Inc., a policy research organization spun-off from the above company, formerly known as Mathematica Policy Research
Philosophiæ Naturalis Principia Mathematica, Newton's book on the basic laws of physics
Mathematica: A World of Numbers... and Beyond, interactive mathematics exhibit (1961) designed by Charles and Ray Eames
Principia Mathematica, Whitehead and Russell's work on axiomatizing mathematics
See also
Mathematics (disambiguation)
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https://en.wikipedia.org/wiki/Oriented%20projective%20geometry
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Oriented projective geometry is an oriented version of real projective geometry.
Whereas the real projective plane describes the set of all unoriented lines through the origin in R3, the oriented projective plane describes lines with a given orientation. There are applications in computer graphics and computer vision where it is necessary to distinguish between rays light being emitted or absorbed by a point.
Elements in an oriented projective space are defined using signed homogeneous coordinates. Let be the set of elements of excluding the origin.
Oriented projective line, : , with the equivalence relation for all .
Oriented projective plane, : , with for all .
These spaces can be viewed as extensions of euclidean space. can be viewed as the union of two copies of , the sets (x,1) and (x,-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise can be viewed as two copies of , (x,y,1) and (x,y,-1), plus one copy of (x,y,0).
An alternative way to view the spaces is as points on the circle or sphere, given by the points (x,y,w) with
x2+y2+w2=1.
Oriented real projective space
Let n be a nonnegative integer. The (analytical model of, or canonical) oriented (real) projective space or (canonical) two-sided projective space is defined as
Here, we use to stand for two-sided.
Alternative models
The straight model
The spherical model
Distance in oriented real projective space
Distances between two points and in can be defined as elements
in .
Oriented complex projective geometry
Let n be a nonnegative integer. The oriented complex projective space is defined as
. Here, we write to stand for the 1-sphere.
See also
Variational analysis
Notes
References
From original Stanford Ph.D. dissertation, Primitives for Computational Geometry, available as .
Nice introduction to oriented projective geometry in chapters 14 and 15. More at author's website. Sherif Ghali.
A. G. Oliveira, P. J. de Rezende, F. P. SelmiDei An Extension of CGAL to the Oriented Projective Plane T2 and its Dynamic Visualization System, 21st Annual ACM Symp. on Computational Geometry, Pisa, Italy, 2005.
Projective geometry
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https://en.wikipedia.org/wiki/List%20of%20mathematic%20operators
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In mathematics, an operator or transform is a function from one space of functions to another. Operators occur commonly in engineering, physics and mathematics. Many are integral operators and differential operators.
In the following L is an operator
which takes a function to another function . Here, and are some unspecified function spaces, such as Hardy space, Lp space, Sobolev space, or, more vaguely, the space of holomorphic functions.
See also
List of transforms
List of Fourier-related transforms
Transfer operator
Fredholm operator
Borel transform
Glossary of mathematical symbols
Operators
Operators
Operators
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https://en.wikipedia.org/wiki/Geometry%20from%20the%20Land%20of%20the%20Incas
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Geometry from the Land of the Incas is a free geometry website funded by advertising, aimed mainly at high school and college age students.
Developed by Antonio Gutierrez, the site uses sound, dynamic geometry, animations, science, and Incan history with the goal of raising students' interest in Euclidean geometry. Numerous problems are presented with step-by-step solutions for each proof.
Around 2003, the website won several minor awards from educational publications, including One of the Top 10 educational Web Sites Canada's SchoolNet's in 2003 , the Knot #284 Canadian Mathematical Society , NCTM Illuminations Web Byte (National Council of Teacher of Mathematics), and one of 30 "Desert Island Theorems" of the book: "The Changing Shape of Geometry Celebrating a Century of Geometry and Geometry Teaching", the Mathematical Association UK.
The author designed the site for teachers, students and parents, and anyone else curious about geometry, mathematics, popular science, geography, Incan history and visual perception.
The site moved to www.gogeometry.com on November 5, 2007 from www.agutie.homestead.com.
Site contents
Geometry from the Land of the Incas is divided into nine major categories: Geometry theorems and problems, Inca Geometry, Quizzes, Puzzles, Quotations, Inspiration, Landscapes, Mindmaps, and Geometric art. This site is loaded with advertisements and very little, if any, interesting exploration of the Inca's understanding of geometry.
Technical requirements
Requires Flash Player 5.0 or higher version, Java Runtime Environment 1.3 or higher version and JavaScript.
This site is best viewed using Internet Explorer version 6.0 or higher, or Firefox version 1.5 or higher and at a display resolution of 1024 × 768 or higher.
References
External links
Go Geometry from the Land of the Incas: New domain
Peruvian educational websites
Mathematics websites
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https://en.wikipedia.org/wiki/List%20of%20Fulham%20F.C.%20records%20and%20statistics
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The following article features the records and statistics of Fulham Football Club, based in Fulham, West London.
Player appearances
There are five Fulham players who have been in the club's starting line-up more than 450 times, all of whom have since retired from football:
Johnny Haynes – 658
Eddie Lowe – 511
Les Barrett – 491
Frank Penn – 459
George Cohen – 459
Current
The players with the most appearances across all competitions who are still at the club as of 2 October 2023 are:
Tim Ream – 300
Tom Cairney – 291
Bobby Decordova-Reid – 177
Harrison Reed – 154
Antonee Robinson – 116
Goalscorers
Eight players have scored 100 or more goals for the club:
Gordon Davies – 178
Johnny Haynes – 158
Bedford Jezzard – 154
Jim Hammond – 150
Graham Leggat – 134
Arthur Stevens – 124
Aleksandar Mitrović – 111
Steve Earle – 108
Current
The three most prolific goalscorers currently at the club as of 2 October 2023 are:
Tom Cairney – 43
Bobby Decordova-Reid – 26
Harry Wilson – 14
Transfers
Highest transfer fees paid
Highest transfer fees received
Player of the season
2021–22: Aleksandar Mitrović
2020–21: Alphonse Areola
2019–20: Aleksandar Mitrović
2018–19: Calum Chambers
2017–18: Tim Ream
2016–17: Tom Cairney
2015–16: Ross McCormack
2014–15: Ross McCormack
2013–14: Ashkan Dejagah
2012–13: Dimitar Berbatov
2011–12: Clint Dempsey
2010–11: Clint Dempsey
2009–10: Zoltán Gera
2008–09: Mark Schwarzer
2007–08: Simon Davies
2006–07: Brian McBride
2005–06: Brian McBride
2004–05: Luís Boa Morte
International squad players
Algeria
Hamer Bouazza
Rafik Halliche
Argentina
Facundo Sava
Australia
Mark Schwarzer
Adrian Leijer
Len Quested
Ahmad Elrich
Ryan Williams
Austria
Michael Madl
Belgium
Philippe Albert
Mousa Dembélé
Denis Odoi
Timothy Castagne
Bermuda
Kyle Lightbourne
Bulgaria
Dimitar Berbatov
Cameroon
Pierre Womé
André-Frank Zambo Anguissa
Canada
Paul Peschisolido
Tomasz Radzinski
Paul Stalteri
Costa Rica
Bryan Ruiz
Czech Republic
Marcel Gecov
Zdeněk Grygera
Jan Laštůvka
Denmark
Leon Andreasen
Claus Jensen
Bjarne Goldbæk
Peter Møller
Joachim Andersen
DR Congo
Neeskens Kebano
Gabriel Zakuani
England
Alan Mullery
Bedford Jezzard
Bernard Joy
Bobby Moore
Bobby Robson
Bobby Zamora
George Cohen
Stan Collymore
Dave Beasant
Johnny Haynes
Zat Knight
Malcolm Macdonald
Paul Parker
Rodney Marsh
Peter Beardsley
Andrew Cole
Wayne Bridge
Paul Konchesky
Danny Murphy
Andy Johnson
David Stockdale
Jim Taylor
Tim Coleman
Albert Barrett
Len Oliver
Danny Shea
Arthur 'Pablo' Stevens
Frank Penn
Johnny Price
Jim Langley
Roy Bentley
Eddie Lowe
Pat Beasley
Allan Clarke
Robert Wilson
Lee Clark
Franky Osborne
Gordon Hoare
Arthur Berry
Johnny 'Budgie' Byrne
Chris Smalling
Joe Bacuzzi
Jim Hammond
Egypt
Hussein Hegazi
Finland
Jari Litmanen
Toni Kallio
Shefki Kuqi
Antti Niemi
Lauri Dalla Valle
France
Alain Goma
Anthony Knockaert
Aboubakar Kamara
Martin Djetou
Steve Marlet
Louis Saha
Philippe
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https://en.wikipedia.org/wiki/Biangular%20coordinates
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In mathematics, biangular coordinates are a coordinate system for the plane where and are two fixed points, and the position of a point P not on the line is determined by the angles and
The sine rule can be used to convert from biangular coordinates to two-center bipolar coordinates.
Applications
Biangular coordinates can be used in geometric modelling and CAD.
See also
Two-center bipolar coordinates
Bipolar coordinates
Sectrix of Maclaurin
References
External links
G. B. M. Zerr Biangular Coordinates, American Mathematical Monthly 17 (2), February 1910
J. C. L. Fish, Coordinates Of Elementary Surveying
George Shoobridge Carr, A synopsis of elementary results in pure mathematics (see page 742)
Coordinate systems
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https://en.wikipedia.org/wiki/Florent%20Bureau
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Florent-Joseph Bureau (1906–1999) was a Belgian mathematician. He was a professor at the University of Liège. He worked on algebraic and differential geometry and the theory of analytical functions. In 1952, he was awarded the Francqui Prize on Exact Sciences.
Academic staff of the University of Liège
20th-century Belgian mathematicians
Walloon people
1906 births
1999 deaths
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https://en.wikipedia.org/wiki/MathCast
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MathCast is a graphical mathematics equation editor. With this computer application, a user can create equations in mathematical notation and use them in documents or web pages. Equations can be rendered into pictures or transformed into MathML.
MathCast features a Rapid Mathline, Equation List Management, and XHTML authoring.
MathCast is a free software application distributed under the GNU General Public License.
External links
MathCast home page
Sourceforge project page
Formula editors
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https://en.wikipedia.org/wiki/Non-abelian%20class%20field%20theory
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In mathematics, non-abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results on abelian extensions of any number field K, to the general Galois extension L/K. While class field theory was essentially known by 1930, the corresponding non-abelian theory has never been formulated in a definitive and accepted sense.
History
A presentation of class field theory in terms of group cohomology was carried out by Claude Chevalley, Emil Artin and others, mainly in the 1940s. This resulted in a formulation of the central results by means of the group cohomology of the idele class group. The theorems of the cohomological approach are independent of whether or not the Galois group G of L/K is abelian. This theory has never been regarded as the sought-after non-abelian theory. The first reason that can be cited for that is that it did not provide fresh information on the splitting of prime ideals in a Galois extension; a common way to explain the objective of a non-abelian class field theory is that it should provide a more explicit way to express such patterns of splitting.
The cohomological approach therefore was of limited use in even formulating non-abelian class field theory. Behind the history was the wish of Chevalley to write proofs for class field theory without using Dirichlet series: in other words to eliminate L-functions. The first wave of proofs of the central theorems of class field theory was structured as consisting of two 'inequalities' (the same structure as in the proofs now given of the fundamental theorem of Galois theory, though much more complex). One of the two inequalities involved an argument with L-functions.
In a later reversal of this development, it was realised that to generalize Artin reciprocity to the non-abelian case, it was essential in fact to seek a new way of expressing Artin L-functions. The contemporary formulation of this ambition is by means of the Langlands program: in which grounds are given for believing Artin L-functions are also L-functions of automorphic representations. As of the early twenty-first century, this is the formulation of the notion of non-abelian class field theory that has widest expert acceptance.
See also
Class field theory
Anabelian geometry
Frobenioid
Langlands correspondences
Class field theory
Notes
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https://en.wikipedia.org/wiki/L%C3%A9on%20Van%20Hove
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Léon Charles Prudent Van Hove (10 February 1924 – 2 September 1990) was a Belgian physicist and a Director General of CERN. He developed a scientific career spanning mathematics, solid state physics, elementary particle and nuclear physics to cosmology.
Biography
Van Hove studied mathematics and physics at the Université Libre de Bruxelles (ULB). In 1946 he received his PhD in mathematics at the ULB. From 1949 to 1954 he worked at the Institute for Advanced Study in Princeton, New Jersey by virtue of his meeting with Robert Oppenheimer. Later he worked at the Brookhaven National Laboratory and was a professor and Director of the Theoretical Physics Institute at the University of Utrecht in the Netherlands. In the 1950s he laid the theoretical foundations for the analysis of inelastic neutron scattering in terms of the dynamic structure factor. In 1958, he was awarded the Francqui Prize in Exact Sciences. In 1959, he received an invitation to become the head of the Theory Division at CERN in Geneva. In 1975 Prof. Van Hove was appointed CERN Director-General, with John Adams, responsible for the research activities of the Organization. The LEP project was proposed during Van Hove's tenure as Director General.
Awards
Francqui Prize, 1958
Dannie Heineman Prize for Mathematical Physics, 1962
Member, American Academy of Arts and Sciences, 1964
Max Planck Medal, 1974
Member, United States National Academy of Sciences, 1980
Member, American Philosophical Society, 1980
There is a square, Square Van Hove, named after Van Hove at CERN, Geneva, Switzerland.
See also
Quark–gluon plasma
Quasielastic scattering
Quasielastic neutron scattering
List of Directors General of CERN
Théophile de Donder
Hilbrand J. Groenewold for the Groenewold–Van Hove theorem
References
External links
Léon Van Hove Biography. Cern official website.
Proc. Am. Phil. Soc. 136, 603 (1992)
Scientific publications of Léon Van Hove on INSPIRE-HEP
1924 births
1990 deaths
Belgian nuclear physicists
Belgian academics
People associated with CERN
Particle physicists
Theoretical physicists
Free University of Brussels (1834–1969) alumni
Mathematical physicists
Foreign associates of the National Academy of Sciences
Winners of the Max Planck Medal
Academic staff of Utrecht University
Members of the American Philosophical Society
Members of the Royal Swedish Academy of Sciences
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https://en.wikipedia.org/wiki/Mathematical%20Grammar%20School
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Mathematical Grammar School (, abbr. "MG" or "MGB"), is a special school for gifted and talented students of mathematics, physics and informatics located in Belgrade, Serbia. It is ranked number one at International Science Olympiads by the number of medals won by its students (more than 400).
The School has developed its own Mathematical Grammar School Curriculum in various mathematics, physics, and IT subjects.
There are approx. 160 professors employed, mostly scientists. One half of the professors come from University of Belgrade staff, Institute of Physics Belgrade, and Mathematical Institute of Serbian Academy of Sciences and Arts. More than half of the professors are former students of the School.
School's staff maintains connections to, collaborates with, and frequently visits world's leading scientific institutions, such as CERN, Joint Institute for Nuclear Research – Dubna, Lomonosov Moscow State University, UC Berkeley, Oxford, Cambridge, Warwick University, Imperial College London.
During the previous decade, students received full scholarships for UC Berkeley, Oxford, Cambridge, Warwick University, Imperial College London, Massachusetts Institute of Technology, MIT, Columbia University, Stanford University, Harvard University, University College London. The rest mostly obtain full scholarships from University of Belgrade.
The School has 550 students, aged 12–19. There are 155 girls, and 395 boys.
The average professors' work experience is 18 years.
Reputation
Mathematical Gymnasium is ranked as number one in the world in the field of mathematics.
The School is famous for its unique results in international competitions and for results its students achieve at later stages of their university education.
The School has approx. 2,000 PhD holders in its alumni, and approx. 6,000 Master of Science degree holders. Its advanced curriculum earned it an esteemed reputation as the breeding ground for future scientists, researchers, and industry leaders.
Mathematical Grammar School is unique for its success in International Science Olympiads in Mathematics, Physics, Informatics, Astronomy, Astrophysics, and Earth Sciences: its students have won more than 400 medals so far, on all 5 continents of the world.
Latest examples include: 12 medals in 2009 in International Science Olympiads only, around 25 medals in total in 2009 alone, 23 medals in 2010 in International Science Olympiads only, around 40 medals in total in 2010 alone.
In 2010, as well as in 2009, Mathematical Gymnasium won the first prize and title of "Absolute Winner" at international competition in Moscow, organized by Special School "Kolmogorov" under Moscow State University – Lomonosov (Специализированный учебно-научный центр Московского государственного университета им.М.В.Ломоносова – Школа им.А.Н.Колмогорова).
The School is ranked as number one school in the world in the field of mathematics, taking the lead from Russian Special School "Kolmogorov", (which operates und
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https://en.wikipedia.org/wiki/Hyperelliptic%20surface
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In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a surface whose Albanese morphism is an elliptic fibration. Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group.
Hyperelliptic surfaces form one of the classes of surfaces of Kodaira dimension 0 in the Enriques–Kodaira classification.
Invariants
The Kodaira dimension is 0.
Hodge diamond:
Classification
Any hyperelliptic surface is a quotient (E×F)/G, where E = C/Λ and F are elliptic curves, and G is a subgroup of F (acting on F by translations). There are seven families of hyperelliptic surfaces as in the following table.
Here ω is a primitive cube root of 1 and i is a primitive 4th root of 1.
Quasi hyperelliptic surfaces
A quasi-hyperelliptic surface is a surface whose canonical divisor is numerically equivalent to zero, the Albanese mapping maps to an elliptic curve, and all its fibers are rational with a cusp. They only exist in characteristics 2 or 3. Their second Betti number is 2, the second Chern number vanishes, and the holomorphic Euler characteristic vanishes. They were classified by , who found six cases in characteristic 3 (in which case 6K= 0) and eight in characteristic 2 (in which case 6K or 4K vanishes).
Any quasi-hyperelliptic surface is a quotient (E×F)/G, where E is a rational curve with one cusp, F is an elliptic curve, and G is a finite subgroup scheme of F (acting on F by translations).
References
- the standard reference book for compact complex surfaces
Complex surfaces
Algebraic surfaces
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https://en.wikipedia.org/wiki/Truncated%20triakis%20tetrahedron
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In geometry, the truncated triakis tetrahedron, or more precisely an order-6 truncated triakis tetrahedron, is a convex polyhedron with 16 faces: 4 sets of 3 pentagons arranged in a tetrahedral arrangement, with 4 hexagons in the gaps.
Construction
It is constructed from a triakis tetrahedron by truncating the order-6 vertices. This creates 4 regular hexagon faces, and leaves 12 mirror-symmetric pentagons.
A topologically similar equilateral polyhedron can be constructed by using 12 regular pentagons with 4 equilateral but nonplanar hexagons, each vertex with internal angles alternating between 108 and 132 degrees.
Topologically, as a near-miss Johnson solid, the four hexagons corresponding to the face planes of a tetrahedron are triambi, with equal edges but alternating angles, while the pentagons only have reflection symmetry.
Full truncation
If all of a triakis tetrahedron's vertices, of both kinds, are truncated, the resulting solid is an irregular icosahedron, whose dual is a trihexakis truncated tetrahedron.
Truncation of only the 3-valence vertices yields the order-3 truncated triakis tetrahedron, which looks like a tetrahedron with each face raised by a low triangular frustum. The dual to that truncation will be the triakis truncated tetrahedron.
Hexakis truncated tetrahedron
The dual of the order-6 Truncated triakis tetrahedron is called a hexakis truncated tetrahedron. It is constructed by a truncated tetrahedron with hexagonal pyramids augmented. If all of the triangles are made regular, the polyhedron becomes a failed Johnson solid, with coplanar triangles in a truncated tetrahedron volume.
See also
Near-miss Johnson solid
Truncated tetrakis cube
Truncated triakis octahedron
Truncated triakis icosahedron
External links
Johnson Solid Near Misses: Number 22
George Hart's Polyhedron generator - "t6kT" (Conway polyhedron notation)
Polyhedra
Truncated tilings
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https://en.wikipedia.org/wiki/Energetic%20space
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In mathematics, more precisely in functional analysis, an energetic space is, intuitively, a subspace of a given real Hilbert space equipped with a new "energetic" inner product. The motivation for the name comes from physics, as in many physical problems the energy of a system can be expressed in terms of the energetic inner product. An example of this will be given later in the article.
Energetic space
Formally, consider a real Hilbert space with the inner product and the norm . Let be a linear subspace of and be a strongly monotone symmetric linear operator, that is, a linear operator satisfying
for all in
for some constant and all in
The energetic inner product is defined as
for all in
and the energetic norm is
for all in
The set together with the energetic inner product is a pre-Hilbert space. The energetic space is defined as the completion of in the energetic norm. can be considered a subset of the original Hilbert space since any Cauchy sequence in the energetic norm is also Cauchy in the norm of (this follows from the strong monotonicity property of ).
The energetic inner product is extended from to by
where and are sequences in Y that converge to points in in the energetic norm.
Energetic extension
The operator admits an energetic extension
defined on with values in the dual space that is given by the formula
for all in
Here, denotes the duality bracket between and so actually denotes
If and are elements in the original subspace then
by the definition of the energetic inner product. If one views which is an element in as an element in the dual via the Riesz representation theorem, then will also be in the dual (by the strong monotonicity property of ). Via these identifications, it follows from the above formula that In different words, the original operator can be viewed as an operator and then is simply the function extension of from to
An example from physics
Consider a string whose endpoints are fixed at two points on the real line (here viewed as a horizontal line). Let the vertical outer force density at each point on the string be , where is a unit vector pointing vertically and Let be the deflection of the string at the point under the influence of the force. Assuming that the deflection is small, the elastic energy of the string is
and the total potential energy of the string is
The deflection minimizing the potential energy will satisfy the differential equation
with boundary conditions
To study this equation, consider the space that is, the Lp space of all square-integrable functions in respect to the Lebesgue measure. This space is Hilbert in respect to the inner product
with the norm being given by
Let be the set of all twice continuously differentiable functions with the boundary conditions Then is a linear subspace of
Consider the operator given by the formula
so the deflection satisfies the equati
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https://en.wikipedia.org/wiki/Link%20Layer%20Topology%20Discovery
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Link Layer Topology Discovery (LLTD) is a proprietary link layer protocol for network topology discovery and quality of service diagnostics. Microsoft developed it as part of the Windows Rally set of technologies. The LLTD protocol operates over both wired (such as Ethernet (IEEE 802.3) or power line communication) as well as wireless networks (such as IEEE 802.11).
LLTD is included in Windows 7, Windows Vista and Windows 10. It is used by their Network Map feature to display a graphical representation of the local area network (LAN) or wireless LAN (WLAN), to which the computer is connected. Windows XP does not contain the LLTD protocol as a standard component and as a result, Windows XP computers do not appear on the Network Map unless the LLTD responder is installed on Windows XP computers. LLTD is available for download for 32-bit editions of Windows XP with Service Pack 2 (as a publicly released update) and for Windows XP with Service Pack 3 (as a hotfix by request). LLTD Responder was not released for Windows XP Professional x64 Edition. A 2006 fall update for the Xbox 360 enabled support for the LLTD protocol.
Being a link layer (or OSI Layer 2) implementation, LLTD operates strictly on a given local network segment. It cannot discover devices across routers, an operation which would require Internet Protocol level routing.
Link Layer Topology Discovery in Windows Vista consists of two components. The LLTD Mapper I/O component is the master module which controls the discovery process and generates the Network Map. Appropriate permissions for this may be configured with Group Policy settings. It can be allowed or disallowed for domains, and private and public networks. The Mapper sends discovery command packets onto the local network segment via a raw network interface socket. The second component of LLTD are the LLTD Responders which answer Mapper requests about their host and possibly other discovered network information.
In addition to illustrating the layout of a network with representative icons for the hosts and interconnecting lines, each device icon may be explored to produce a popup information box summarizing important network and host parameters, such as MAC address and IP address (both IPv4 and IPv6). Icons are labeled with the hostnames (or first component of their fully qualified domain names), or a representative name of the function of the device, e.g., "gateway". If the device has reported the presence of a management Web interface, clicking on the icon will open a HTTP session to the host.
The LLTD responder for Windows XP only supports reporting of IPv4 addresses and not IPv6.
A royalty free Linux sample implementation of the LLTD responder is available from Microsoft as part of the Windows Rally Development Kit. Using LLTD specifications requires signing a Microsoft Windows Rally license agreement.
There also exists a Perl implementation, using Net::Frame, available via CPAN.
See also
Windows Vista networking te
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https://en.wikipedia.org/wiki/Two-center%20bipolar%20coordinates
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In mathematics, two-center bipolar coordinates is a coordinate system based on two coordinates which give distances from two fixed centers and . This system is very useful in some scientific applications (e.g. calculating the electric field of a dipole on a plane).
Transformation to Cartesian coordinates
When the centers are at and , the transformation to Cartesian coordinates from two-center bipolar coordinates is
Transformation to polar coordinates
When x > 0, the transformation to polar coordinates from two-center bipolar coordinates is
where is the distance between the poles (coordinate system centers).
Applications
Polar plotters use two-center bipolar coordinates to describe the drawing paths required to draw a target image.
See also
Bipolar coordinates
Biangular coordinates
Lemniscate of Bernoulli
Oval of Cassini
Cartesian oval
Ellipse
References
Two-center bipolar coordinates
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https://en.wikipedia.org/wiki/Ludwig%20Hopf
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Ludwig Hopf (23 October 1884 in Nürnberg, Germany – 23 December 1939 in Dublin) was a German-Jewish theoretical physicist who made contributions to mathematics, special relativity, hydrodynamics, and aerodynamics. Early in his career he was the assistant to and a collaborator and co-author with Albert Einstein.
Biography
Hopf was born into a family of prominent hops merchants and municipal counselors in Nürnberg, Germany, the son of Elise (née Josephthal) and Hans Hopf. From 1902-1909 he studied math and physics at the Universities of Munich and Berlin.
Hopf studied under Arnold Sommerfeld at the University of Munich, where he received his Ph.D. in 1909, on the topic of hydrodynamics. Shortly after this, Sommerfeld introduced Hopf to Albert Einstein at a physics conference in Salzburg. Later that year, Einstein, needing an assistant at the University of Zurich, hired Hopf; it was an added bonus that Hopf was a talented pianist, since Einstein played the violin and liked to play duets. Hopf was an ardent fan of psychoanalysis, had studied Freud and, once in Zurich, attached himself to Freud's ex-disciple Carl Jung. Hopf introduced Einstein to Jung, and Einstein returned to Jung's house several times over the years. In 1910, Hopf collaborated and published with Einstein two papers on classical statistical aspects of radiation. Hopf’s collaboration with Sommerfeld on integral representations of Bessel Functions resulted in the publication of a paper in 1911. Also in that year, Hopf accompanied Einstein to the Karl-Ferdinand University in Prague; however, he did not stay with Einstein long – due to “unsanitary conditions” in Prague.
In 1912, Hopf married Alice Goldschmidt, with whom he had five sons and a daughter.
During World War I, Hopf contributed to the design of military aircraft. In 1921, he accepted a position at the Rheinisch-Westfälische Technische Hochschule Aachen (RWTH Aachen University), a leading technical university in Germany, where he eventually became a professor in hydrodynamics and aerodynamics. It was during his tenure at Aachen that he made a contribution to the Handbuch der Physik and co-authored a “highly esteemed” book on aerodynamics.
In 1933, with the Nazis coming to power in Germany, Hopf was put on leave at Aachen due to his being a Jew, and in 1934 lost his position entirely.
Hopf remained in Germany until 1939 and escaped the Nazi regime only at the last minute. The SS was seeking to arrest him and were thwarted by his son Arnold posing as his father. Arnold was arrested and sent to the Buchenwald concentration camp, from which he was able to escape after 3–4 weeks and emigrate to Kenya. Ludwig left Germany for Great Britain with his wife and three of his children, taking a research position at Cambridge. He moved to Dublin in July 1939 to assume a professorship of mathematics at Trinity College.
Shortly after taking up his duties at Trinity, Hopf became seriously ill and died of thyroid failure
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https://en.wikipedia.org/wiki/G%C3%B6del%20numbering%20for%20sequences
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In mathematics, a Gödel numbering for sequences provides an effective way to represent each finite sequence of natural numbers as a single natural number. While a set theoretical embedding is surely possible, the emphasis is on the effectiveness of the functions manipulating such representations of sequences: the operations on sequences (accessing individual members, concatenation) can be "implemented" using total recursive functions, and in fact by primitive recursive functions.
It is usually used to build sequential “data types” in arithmetic-based formalizations of some fundamental notions of mathematics. It is a specific case of the more general idea of Gödel numbering. For example, recursive function theory can be regarded as a formalization of the notion of an algorithm, and can be regarded as a programming language to mimic lists by encoding a sequence of natural numbers in a single natural number.
Gödel numbering
Besides using Gödel numbering to encode unique sequences of symbols into unique natural numbers (i.e. place numbers into mutually exclusive or one-to-one correspondence with the sequences), we can use it to encode whole “architectures” of sophisticated “machines”. For example, we can encode Markov algorithms, or Turing machines into natural numbers and thereby prove that the expressive power of recursive function theory is no less than that of the former machine-like formalizations of algorithms.
Accessing members
Any such representation of sequences should contain all the information as in the original sequence—most importantly, each individual member must be retrievable. However, the length does not have to match directly; even if we want to handle sequences of different length, we can store length data as a surplus member, or as the other member of an ordered pair by using a pairing function.
We expect that there is an effective way for this information retrieval process in form of an appropriate total recursive function. We want to find a totally recursive function f with the property that
for all n and for any n-length sequence of natural numbers , there exists an appropriate natural number a, called the Gödel number of the sequence, such that for all i where , .
There are effective functions which can retrieve each member of the original sequence from a Gödel number of the sequence. Moreover, we can define some of them in a constructive way, so we can go well beyond mere proofs of existence.
Gödel's β-function lemma
By an ingenious use of the Chinese remainder theorem, we can constructively define such a recursive function (using simple number-theoretical functions, all of which can be defined in a total recursive way) fulfilling the specifications given above. Gödel defined the function using the Chinese remainder theorem in his article written in 1931. This is a primitive recursive function.
Thus, for all n and for any n-length sequence of natural numbers , there exists an appropriate natural number a, ca
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https://en.wikipedia.org/wiki/Pierre%20van%20Moerbeke
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Pierre van Moerbeke (born 1 October 1944 in Leuven, Belgium) is a Belgian mathematician. He studied mathematics at the Catholic University of Leuven, where he received his degree in 1966. He then obtained a PhD in mathematics at Rockefeller University, New York City (1972). He is a professor of mathematics at Brandeis University (United States) and the UCL. He studies non-linear differential equations and partial differential equations, with soliton behavior. In 1988, he was awarded the Francqui Prize on Exact Sciences.
See also
Kac-van Moerbeke lattice
External links
Official Webpage
Alternate webpage
Van Moerbeke, Pierre
Van Moerbeke, Pierre
Van Moerbeke, Pierre
Van Moerbeke, Pierre
Van Moerbeke, Pierre
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https://en.wikipedia.org/wiki/New%20home%20sales
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New home sales is an economic indicator which records sales of newly constructed residences in the United States of America.
The United States Census Bureau publishes new home sales statistics monthly on their website. Statistics are reported as unadjusted monthly rates and seasonally adjusted annual rates.
Economic significance
Because new home sales trigger consumption, they have significant market impact upon release. New home sales also serve as a good indicator of economic turning points due to its consumer income sensitivity. Generally, when economic conditions slow down, new home sales serves as an early indicator of such a depression.
Limitations
Several cautions apply when interpreting new home sales statistics:
Statistics exclude any new houses that were not built for immediate sale. For example, in the situation where a purchaser commissions a builder to build a house on a lot that the purchaser already owns, this housing unit would not be included in the statistics. Other construction statistics, such as Permits, starts and completions, do include virtually all new residential construction.
Sales are reported as of the month that a customer signs a sales contract or the builder accepts a deposit. The house can be in any stage of construction.
Sales are not reduced to account for sales contracts which are subsequently cancelled by the customer or the builder. However, in those situations where a cancellation occurs, the house is not re-counted upon a subsequent sale to another customer.
See also
Economic reports
FRED (Federal Reserve Economic Data)
External links
New Residential Sales definitions and statistics from the U.S. Census Bureau
Economic indicators
Housing in the United States
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https://en.wikipedia.org/wiki/Simon%20model
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In applied probability theory, the Simon model is a class of stochastic models that results in a power-law distribution function. It was proposed by Herbert A. Simon to account for the wide range of empirical distributions following a power-law. It models the dynamics of a system of elements with associated counters (e.g., words and their frequencies in texts, or nodes in a network and their connectivity ). In this model the dynamics of the system is based on constant growth via addition of new elements (new instances of words) as well as incrementing the counters (new occurrences of a word) at a rate proportional to their current values.
Description
To model this type of network growth as described above, Bornholdt and Ebel considered a network with nodes, and each node with connectivities , . These nodes
form classes of nodes with identical connectivity .
Repeat the following steps:
(i) With probability add a new node and attach a link to it from an arbitrarily chosen node.
(ii) With probability add one link from an arbitrary node to a node of class chosen with a probability proportional to .
For this stochastic process, Simon found a stationary solution exhibiting power-law scaling, , with exponent
Properties
(i) Barabási-Albert (BA) model can be mapped to the subclass of Simon's model, when using the simpler probability for a node being
connected to another node with connectivity (same as the preferential attachment at BA model). In other words, the Simon model describes a general class of stochastic processes that can result in a scale-free network, appropriate to capture Pareto and Zipf's laws.
(ii) The only free parameter of the model reflects the relative
growth of number of nodes versus the number of links. In general has small values; therefore, the scaling exponents can be predicted to be . For instance, Bornholdt and Ebel studied the linking dynamics of World Wide Web, and predicted the scaling exponent as , which was consistent with observation.
(iii) The interest in the scale-free model comes from its ability to describe the topology of complex networks. The Simon model does not have an underlying network structure, as it was designed to describe events whose frequency follows a power-law. Thus network measures going beyond the degree distribution such
as the average path length, spectral properties, and clustering coefficient, cannot be obtained from this mapping.
The Simon model is related to generalized scale-free models with growth and preferential attachment properties. For more reference, see.
References
Power laws
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https://en.wikipedia.org/wiki/Mean%20absolute%20error
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In statistics, mean absolute error (MAE) is a measure of errors between paired observations expressing the same phenomenon. Examples of Y versus X include comparisons of predicted versus observed, subsequent time versus initial time, and one technique of measurement versus an alternative technique of measurement. MAE is calculated as the sum of absolute errors divided by the sample size:It is thus an arithmetic average of the absolute errors , where is the prediction and the true value. Alternative formulations may include relative frequencies as weight factors. The mean absolute error uses the same scale as the data being measured. This is known as a scale-dependent accuracy measure and therefore cannot be used to make comparisons between predicted values that use different scales. The mean absolute error is a common measure of forecast error in time series analysis, sometimes used in confusion with the more standard definition of mean absolute deviation. The same confusion exists more generally.
Quantity disagreement and allocation disagreement
In remote sensing the MAE is sometimes expressed as the sum of two components: quantity disagreement and allocation disagreement. Quantity disagreement is the absolute value of the mean error:Allocation disagreement is MAE minus quantity disagreement.
It is also possible to identify the types of difference by looking at an plot. Quantity difference exists when the average of the X values does not equal the average of the Y values. Allocation difference exists if and only if points reside on both sides of the identity line.
Related measures
The mean absolute error is one of a number of ways of comparing forecasts with their eventual outcomes. Well-established alternatives are the mean absolute scaled error (MASE) and the mean squared error. These all summarize performance in ways that disregard the direction of over- or under- prediction; a measure that does place emphasis on this is the mean signed difference.
Where a prediction model is to be fitted using a selected performance measure, in the sense that the least squares approach is related to the mean squared error, the equivalent for mean absolute error is least absolute deviations.
MAE is not identical to root-mean square error (RMSE), although some researchers report and interpret it that way. The MAE is conceptually simpler and also easier to interpret than RMSE: it is simply the average absolute vertical or horizontal distance between each point in a scatter plot and the Y=X line. In other words, MAE is the average absolute difference between X and Y. Furthermore, each error contributes to MAE in proportion to the absolute value of the error. This is in contrast to RMSE which involves squaring the differences, so that a few large differences will increase the RMSE to a greater degree than the MAE.
Optimality property
The mean absolute error of a real variable c with respect to the random variable X isProvided that the probability distr
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https://en.wikipedia.org/wiki/Melnikov%20distance
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In mathematics, the Melnikov method is a tool to identify the existence of chaos in a class of dynamical systems under periodic perturbation.
Introduction
The Melnikov method is used in many cases to predict the occurrence of chaotic orbits in non-autonomous smooth nonlinear systems under periodic perturbation. According to the method, it is possible to construct a function called the "Melnikov function" which can be used to predict either regular or chaotic behavior of a dynamical system. Thus, the Melnikov function will be used to determine a measure of distance between stable and unstable manifolds in the Poincaré map. Moreover, when this measure is equal to zero, by the method, those manifolds crossed each other transversally and from that crossing the system will become chaotic.
This method appeared in 1890 by H. Poincaré and by V. Melnikov in 1963 and could be called the "Poincaré-Melnikov Method". Moreover, it was described by several textbooks as Guckenheimer & Holmes,Kuznetsov, S. Wiggins, Awrejcewicz & Holicke and others. There are many applications for Melnikov distance as it can be used to predict chaotic vibrations. In this method, critical amplitude is found by setting the distance between homoclinic orbits and stable manifolds equal to zero. Just like in Guckenheimer & Holmes where they were the first who based on the KAM theorem, determined a set of parameters of relatively weak perturbed Hamiltonian systems of two-degrees-of-freedom, at which homoclinic bifurcation occurred.
The Melnikov distance
Consider the following class of systems given by
or in vector form
where , , and
Assume that system (1) is smooth on the region of interest, is a small perturbation parameter and is a periodic vector function in with the period .
If , then there is an unperturbed system
From this system (3), looking at the phase space in Figure 1, consider the following assumptions
A1 - The system has a hyperbolic fixed point , connected to itself by a homoclinic orbit
A2 - The system is filled inside by a continuous family of periodic orbits of period with where
To obtain the Melnikov function, some tricks have to be used, for example, to get rid of the time dependence and to gain geometrical advantages new coordinate has to be used that is cyclic type given by Then, the system (1) could be rewritten in vector form as follows
Hence, looking at Figure 2, the three-dimensional phase space where and has the hyperbolic fixed point of the unperturbed system becoming a periodic orbit The two-dimensional stable and unstable manifolds of by and are denoted, respectively. By the assumption and coincide along a two-dimensional homoclinic manifold. This is denoted by where is the time of flight from a point to the point on the homoclinic connection.
In the Figure 3, for any point a vector is constructed , normal to the as follows Thus varying and serve to move to every point on
Splitting of stable and unstable manifolds
If is
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https://en.wikipedia.org/wiki/Hanover%20High%20School%20%28Pennsylvania%29
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Hanover Senior High School is located at 401 Moul Ave, Hanover, Pennsylvania. It is part of the Hanover Public School District. According to the National Center for Education Statistics, in 2018–2019, the school reported an enrollment of 481 pupils in grades 9th through 12th. The school employed 35.56 full-time-equivalent teachers, yielding a student–teacher ratio of 13.53:1.The school's colors are orange and black, and the mascot is the Nighthawk.
Extracurriculars
The high school's students have access to a variety of clubs, activities and an extensive sports program.
Sports
The District funds:
Boys
Baseball - AA
Basketball- AA
Football - AA
Golf - AA
Soccer - A
Tennis - AA
Track and Field - AA
Wrestling - AA
Girls
Basketball - AA
Field Hockey - AA
Soccer (Fall) - A
Softball - AA
Girls' Tennis - AA
Track and Field - AA
Volleyball - AA
Middle School Sports
Boys
Basketball
Cross Country
Football
Soccer
Track and Field
Wrestling
Girls
Basketball
Cross Country
Field Hockey
Soccer
Track and Field
Volleyball
References
Public high schools in Pennsylvania
Hanover, Pennsylvania
Schools in York County, Pennsylvania
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https://en.wikipedia.org/wiki/Bicycle%20and%20motorcycle%20geometry
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Bicycle and motorcycle geometry is the collection of key measurements (lengths and angles) that define a particular bike configuration. Primary among these are wheelbase, steering axis angle, fork offset, and trail. These parameters have a major influence on how a bike handles.
Wheelbase
The wheelbase is the horizontal distance between the centers (or the ground contact points) of the front and rear wheels. Wheelbase is a function of rear frame length, steering axis angle, and fork offset. It is similar to the term wheelbase used for automobiles and trains.
Wheelbase has a major influence on the longitudinal stability of a bike, along with the height of the center of mass of the combined bike and rider. Short bikes are much more suitable for performing wheelies and stoppies.
Steering axis angle
The steering axis angle is called caster angle when measured from vertical axis or head angle when measured from horizontal axis. The steering axis is the axis about which the steering mechanism (fork, handlebars, front wheel, etc.) pivots. The steering axis angle usually matches the angle of the head tube.
Bicycle head angle
In bicycles, the steering axis angle is measured from the horizontal and called the head angle; a 90° head angle would be vertical. For example, Lemond offers:
a 2007 Filmore, designed for the track, with a head angle that varies from 72.5° to 74° depending on frame size
a 2006 Tete de Course, designed for road racing, with a head angle that varies from 71.25° to 74°, depending on frame size.
Due to front fork suspension, modern mountain bikes—as opposed to road bikes—tend to have slacker head tube angles, generally around 70°, although they can be as low as 62° (depending on frame geometry setting).
At least one manufacturer, Cane Creek, offers an after-market threadless headset that enables changing the head angle.
Motorcycle rake angle
In motorcycles, the steering axis angle is measured from the vertical and called the caster angle, rake angle, or just rake; a 0° rake is therefore vertical. For example, Moto Guzzi offers:
a 2007 Breva V 1100 with a rake of 25°30′ (25.5 degrees)
a 2007 Nevada Classic 750 with a rake of 27.5°
Fork offset
The fork offset is the perpendicular distance from the steering axis to the center of the front wheel.
In bicycles, fork offset is also called fork rake. Road racing bicycle forks have an offset of .
The offset may be implemented by curving the forks, adding a perpendicular tab at their lower ends, offsetting the fork blade sockets of the fork crown ahead of the steerer, or by mounting the forks into the crown at an angle to the steer tube. The development of forks with curves is attributed to George Singer.
In motorcycles with telescopic fork tubes, fork offset can be implemented by either an offset in the triple tree, adding a triple tree rake (usually measured in degrees from 0) to the fork tubes as they mount into the triple tree, or a combination of the two. Other, less-common motorcy
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https://en.wikipedia.org/wiki/List%20of%20bioinformatics%20companies
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This is a list of bioinformatics companies that have articles at Wikipedia:
Applied Maths provides the software suite BioNumerics
Astrid Research
BIOBASE
BioBam Bioinformatics creator of Blast2GO
Biomax Informatics AG bioinformatics services.
Biovia (formerly Accelrys).
Chemical Computing Group MOE software for structural modelling
CLC Bio Bioinformatics workbenches.
DNASTAR provides software for sequence editing and annotation, primer and clone design; sequence assembly & analysis; and protein sequence analysis and structure prediction.
Gene Codes Corporation
Genedata software for data analysis and storage.
GeneTalk web-based services.
GenoCAD
Genomatix
Genostar provides streamlined bioinformatics.
Inte:Ligand
Integromics
Invitae
Invitrogen creator of Vector NTI
Leidos Biomedical Research Inc. formerly SAIC. Services are aimed at the Federal Government market.
MacVector
QIAGEN Silicon Valley (formerly Ingenuity Systems)
Qlucore
Phalanx Biotech Group
Seqera Labs
SimBioSys created the eHITS software
SRA International services aimed at the Federal Government market.
Strand Life Sciences
TimeLogic offers DeCypher FPGA-accelerated BLAST, Smith-Waterman, HMMER and other sequence search tools.
Bioinformatics companies
Lists of companies
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https://en.wikipedia.org/wiki/Solving%20quadratic%20equations%20with%20continued%20fractions
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In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is
where a ≠ 0.
The quadratic equation on a number can be solved using the well-known quadratic formula, which can be derived by completing the square. That formula always gives the roots of the quadratic equation, but the solutions are expressed in a form that often involves a quadratic irrational number, which is an algebraic fraction that can be evaluated as a decimal fraction only by applying an additional root extraction algorithm.
If the roots are real, there is an alternative technique that obtains a rational approximation to one of the roots by manipulating the equation directly. The method works in many cases, and long ago it stimulated further development of the analytical theory of continued fractions.
Simple example
Here is a simple example to illustrate the solution of a quadratic equation using continued fractions. We begin with the equation
and manipulate it directly. Subtracting one from both sides we obtain
This is easily factored into
from which we obtain
and finally
Now comes the crucial step. We substitute this expression for x back into itself, recursively, to obtain
But now we can make the same recursive substitution again, and again, and again, pushing the unknown quantity x as far down and to the right as we please, and obtaining in the limit the infinite continued fraction
By applying the fundamental recurrence formulas we may easily compute the successive convergents of this continued fraction to be 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, ..., where each successive convergent is formed by taking the numerator plus the denominator of the preceding term as the denominator in the next term, then adding in the preceding denominator to form the new numerator. This sequence of denominators is a particular Lucas sequence known as the Pell numbers.
Algebraic explanation
We can gain further insight into this simple example by considering the successive powers of
That sequence of successive powers is given by
and so forth. Notice how the fractions derived as successive approximants to appear in this geometric progression.
Since 0 < ω < 1, the sequence {ωn} clearly tends toward zero, by well-known properties of the positive real numbers. This fact can be used to prove, rigorously, that the convergents discussed in the simple example above do in fact converge to , in the limit.
We can also find these numerators and denominators appearing in the successive powers of
The sequence of successive powers {ω−n} does not approach zero; it grows without limit instead. But it can still be used to obtain the convergents in our simple example.
Notice also that the set obtained by forming all the combinations a + b, where a and b are integers, is an example of an object known in abstract algebra as a ring, and more specifically as an integral domain. The number ω is a unit in that integral domain. See also algebraic number
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https://en.wikipedia.org/wiki/Ivan%20Ivanov%20%28mathematician%29
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Ivan Ivanovich Ivanov (; 11 August 1862 – 17 December 1939) was a Russian-Soviet mathematician who worked in the field of number theory. Together with Georgy Voronoy he continued Pafnuty Chebyshev's work on the subject.
Life and work
Ivanov was born in Saint Petersburg, Russia. He finished his studies in mathematics at Saint Petersburg University with his candidate thesis, "About prime numbers". In 1891 there followed his master thesis "integral complex numbers", and in 1901 his doctoral thesis, "About some questions in connection with the number of prime numbers".
Starting in 1891, Ivanov lectured at St. Petersburg University; from 1896, he lectured at the women's university, and after 1902 at Saint Petersburg Polytechnical University.
In 1924 Ivanov was elected corresponding member of the Russian Academy of Sciences.
References
External links
Иванов, Иван Иванович - dic.academic.ru
1862 births
1939 deaths
Mathematicians from the Russian Empire
Soviet mathematicians
Number theorists
Mathematicians from Saint Petersburg
Saint Petersburg State University alumni
Burials at Bogoslovskoe Cemetery
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https://en.wikipedia.org/wiki/Tennis%20male%20players%20statistics
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Since 1990, the biggest events in men's tennis have been the four Grand Slam tournaments, the ATP Finals and the ATP Masters tournaments, in addition to the Grand Slam Cup between 1990–99. From 1983 to 1990, men's tennis had a very strong tradition and clear hierarchy of tournaments: the Grand Slam tournaments, including Wimbledon, the US Open, the French Open, and the Australian Open; the season-ending Masters Grand Prix; and the Davis Cup. Before 1983, however, and in particular before the start of the Open Era in 1968, the hierarchy of professional tournaments changed virtually every year. For example, in 1934, the U.S. Pro was a high-class tournament with all the best players, but just two years later, the same tournament was ordinary because only professional teachers (no leading touring pros) entered the event.
Professional tennis before the Open Era
Before the start of the Open Era and in addition to numerous small tournaments and head-to-head tours between the leading professionals, there were a few major professional tournaments that stood out during different periods:
Some survived sporadically because of financial collapses and others temporarily stood out when other important tournaments were not held:
Bristol Cup (held on the Côte d'Azur (French Riviera) at Cannes or Menton or Beaulieu) from 1920 to 1932.
Queen's Club Pro (in the 1927 and 1928).
International Pro Championship of Britain in Southport in the 1930s.
World Pro Championships in Berlin in the 1930s.
U.S Pro Hardcourt in Los Angeles, California in 1945 (the only significant professional tournament that year).
Philadelphia Pro 1950–1952.
Tournament of Champions, held at Forest Hills 1957, 1958, 1959, and also held in Australia at White City (1957, 1959) and at Kooyong (1958).
Masters Pro Round Robin in Los Angeles in 1956, 1957 and 1958.
Australian Pro in 1954, 1957, and 1958.
Madison Square Garden Pro in 1966 and 1967.
Wimbledon Pro in 1967.
There were a few team events modeled on the Davis Cup, such as the Bonnardel Cup in the 1930s and the Kramer Cup from 1961 through 1963.
Three traditional "championship tournaments" survived into the Open Era, often having all the leading players but sometimes having very depleted fields.
The most prestigious of the three was generally the London Indoor Professional Championship. Played between 1934 and 1990 at Wembley Arena in England, it was unofficially usually considered the world championship until 1967.
The oldest of the three was the United States Professional Championship, played between 1927 and 1999. From 1954 through 1962, this tournament was played indoors in Cleveland and was called the "World Professional Championships".
The third major tournament was the French Professional Championship, played usually at Roland Garros from 1934 (perhaps before but the data are unclear) through 1968. The British and American championships continued into the Open Era but soon devolved to the status of minor tournaments.
Because of
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https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg%20method
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In mathematics, the Runge–Kutta–Fehlberg method (or Fehlberg method) is an algorithm in numerical analysis for the numerical solution of ordinary differential equations. It was developed by the German mathematician Erwin Fehlberg and is based on the large class of Runge–Kutta methods.
The novelty of Fehlberg's method is that it is an embedded method from the Runge–Kutta family, meaning that identical function evaluations are used in conjunction with each other to create methods of varying order and similar error constants. The method presented in Fehlberg's 1969 paper has been dubbed the RKF45 method, and is a method of order O(h4) with an error estimator of order O(h5). By performing one extra calculation, the error in the solution can be estimated and controlled by using the higher-order embedded method that allows for an adaptive stepsize to be determined automatically.
Butcher tableau for Fehlberg's 4(5) method
Any Runge–Kutta method is uniquely identified by its Butcher tableau. The embedded pair proposed by Fehlberg
The first row of coefficients at the bottom of the table gives the fifth-order accurate method, and the second row gives the fourth-order accurate method.
Implementing an RK4(5) Algorithm
The coefficients found by Fehlberg for Formula 1 (derivation with his parameter α2=1/3) are given in the table below, using array indexing of base 1 instead of base 0 to be compatible with most computer languages:
The coefficients in the below table do not work.
Fehlberg outlines a solution to solving a system of n differential equations of the form:
to iterative solve for
where h is an adaptive stepsize to be determined algorithmically:
The solution is the weighted average of six increments, where each increment is the product of the size of the interval, , and an estimated slope specified by function f on the right-hand side of the differential equation.
Then the weighted average is:
The estimate of the truncation error is:
At the completion of the step, a new stepsize is calculated:
If , then replace with and repeat the step. If , then the step is completed. Replace with for the next step.
The coefficients found by Fehlberg for Formula 2 (derivation with his parameter α2 = 3/8) are given in the table below, using array indexing of base 1 instead of base 0 to be compatible with most computer languages:
In another table in Fehlberg, coefficients for an RKF4(5) derived by D. Sarafyan are given:
See also
List of Runge–Kutta methods
Numerical methods for ordinary differential equations
Runge–Kutta methods
Notes
References
Fehlberg, Erwin (1968) Classical fifth-, sixth-, seventh-, and eighth-order Runge-Kutta formulas with stepsize control. NASA Technical Report 287. https://ntrs.nasa.gov/api/citations/19680027281/downloads/19680027281.pdf
Fehlberg, Erwin (1969) Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems. Vol. 315. National aeronautics and
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https://en.wikipedia.org/wiki/Finite-rank%20operator
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In functional analysis, a branch of mathematics, a finite-rank operator is a bounded linear operator between Banach spaces whose range is finite-dimensional.
Finite-rank operators on a Hilbert space
A canonical form
Finite-rank operators are matrices (of finite size) transplanted to the infinite dimensional setting. As such, these operators may be described via linear algebra techniques.
From linear algebra, we know that a rectangular matrix, with complex entries, has rank if and only if is of the form
Exactly the same argument shows that an operator on a Hilbert space is of rank if and only if
where the conditions on are the same as in the finite dimensional case.
Therefore, by induction, an operator of finite rank takes the form
where and are orthonormal bases. Notice this is essentially a restatement of singular value decomposition. This can be said to be a canonical form of finite-rank operators.
Generalizing slightly, if is now countably infinite and the sequence of positive numbers accumulate only at , is then a compact operator, and one has the canonical form for compact operators.
If the series is convergent, is a trace class operator.
Algebraic property
The family of finite-rank operators on a Hilbert space form a two-sided *-ideal in , the algebra of bounded operators on . In fact it is the minimal element among such ideals, that is, any two-sided *-ideal in must contain the finite-rank operators. This is not hard to prove. Take a non-zero operator , then for some . It suffices to have that for any , the rank-1 operator that maps to lies in . Define to be the rank-1 operator that maps to , and analogously. Then
which means is in and this verifies the claim.
Some examples of two-sided *-ideals in are the trace-class, Hilbert–Schmidt operators, and compact operators. is dense in all three of these ideals, in their respective norms.
Since any two-sided ideal in must contain , the algebra is simple if and only if it is finite dimensional.
Finite-rank operators on a Banach space
A finite-rank operator between Banach spaces is a bounded operator such that its range is finite dimensional. Just as in the Hilbert space case, it can be written in the form
where now , and are bounded linear functionals on the space .
A bounded linear functional is a particular case of a finite-rank operator, namely of rank one.
References
Operator theory
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https://en.wikipedia.org/wiki/Probability%20integral%20transform
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In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to random variables having a standard uniform distribution. This holds exactly provided that the distribution being used is the true distribution of the random variables; if the distribution is one fitted to the data, the result will hold approximately in large samples.
The result is sometimes modified or extended so that the result of the transformation is a standard distribution other than the uniform distribution, such as the exponential distribution.
The transform was introduced by Ronald Fisher in his 1932 edition of the book Statistical Methods for Research Workers.
Applications
One use for the probability integral transform in statistical data analysis is to provide the basis for testing whether a set of observations can reasonably be modelled as arising from a specified distribution. Specifically, the probability integral transform is applied to construct an equivalent set of values, and a test is then made of whether a uniform distribution is appropriate for the constructed dataset. Examples of this are P–P plots and Kolmogorov–Smirnov tests.
A second use for the transformation is in the theory related to copulas which are a means of both defining and working with distributions for statistically dependent multivariate data. Here the problem of defining or manipulating a joint probability distribution for a set of random variables is simplified or reduced in apparent complexity by applying the probability integral transform to each of the components and then working with a joint distribution for which the marginal variables have uniform distributions.
A third use is based on applying the inverse of the probability integral transform to convert random variables from a uniform distribution to have a selected distribution: this is known as inverse transform sampling.
Statement
Suppose that a random variable has a continuous distribution for which the cumulative distribution function (CDF) is Then the random variable defined as
has a standard uniform distribution.
Equivalently, if is the uniform measure on , the distribution of on is the pushforward measure .
Proof
Given any random continuous variable , define . Given , if exists (i.e., if there exists a unique such that ), then:
If does not exist, then it can be replaced in this proof by the function , where we define , , and for , with the same result that . Thus, is just the CDF of a random variable, so that has a uniform distribution on the interval .
Examples
For a first, illustrative example, let be a random variable with a standard normal distribution . Then its CDF is
where is the error function. Then the new random variable defined by is uniformly distributed.
As second example, if has an exponential distribution
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https://en.wikipedia.org/wiki/Lexis%20ratio
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The Lexis ratio is used in statistics as a measure which seeks to evaluate differences between the statistical properties of random mechanisms where the outcome is two-valued — for example "success" or "failure", "win" or "lose". The idea is that the probability of success might vary between different sets of trials in different situations. This ratio is not much used currently having been largely replaced by the use of the chi-squared test in testing for the homogeneity of samples.
This measure compares the between-set variance of the sample proportions (evaluated for each set) with what the variance should be if there were no difference between in the true proportions of success across the different sets. Thus the measure is used to evaluate how data compares to a fixed-probability-of-success Bernoulli distribution. The term "Lexis ratio" is sometimes referred to as L or Q, where
Where is the (weighted) sample variance derived from the observed proportions of success in sets in "Lexis trials" and is the variance computed from the expected Bernoulli distribution on the basis of the overall average proportion of success. Trials where L falls significantly above or below 1 are known as supernormal and subnormal, respectively.
This ratio ( Q ) is a measure that can be used to distinguish between three types of variation in sampling for attributes: Bernoullian, Lexian and Poissonian. The Lexis ratio is sometimes also referred to as L.
Definition
Let there be k samples of size n1, n3, n3, ... , nk and these samples have the proportion of the attribute being examined of p1, p2, p3, ..., pk respectively. Then the Lexis ratio is
If the Lexis ratio is significantly below 1, the sampling is referred to as Poissonian (or subnormal); it is equal to 1 the sampling is referred to as Bernoullian (or normal); and if it is above 1 it is referred to as Lexian (or supranormal).
Chuprov showed in 1922 that in the case of statistical homogeneity
and
where E() is the expectation and var() is the variance. The formula for the variance is approximate and holds only for large values of n.
An alternative definition is
here is the (weighted) sample variance derived from the observed proportions of success in sets in "Lexis trials" and is the variance computed from the expected Bernoulli distribution on the basis of the overall average proportion of success.
Lexis variation
A closely related concept is the Lexis variation. Let k samples each of size n be drawn at random. Let the probability of success (p) be constant and let the actual probability of success in the kth sample be p1, p2, ... , pk.
The average probability of success (p) is
The variance in the number of successes is
where var( pi ) is the variance of the pi.
If all the pi are equal the sampling is said to be Bernoullian; where the pi differ the sampling is said to be Lexian and the dispersion is said to be supranormal.
Lexian sampling occurs in sampling from non homogen
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https://en.wikipedia.org/wiki/Mian%E2%80%93Chowla%20sequence
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In mathematics, the Mian–Chowla sequence is an integer sequence defined
recursively in the following way. The sequence starts with
Then for , is the smallest integer such that every pairwise sum
is distinct, for all and less than or equal to .
Properties
Initially, with , there is only one pairwise sum, 1 + 1 = 2. The next term in the sequence, , is 2 since the pairwise sums then are 2, 3 and 4, i.e., they are distinct. Then, can't be 3 because there would be the non-distinct pairwise sums 1 + 3 = 2 + 2 = 4. We find then that , with the pairwise sums being 2, 3, 4, 5, 6 and 8. The sequence thus begins
1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, 401, 475, ... .
Similar sequences
If we define , the resulting sequence is the same except each term is one less (that is, 0, 1, 3, 7, 12, 20, 30, 44, 65, 80, 96, ... ).
History
The sequence was invented by Abdul Majid Mian and Sarvadaman Chowla.
References
S. R. Finch, Mathematical Constants, Cambridge (2003): Section 2.20.2
R. K. Guy Unsolved Problems in Number Theory, New York: Springer (2003)
Integer sequences
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https://en.wikipedia.org/wiki/U%C5%9Fak%20Airport
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Uşak Airport is the main airport of the city of Uşak in the Aegean Region of Turkey.
Statistics
External links
Official website
References
Airport
Airports in Turkey
Buildings and structures in Uşak Province
Transport in Uşak Province
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https://en.wikipedia.org/wiki/Religion%20in%20Austria
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Religion in Austria is predominantly Christianity, adhered to by 68.2% of the country's population according to the 2021 national survey conducted by Statistics Austria. Among Christians, 80.9% were Catholics, 7.2% were Orthodox Christians (mostly belonging to the Eastern Orthodox Church), 5.6% were Protestants, while the remaining 6.2% were other Christians, belonging to other denominations of the religion or not affiliated to any denomination. In the same census, 8.3% of the Austrians declared that their religion was Islam, 1.2% declared to believe in other non-Christian religions (including Buddhism, Hindusim, Judaism and others), and 22.4% declared they did not belong to any religion, denomination or religious community.
According to church membership data, in 2021 53.9% of the population were Roman Catholics and 3.0% adhered to Protestant churches.
Austria was historically a strongly Catholic country, having been the centre of the Habsburg monarchy (1273–1918) which championed Roman Catholicism. Although in the 16th century many Austrians converted to Protestantism, Lutheranism in particular, as the Protestant Reformation (begun in 1517) was spreading across Europe, the Habsburgs enacted measures of Counter-Reformation as early as 1527 and harshly repressed Austrian Protestantism, albeit a minority of Austrians remained Protestant. A few decades after the fall of the Habsburg monarchy at the end of the World War I, and the transformation of Austria into a federal republic, at least since the 1970s there has been a decline of Christianity (with the exception of Orthodox churches) and a proliferation of other religions, a process which has been particularly pronounced in the capital state of Vienna.
Between the censuses of 1971 and 2021, Christianity declined from 93.8% to 68.2% of the Austrian population (Catholicism from 87.4% to 55.2%, and Protestantism from 6% to 3.8%, while Orthodox Christianity grew from 2.2% to 4.9% between 2001 and 2021). During the same timespan, Islam grew from being the religion of 0.2% to 8.3% of the Austrian population, and the proportion of people neither affiliating with nor belonging to any religion grew from 4.3% to 22.4%.
Demographics
Census statistics, 1921–2021
Line chart of the trends, 1951–2021
Census statistics 1951–2021:
Religion by federal state
History
The Protestant Reformation spread from northern Germany to Austria. By the Council of Trent in 1545, almost half of the Austrian population had converted to Lutheranism, while a minority also endorsed Calvinism. Eastern Austria was more affected by this phenomenon than western Austria. After 1545, Austria was recatholicized in the Counter Reformation. The Habsburgs imposed a strict regime to restore the influence of the Catholic Church among Austrians and their campaign proved successful. The Habsburgs for a long time viewed themselves as the vanguard of Catholicism, while all the other Christian confessions and religions were repressed.
In 1
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https://en.wikipedia.org/wiki/Journal%20of%20Number%20Theory
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The Journal of Number Theory (JNT) is a bimonthly peer-reviewed scientific journal covering all aspects of number theory. The journal was established in 1969 by R.P. Bambah, P. Roquette, A. Ross, A. Woods, and H. Zassenhaus (Ohio State University). It is currently published monthly by Elsevier and the editor-in-chief is Dorian Goldfeld (Columbia University). According to the Journal Citation Reports, the journal has a 2020 impact factor of 0.72.
References
External links
Number theory
Mathematics journals
Academic journals established in 1969
Elsevier academic journals
Monthly journals
English-language journals
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https://en.wikipedia.org/wiki/NCAA%20Division%20I%20men%27s%20lacrosse%20records
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NCAA Division I men's lacrosse records listed here are primarily records compiled by the NCAA's Director of Statistics office.
Included in this consolidation are the NCAA men's Division I individual single-season and career leaders. Official NCAA men's lacrosse records did not begin until the 1971 season and are based on information submitted to the NCAA statistics service by institutions participating in the weekly statistics rankings, which started in 1996. Career records include players who played at least three seasons (in a four-season career) or two (in a three-season career) in Division I during the era of official NCAA statistics. In statistical rankings, the rounding of percentages and/or averages may indicate ties where none exist. In these cases, the numerical order of the rankings is accurate.
Career leaders
Points
[a] Granted a fifth season of eligibility
[b] Lehigh records have Cameron with 308 career points, while NCAA record book shows Cameron with 307 career points.
[c] Zach Greer's career points mark of 353 points is not officially recognized by the NCAA. Greer was granted a fifth season of eligibility and Bryant was considered a reclassifying institution that year. The NCAA lists Greer's career points as 285, though he scored 42 goals with 26 assists for 68 points in 2009, for a total of 353 career points.
Points per game
[a] Not recognized by the NCAA
Goals
[a] Zach Greer's career goals of 248 are not officially recognized by the NCAA, because Greer was granted a fifth season of eligibility and Bryant was considered a reclassifying institution. Greer scored 42 goals in 2009 for Bryant.
Source:
Goals per game
Assists
[a] Lehigh record books show Cameron with 186 career assists while NCAA records have Cameron with 185.
[b] Granted a fifth season of eligibility
Assists per game
Single-season leaders
Points
Points per game in one season
[a] - Not recognized by the NCAA
Goals in one season
Goals per game in one season
[a] Not recognized by the NCAA
Assists in one season
Assists per game
[a] Not recognized by the NCAA
Most Wins and National Titles by a program
The NCAA does not officially recognize lacrosse records prior to 1971, and the USILA does not maintain a database of lacrosse records. USILA era lacrosse records, nonetheless, have been included below. National titles include all NCAA, USILA, all divisions.
Current NCAA Division I lacrosse programs with 480 or more wins through 2022:
Winningest coaches
See also
NCAA Division I Men's Lacrosse Championship
United States Intercollegiate Lacrosse Association
Wingate Memorial Trophy
Division I men's lacrosse records
References
Division I Men's Lacrosse Records through 2020
NCAA lacrosse
College sports records and statistics in the United States
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https://en.wikipedia.org/wiki/Anthony%20Santasiere
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Anthony Edward Santasiere (December 9, 1904January 13, 1977) was an American chess master and chess writer, who also wrote extensively on non-chess topics. Santasiere was a middle school mathematics teacher by profession. Santasiere won the 1945 U.S. Open Chess Championship, four New York State championships, and six Marshall Chess Club championships. He competed in four U.S. Chess Championships, with his best finish being a tie for third in 1946. He was a chess organizer.
Early life, education, professional career
Santasiere, of French and Italian ancestry, was born and raised in New York City, the 12th of 13 children, and grew up in extreme poverty. He graduated from City College with a degree in mathematics. His studies there were financed by Alrick Man, a wealthy chess enthusiast who had served as president of the Marshall Chess Club. Santasiere represented CCNY in intercollegiate chess. Following graduation, beginning in 1927, Santasiere taught mathematics at the Angelo Patri Middle School in the Bronx. He also taught mathematics and home room at P.S. 92 in the Bronx. He retired to South Florida in 1965.
Chess career
Santasiere wrote extensively on chess in the magazine American Chess Bulletin, from 1930 to 1963; he served as Games Editor, working with Editor Hermann Helms. The chess opening Santasiere's Folly (1.Nf3 d5 2.b4), was originated and developed by him, and is named for him. Santasiere was also an expert in the Reti Opening, the King's Gambit, and the Vienna Game.
Metropolitan competitor
In the 1930s, Santasiere defeated Albert Simonson by (+3 -1) in a match, and also defeated Fred Reinfeld in match play by (+3 =3 -0). Santasiere competed in 34 consecutive Marshall Chess Club Championships, and represented the Marshall Club for 37 consecutive seasons in the Metropolitan Chess League.
1920s
In 1922, at age 17, Santasiere won the first of his six Marshall Chess Club Championships. In 1923, Santasiere tied for 13th/14th place at Lake Hopatcong (9th American Chess Congress, Frank Marshall and Abraham Kupchik won). In 1924, he took third place, behind Marshall and Carlos Torre, at New York. In 1927, he tied for third/fourth at New York (Albert Pinkus won). In 1927, he tied for fourth through sixth place at Rome, New York (New York State Championship; Rudolph Smirka won). In 1928, Santasiere won at Buffalo (New York State Championship). In 1929, he took third place, behind Herman Steiner and Jacob Bernstein, at Buffalo (New York State Championship).
1930s
In 1930, he tied for first with Norman Lessing at Utica (New York State Championship). In 1931, he took seventh place in New York (José Raúl Capablanca won). In 1931, he tied for third/fourth at Rome (New York State Championship; Fred Reinfeld won). In 1934, he tied for ninth/tenth at Syracuse (Samuel Reshevsky won). In 1935, he took seventh at Milwaukee (U.S. Open); (Reuben Fine won). In 1938, he tied for 10th/11th at New York (second US Championship; Reshevsky won). In 1938, h
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https://en.wikipedia.org/wiki/Jim%20Hefferon
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Jim Hefferon (born October 12, 1958) is a Professor of Mathematics at Saint Michael's College. He is known for his award-winning textbook on linear algebra that is available for free
download, with LaTeX source, and for his activity in the TeX community.
Early life
Jim Hefferon grew up in Connecticut and attended the University of Connecticut, where he obtained a PhD in mathematics in recursion theory as a student of Manuel Lerman.
Saint Michael's days
Jim Hefferon moved to Vermont in 1990 to take a job at Saint Michael's College. He became an active member of the Linux community, including founding the Vermont Area Group of Unix Enthusiasts.
Textbooks
In 2020, for his open-content undergraduate textbook Linear Algebra, Hefferon won the Daniel Solow Author's Award of the Mathematical Association of America, with the award citation noting the book's "clear writing style, tremendous variety of exercises, amenability to use with active learning strategies, and […] careful attention to detail" and its status as one of "the most successful and the most popular" open textbooks. Since 1996, Hefferon's Linear Algebra has been available for free download on the World Wide Web under the GNU Free Documentation License or a Creative Commons license. , the book is in its fourth edition and is published by Orthogonal Publishing L3C.
Other textbooks of Hefferon's, made available under the same terms, are an inquiry-based Introduction to Proofs and a textbook on computer science, Theory of Computation.
TeX connection
Hefferon is a member of the board of directors of the TeX Users Group (TUG), serving from 2019 to 2023. He previously had been a member of the board from 2003 to 2017, serving as vice-president of TUG from 2011 until 2016, when he became acting president of TUG when the board of directors suspended the previous president, Kaveh Bazargan.
In 1999 Jim became one of the core maintainers of the TeX archive CTAN, running one of three core CTAN archive sites until 2011.
Other interests
Jim is a Ham radio enthusiast, holding the Extra Class license KE1AZ, and is active with Morse code.
References
External links
Hefferon's book Linear Algebra
Home page at Saint Michael's College
Jim Hefferon - Interview - TeX Users Group
1958 births
Living people
American textbook writers
American male non-fiction writers
20th-century American mathematicians
21st-century American mathematicians
Saint Michael's College faculty
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https://en.wikipedia.org/wiki/Negligible%20function
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For a similar term, please see negligible set. (disambiguation)
In mathematics, a negligible function is a function such that for every positive integer c there exists an integer Nc such that for all x > Nc,
Equivalently, we may also use the following definition.
A function is negligible, if for every positive polynomial poly(·) there exists an integer Npoly > 0 such that for all x > Npoly
History
The concept of negligibility can find its trace back to sound models of analysis. Though the concepts of "continuity" and "infinitesimal" became important in mathematics during Newton and Leibniz's time (1680s), they were not well-defined until the late 1810s. The first reasonably rigorous definition of continuity in mathematical analysis was due to Bernard Bolzano, who wrote in 1817 the modern definition of continuity. Later Cauchy, Weierstrass and Heine also defined as follows (with all numbers in the real number domain ):
(Continuous function) A function is continuous at if for every , there exists a positive number such that implies
This classic definition of continuity can be transformed into the
definition of negligibility in a few steps by changing parameters used in the definition. First, in the case with , we must define the concept of "infinitesimal function":
(Infinitesimal) A continuous function is infinitesimal (as goes to infinity) if for every there exists such that for all
Next, we replace by the functions where or by where is a positive polynomial. This leads to the definitions of negligible functions given at the top of this article. Since the constants can be expressed as with a constant polynomial, this shows that infinitesimal functions are a subset of negligible functions.
Use in cryptography
In complexity-based modern cryptography, a security scheme is
provably secure if the probability of security failure (e.g.,
inverting a one-way function, distinguishing cryptographically strong pseudorandom bits from truly random bits) is negligible in terms of the input = cryptographic key length . Hence comes the definition at the top of the page because key length must be a natural number.
Nevertheless, the general notion of negligibility doesn't require that the input parameter is the key length . Indeed, can be any predetermined system metric and corresponding mathematical analysis would illustrate some hidden analytical behaviors of the system.
The reciprocal-of-polynomial formulation is used for the same reason that computational boundedness is defined as polynomial running time: it has mathematical closure properties that make it tractable in the asymptotic setting (see #Closure properties). For example, if an attack succeeds in violating a security condition only with negligible probability, and the attack is repeated a polynomial number of times, the success probability of the overall attack still remains negligible.
In practice one might want to have more concrete functions bounding the adve
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https://en.wikipedia.org/wiki/George%20Langdale
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George Richmond Langdale (11 March 1916 – 24 April 2002) was a schoolmaster, writer on mathematics and an English cricketer who played for Derbyshire from 1936 to 1937 and for Somerset from 1946 to 1949. He also played for the minor counties Norfolk and Berkshire.
Langdale was born in Thornaby-on-Tees in Yorkshire. He first played cricket for Derbyshire in the 1936 season taking part in three games and helping them to capture their first and only County Championship victory. In the 1937 season he played one first team game and then for the Derbyshire second team. On 1 September 1938 Langdale became a teacher at the City of Norwich School
and during 1939 he played for minor county Norfolk. During the Second World War he played occasional matches including one for Anti-Aircraft Command against Balloon command, and another for the Army against the Royal Australian Air Force.
In 1946 Langdale left Norwich and began playing for Somerset. In his first match for his new county, he took five Warwickshire wickets in an innings for 30 runs and in his next, two months later, against Yorkshire at Taunton, batting at No 8, he scored 146 to enable Somerset to reach 508, though the match was drawn. He continued in the next three seasons and played for Somerset for the last time in 1949. In 1950 he appeared in a match for Sandhurst Wanderers in the Netherlands. In 1952 Langdale started playing for Berkshire as captain, and his final first-class match was in 1953 for the Minor Counties against Australia. In the same Minor Counties season, playing for Berkshire he took all 10 Dorset wickets in an innings for 25 runs. He captained Berkshire again from 1956 to 1959 and continued playing for the team until 1963. Langdale was a right-arm offbreak bowler and took 23 wickets first-class wickets with an average of 40.82 and a best performance of 5–30. He was a left-handed batsman and remained a lower-middle order batsman throughout his first-class career. He played 42 innings in 25 first-class matches with an average of 18.12 and a top score of 146.
Langdale published a number of papers on mathematical subjects and teaching, often using cricket examples. He was a senior lecturer at Welbeck College, for the Ministry of Defence. In the 1982 New Year Honours he was awarded the OBE.
Langdale died in Holbeck, Leeds in at the age of 86.
Publications
Wisden's Cricketers' Almanack and the teaching of statistics. The Mathematical Gazette, Vol. 38, No. 324, 118–120. May 1954.
The Slide Rule The Mathematical Gazette, May 1959 p113
A Simple Course on Astronautics The Mathematical Gazette, Vol. 47, No. 360 May 1963, pp. 107–113
Square ball The Mathematical Gazette, October 1974 p216,
References
English cricketers
Derbyshire cricketers
Somerset cricketers
Minor Counties cricketers
1916 births
2002 deaths
Officers of the Order of the British Empire
People from Thornaby-on-Tees
Cricketers from County Durham
Berkshire cricketers
Norfolk cricketers
Berkshire cricket captains
Cr
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