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https://en.wikipedia.org/wiki/Arpad%20Va%C5%A1 | Arpad Vaš (; born 31 July 1989) is a Slovenian footballer who plays as a midfielder for Rudar Mursko Središće in Croatia.
Career statistics
Source: NZS (league games only)
References
External links
NZS profile
1989 births
Living people
People from Lendava
Slovenian people of Hungarian descent
Slovenian men's footballers
Slovenian expatriate men's footballers
Slovenia men's youth international footballers
Men's association football midfielders
NK Nafta Lendava players
ND Mura 05 players
NK Aluminij players
NK Zavrč players
NŠ Mura players
Slovenian PrvaLiga players
Slovenian Second League players
Second Football League (Croatia) players
Slovenian expatriate sportspeople in Croatia
Expatriate men's footballers in Croatia |
https://en.wikipedia.org/wiki/Multibrot%20set | In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a portmanteau of multiple and Mandelbrot set. The same can be applied to the Julia set, this being called Multijulia set.
where d ≥ 2. The exponent d may be further generalized to negative and fractional values.
Examples
The case of
is the classic Mandelbrot set from which the name is derived.
The sets for other values of d also show fractal images when they are plotted on the complex plane.
Each of the examples of various powers d shown below is plotted to the same scale. Values of c belonging to the set are black. Values of c that have unbounded value under recursion, and thus do not belong in the set, are plotted in different colours, that show as contours, depending on the number of recursions that caused a value to exceed a fixed magnitude in the Escape Time algorithm.
Positive powers
The example is the original Mandelbrot set. The examples for are often called multibrot sets. These sets include the origin and have fractal perimeters, with rotational symmetry.
Negative powers
When d is negative the set appears to surround but does not include the origin, However this is just an artifact of the fixed maximum radius allowed by the Escape Time algorithm, and is not a limit of the sets that actually have a shape in the middle with an no hole (You can see this by using the Lyapunov exponent [No hole because the origin diverges to undefined not infinity because the origin {0 or 0+0i} taken to a negative power becomes undefined]). There is interesting complex behaviour in the contours between the set and the origin, in a star-shaped area with rotational symmetry. The sets appear to have a circular perimeter, however this is just an artifact of the fixed maximum radius allowed by the Escape Time algorithm, and is not a limit of the sets that actually extend in all directions to infinity.
Fractional powers
Rendering along the exponent
An alternative method is to render the exponent along the vertical axis. This requires either fixing the real or the imaginary value, and rendering the remaining value along the horizontal axis. The resulting set rises vertically from the origin in a narrow column to infinity. Magnification reveals increasing complexity. The first prominent bump or spike is seen at an exponent of 2, the location of the traditional Mandelbrot set at its cross-section. The third image here renders on a plane that is fixed at a 45-degree angle between the real and imaginary axes.
Rendering images
All the above images are rendered using an Escape Time algorithm that identifies points outside the set in a simple way. Much greater fractal detail is revealed by plotting the Lyapunov exponent, as shown by the example below. The Lyapunov exponent is the error growth-rate of a given sequence. |
https://en.wikipedia.org/wiki/Exact%20statistics | Exact statistics, such as that described in exact test, is a branch of statistics that was developed to provide more accurate results pertaining to statistical testing and interval estimation by eliminating procedures based on asymptotic and approximate statistical methods. The main characteristic of exact methods is that statistical tests and confidence intervals are based on exact probability statements that are valid for any sample size.
Exact statistical methods help avoid some of the unreasonable assumptions of traditional statistical methods, such as the assumption of equal variances in classical ANOVA. They also allow exact inference on variance components of mixed models.
When exact p-values and confidence intervals are computed under a certain distribution, such as the normal distribution, then the underlying methods are referred to as exact parametric methods. The exact methods that do not make any distributional assumptions are referred to as exact nonparametric methods. The latter has the advantage of making fewer assumptions whereas, the former tend to yield more powerful tests when the distributional assumption is reasonable. For advanced methods such as higher-way ANOVA regression analysis, and mixed models, only exact parametric methods are available.
When the sample size is small, asymptotic results given by some traditional methods may not be valid. In such situations, the asymptotic p-values may differ substantially from the exact p-values. Hence asymptotic and other approximate results may lead to unreliable and misleading conclusions.
The approach
All classical statistical procedures are constructed using statistics which depend only on observable random vectors, whereas generalized estimators, tests, and confidence intervals used in exact statistics take advantage of the observable random vectors and the observed values both, as in the Bayesian approach but without having to treat constant parameters as random variables. For example, in sampling from a normal population with mean and variance , suppose and are the sample mean and the sample variance. Then, defining Z and U thus:
and that
.
Now suppose the parameter of interest is the coefficient of variation, . Then, we can easily perform exact tests and exact confidence intervals for based on the generalized statistic
,
where is the observed value of and is the observed value of . Exact inferences on based on probabilities and expected values of are possible because its distribution and the observed value are both free of nuisance parameters.
Generalized p-values
Classical statistical methods do not provide exact tests to many statistical problems such as testing Variance Components and ANOVA under unequal variances. To rectify this situation, the generalized p-values are defined as an extension of the classical p-values so that one can perform tests based on exact probability statements valid for any sample size.
See also
Fisher's exact test
Optimal |
https://en.wikipedia.org/wiki/Greg%20Tabor | Greg Steven Tabor (born May 21, 1961) is a former right-handed Major League Baseball second baseman and pinch runner who played for the Texas Rangers in 1987.
Baseball career and statistics
Drafted by the Rangers 10th overall in the January Regular phase of the 1981 amateur draft, Tabor split time with the GCL Rangers and Asheville Tourists that year. Combined, he hit .193 in 161 at-bats that season.
In 1982, Tabor played for the Burlington Rangers and Tulsa Drillers, he hit .240 with 32 stolen bases in 392 at-bats. He spent 1983 with the Drillers, hitting .268 with 30 stolen bases in 370 at-bats. Again with the Drillers in 1984, he hit .299 with 22 stolen bases in 462 at-bats.
Tabor spent all of 1985, 1986 and 1987 with the Oklahoma City 89ers. With the 89ers in 1985, he hit .222 in 81 at-bats. He hit .284 in 401 at-bats in 1986, and in 1987 he hit .303 with 22 stolen bases in 528 at-bats. He made his major league debut that year on September 10. He pinch ran for Larry Parrish against the California Angels, and in his sole at-bat he popped out. He did however score a run in his first game. Tabor appeared in nine games in 1987, collecting one hit in nine at-bats for a .111 batting average. Used often as a pinch runner, he scored four runs. He played his final game on October 4.
On March 17, 1988, he was traded to the Chicago Cubs with Ray Hayward for Dave Meier. Tabor and Paul Noce battled for the 25th roster spot in spring training of 1988, however the Cubs signed Angel Salazar, and gave him the 25th spot. He played in 130 games for their Triple-A team the Iowa Cubs in 1988, hitting .267, walking only 19 times in 469 at-bats.
Overall, he hit .272 with 145 stolen bases in 800 minor league games. He scored 404 runs and drove 310 in.
References
External links
Baseball Reference
The Baseball Cube
1961 births
Living people
Chabot Gladiators baseball players
Major League Baseball second basemen
Texas Rangers players
Sportspeople from Castro Valley, California
Baseball players from Alameda County, California |
https://en.wikipedia.org/wiki/Higher-dimensional%20algebra | In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.
Higher-dimensional categories
A first step towards defining higher dimensional algebras is the concept of 2-category of higher category theory, followed by the more 'geometric' concept of double category.
A higher level concept is thus defined as a category of categories, or super-category, which generalises to higher dimensions the notion of category – regarded as any structure which is an interpretation of Lawvere's axioms of the elementary theory of abstract categories (ETAC). Thus, a supercategory and also a super-category, can be regarded as natural extensions of the concepts of meta-category, multicategory, and multi-graph, k-partite graph, or colored graph (see a color figure, and also its definition in graph theory).
Supercategories were first introduced in 1970, and were subsequently developed for applications in theoretical physics (especially quantum field theory and topological quantum field theory) and mathematical biology or mathematical biophysics.
Other pathways in higher-dimensional algebra involve: bicategories, homomorphisms of bicategories, variable categories (also known as indexed or parametrized categories), topoi, effective descent, and enriched and internal categories.
Double groupoids
In higher-dimensional algebra (HDA), a double groupoid is a generalisation of a one-dimensional groupoid to two dimensions, and the latter groupoid can be considered as a special case of a category with all invertible arrows, or morphisms.
Double groupoids are often used to capture information about geometrical objects such as higher-dimensional manifolds (or n-dimensional manifolds). In general, an n-dimensional manifold is a space that locally looks like an n-dimensional Euclidean space, but whose global structure may be non-Euclidean.
Double groupoids were first introduced by Ronald Brown in Double groupoids and crossed modules (1976), and were further developed towards applications in nonabelian algebraic topology. A related, 'dual' concept is that of a double algebroid, and the more general concept of R-algebroid.
Nonabelian algebraic topology
See Nonabelian algebraic topology
Applications
Theoretical physics
In quantum field theory, there exist quantum categories. and quantum double groupoids. One can consider quantum double groupoids to be fundamental groupoids defined via a 2-functor, which allows one to think about the physically interesting case of quantum fundamental groupoids (QFGs) in terms of the bicategory Span(Groupoids), and then constructing 2-Hilbert spaces and 2-linear maps for manifolds and cobordisms. At the next step, one obtains cobordisms with corners via natural transformations of such 2-functors. A claim was then made that, with the gauge group SU(2), "the extended TQFT, or ETQFT, gives a theory |
https://en.wikipedia.org/wiki/Bhattacharya | Bhattacharya or Bhattacharyya may refer to
Bhattacharya (surname)
Bhattacharyya angle in statistics
Bhattacharyya distance in statistics
8348 Bhattacharyya, an asteroid
Harasankar Bhattacharya Institute of Technology and Mining in West Bengal, India
Nani Bhattacharya Smarak Mahavidyalaya, a general degree college in Mangalbari, India |
https://en.wikipedia.org/wiki/Gladesmore%20Community%20School | Gladesmore Community School is a coeducational secondary school located in Tottenham, London, England.
The school's specialisms include Gifted and Talented, Mathematics and Computing, and Applied Learning.
History
Gladesmore combines a number of previous schools in the vicinity:
Crowland Road School opened in 1911 and became Crowland Secondary Modern in 1946.
The Crowland School buildings then became part of the new Markfield County Secondary, which was founded in 1956.
Drayton School then replaced Markfield School, opening in Gladesmore Road in 1967.
Gladesmore was a poor school in the 1970s-90s, but when Tony Hartney became Headteacher in 1999, the school was transformed from Poor to Outstanding, receiving extremely positive reviews from Ofsted, it is rated as exemplary gaining three consecutive judgements of outstanding in all categories. In 2006, Mr Hartney was appointed a CBE for his work at the school.
Gladesmore received the Queen's Award for Service on 29 June 2011.
Prefecture
Pupils may apply to become a prefect in year 10, stating their qualities and suitability. In year 11, a head boy and head girl are appointed, along with a deputy for each. Students may vote for a boy and a girl in their year to become a part of the School Council. Gladesmore embraces the rich diversity of its community and plays a strong role in promoting improvements. The ethos of the school is extremely positive, friendly and uplifting. students and staff relate very well to each other and enjoy a 'family' atmosphere.
Value Life
In 2003, Gladesmore students founded the Value Life campaign, aiming to teach students how to stay safe and make the most of their life. It tackles gun and knife crime. This evolved into a series of large events, such as a carnival, a march, a music video and a short film.
Value Life was supported by many officials, such as Queen Elizabeth II, Boris Johnson & David Cameron. The campaign won the Philip Lawrence Award in 2008.
Everybody Dreams
In 2011, Gladesmore founded the Everybody Dreams campaign, which aims to improve the reputation of Tottenham after the 2011 England Riots. This included the release of a song performed by pupils at the school. It was supported by people like Leona Lewis, Dave Stewart David Lammy, Boris Johnson, Westlife's Mark Feehily, Ricky Gervais, Jessica Ennis and Wretch 32. The song reached number 33 in the iTunes Chart.
Notable former pupils
Chip (formerly Chipmunk), solo grime artist, who used the school to shoot the video for his single "Chip Diddy Chip".
Emmanuel Frimpong, former footballer at Arsenal.
Professor Green, rapper.
Wendell Richardson, lead guitarist with Osibisa.
Gabriel Zakuani, footballer.
Steve Zakuani, footballer.
Bob Bradbury, musician, founding member singer and guitarist of 1970s glam rock band, Hello.
Elijah Quashie, "The Chicken Connoisseur", food critic and author.
References
External links
School website
Everybody Dreams website
Value Life website
Aerial view
Ofsted page |
https://en.wikipedia.org/wiki/Bailey%20pair | In mathematics, a Bailey pair is a pair of sequences satisfying certain relations, and a Bailey chain is a sequence of Bailey pairs. Bailey pairs were introduced by while studying the second proof Rogers 1917 of the Rogers–Ramanujan identities, and Bailey chains were introduced by .
Definition
The q-Pochhammer symbols are defined as:
A pair of sequences (αn,βn) is called a Bailey pair if they are related by
or equivalently
Bailey's lemma
Bailey's lemma states that if (αn,βn) is a Bailey pair, then so is (α'n,β'n) where
In other words, given one Bailey pair, one can construct a second using the formulas above. This process can be iterated to produce an infinite sequence of Bailey pairs, called a Bailey chain.
Examples
An example of a Bailey pair is given by
gave a list of 130 examples related to Bailey pairs.
References
Special functions
Q-analogs |
https://en.wikipedia.org/wiki/Park%20Gwang-min | Park Gwang-Min (; born 14 May 1982) is a South Korean football midfielder.
He has played for Seongnam Ilhwa Chunma and Gwangju Sangmu in K-League.
Career statistics
External links
K-League Player Record
Korean FA Cup match result
1982 births
Living people
Men's association football forwards
South Korean men's footballers
Seongnam FC players
Gimcheon Sangmu FC players
K League 1 players |
https://en.wikipedia.org/wiki/Dmitry%20Belorukov | Dmitry Aleksandrovich Belorukov (; born 24 March 1983) is a Russian football coach and a former player who played as a centre-back. He is the manager of FC Dynamo Saint Petersburg.
Career statistics
Club
Notes
External links
Profile on Official FC Amkar Website
1983 births
Footballers from Saint Petersburg
Living people
Russian men's footballers
Russia men's under-21 international footballers
Men's association football defenders
FC Anzhi Makhachkala players
FC Amkar Perm players
FC Dynamo Moscow players
FC Zenit-2 Saint Petersburg players
FC Petrotrest players
Russian Premier League players
Russian First League players
Russian Second League players
Russian football managers
FC Dynamo Saint Petersburg managers |
https://en.wikipedia.org/wiki/Aleksei%20Popov%20%28footballer%2C%20born%201978%29 | Aleksei Vladislavovich Popov (; born 7 July 1978) is a football coach and a former player. Born in Russia, he played for the Kazakhstan national team.
Career statistics
Club
International
Statistics accurate as of match played 12 October 2010
Honours
Amkar Perm
Russian Second Division Ural (1): 1998
Russian First Division (1): 2003
Rubin Kazan
Russian Premier League (2): 2008, 2009
References
External links
Player page on the official FC Rubin Kazan website
1978 births
Living people
Footballers from Perm, Russia
Kazakhstani men's footballers
Kazakhstani football managers
Kazakhstan men's international footballers
Russian men's footballers
Russian football managers
Kazakhstani people of Bulgarian descent
Kazakhstani people of Russian descent
Russian people of Bulgarian descent
Men's association football defenders
FC Amkar Perm players
FC Rubin Kazan players
Russian Premier League players
FC Zvezda Perm players |
https://en.wikipedia.org/wiki/Ivan%20Cherenchikov | Ivan Andreyevich Cherenchikov (; born 25 August 1984) is a Russian football coach and a former player. He is the manager of FC Amkar Perm.
Career statistics
External links
Profile on Official FC Amkar Website
1984 births
People from Ozyorsk, Chelyabinsk Oblast
Footballers from Chelyabinsk Oblast
Living people
Russian men's footballers
Russia men's under-21 international footballers
Men's association football midfielders
Men's association football defenders
FC Amkar Perm players
FC Baltika Kaliningrad players
Russian Premier League players
Russian First League players
Russian football managers
FC Amkar Perm managers |
https://en.wikipedia.org/wiki/Kummer%27s%20conjecture | In mathematics, Kummer's conjecture is either of two the conjectures made by Ernst Eduard Kummer:
The Kummer–Vandiver conjecture about class numbers of cyclotomic fields
Kummer's conjecture about the Kummer sum |
https://en.wikipedia.org/wiki/Algebra%20tile | Algebra tiles are mathematical manipulatives that allow students to better understand ways of algebraic thinking and the concepts of algebra. These tiles have proven to provide concrete models for elementary school, middle school, high school, and college-level introductory algebra students. They have also been used to prepare prison inmates for their General Educational Development (GED) tests. Algebra tiles allow both an algebraic and geometric approach to algebraic concepts. They give students another way to solve algebraic problems other than just abstract manipulation. The National Council of Teachers of Mathematics (NCTM) recommends a decreased emphasis on the memorization of the rules of algebra and the symbol manipulation of algebra in their Curriculum and Evaluation Standards for Mathematics. According to the NCTM 1989 standards "[r]elating models to one another builds a better understanding of each".
Examples
Solving linear equations using addition
The linear equation can be modeled with one positive tile and eight negative unit tiles on the left side of a piece of paper and six positive unit tiles on the right side. To maintain equality of the sides, each action must be performed on both sides. For example, eight positive unit tiles can be added to both sides. Zero pairs of unit tiles are removed from the left side, leaving one positive tile. The right side has 14 positive unit tiles, so .
Solving linear equations using subtraction
The equation can be modeled with one positive tile and seven positive unit tiles on the left side and 10 positive unit tiles on the right side. Rather than adding the same number of tiles to both sides, the same number of tiles can be subtracted from both sides. For example, seven positive unit tiles can be removed from both sides. This leaves one positive tile on the left side and three positive unit tiles on the right side, so .
Multiplying polynomials
When using algebra tiles to multiply a monomial by a monomial, the student must first set up a rectangle where the length of the rectangle is the one monomial and then the width of the rectangle is the other monomial, similar to when one multiplies integers using algebra tiles. Once the sides of the rectangle are represented by the algebra tiles, one would then try to figure out which algebra tiles would fill in the rectangle. For instance, if one had x×x, the only algebra tile that would complete the rectangle would be x2, which is the answer.
Multiplication of binomials is similar to multiplication of monomials when using the algebra tiles . Multiplication of binomials can also be thought of as creating a rectangle where the factors are the length and width. As with the monomials, one would set up the sides of the rectangle to be the factors and then fill in the rectangle with the algebra tiles. This method of using algebra tiles to multiply polynomials is known as the area model and it can also be applied to multiplying monomials and binomial |
https://en.wikipedia.org/wiki/Timeline%20of%20mathematics | This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.
Rhetorical stage
Before 1000 BC
ca. 70,000 BC – South Africa, ochre rocks adorned with scratched geometric patterns (see Blombos Cave).
ca. 35,000 BC to 20,000 BC – Africa and France, earliest known prehistoric attempts to quantify time (see Lebombo bone).
c. 20,000 BC – Nile Valley, Ishango bone: possibly the earliest reference to prime numbers and Egyptian multiplication.
c. 3400 BC – Mesopotamia, the Sumerians invent the first numeral system, and a system of weights and measures.
c. 3100 BC – Egypt, earliest known decimal system allows indefinite counting by way of introducing new symbols.
c. 2800 BC – Indus Valley Civilisation on the Indian subcontinent, earliest use of decimal ratios in a uniform system of ancient weights and measures, the smallest unit of measurement used is 1.704 millimetres and the smallest unit of mass used is 28 grams.
2700 BC – Egypt, precision surveying.
2400 BC – Egypt, precise astronomical calendar, used even in the Middle Ages for its mathematical regularity.
c. 2000 BC – Mesopotamia, the Babylonians use a base-60 positional numeral system, and compute the first known approximate value of π at 3.125.
c. 2000 BC – Scotland, carved stone balls exhibit a variety of symmetries including all of the symmetries of Platonic solids, though it is not known if this was deliberate.
1800 BC – Egypt, Moscow Mathematical Papyrus, finding the volume of a frustum.
c. 1800 BC – Berlin Papyrus 6619 (Egypt, 19th dynasty) contains a quadratic equation and its solution.
1650 BC – Rhind Mathematical Papyrus, copy of a lost scroll from around 1850 BC, the scribe Ahmes presents one of the first known approximate values of π at 3.16, the first attempt at squaring the circle, earliest known use of a sort of cotangent, and knowledge of solving first order linear equations.
The earliest recorded use of combinatorial techniques comes from problem 79 of the Rhind papyrus which dates to the 16th century BCE.
Syncopated stage
1st millennium BC
c. 1000 BC – Simple fractions used by the Egyptians. However, only unit fractions are used (i.e., those with 1 as the numerator) and interpolation tables are used to approximate the values of the other fractions.
first half of 1st millennium BC – Vedic India – Yajnavalkya, in his Shatapatha Brahmana, describes the motions of the Sun and the Moon, and advances a 95-year cycle to synchronize the motions of the Sun and the Moon.
800 BC – Baudhayana, author of the Baudhayana Shulba Sutra, a Vedic Sanskrit geometric text, cont |
https://en.wikipedia.org/wiki/2006%20Djurg%C3%A5rdens%20IF%20season | Djurgården competed in the 2006 season in the Allsvenskan, Svenska Cupen and UEFA Champions League
Squad information
Squad
Player statistics
Appearances for competitive matches only.
|}
Goals
Total
Allsvenskan
Svenska Cupen
Champions League
Competitions
Overall
Allsvenskan
League table
Results summary
Matches
Hammarby-Djurgården ended in the beginning of second half because fan riots from the Hammarby supporters. The result was 3-0 and it became also the result of the game. Hammarby also lost 6 points because of this.
Svenska Cupen
Champions League
2nd qualifying round
Ružomberok won 3–2 on aggregate.
Royal League
Group stage
Quarter-finals
Semifinals
Friendlies
Djurgårdens IF Fotboll seasons
Djurgarden |
https://en.wikipedia.org/wiki/Lonely%20runner%20conjecture | In number theory, specifically the study of Diophantine approximation, the lonely runner conjecture is a conjecture about the long-term behavior of runners on a circular track. It states that runners on a track of unit length, with constant speeds all distinct from one another, will each be lonely at some time—at least units away from all others.
The conjecture was first posed in 1967 by German mathematician Jörg M. Wills, in purely number-theoretic terms, and independently in 1974 by T. W. Cusick; its illustrative and now-popular formulation dates to 1998. The conjecture is known to be true for 7 runners or less, but the general case remains unsolved. Implications of the conjecture include solutions to view-obstruction problems and bounds on properties, related to chromatic numbers, of certain graphs.
Formulation
Consider runners on a circular track of unit length. At the initial time , all runners are at the same position and start to run; the runners' speeds are constant, all distinct, and may be negative. A runner is said to be lonely at time if they are at a distance (measured along the circle) of at least from every other runner. The lonely runner conjecture states that each runner is lonely at some time, no matter the choice of speeds.
This visual formulation of the conjecture was first published in 1998. In many formulations, including the original by Jörg M. Wills, some simplifications are made. The runner to be lonely is stationary at 0 (with zero speed), and therefore other runners, with nonzero speeds, are considered. The moving runners may be further restricted to positive speeds only: by symmetry, runners with speeds and have the same distance from 0 at all times, and so are essentially equivalent. Proving the result for any stationary runner implies the general result for all runners, since they can be made stationary by subtracting their speed from all runners, leaving them with zero speed. The conjecture then states that, for any collection of positive, distinct speeds, there exists some time such that
where denotes the fractional part of . Interpreted visually, if the runners are running counterclockwise, the middle term of the inequality is the distance from the origin to the th runner at time , measured counterclockwise. This convention is used for the rest of this article. Wills' conjecture was part of his work in Diophantine approximation, the study of how closely fractions can approximate irrational numbers.
Implications
Suppose is a -hypercube of side length in -dimensional space (). Place a centered copy of at every point with half-integer coordinates. A ray from the origin may either miss all of the copies of , in which case there is a (infinitesimal) gap, or hit at least one copy. made an independent formulation of the lonely runner conjecture in this context; the conjecture implies that there are gaps if and only if , ignoring rays lying in one of the coordinate hyperplanes. For example, placed i |
https://en.wikipedia.org/wiki/Fr%C3%A9chet%20distance | In mathematics, the Fréchet distance is a measure of similarity between curves that takes into account the location and ordering of the points along the curves. It is named after Maurice Fréchet.
Intuitive definition
Imagine a person traversing a finite curved path while walking their dog on a leash, with the dog traversing a separate finite curved path. Each can vary their speed to keep slack in the leash, but neither can move backwards. The Fréchet distance between the two curves is the length of the shortest leash sufficient for both to traverse their separate paths from start to finish. Note that the definition is symmetric with respect to the two curves—the Fréchet distance would be the same if the dog were walking its owner.
Formal definition
Let be a metric space. A curve in is a continuous map from the unit interval into , i.e., . A reparameterization of is a continuous, non-decreasing, surjection .
Let and be two given curves in . Then, the Fréchet distance between and is defined as the infimum over all reparameterizations and of of the maximum over all of the distance in between and . In mathematical notation, the Fréchet distance is
where is the distance function of .
Informally, we can think of the parameter as "time". Then, is the position of the dog and is the position of the dog's owner at time (or vice versa). The length of the leash between them at time is the distance between and . Taking the infimum over all possible reparametrizations of corresponds to choosing the walk along the given paths where the maximum leash length is minimized. The restriction that and be non-decreasing means that neither the dog nor its owner can backtrack.
The Fréchet metric takes into account the flow of the two curves because the pairs of points whose distance contributes to the Fréchet distance sweep continuously along their respective curves. This makes the Fréchet distance a better measure of similarity for curves than alternatives, such as the Hausdorff distance, for arbitrary point sets. It is possible for two curves to have small Hausdorff distance but large Fréchet distance.
The Fréchet distance and its variants find application in several problems, from morphing and handwriting recognition to protein structure alignment. Alt and Godau were the first to describe a polynomial-time algorithm to compute the Fréchet distance between two polygonal curves in Euclidean space, based on the principle of parametric search. The running time of their algorithm is for two polygonal curves with m and n segments.
The free-space diagram
An important tool for calculating the Fréchet distance of two curves is the free-space diagram, which was introduced by Alt and Godau.
The free-space diagram between two curves for a given distance threshold ε is a two-dimensional region in the parameter space that consist of all point pairs on the two curves at distance at most ε:
The Fréchet distance is at most ε if and only if th |
https://en.wikipedia.org/wiki/Acyclic%20model | In algebraic topology, a discipline within mathematics, the acyclic models theorem can be used to show that two homology theories are isomorphic. The theorem was developed by topologists Samuel Eilenberg and Saunders MacLane. They discovered that, when topologists were writing proofs to establish equivalence of various homology theories, there were numerous similarities in the processes. Eilenberg and MacLane then discovered the theorem to generalize this process.
It can be used to prove the Eilenberg–Zilber theorem; this leads to the idea of the model category.
Statement of the theorem
Let be an arbitrary category and be the category of chain complexes of -modules over some ring . Let be covariant functors such that:
for .
There are for such that has a basis in , so is a free functor.
is - and -acyclic at these models, which means that for all and all .
Then the following assertions hold:
Every natural transformation induces a natural chain map .
If are natural transformations, are natural chain maps as before and for all models , then there is a natural chain homotopy between and .
In particular the chain map is unique up to natural chain homotopy.
Generalizations
Projective and acyclic complexes
What is above is one of the earliest versions of the theorem. Another
version is the one that says that if is a complex of
projectives in an abelian category and is an acyclic
complex in that category, then any map extends to a chain map , unique up to
homotopy.
This specializes almost to the above theorem if one uses the functor category as the abelian category. Free functors are projective objects in that category. The morphisms in the functor category are natural transformations, so the constructed chain maps and homotopies are all natural. The difference is that in the above version, being acyclic is a stronger assumption than being acyclic only at certain objects.
On the other hand, the above version almost implies this version by letting a category with only one object. Then the free functor is basically just a free (and hence projective) module. being acyclic at the models (there is only one) means nothing else than that the complex is acyclic.
Acyclic classes
There is a grand theorem that unifies both of the above. Let be an abelian category (for example, or ). A class of chain complexes over will be called an acyclic class provided that:
The 0 complex is in .
The complex belongs to if and only if the suspension of does.
If the complexes and are homotopic and , then .
Every complex in is acyclic.
If is a double complex, all of whose rows are in , then the total complex of belongs to .
There are three natural examples of acyclic classes, although doubtless others exist. The first is that of homotopy contractible complexes. The second is that of acyclic complexes. In functor categories (e.g. the category of all functors from topological spaces to abelian groups), there is a class o |
https://en.wikipedia.org/wiki/Decagram%20%28geometry%29 | In geometry, a decagram is a 10-point star polygon. There is one regular decagram, containing the vertices of a regular decagon, but connected by every third point. Its Schläfli symbol is {10/3}.
The name decagram combines a numeral prefix, deca-, with the Greek suffix -gram. The -gram suffix derives from γραμμῆς (grammēs) meaning a line.
Regular decagram
For a regular decagram with unit edge lengths, the proportions of the crossing points on each edge are as shown below.
Applications
Decagrams have been used as one of the decorative motifs in girih tiles.
Isotoxal variations
An isotoxal polygon has two vertices and one edge. There are isotoxal decagram forms, which alternates vertices at two radii. Each form has a freedom of one angle. The first is a variation of a double-wound of a pentagon {5}, and last is a variation of a double-wound of a pentagram {5/2}. The middle is a variation of a regular decagram, {10/3}.
Related figures
A regular decagram is a 10-sided polygram, represented by symbol {10/n}, containing the same vertices as regular decagon. Only one of these polygrams, {10/3} (connecting every third point), forms a regular star polygon, but there are also three ten-vertex polygrams which can be interpreted as regular compounds:
{10/5} is a compound of five degenerate digons 5{2}
{10/4} is a compound of two pentagrams 2{5/2}
{10/2} is a compound of two pentagons 2{5}.
{10/2} can be seen as the 2D equivalent of the 3D compound of dodecahedron and icosahedron and 4D compound of 120-cell and 600-cell; that is, the compound of two pentagonal polytopes in their respective dual positions.
{10/4} can be seen as the two-dimensional equivalent of the three-dimensional compound of small stellated dodecahedron and great dodecahedron or compound of great icosahedron and great stellated dodecahedron through similar reasons. It has six four-dimensional analogues, with two of these being compounds of two self-dual star polytopes, like the pentagram itself; the compound of two great 120-cells and the compound of two grand stellated 120-cells. A full list can be seen at Polytope compound#Compounds with duals.
Deeper truncations of the regular pentagon and pentagram can produce intermediate star polygon forms with ten equally spaced vertices and two edge lengths that remain vertex-transitive (any two vertices can be transformed into each other by a symmetry of the figure).
See also
References
10 (number)
10 |
https://en.wikipedia.org/wiki/1%2052%20honeycomb | {{DISPLAYTITLE:1 52 honeycomb}}
In geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. It contains 142 and 151 facets, in a birectified 8-simplex vertex figure. It is the final figure in the 1k2 polytope family.
Construction
It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing the node on the end of the 2-length branch leaves the 8-demicube, 151.
Removing the node on the end of the 5-length branch leaves the 142.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 8-simplex, 052.
Related polytopes and honeycombs
See also
521 honeycomb
251 honeycomb
References
Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, (Chapter 3: Wythoff's Construction for Uniform Polytopes)
Coxeter Regular Polytopes (1963), Macmillan Company
Regular Polytopes, Third edition, (1973), Dover edition, (Chapter 5: The Kaleidoscope)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, GoogleBook
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
9-polytopes |
https://en.wikipedia.org/wiki/2%2051%20honeycomb | {{DISPLAYTITLE:2 51 honeycomb}}
In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation. It is composed of 241 polytope and 8-simplex facets arranged in an 8-demicube vertex figure. It is the final figure in the 2k1 family.
Construction
It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing the node on the short branch leaves the 8-simplex.
Removing the node on the end of the 5-length branch leaves the 241.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 8-demicube, 151.
The edge figure is the vertex figure of the vertex figure. This makes the rectified 7-simplex, 051.
Related polytopes and honeycombs
References
Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, (Chapter 3: Wythoff's Construction for Uniform Polytopes)
Coxeter Regular Polytopes (1963), Macmillan Company
Regular Polytopes, Third edition, (1973), Dover edition, (Chapter 5: The Kaleidoscope)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
9-polytopes |
https://en.wikipedia.org/wiki/Encyclopedia%20of%20Cryptography%20and%20Security | The Encyclopedia of Cryptography and Security is a comprehensive work on Cryptography for both information security professionals and experts in the fields of Computer Science, Applied Mathematics, Engineering, Information Theory, Data Encryption, etc. It consists of 460 articles in alphabetical order and is available electronically and in print. The Encyclopedia has a representative Advisory Board consisting of 18 leading international specialists.
Topics include but are not limited to authentication and identification, copy protection, cryptoanalysis and security, factorization algorithms and primality tests, cryptographic protocols, key management, electronic payments and digital certificates, hash functions and MACs, elliptic curve cryptography, quantum cryptography and web security.
The style of the articles is of explanatory character and can be used for undergraduate or graduate courses.
Advisory board members
Carlisle Adams, Entrust, Inc.
Friedrich Bauer, Technische Universität München
Gerrit Bleumer, Francotyp-Postalia
Dan Boneh, Stanford University
Pascale Charpin, INRIA-Rocquencourt
Claude Crepeau, McGill University
Yvo G. Desmedt, University College London (University of London)
Grigory Kabatiansky, Institute for Information Transmission Problems
Burt Kaliski, RSA Security
Peter Landrock, University of Aarhus
Patrick Drew McDaniel, Penn State University
Alfred Menezes, University of Waterloo
David Naccache, Gemplus
Christof Paar, Ruhr-Universität Bochum
Bart Preneel, Katholieke Universiteit Leuven
Jean-Jacques Quisquater, Université Catholique de Louvain
Kazue Sako, NEC Corporation
Berry Schoenmakers, Technische Universiteit Eindhoven
References
Cryptography publications
Cryptography
Specialized encyclopedias |
https://en.wikipedia.org/wiki/Aleksandr%20Minchenkov | Aleksandr Viktorovich Minchenkov (; born 13 January 1989) is a Russian football coach and a former player. He works as a coach at the academy of FC Chertanovo Moscow.
Career statistics
Club
References
Russian men's footballers
Living people
Footballers from Moscow
1989 births
FC Lokomotiv Moscow players
Russian Premier League players
FC Mordovia Saransk players
FC Baltika Kaliningrad players
Men's association football forwards
FC Dynamo Bryansk players |
https://en.wikipedia.org/wiki/Vertex%20enumeration%20problem | In mathematics, the vertex enumeration problem for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry, is the problem of determination of the object's vertices given some formal representation of the object. A classical example is the problem of enumeration of the vertices of a convex polytope specified by a set of linear inequalities:
where A is an m×n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants. The inverse (dual) problem of finding the bounding inequalities given the vertices is called facet enumeration (see convex hull algorithms).
Computational complexity
The computational complexity of the problem is a subject of research in computer science. For unbounded polyhedra, the problem is known to be NP-hard, more precisely, there is no algorithm that runs in polynomial time in the combined input-output size, unless P=NP.
A 1992 article by David Avis and Komei Fukuda presents a reverse-search algorithm which finds the v vertices of a polytope defined by a nondegenerate system of n inequalities in d dimensions (or, dually, the v facets of the convex hull of n points in d dimensions, where each facet contains exactly d given points) in time O(ndv) and space O(nd). The v vertices in a simple arrangement of n hyperplanes in d dimensions can be found in O(n2dv) time and O(nd) space complexity. The Avis–Fukuda algorithm adapted the criss-cross algorithm for oriented matroids.
Notes
References
Geometric algorithms
Linear programming
Polyhedral combinatorics
Polyhedra
Discrete geometry
Enumerative combinatorics
Mathematical problems
Computational geometry |
https://en.wikipedia.org/wiki/One-seventh%20area%20triangle | In plane geometry, a triangle ABC contains a triangle having one-seventh of the area of ABC, which is formed as follows: the sides of this triangle lie on cevians p, q, r where
p connects A to a point on BC that is one-third the distance from B to C,
q connects B to a point on CA that is one-third the distance from C to A,
r connects C to a point on AB that is one-third the distance from A to B.
The proof of the existence of the one-seventh area triangle follows from the construction of six parallel lines:
two parallel to p, one through C, the other through q.r
two parallel to q, one through A, the other through r.p
two parallel to r, one through B, the other through p.q.
The suggestion of Hugo Steinhaus is that the (central) triangle with sides p,q,r be reflected in its sides and vertices. These six extra triangles partially cover ABC, and leave six overhanging extra triangles lying outside ABC. Focusing on the parallelism of the full construction (offered by Martin Gardner through James Randi’s on-line magazine), the pair-wise congruences of overhanging and missing pieces of ABC is evident. As seen in the graphical solution, six plus the original equals the whole triangle ABC.
An early exhibit of this geometrical construction and area computation was given by Robert Potts in 1859 in his Euclidean geometry textbook.
According to Cook and Wood (2004), this triangle puzzled Richard Feynman in a dinner conversation; they go on to give four different proofs.
A more general result is known as Routh's theorem.
References
H. S. M. Coxeter (1969) Introduction to Geometry, page 211, John Wiley & Sons.
Objects defined for a triangle
Articles containing proofs
Area
Affine geometry |
https://en.wikipedia.org/wiki/Weakly%20symmetric%20space | In mathematics, a weakly symmetric space is a notion introduced by the Norwegian mathematician Atle Selberg in the 1950s as a generalisation of symmetric space, due to Élie Cartan. Geometrically the spaces are defined as complete Riemannian manifolds such that any two points can be exchanged by an isometry, the symmetric case being when the isometry is required to have period two. The classification of weakly symmetric spaces relies on that of periodic automorphisms of complex semisimple Lie algebras. They provide examples of Gelfand pairs, although the corresponding theory of spherical functions in harmonic analysis, known for symmetric spaces, has not yet been developed.
References
Differential geometry
Riemannian geometry
Lie groups
Homogeneous spaces
Harmonic analysis |
https://en.wikipedia.org/wiki/Special%20linear%20Lie%20algebra | In mathematics, the special linear Lie algebra of order n (denoted or ) is the Lie algebra of matrices with trace zero and with the Lie bracket . This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras. The Lie group that it generates is the special linear group.
Applications
The Lie algebra is central to the study of special relativity, general relativity and supersymmetry: its fundamental representation is the so-called spinor representation, while its adjoint representation generates the Lorentz group SO(3,1) of special relativity.
The algebra plays an important role in the study of chaos and fractals, as it generates the Möbius group SL(2,R), which describes the automorphisms of the hyperbolic plane, the simplest Riemann surface of negative curvature; by contrast, SL(2,C) describes the automorphisms of the hyperbolic 3-dimensional ball.
Representation theory
Representation theory of sl2ℂ
The Lie algebra is a three-dimensional complex Lie algebra. Its defining feature is that it contains a basis satisfying the commutation relations
, , and .
This is a Cartan-Weyl basis for .
It has an explicit realization in terms of two-by-two complex matrices with zero trace:
, , .
This is the fundamental or defining representation for .
The Lie algebra can be viewed as a subspace of its universal enveloping algebra and, in , there are the following commutator relations shown by induction:
,
.
Note that, here, the powers , etc. refer to powers as elements of the algebra U and not matrix powers. The first basic fact (that follows from the above commutator relations) is:
From this lemma, one deduces the following fundamental result:
The first statement is true since either is zero or has -eigenvalue distinct from the eigenvalues of the others that are nonzero. Saying is a -weight vector is equivalent to saying that it is simultaneously an eigenvector of ; a short calculation then shows that, in that case, the -eigenvalue of is zero: . Thus, for some integer , and in particular, by the early lemma,
which implies that . It remains to show is irreducible. If is a subrepresentation, then it admits an eigenvector, which must have eigenvalue of the form ; thus is proportional to . By the preceding lemma, we have is in and thus .
As a corollary, one deduces:
If has finite dimension and is irreducible, then -eigenvalue of v is a nonnegative integer and has a basis .
Conversely, if the -eigenvalue of is a nonnegative integer and is irreducible, then has a basis ; in particular has finite dimension.
The beautiful special case of shows a general way to find irreducible representations of Lie algebras. Namely, we divide the algebra to three subalgebras "h" (the Cartan Subalgebra), "e", and "f", which behave approximately like their namesakes in . Namely, in an irreducible representation, we have a "highest" eigenvector of "h", on which "e" acts by zero. The basis of the irreduci |
https://en.wikipedia.org/wiki/Song%20Tae-lim | Song Tae-Lim (, Hanja: 宋泰林; born 20 February 1984) is a South Korean football defender, who currently plays for Shenyang Dongjin in China League One.
Club career statistics
Last update: 28 April 2010
Interntiaonl goals
Henan Jianye
External links
1984 births
Living people
South Korean men's footballers
South Korean expatriate men's footballers
Jeonnam Dragons players
Busan IPark players
Henan F.C. players
Goyang KB Kookmin Bank FC players
Shenyang Dongjin F.C. players
Chinese Super League players
China League One players
K League 1 players
Korea National League players
Expatriate men's footballers in China
South Korean expatriate sportspeople in China
Chung-Ang University alumni
Men's association football midfielders
People from Geoje
Footballers from South Gyeongsang Province |
https://en.wikipedia.org/wiki/Linear%20inequality | In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality:
< less than
> greater than
≤ less than or equal to
≥ greater than or equal to
≠ not equal to
A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign.
Linear inequalities of real numbers
Two-dimensional linear inequalities
Two-dimensional linear inequalities are expressions in two variables of the form:
where the inequalities may either be strict or not. The solution set of such an inequality can be graphically represented by a half-plane (all the points on one "side" of a fixed line) in the Euclidean plane. The line that determines the half-planes (ax + by = c) is not included in the solution set when the inequality is strict. A simple procedure to determine which half-plane is in the solution set is to calculate the value of ax + by at a point (x0, y0) which is not on the line and observe whether or not the inequality is satisfied.
For example, to draw the solution set of x + 3y < 9, one first draws the line with equation x + 3y = 9 as a dotted line, to indicate that the line is not included in the solution set since the inequality is strict. Then, pick a convenient point not on the line, such as (0,0). Since 0 + 3(0) = 0 < 9, this point is in the solution set, so the half-plane containing this point (the half-plane "below" the line) is the solution set of this linear inequality.
Linear inequalities in general dimensions
In Rn linear inequalities are the expressions that may be written in the form
or
where f is a linear form (also called a linear functional), and b a constant real number.
More concretely, this may be written out as
or
Here are called the unknowns, and are called the coefficients.
Alternatively, these may be written as
or
where g is an affine function.
That is
or
Note that any inequality containing a "greater than" or a "greater than or equal" sign can be rewritten with a "less than" or "less than or equal" sign, so there is no need to define linear inequalities using those signs.
Systems of linear inequalities
A system of linear inequalities is a set of linear inequalities in the same variables:
Here are the unknowns, are the coefficients of the system, and are the constant terms.
This can be concisely written as the matrix inequality
where A is an m×n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants.
In the above systems both strict and non-strict inequalities may be used.
Not all systems of linear inequalities have solutions.
Variables can be eliminated from systems of linear inequalities using Fourier–Motzkin elimination.
Applications
Polyhedra
The set of solutions of a real linear inequality constitutes a half-space of the 'n'-dimensional real space, one of the two defined by the corresponding linear equation.
The set of solutions of a |
https://en.wikipedia.org/wiki/Clausen%27s%20formula | In mathematics, Clausen's formula, found by , expresses the square of a Gaussian hypergeometric series as a generalized hypergeometric series. It states
In particular it gives conditions for a hypergeometric series to be positive. This can be used to prove
several inequalities, such as the Askey–Gasper inequality used in the proof of de Branges's theorem.
References
For a detailed proof of Clausen's formula:
Special functions |
https://en.wikipedia.org/wiki/Askey%E2%80%93Wilson%20polynomials | In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomials introduced by as q-analogs of the Wilson polynomials. They include many of the other orthogonal polynomials in 1 variable as special or limiting cases, described in the Askey scheme. Askey–Wilson polynomials are the special case of Macdonald polynomials (or Koornwinder polynomials) for the non-reduced affine root system of type (), and their 4 parameters , , , correspond to the 4 orbits of roots of this root system.
They are defined by
where is a basic hypergeometric function, , and is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values of .
Proof
This result can be proven since it is known that
and using the definition of the q-Pochhammer symbol
which leads to the conclusion that it equals
See also
Askey scheme
References
Q-analogs
Hypergeometric functions
Orthogonal polynomials |
https://en.wikipedia.org/wiki/2%2022%20honeycomb | {{DISPLAYTITLE:2 22 honeycomb}}
In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32,2}. It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex.
Its vertex arrangement is the E6 lattice, and the root system of the E6 Lie group so it can also be called the E6 honeycomb.
Construction
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 6-dimensional space.
The facet information can be extracted from its Coxeter–Dynkin diagram, .
Removing a node on the end of one of the 2-node branches leaves the 221, its only facet type,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 122, .
The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t2{34}, .
The face figure is the vertex figure of the edge figure, here being a triangular duoprism, {3}×{3}, .
Kissing number
Each vertex of this tessellation is the center of a 5-sphere in the densest known packing in 6 dimensions, with kissing number 72, represented by the vertices of its vertex figure 122.
E6 lattice
The 222 honeycomb's vertex arrangement is called the E6 lattice.
The E62 lattice, with [[3,3,32,2]] symmetry, can be constructed by the union of two E6 lattices:
∪
The E6* lattice (or E63) with [[3,32,2,2]] symmetry. The Voronoi cell of the E6* lattice is the rectified 122 polytope, and the Voronoi tessellation is a bitruncated 222 honeycomb. It is constructed by 3 copies of the E6 lattice vertices, one from each of the three branches of the Coxeter diagram.
∪ ∪ = dual to .
Geometric folding
The group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional 16-cell honeycomb.
Related honeycombs
The 222 honeycomb is one of 127 uniform honeycombs (39 unique) with symmetry. 24 of them have doubled symmetry [[3,3,32,2]] with 2 equally ringed branches, and 7 have sextupled (3!) symmetry [[3,32,2,2]] with identical rings on all 3 branches. There are no regular honeycombs in the family since its Coxeter diagram a nonlinear graph, but the 222 and birectified 222 are isotopic, with only one type of facet: 221, and rectified 122 polytopes respectively.
Birectified 222 honeycomb
The birectified 222 honeycomb , has rectified 1 22 polytope facets, , and a proprism {3}×{3}×{3} vertex figure.
Its facets are centered on the vertex arrangement of E6* lattice, as:
∪ ∪
Construction
The facet information can be extracted from its Coxeter–Dynkin diagram, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes a proprism {3}×{3}×{3}, .
Removing a node on the end of one of the 3-node branches leaves the 122, its only facet type, .
Removing a second end node defines 2 types of 5-faces: birectified 5-simplex, 022 and birectified 5- |
https://en.wikipedia.org/wiki/Wilson%20polynomials | In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by
that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials.
They are defined in terms of the generalized hypergeometric function and the Pochhammer symbols by
See also
Askey–Wilson polynomials are a q-analogue of Wilson polynomials.
References
Hypergeometric functions
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Hahn%20polynomials | In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 and rediscovered by Wolfgang Hahn . The Hahn class is a name for special cases of Hahn polynomials, including Hahn polynomials, Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials. Sometimes the Hahn class is taken to include limiting cases of these polynomials, in which case it also includes the classical orthogonal polynomials.
Hahn polynomials are defined in terms of generalized hypergeometric functions by
give a detailed list of their properties.
If , these polynomials are identical to the discrete Chebyshev polynomials except for a scale factor.
Closely related polynomials include the dual Hahn polynomials Rn(x;γ,δ,N), the continuous Hahn polynomials pn(x,a,b, , ), and the continuous dual Hahn polynomials Sn(x;a,b,c). These polynomials all have q-analogs with an extra parameter q, such as the q-Hahn polynomials Qn(x;α,β, N;q), and so on.
Orthogonality
where δx,y is the Kronecker delta function and the weight functions are
and
.
Relation to other polynomials
Racah polynomials are a generalization of Hahn polynomials
References
Special hypergeometric functions
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Charlier%20polynomials | In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier.
They are given in terms of the generalized hypergeometric function by
where are generalized Laguerre polynomials. They satisfy the orthogonality relation
They form a Sheffer sequence related to the Poisson process, similar to how Hermite polynomials relate to the Brownian motion.
See also
Wilson polynomials, a generalization of Charlier polynomials.
References
C. V. L. Charlier (1905–1906) Über die Darstellung willkürlicher Funktionen, Ark. Mat. Astr. och Fysic 2, 20.
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Universally%20Baire%20set | In the mathematical field of descriptive set theory, a set of real numbers (or more generally a subset of the Baire space or Cantor space) is called universally Baire if it has a certain strong regularity property. Universally Baire sets play an important role in Ω-logic, a very strong logical system invented by W. Hugh Woodin and the centerpiece of his argument against the continuum hypothesis of Georg Cantor.
Definition
A subset A of the Baire space is universally Baire if it has the following equivalent properties:
For every notion of forcing, there are trees T and U such that A is the projection of the set of all branches through T, and it is forced that the projections of the branches through T and the branches through U are complements of each other.
For every compact Hausdorff space Ω, and every continuous function f from Ω to the Baire space, the preimage of A under f has the property of Baire in Ω.
For every cardinal λ and every continuous function f from λω to the Baire space, the preimage of A under f has the property of Baire.
References
Descriptive set theory |
https://en.wikipedia.org/wiki/Gosper%27s%20algorithm | In mathematics, Gosper's algorithm, due to Bill Gosper, is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has a(1) + ... + a(n) = S(n) − S(0), where S(n) is a hypergeometric term (i.e., S(n + 1)/S(n) is a rational function of n); then necessarily a(n) is itself a hypergeometric term, and given the formula for a(n) Gosper's algorithm finds that for S(n).
Outline of the algorithm
Step 1: Find a polynomial p such that, writing b(n) = a(n)/p(n), the ratio b(n)/b(n − 1) has the form q(n)/r(n) where q and r are polynomials and no q(n) has a nontrivial factor with r(n + j) for j = 0, 1, 2, ... . (This is always possible, whether or not the series is summable in closed form.)
Step 2: Find a polynomial ƒ such that S(n) = q(n + 1)/p(n) ƒ(n) a(n). If the series is summable in closed form then clearly a rational function ƒ with this property exists; in fact it must always be a polynomial, and an upper bound on its degree can be found. Determining ƒ (or finding that there is no such ƒ) is then a matter of solving a system of linear equations.
Relationship to Wilf–Zeilberger pairs
Gosper's algorithm can be used to discover Wilf–Zeilberger pairs, where they exist. Suppose that F(n + 1, k) − F(n, k) = G(n, k + 1) − G(n, k) where F is known but G is not. Then feed a(k) := F(n + 1, k) − F(n, k) into Gosper's algorithm. (Treat this as a function of k whose coefficients happen to be functions of n rather than numbers; everything in the algorithm works in this setting.) If it successfully finds S(k) with S(k) − S(k − 1) = a(k), then we are done: this is the required G. If not, there is no such G.
Definite versus indefinite summation
Gosper's algorithm finds (where possible) a hypergeometric closed form for the indefinite sum of hypergeometric terms. It can happen that there is no such closed form, but that the sum over all n, or some particular set of values of n, has a closed form. This question is only meaningful when the coefficients are themselves functions of some other variable. So, suppose a(n,k) is a hypergeometric term in both n and k: that is, a(n, k)/a(n − 1,k) and a(n, k)/a(n, k − 1) are rational functions of n and k. Then Zeilberger's algorithm and Petkovšek's algorithm may be used to find closed forms for the sum over k of a(n, k).
History
Bill Gosper discovered this algorithm in the 1970s while working on the Macsyma computer algebra system at SAIL and MIT.
References
Computer algebra
Hypergeometric functions |
https://en.wikipedia.org/wiki/Igor%20Shevchenko | Igor Vadimovich Shevchenko (; born 2 February 1985) is a Russian former footballer. He played as a left midfielder.
Career statistics
Club
Notes
External links
Player page on the official Luch-Energiya website
1985 births
Footballers from Samara, Russia
Living people
Russian men's footballers
Russia men's under-21 international footballers
PFC Krylia Sovetov Samara players
FC Fakel Voronezh players
FC Anzhi Makhachkala players
FC Luch Vladivostok players
FC Akhmat Grozny players
FC Kuban Krasnodar players
FC Sibir Novosibirsk players
Russian Premier League players
FC Zhemchuzhina-Sochi players
FC Torpedo Moscow players
FC Ufa players
FC Arsenal Tula players
Men's association football midfielders
FC Yenisey Krasnoyarsk players |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20France | In the NUTS (Nomenclature of Territorial Units for Statistics) codes of France (FR), the three levels are:
History
Up until 2016, the first level NUTS regions of France consisted of Ile de France, Bassin Parisien, Nord-Pas-de-Calais, Est, Ouest, Sud-Ouest, Centre-Est,
Mediterranee and the Departement d'Outre Mer. The Departement d'Outre Mer consisted of all the overseas departments of France, while the remaining eight statistical regions were made up of the 22 regions of France.
A law passed in 2014 by the French parliament reduced the number of metropolitan regions in the country from 22 to 13. The decrease took effect from 1 January 2016. As a result of these changes in the regions of France, the first level NUTS statistical regions were altered to reflect the changes. The number of first level regions was increased to 14 so that each of the 13 metropolitan regions of France became a separate first level statistical region.
Départements d'outre mer
Although the Départements d'outre mer, as integrated departments of France, have always been a part of the European Union and its predecessors, they were only officially included as a permanent NUTS statistical area of France in 1989. When they did appear in previous statistics the first and second level NUTS areas where one and the same, from 1989, Guadeloupe, Martinique, French Guiana and La Réunion were designated as separate level 2 regions.
NUTS codes
Local administrative units
Below the NUTS levels, the two LAU (Local Administrative Units) levels are:
The LAU codes of France can be downloaded here: See also
Subdivisions of France
ISO 3166-2 codes of France
FIPS region codes of France
Notes
References
Sources
FRANC - NUTS level 1 to 3 (Commission Regulation (EU) 2016/2066, 21 November 2016), Official Journal of the European Union, pdf, page 26
List of current NUTS codes (2016) with previously used codes, simap.ted.europa.eu History of NUTS, ec.europa.eu/eurostat Overview map of EU Countries - NUTS level 1 (2010) Correspondence between the NUTS levels and the national administrative units (2016), ec.europa.eu/eurostat''
France
Nuts |
https://en.wikipedia.org/wiki/3%2031%20honeycomb | {{DISPLAYTITLE:3 31 honeycomb}}
In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.
Construction
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing the node on the short branch leaves the 6-simplex facet:
Removing the node on the end of the 3-length branch leaves the 321 facet:
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 231 polytope.
The edge figure is determined by removing the ringed node and ringing the neighboring node. This makes 6-demicube (131).
The face figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-simplex (031).
The cell figure is determined by removing the ringed node of the face figure and ringing the neighboring nodes. This makes tetrahedral prism {}×{3,3}.
Kissing number
Each vertex of this tessellation is the center of a 6-sphere in the densest known packing in 7 dimensions; its kissing number is 126, represented by the vertices of its vertex figure 231.
E7 lattice
The 331 honeycomb's vertex arrangement is called the E7 lattice.
contains as a subgroup of index 144. Both and can be seen as affine extension from from different nodes:
The E7 lattice can also be expressed as a union of the vertices of two A7 lattices, also called A72:
= ∪
The E7* lattice (also called E72) has double the symmetry, represented by [[3,33,3]]. The Voronoi cell of the E7* lattice is the 132 polytope, and voronoi tessellation the 133 honeycomb. The E7* lattice is constructed by 2 copies of the E7 lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A7* lattices, also called A74:
∪ = ∪ ∪ ∪ = dual of .
Related honeycombs
It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.
See also
8-polytope
133 honeycomb
References
H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, (Chapter 3: Wythoff's Construction for Uniform Polytopes)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, GoogleBook
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
R. T. Worley, The Voronoi Region of E7*. SIAM J. Discrete Math., 1.1 (1988), 134-141.
p124-125, 8.2 The 7-dimensinoal lattices: E7 and E7*
8-polytopes |
https://en.wikipedia.org/wiki/1%2033%20honeycomb | {{DISPLAYTITLE:1 33 honeycomb}}
In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol {3,33,3}, and is composed of [[1 32 polytope|132]] facets.
Construction
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing a node on the end of one of the 3-length branch leaves the 132, its only facet type.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex, 033.
The edge figure is determined by removing the ringed nodes of the vertex figure and ringing the neighboring node. This makes the tetrahedral duoprism, {3,3}×{3,3}.
Kissing number
Each vertex of this polytope corresponds to the center of a 6-sphere in a moderately dense sphere packing, in which each sphere is tangent to 70 others; the best known for 7 dimensions (the kissing number) is 126.
Geometric folding
The group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.
E7* lattice
contains as a subgroup of index 144. Both and can be seen as affine extension from from different nodes:
The E7* lattice (also called E72) has double the symmetry, represented by [[3,33,3]]. The Voronoi cell of the E7* lattice is the 132 polytope, and voronoi tessellation the 133 honeycomb. The E7* lattice is constructed by 2 copies of the E7 lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A7* lattices, also called A74:
∪ = ∪ ∪ ∪ = dual of .
Related polytopes and honeycombs
The 133 is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The final is a noncompact hyperbolic honeycomb, 134.
Rectified 133 honeycomb
The rectified 133 or 0331, Coxeter diagram has facets and , and vertex figure .
See also
8-polytope
331 honeycomb
Notes
References
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, (Chapter 3: Wythoff's Construction for Uniform Polytopes)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3–45]
8-polytopes |
https://en.wikipedia.org/wiki/Aleksandr%20Dantsev | Aleksandr Alekseyevich Dantsev (; born 14 October 1984) is a Russian football coach and a former player who played as a left-back. He is the manager of Zenit Penza.
Career statistics
Club
Notes
External links
1984 births
People from Kamensk-Shakhtinsky
Living people
Russian men's footballers
Russia men's under-21 international footballers
Men's association football midfielders
FC Rostov players
FC Khimki players
FC Luch Vladivostok players
Russian Premier League players
FC Ural Yekaterinburg players
Footballers from Rostov Oblast |
https://en.wikipedia.org/wiki/Racah%20polynomials | In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients.
The Racah polynomials were first defined by and are given by
Orthogonality
when ,
where is the Racah polynomial,
is the Kronecker delta function and the weight functions are
and
is the Pochhammer symbol.
Rodrigues-type formula
where is the backward difference operator,
Generating functions
There are three generating functions for
when or
when or
when or
Connection formula for Wilson polynomials
When
where are Wilson polynomials.
q-analog
introduced the q-Racah polynomials defined in terms of basic hypergeometric functions by
They are sometimes given with changes of variables as
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Q-Racah%20polynomials | In mathematics, the q-Racah polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties.
Definition
The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by
They are sometimes given with changes of variables as
Relation to other polynomials
q-Racah polynomials→Racah polynomials
References
Orthogonal polynomials
Q-analogs
Special hypergeometric functions |
https://en.wikipedia.org/wiki/Hobby%E2%80%93Rice%20theorem | In mathematics, and in particular the necklace splitting problem, the Hobby–Rice theorem is a result that is useful in establishing the existence of certain solutions. It was proved in 1965 by Charles R. Hobby and John R. Rice; a simplified proof was given in 1976 by A. Pinkus.
The theorem
Define a partition of the interval [0,1] as a division of the interval into subintervals by as an increasing sequence of numbers:
Define a signed partition as a partition in which each subinterval has an associated sign :
The Hobby–Rice theorem says that for every n continuously integrable functions:
there exists a signed partition of [0,1] such that:
(in other words: for each of the n functions, its integral over the positive subintervals equals its integral over the negative subintervals).
Application to fair division
The theorem was used by Noga Alon in the context of necklace splitting in 1987.
Suppose the interval [0,1] is a cake. There are n partners and each of the n functions is a value-density function of one partner. We want to divide the cake into two parts such that all partners agree that the parts have the same value. This fair-division challenge is sometimes referred to as the consensus-halving problem. The Hobby–Rice theorem implies that this can be done with n cuts.
References
Theorems in measure theory
Fair division
Combinatorics on words
Theorems in analysis |
https://en.wikipedia.org/wiki/Koornwinder%20polynomials | In mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of orthogonal polynomials in several variables, introduced by Koornwinder and I. G. Macdonald, that generalize the Askey–Wilson polynomials. They are the Macdonald polynomials attached to the non-reduced affine root system of type (C, Cn), and in particular satisfy analogues of Macdonald's conjectures. In addition Jan Felipe van Diejen showed that the Macdonald polynomials associated to any classical root system can be expressed as limits or special cases of Macdonald-Koornwinder polynomials and found complete sets of concrete commuting difference operators diagonalized by them. Furthermore, there is a large class of interesting families of multivariable orthogonal polynomials associated with classical root systems which are degenerate cases of the Macdonald-Koornwinder polynomials. The Macdonald-Koornwinder polynomials have also been studied with the aid of affine Hecke algebras.
The Macdonald-Koornwinder polynomial in n variables associated to the partition λ is the unique Laurent polynomial invariant under permutation and inversion of variables, with leading monomial xλ, and orthogonal with respect to the density
on the unit torus
,
where the parameters satisfy the constraints
and (x;q)∞ denotes the infinite q-Pochhammer symbol.
Here leading monomial xλ means that μ≤λ for all terms xμ with nonzero coefficient, where μ≤λ if and only if μ1≤λ1, μ1+μ2≤λ1+λ2, …, μ1+…+μn≤λ1+…+λn.
Under further constraints that q and t are real and that a, b, c, d are real or, if complex, occur in conjugate pairs, the given density is positive.
Citations
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Paiguano | Paiguano () or Paihuano () is a small agricultural town and commune in the Elqui Province of the Coquimbo Region of Chile.
Demographics
According to the 2002 census of the National Statistics Institute, Paiguano had 4,168 inhabitants (2,145 men and 2,023 women), making the commune an entirely rural area. The population grew by 10.5% (396 persons) between the 1992 and 2002 censuses.
Administration
As a commune, Paiguano is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years.
Within the electoral divisions of Chile, Paiguano is represented in the Chamber of Deputies by Mr. Mario Bertolino (RN) and Marcelo Díaz (PS) as part of the 7th electoral district, (together with La Serena, La Higuera, Vicuña and Andacollo). The commune is represented in the Senate by Evelyn Matthei Fornet (UDI) and Jorge Pizarro Soto (PDC) as part of the 4th senatorial constituency (Coquimbo Region).
Patriotic Celebrations
Paiguano is one of the main locations to celebrate Chile's patriotic holiday on September 18.
See also
List of towns in Chile
External links
Ilustre Municipalidad de Paihuano
Turismo Valle del Elqui.
References
Geography of Coquimbo Region
Communes of Chile
Populated places in Elqui Province |
https://en.wikipedia.org/wiki/Tom%20H.%20Koornwinder | Tom H. Koornwinder (born 19 September 1943, in Rotterdam) is a Dutch mathematician at the Korteweg-de Vries Institute for Mathematics who introduced Koornwinder polynomials.
See also
Askey–Bateman project
References
Curriculum Vitae
home page
brief bio
1943 births
Living people
20th-century Dutch mathematicians
21st-century Dutch mathematicians
Leiden University alumni
University of Amsterdam alumni
Academic staff of the University of Amsterdam
Scientists from Rotterdam |
https://en.wikipedia.org/wiki/Narvik%20S%C4%B1rxayev | Narvik Zagidinoviç Sırxayev (; born 16 March 1974) is a former Soviet and Azerbaijani footballer and a football coach of Lezgin origin. He also holds Russian citizenship.
National team statistics
International goals
Honors
Russian Super Cup: 2003
References
1974 births
People from Suleyman-Stalsky District
Azerbaijani people of Lezgian descent
Living people
Soviet men's footballers
Azerbaijani men's footballers
Azerbaijani expatriate men's footballers
Azerbaijan men's international footballers
Russian Premier League players
FC Anzhi Makhachkala players
FC Lokomotiv Moscow players
FC Moscow players
FC Akhmat Grozny players
Expatriate men's footballers in Russia
Azerbaijani football managers
Soviet Azerbaijani people
Men's association football midfielders
FC Dynamo Makhachkala players
Sportspeople from Dagestan |
https://en.wikipedia.org/wiki/Q-Charlier%20polynomials | In mathematics, the q-Charlier polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition
The polynomials are given in terms of the basic hypergeometric function by
References
Orthogonal polynomials
Q-analogs
Special hypergeometric functions |
https://en.wikipedia.org/wiki/Meixner%20polynomials | In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by . They are given in terms of binomial coefficients and the (rising) Pochhammer symbol by
See also
Kravchuk polynomials
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Beno%20Eckmann | Beno Eckmann (31 March 1917 – 25 November 2008) was a Swiss mathematician who made contributions to algebraic topology, homological algebra, group theory, and differential geometry.
Life
Born in Bern, Eckmann received his master's degree from Eidgenössische Technische Hochschule Zürich (ETH) in 1939. Later, he studied there under Heinz Hopf, obtaining his Ph.D. in 1941. Eckmann was the 2008 recipient of the Albert Einstein Medal.
Legacy
Calabi–Eckmann manifolds, Eckmann–Hilton duality, the Eckmann–Hilton argument, and the Eckmann–Shapiro lemma are named after Eckmann.
Family
Eckmann's son is mathematical physicist Jean-Pierre Eckmann.
References
External links
Biography of Beno Eckmann
1917 births
2008 deaths
People from Bern
20th-century Swiss mathematicians
21st-century Swiss mathematicians
ETH Zurich alumni
Academic staff of ETH Zurich
Topologists
Albert Einstein Medal recipients |
https://en.wikipedia.org/wiki/M%C3%B6bius%20configuration | In geometry, the Möbius configuration or Möbius tetrads is a certain configuration in Euclidean space or projective space, consisting of two tetrahedra that are mutually inscribed: each vertex of one tetrahedron lies on a face plane of the other tetrahedron and vice versa. Thus, for the resulting system of eight points and eight planes, each point lies on four planes (the three planes defining it as a vertex of a tetrahedron and the fourth plane from the other tetrahedron that it lies on), and each plane contains four points (the three tetrahedron vertices of its face, and the vertex from the other tetrahedron that lies on it).
Möbius's theorem
The configuration is named after August Ferdinand Möbius, who in 1828 proved that, if two tetrahedra have the property that seven of their vertices lie on corresponding face planes of the other tetrahedron, then the eighth vertex also lies on the plane of its corresponding face, forming a configuration of this type. This incidence theorem is true more generally in a three-dimensional projective space if and only if Pappus's theorem holds for that space (Reidemeister, Schönhardt), and it is true for a three-dimensional space modeled on a division ring if and only if the ring satisfies the commutative law and is therefore a field (Al-Dhahir). By projective duality, Möbius' result is equivalent to the statement that, if seven of the eight face planes of two tetrahedra contain the corresponding vertices of the other tetrahedron, then the eighth face plane also contains the same vertex.
Construction
describes a simple construction for the configuration. Beginning with an arbitrary point p in Euclidean space, let A, B, C, and D be four planes through p, no three of which share a common intersection line, and place the six points q, r, s, t, u, and v on the six lines formed by pairwise intersection of these planes in such a way that no four of these points are coplanar. For each of the planes A, B, C, and D, four of the seven points p, q, r, s, t, u, and v lie on that plane and three are disjointed from it; form planes A’, B’, C’, and D’ through the triples of points disjoint from A, B, C, and D respectively. Then, by the dual form of Möbius' theorem, these four new planes meet in a single point w. The eight points p, q, r, s, t, u, v, and w and the eight planes A, B, C, D, A’, B’, C’, and D’ form an instance of Möbius' configuration.
Related constructions
state (without references) that there are five configurations having eight points and eight planes with four points on every plane and four planes through every point that are realisable in three-dimensional
Euclidean space: such configurations have the shorthand notation . They must have obtained their information from the article by . This actually states, depending upon results by , , and , that there are five configurations with the property that at most two planes have two points in common, and dually at most two points are common to two planes. (T |
https://en.wikipedia.org/wiki/Abstract%20algebra | In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in pedagogy.
Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory gives a unified framework to study properties and constructions that are similar for various structures.
Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the variety of groups.
History
Before the nineteenth century, algebra was defined as the study of polynomials. Abstract algebra came into existence during the nineteenth century as more complex problems and solution methods developed. Concrete problems and examples came from number theory, geometry, analysis, and the solutions of algebraic equations. Most theories that are now recognized as parts of abstract algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. This unification occurred in the early decades of the 20th century and resulted in the formal axiomatic definitions of various algebraic structures such as groups, rings, and fields. This historical development is almost the opposite of the treatment found in popular textbooks, such as van der Waerden's Moderne Algebra, which start each chapter with a formal definition of a structure and then follow it with concrete examples.
Elementary algebra
The study of polynomial equations or algebraic equations has a long history. , the Babylonians were able to solve quadratic equations specified as word problems. This word problem stage is classified as rhetorical algebra and was the dominant approach up to the 16th century. Muhammad ibn Mūsā al-Khwārizmī originated the word "algebra" in 830 AD, but his work was entirely rhetorical algebra. Fully symbolic algebra did not appear until François Viète's 1591 New Algebra, and even this had some spelled out words that were given symbols in Descartes's 1637 La Géométrie. The formal study of solving symbolic equations led Leonhard Euler to accept what were then considered "nonsense" roots such as negative numbers and imaginary numbers, in the late 18th century. However, European mathematicians, for the most part, resisted these co |
https://en.wikipedia.org/wiki/K-Poincar%C3%A9%20group | In physics and mathematics, the κ-Poincaré group, named after Henri Poincaré, is a quantum group, obtained by deformation of the Poincaré group into a Hopf algebra.
It is generated by the elements and with the usual constraint:
where is the Minkowskian metric:
The commutation rules reads:
In the (1 + 1)-dimensional case the commutation rules between and are particularly simple. The Lorentz generator in this case is:
and the commutation rules reads:
The coproducts are classical, and encode the group composition law:
Also the antipodes and the counits are classical, and represent the group inversion law and the map to the identity:
The κ-Poincaré group is the dual Hopf algebra to the K-Poincaré algebra, and can be interpreted as its “finite” version.
References
Hopf algebras
Mathematical physics |
https://en.wikipedia.org/wiki/Noureddine%20Kacemi | Noureddine Kacemi () is a Moroccan former football defender. He last played for FAR Rabat.
Kacemi played for Morocco at the 2000 Summer Olympics.
Career statistics
International goals
References
External links
1977 births
Living people
Moroccan men's footballers
Moroccan expatriate men's footballers
Morocco men's under-20 international footballers
Morocco men's international footballers
2006 Africa Cup of Nations players
Olympic footballers for Morocco
Footballers at the 2000 Summer Olympics
Raja CA players
FC Istres players
Grenoble Foot 38 players
Ligue 1 players
Ligue 2 players
AS FAR (football club) players
Expatriate men's footballers in France
Men's association football defenders
People from Mohammedia
SCC Mohammédia players |
https://en.wikipedia.org/wiki/Experiment%20%28probability%20theory%29 | In probability theory, an experiment or trial (see below) is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be random if it has more than one possible outcome, and deterministic if it has only one. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as a Bernoulli trial.
When an experiment is conducted, one (and only one) outcome results— although this outcome may be included in any number of events, all of which would be said to have occurred on that trial. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess the empirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods of statistical analysis.
Experiments and trials
Random experiments are often conducted repeatedly, so that the collective results may be subjected to statistical analysis. A fixed number of repetitions of the same experiment can be thought of as a composed experiment, in which case the individual repetitions are called trials. For example, if one were to toss the same coin one hundred times and record each result, each toss would be considered a trial within the experiment composed of all hundred tosses.
Mathematical description
A random experiment is described or modeled by a mathematical construct known as a probability space. A probability space is constructed and defined with a specific kind of experiment or trial in mind.
A mathematical description of an experiment consists of three parts:
A sample space, Ω (or S), which is the set of all possible outcomes.
A set of events , where each event is a set containing zero or more outcomes.
The assignment of probabilities to the events—that is, a function P mapping from events to probabilities.
An outcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complicated events are used to characterize groups of outcomes. The collection of all such events is a sigma-algebra . Finally, there is a need to specify each event's likelihood of happening; this is done using the probability measure function, P.
Once an experiment is designed and established, ω from the sample space Ω, all the events in that contain the selected outcome ω (recall that each event is a subset of Ω) are said to “have occurred”. The probability function P is defined in such a way that, if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would approach agreement with the values P assigns them.
As a simple experiment, we may flip a coin twice. The sample space (where the order of the two flips is relevant) is {(H, T), (T, H), (T, T), (H, H)} where "H" means "heads" and "T" means "tails". Note that each of (H, T), (T, H), ... are possible outcomes of the experiment. We m |
https://en.wikipedia.org/wiki/Barnes%20integral | In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. They were introduced by . They are closely related to generalized hypergeometric series.
The integral is usually taken along a contour which is a deformation of the imaginary axis passing to the right of all poles of factors of the form Γ(a + s) and to the left of all poles of factors of the form Γ(a − s).
Hypergeometric series
The hypergeometric function is given as a Barnes integral by
see also . This equality can be obtained by moving the contour to the right while picking up the residues at s = 0, 1, 2, ... . for , and by analytic continuation elsewhere. Given proper convergence conditions, one can relate more general Barnes' integrals and generalized hypergeometric functions pFq in a similar way .
Barnes lemmas
The first Barnes lemma states
This is an analogue of Gauss's 2F1 summation formula, and also an extension of Euler's beta integral. The integral in it is sometimes called Barnes's beta integral.
The second Barnes lemma states
where e = a + b + c − d + 1. This is an analogue of Saalschütz's summation formula.
q-Barnes integrals
There are analogues of Barnes integrals for basic hypergeometric series, and many of the other results can also be extended to this case .
References
(there is a 2008 paperback with )
Special functions
Hypergeometric functions |
https://en.wikipedia.org/wiki/Residual%20property%20%28mathematics%29 | In the mathematical field of group theory, a group is residually X (where X is some property of groups) if it "can be recovered from groups with property X".
Formally, a group G is residually X if for every non-trivial element g there is a homomorphism h from G to a group with property X such that .
More categorically, a group is residually X if it embeds into its pro-X completion (see profinite group, pro-p group), that is, the inverse limit of the inverse system consisting of all morphisms from G to some group H with property X.
Examples
Important examples include:
Residually finite
Residually nilpotent
Residually solvable
Residually free
References
Infinite group theory
Properties of groups |
https://en.wikipedia.org/wiki/Ofer%20Lellouche | Ofer Lellouche (, born 19 April 1947 in Tunis) is an Israeli painter, sculptor, etcher and video artist.
Biography
Lellouche was born in Tunisia in 1947. He studied mathematics and physics in Paris at Saint Louis College. In 1966, two months before he was scheduled to graduate, he ran away to Kibbutz Yehiam in Israel. In 1968, during his service in the Israel Defense Forces, he contracted hepatitis and began to paint while recovering. He began his formal art training at the Avni Institute of Art and Design in Tel Aviv under the abstract lyrical painter Yehezkiel Streichman.
He returned to Paris to study with the sculptor César Baldaccini (1921-1998) and earn a master's degree in literature with a thesis on Stéphane Mallarmé. In the late 1970s, he worked in video art and painted self-portraits. During the coming years, he drew and etched self-portraits, often in violent industrial colors.
In 1979, he produced several videos related to the subject of the mirror. In the early 1980s, he began painting landscapes in addition to self-portraits. His 1987 painting Figure in a Landscape" was exhibited at the 19th São Paulo Art Biennial.
In the early 1990s, Lellouche produced more than 600 etchings, illustrated Stéphane Mallarmé's poem, "Un coup de dés jamais n'abolira le hasard", and published the books "Panim" (faces) and "Ein Karem". He also produced large-format paintings, which he called the "Atelier César" in homage to his former teacher. In 1991, he returned to Paris and visited the location of César's studio, where he found some clay models on their bases and decided to make a series of works that would remind him of what he had seen. Since the late 1990s, he has been engaged primarily in sculpture and etching.
References
Ofer Lellouche: Self Portrait. Tel Aviv: Gordon Gallery, 1980.
Omer, Mordechai: Ofer Lellouche: Self-Portrait, 1977-2001. Tel Aviv: Tel Aviv Museum of Art, 2001.
Restany, Pierre: Ofer Lellouche: The Hand that Thinks. Tel Aviv: Tel Aviv Museum of Art, 2001.
Lellouche, Ofer: Between Etching and Sculpture. The Open Museum: Industrial Parks Tefen and Omer, 2005.
Ofer Lellouche: Head II. Tel Aviv: Tel Aviv Museum of Art, 2012.
Bartos, Ron: Ofer Lellouche: Nine''. Tel Aviv: Zemack Gallery, 2013.
External links
Official website
Head For Meina
Ofer Lellouche Self Portrait on a Transparent Mirror
Video interview with Ofer Lellouche on Art-in-Process.com (Hebrew, Russian translation)
Israeli Jews
Jewish sculptors
Israeli sculptors
Modern sculptors
Jewish painters
Israeli people of Tunisian-Jewish descent
Israeli printmakers
1947 births
Living people |
https://en.wikipedia.org/wiki/Choi%20Jong-hyuk | Choi Jong-Hyuk (; born 3 September 1984) is a South Korean football midfielder.
Career statistics
External links
K-League Player Record
1984 births
Living people
Men's association football midfielders
South Korean men's footballers
Daegu FC players
K League 1 players |
https://en.wikipedia.org/wiki/Mikhail%20Postnikov | Mikhail Mikhailovich Postnikov (; 27 October 1927 – 27 May 2004) was a Soviet mathematician, known for his work in algebraic and differential topology.
Biography
He was born in Shatura, near Moscow. He received his Ph.D. from Moscow State University under the direction of Lev Pontryagin, and then became a professor at this university. He died in Moscow.
Selected publications
Fundamentals of Galois theory, 1961; 142 pp. Dover reprint, 2004
The variational theory of geodesics, Translated from the Russian by Scripta Technica, Inc. Edited by Bernard R. Gelbaum. Saunders, Philadelphia, Pa., 1967; 200 pp.
Linear algebra and differential geometry. Translated from the Russian by Vladimir Shokurov. Mir Publishers, 1982; 319 pp.
Smooth manifolds, Mir Publishers, 1989; 511 pp.
Geometry VI: Riemannian Geometry, Springer, 2001, 504 pp.
French
Leçons de géométrie : Semestre I : Géométrie analytique, Éditions Mir, 1981; 279 pp.
Leçons de géométrie : Semestre II : Algèbre linéaire et géométrie différentielle, Éditions Mir, 1981; 263 pp.
Leçons de géométrie : Semestre III : Variétés différentiables, Éditions Mir, 1990; 431 pp.
Leçons de géométrie : Semestre IV : Géométrie différentielle, Éditions Mir, 1990; 439 pp.
Leçons de géométrie : Semestre V : Groupes et algèbres de Lie, Éditions Mir, 1985; 374 pp.
See also
Postnikov system
Postnikov square
References
Bibliography
1927 births
2004 deaths
20th-century Russian mathematicians
Topologists
Soviet mathematicians
Perm State University alumni
Moscow State University alumni
People from Shatursky District |
https://en.wikipedia.org/wiki/Philosophy%20of%20statistics | The philosophy of statistics involves the meaning, justification, utility, use and abuse of statistics and its methodology, and ethical and epistemological issues involved in the consideration of choice and interpretation of data and methods of statistics.
Topics of interest
Foundations of statistics involves issues in theoretical statistics, its goals and optimization methods to meet these goals, parametric assumptions or lack thereof considered in nonparametric statistics, model selection for the underlying probability distribution, and interpretation of the meaning of inferences made using statistics, related to the philosophy of probability and the philosophy of science. Discussion of the selection of the goals and the meaning of optimization, in foundations of statistics, are the subject of the philosophy of statistics. Selection of distribution models, and of the means of selection, is the subject of the philosophy of statistics, whereas the mathematics of optimization is the subject of nonparametric statistics.
David Cox makes the point that any kind of interpretation of evidence is in fact a statistical model, although it is known through Ian Hacking's work that many are ignorant of this subtlety.
Issues arise involving sample size, such as cost and efficiency, are common, such as in polling and pharmaceutical research.
Extra-mathematical considerations in the design of experiments and accommodating these issues arise in most actual experiments.
The motivation and justification of data analysis and experimental design, as part of the scientific method are considered.
Distinctions between induction and logical deduction relevant to inferences from data and evidence arise, such as when frequentist interpretations are compared with degrees of certainty derived from Bayesian inference. However, the difference between induction and ordinary reasoning is not generally appreciated.
Leo Breiman exposed the diversity of thinking in his article on 'The Two Cultures', making the point that statistics has several kinds of inference to make, modelling and prediction amongst them.
Issues in the philosophy of statistics arise throughout the history of statistics. Causality considerations arise with interpretations of, and definitions of, correlation, and in the theory of measurement.
Objectivity in statistics is often confused with truth whereas it is better understood as replicability, which then needs to be defined in the particular case. Theodore Porter develops this as being the path pursued when trust has evaporated, being replaced with criteria.
Ethics associated with epistemology and medical applications arise from potential abuse of statistics, such as selection of method or transformations of the data to arrive at different probability conclusions for the same data set. For example, the meaning of applications of a statistical inference to a single person, such as one single cancer patient, when there is no frequentist interpret |
https://en.wikipedia.org/wiki/Bilateral%20hypergeometric%20series | In mathematics, a bilateral hypergeometric series is a series Σan summed over all integers n, and such that the ratio
an/an+1
of two terms is a rational function of n. The definition of the generalized hypergeometric series is similar, except that the terms with negative n must vanish; the bilateral series will in general have infinite numbers of non-zero terms for both positive and negative n.
The bilateral hypergeometric series fails to converge for most rational functions, though it can be analytically continued to a function defined for most rational functions. There are several summation formulas giving its values for special values where it does converge.
Definition
The bilateral hypergeometric series pHp is defined by
where
is the rising factorial or Pochhammer symbol.
Usually the variable z is taken to be 1, in which case it is omitted from the notation.
It is possible to define the series pHq with different p and q in a similar way, but this either fails to converge or can be reduced to the usual hypergeometric series by changes of variables.
Convergence and analytic continuation
Suppose that none of the variables a or b are integers, so that all the terms of the series are finite and non-zero. Then the terms with n<0 diverge if |z| <1, and the terms with n>0 diverge if |z| >1, so the series cannot converge unless |z|=1. When |z|=1, the series converges if
The bilateral hypergeometric series can be analytically continued to a multivalued meromorphic function of several variables whose singularities are
branch points at z = 0 and z=1 and simple poles at ai = −1, −2,... and bi = 0, 1, 2, ...
This can be done as follows. Suppose that none of the a or b variables are integers. The terms with n positive converge for |z| <1 to a function satisfying an inhomogeneous linear equation with singularities at z = 0 and z=1, so can be continued to a multivalued function with these points as branch points. Similarly the terms with n negative converge for |z| >1 to a function satisfying an inhomogeneous linear equation with singularities at z = 0 and z=1, so can also be continued to a multivalued function with these points as branch points. The sum of these functions gives the analytic continuation of the bilateral hypergeometric series to all values of z other than 0 and 1, and satisfies a linear differential equation in z similar to the hypergeometric differential equation.
Summation formulas
Dougall's bilateral sum
This is sometimes written in the equivalent form
Bailey's formula
gave the following generalization of Dougall's formula:
where
See also
Basic bilateral hypergeometric series
References
(there is a 2008 paperback with )
Hypergeometric functions |
https://en.wikipedia.org/wiki/List%20of%20probability%20journals | This is a list of peer-reviewed scientific journals published in the field of probability.
Advances in Applied Probability
ALEA - Latin American Journal of Probability and Mathematical Statistics
Annales de l’Institut Henri Poincaré
Annals of Applied Probability
Annals of Probability
Bernoulli
Brazilian Journal of Probability and Statistics
Combinatorics, Probability and Computing
Communications on Stochastic Analysis
Electronic Communications in Probability
Electronic Journal of Probability
ESAIM: Probability and Statistics
Finance and Stochastics
Journal of Applied Probability
Journal of Theoretical Probability
Markov Processes and Related Fields
Methodology and Computing in Applied Probability
Modern Stochastics: Theory and Applications
Probability and Mathematical Statistics
Probability in the Engineering and Informational Sciences
Probability Surveys
Probability Theory and Related Fields
Queueing Systems
Random Matrices: Theory and Applications
Random Operators and Stochastic Equations
Random Structures & Algorithms
Stochastics: An International Journal of Probability and Stochastic Processes
Statistics & Probability Letters
Stochastic Analysis and Applications
Stochastics and Dynamics
Stochastic Models
Stochastic Processes and their Applications
Stochastic Systems
Theory of Probability and Its Applications
Theory of Probability and Mathematical Statistics
Theory of Stochastic Processes
See also
List of scientific journals
List of statistics journals
List of mathematics journals
Journals
Probability |
https://en.wikipedia.org/wiki/Han%20Dong-jin | Han Dong-Jin (born August 25, 1979) is a South Korean footballer. He currently plays for Jeju United.
Club career statistics
External links
1979 births
Living people
Men's association football goalkeepers
South Korean men's footballers
Jeju United FC players
Gimcheon Sangmu FC players
K League 1 players
People from Wonju
Sangji University alumni
Footballers from Gangwon Province, South Korea |
https://en.wikipedia.org/wiki/Filipinos%20in%20Indonesia | Filipinos in Indonesia were estimated to number 7,400 individuals as of 2022, according to the statistics of the Philippine government. Most are based in Jakarta, though there is also a community in Surabaya and other major cities in Indonesia. This represented growth of nearly five times over the government's 1998 estimate of 1,046 individuals.
Employment
Unlike many other overseas Filipino communities, Filipinos in Indonesia consist largely of skilled professionals, especially in the advertising industries and as teachers in international schools where their English skills are most needed. 20% also work in finance, especially as accountants.
Some Filipinos also work as fisherman on Indonesian waters. However, some have fished illegally and have faced a crackdown with the consequence of deportation by Indonesian authorities.
Inter-ethnic relations
Filipinos in Indonesia generally maintain good interethnic relations with their Indonesian neighbours, with whom they feel culturally closer than Europeans or Americans; Indonesians stereotype Filipinos as being gregarious and cheerful. However, there are fears that Filipinos in Indonesia may become the targets of kidnappings by local militant groups such as Jemaah Islamiyah in an attempt to secure the release of JI members imprisoned in Philippine jails.
Community
Filipinos in Indonesia have formed eight different community associations, including three sports teams, one teachers' association, and two Christian groups. The annual Philippine Independence Day celebrations attract numerous participants.
See also
Indonesians in the Philippines
Indonesia–Philippines relations
References
External links
Dahil Sa'Yo, a publication aimed at Filipinos in Indonesia
Indonesia
Indonesia
Immigration to Indonesia
Expatriates in Indonesia
Indonesia–Philippines relations |
https://en.wikipedia.org/wiki/Parametric%20family | In mathematics and its applications, a parametric family or a parameterized family is a family of objects (a set of related objects) whose differences depend only on the chosen values for a set of parameters.
Common examples are parametrized (families of) functions, probability distributions, curves, shapes, etc.
In probability and its applications
For example, the probability density function of a random variable may depend on a parameter . In that case, the function may be denoted to indicate the dependence on the parameter . is not a formal argument of the function as it is considered to be fixed. However, each different value of the parameter gives a different probability density function. Then the parametric family of densities is the set of functions , where denotes the parameter space, the set of all possible values that the parameter can take. As an example, the normal distribution is a family of similarly-shaped distributions parametrized by their mean and their variance.
In decision theory, two-moment decision models can be applied when the decision-maker is faced with random variables drawn from a location-scale family of probability distributions.
In algebra and its applications
In economics, the Cobb–Douglas production function is a family of production functions parametrized by the elasticities of output with respect to the various factors of production.
In algebra, the quadratic equation, for example, is actually a family of equations parametrized by the coefficients of the variable and of its square and by the constant term.
See also
Indexed family
References
Mathematical terminology
Theory of probability distributions |
https://en.wikipedia.org/wiki/Larry%20V.%20Hedges | Larry Vernon Hedges is a researcher in statistical methods for meta-analysis and evaluation of education policy. He is Professor of Statistics and Education and Social Policy, Institute for Policy Research, Northwestern University. Previously, he was the Stella M. Rowley Distinguished Service Professor of Education, Sociology, Psychology, and Public Policy Studies at the University of Chicago. He is a member of the National Academy of Education and a fellow of the American Academy of Arts and Sciences, the American Educational Research Association, the American Psychological Association, and the American Statistical Association. In 2018, he received the Yidan Prize for Education Research, the world's most prestigious and largest education prize, i.e. USD four million.
He has authored a number of articles and books on statistical methods for meta-analysis, which is the use of statistical methods for combining results from different studies. He also suggested several estimators for effect sizes and derived their properties. He carried out research on the relation of resources available to schools and student achievement, most notably the relation between class size and achievement.
Bibliography
References
External links
Living people
American statisticians
American social scientists
Northwestern University faculty
University of Chicago faculty
Stanford University alumni
Fellows of the American Statistical Association
Fellows of the American Academy of Arts and Sciences
Fellows of the American Psychological Association
Educational researchers
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Extreme%20point%20%28disambiguation%29 | An extreme point, in mathematics, is a point in a convex set which does not lie in any open line segment joining two points in the set.
Extreme point or extremal point may also refer to:
A point where some function attains its extremum
A leaf vertex of a tree in graph theory
Extreme points of Earth, points of land that extend farther in one direction than any other part of that land
Physical geography |
https://en.wikipedia.org/wiki/Hexagonal%20pyramid | In geometry, a hexagonal pyramid or hexacone is a pyramid with a hexagonal base upon which are erected six isosceles triangular faces that meet at a point (the apex). Like any pyramid, it is self-dual.
A right hexagonal pyramid with a regular hexagon base has C6v symmetry.
A right regular pyramid is one which has a regular polygon as its base and whose apex is "above" the center of the base, so that the apex, the center of the base and any other vertex form a right triangle.
Vertex coordinates
A hexagonal pyramid of edge length 1 has the following vertices:
These coordinates are a subset of the vertices of the regular triangular tiling.
Representations
A hexagonal pyramid has the following Coxeter diagrams:
ox6oo&#x (full symmetry)
ox3ox&#x (generally a ditrigonal pyramid)
Related polyhedra
See also
Bipyramid, prism and antiprism
External links
Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
Conway Notation for Polyhedra Try: "Y6"
Hexagonal pyramid - Polytope Wiki
Pyramids and bipyramids
Self-dual polyhedra
Prismatoid polyhedra |
https://en.wikipedia.org/wiki/Willard%20L.%20Miranker | Willard L. Miranker (March 8, 1932 – April 28, 2011) was an American mathematician and computer scientist, known for his contributions to applied mathematics and numerical mathematics.
Raised in Brooklyn, New York, he earned Bachelor of Arts (1952), Master of Science (1953) and Ph.D. (1956) from the Courant Institute at New York University, the latter on the thesis The Asymptotic Theory of Solutions of U + (K2)U = 0 advised by Joseph Keller.
He then worked for the mathematics department at Bell Labs (1956–1958) before joining IBM Research (1961). After retirement from IBM, he joined the computer science faculty at Yale University (1989) as research faculty.
He also held professor affiliations at California Institute of Technology (1963), Hebrew University of Jerusalem (1968), Yale University (1973), University of Paris-Sud (1974), City University of New York (1966–1967) and New York University (1970–1973).
Miranker's work includes articles and books on stiff differential equations, interval arithmetic, analog computing, and neural networks and the modeling of consciousness.
Miranker was also an accomplished and prolific painter. Over the course of his life, Willard Miranker painted ~4000 watercolors/aquarelles and ~200 oil paintings, many of which are displayed online. He exhibited internationally in New York City, Paris and Bonn.
Awards
Fellow of the American Association for the Advancement of Science
References
External links
Paintings by Will Miranker
short page at Yale University
FindaGrave entry
1932 births
2011 deaths
20th-century American mathematicians
21st-century American mathematicians
American computer scientists
Courant Institute of Mathematical Sciences alumni
Scientists at Bell Labs
People from Brooklyn
IBM employees
Yale University faculty
California Institute of Technology faculty
Academic staff of the Hebrew University of Jerusalem
Academic staff of Paris-Sud University
City University of New York faculty
Mathematicians from New York (state) |
https://en.wikipedia.org/wiki/Robert%20Osserman | Robert "Bob" Osserman (December 19, 1926 – November 30, 2011) was an American mathematician who worked in geometry. He is specially remembered for his work on the theory of minimal surfaces.
Raised in Bronx, he went to Bronx High School of Science (diploma, 1942) and New York University. He earned a Ph.D. in 1955 from Harvard University with the thesis Contributions to the Problem of Type (on Riemann surfaces) supervised by Lars Ahlfors.
He joined Stanford University in 1955. He joined the Mathematical Sciences Research Institute in 1990.
He worked on geometric function theory, differential geometry, the two integrated in a theory of minimal surfaces, isoperimetric inequality, and other issues in the areas of astronomy, geometry, cartography and complex function theory.
Osserman was the head of mathematics at Office of Naval Research, a Fulbright Lecturer at the University of Paris and Guggenheim Fellow at the University of Warwick. He edited numerous books and promoted mathematics, such as in interviews with celebrities Steve Martin and Alan Alda.
He was an invited speaker at the International Congress of Mathematicians (ICM) of 1978 in Helsinki.
He received the Lester R. Ford Award (1980) of the Mathematical Association of America for his popular science writings.
H. Blaine Lawson, David Allen Hoffman and Michael Gage were Ph.D. students of his.
Robert Osserman died on Wednesday, November 30, 2011 at his home.
Mathematical contributions
The Keller–Osserman problem
Osserman's most widely cited research article, published in 1957, dealt with the partial differential equation
He showed that fast growth and monotonicity of is incompatible with the existence of global solutions. As a particular instance of his more general result:
Osserman's method was to construct special solutions of the PDE which would facilitate application of the maximum principle. In particular, he showed that for any real number there exists a rotationally symmetric solution on some ball which takes the value at the center and diverges to infinity near the boundary. The maximum principle shows, by the monotonicity of , that a hypothetical global solution would satisfy for any and any , which is impossible.
The same problem was independently considered by Joseph Keller, who was drawn to it for applications in electrohydrodynamics. Osserman's motivation was from differential geometry, with the observation that the scalar curvature of the Riemannian metric on the plane is given by
An application of Osserman's non-existence theorem then shows:
By a different maximum principle-based method, Shiu-Yuen Cheng and Shing-Tung Yau generalized the Keller–Osserman non-existence result, in part by a generalization to the setting of a Riemannian manifold. This was, in turn, an important piece of one of their resolutions of the Calabi–Jörgens problem on rigidity of affine hyperspheres with nonnegative mean curvature.
Non-existence for the minimal surface system in hi |
https://en.wikipedia.org/wiki/Systole%20%28disambiguation%29 | Systole may refer to:
Systole (medicine), a term describing the contraction of the heart
Systolic array, a term used in computer architecture
Systolic geometry, a term used in mathematics
In mathematics, Systoles of surfaces are systolic inequalities for curves on surfaces
Also see Introduction to systolic geometry |
https://en.wikipedia.org/wiki/Sergei%20Kozko | Sergei Viktorovich Kozko (; born 12 April 1975) is a Russian football coach and a former player who is currently a goalkeeping coach with FC Rubin Kazan.
Career statistics
External links
Player page on the official FC Rubin Kazan website
1975 births
Living people
Russian men's footballers
PFC Dynamo Stavropol players
FC Moscow players
FC Rubin Kazan players
Russian Premier League players
Men's association football goalkeepers
FC Khimki players
FC Narzan Kislovodsk players |
https://en.wikipedia.org/wiki/Yevgeni%20Balyaikin | Yevgeni Viktorovich Balyaikin (; born 19 May 1988) is a Russian former footballer who played as central midfielder.
Club career
Career statistics
Notes
International career
Balyaikin was a part of the Russia U21 side that was competing in the 2011 European Under-21 Championship qualification.
References
External links
Player page on the official FC Rubin Kazan website
1988 births
People from Bratsk
Living people
Russian men's footballers
Russia men's under-21 international footballers
Russia men's B international footballers
Men's association football midfielders
FC Rubin Kazan players
FC Tom Tomsk players
PFC Krylia Sovetov Samara players
FC SKA-Khabarovsk players
FC Ararat Yerevan players
FC Metalurgi Rustavi players
Russian Premier League players
Russian First League players
Armenian Premier League players
Russian expatriate men's footballers
Expatriate men's footballers in Armenia
Expatriate men's footballers in Georgia (country)
Russian expatriate sportspeople in Armenia
Russian expatriate sportspeople in Georgia (country)
Footballers from Irkutsk Oblast
FC Sibiryak Bratsk players |
https://en.wikipedia.org/wiki/Set%20operation | Set operation may have one of the following meanings.
Any operation with sets
Set operation (Boolean), Boolean set operations in the algebra of sets
Set operations (SQL), type of operation in SQL
Fuzzy set operations, a generalization of crisp sets for fuzzy sets
See also
Set (disambiguation)
Set theory |
https://en.wikipedia.org/wiki/Graph%20continuous%20function | In mathematics, and in particular the study of game theory, a function is graph continuous if it exhibits the following properties. The concept was originally defined by Partha Dasgupta and Eric Maskin in 1986 and is a version of continuity that finds application in the study of continuous games.
Notation and preliminaries
Consider a game with agents with agent having strategy ; write for an N-tuple of actions (i.e. ) and as the vector of all agents' actions apart from agent .
Let be the payoff function for agent .
A game is defined as .
Definition
Function is graph continuous if for all there exists a function such that is continuous at .
Dasgupta and Maskin named this property "graph continuity" because, if one plots a graph of a player's payoff as a function of his own strategy (keeping the other players' strategies fixed), then a graph-continuous payoff function will result in this graph changing continuously as one varies the strategies of the other players.
The property is interesting in view of the following theorem.
If, for , is non-empty, convex, and compact; and if is quasi-concave in , upper semi-continuous in , and graph continuous, then the game possesses a pure strategy Nash equilibrium.
References
Partha Dasgupta and Eric Maskin 1986. "The existence of equilibrium in discontinuous economic games, I: theory". The Review of Economic Studies, 53(1):1–26
Game theory
Theory of continuous functions |
https://en.wikipedia.org/wiki/Lie%20product%20formula | In mathematics, the Lie product formula, named for Sophus Lie (1875), but also widely called the Trotter product formula, named after Hale Trotter, states that for arbitrary m × m real or complex matrices A and B,
where eA denotes the matrix exponential of A. The Lie–Trotter product formula and the Trotter–Kato theorem extend this to certain unbounded linear operators A and B.
This formula is an analogue of the classical exponential law
which holds for all real or complex numbers x and y. If x and y are replaced with matrices A and B, and the exponential replaced with a matrix exponential, it is usually necessary for A and B to commute for the law to still hold. However, the Lie product formula holds for all matrices A and B, even ones which do not commute.
The Lie product formula is conceptually related to the Baker–Campbell–Hausdorff formula, in that both are replacements, in the context of noncommuting operators, for the classical exponential law.
The formula has applications, for example, in the path integral formulation of quantum mechanics. It allows one to separate the Schrödinger evolution operator (propagator) into alternating increments of kinetic and potential operators (the Suzuki–Trotter decomposition, after Trotter and Masuo Suzuki). The same idea is used in the construction of splitting methods for the numerical solution of differential equations. Moreover, the Lie product theorem is sufficient to prove the Feynman–Kac formula.
The Trotter–Kato theorem can be used for approximation of linear C0-semigroups.
See also
Time-evolving block decimation
References
Sophus Lie and Friedrich Engel (1888, 1890, 1893). Theorie der Transformationsgruppen (1st edition, Leipzig; 2nd edition, AMS Chelsea Publishing, 1970)
.
, pp. 99.
Matrix theory
Lie groups |
https://en.wikipedia.org/wiki/Superellipsoid | In mathematics, a superellipsoid (or super-ellipsoid) is a solid whose horizontal sections are superellipses (Lamé curves) with the same squareness parameter , and whose vertical sections through the center are superellipses with the squareness parameter . It is a generalization of an ellipsoid, which is a special case when .
Superellipsoids as computer graphics primitives were popularized by Alan H. Barr (who used the name "superquadrics" to refer to both superellipsoids and supertoroids). In modern computer vision and robotics literatures, superquadrics and superellipsoids are used interchangeably, since superellipsoids are the most representative and widely utilized shape among all the superquadrics.
Superellipsoids have an rich shape vocabulary, including cuboids, cylinders, ellipsoids, octahedra and their intermediates. It becomes an important geometric primitive widely used in computer vision, robotics, and physical simulation. The main advantage of describing objects and envirionment with superellipsoids is its conciseness and expressiveness in shape. Furthermore, a closed-form expression of the Minkowski sum between two superellipsoids is available. This makes it a desirable geometric primitive for robot grasping, collision detection, and motion planning. Useful tools and algorithms for superquadric visualization, sampling, and recovery are open-sourced here.
Special cases
A handful of notable mathematical figures can arise as special cases of superellipsoids given the correct set of values, which are depicted in the above graphic:
Cylinder
Sphere
Steinmetz solid
Bicone
Regular octahedron
Cube, as a limiting case where the exponents tend to infinity
Piet Hein's supereggs are also special cases of superellipsoids.
Formulas
Basic (normalized) superellipsoid
The basic superellipsoid is defined by the implicit function
The parameters and are positive real numbers that control the squareness of the shape.
The surface of the superellipsoid is defined by the equation:
For any given point , the point lies inside the superellipsoid if , and outside if .
Any "parallel of latitude" of the superellipsoid (a horizontal section at any constant z between -1 and +1) is a Lamé curve with exponent , scaled by , which is
Any "meridian of longitude" (a section by any vertical plane through the origin) is a Lamé curve with exponent , stretched horizontally by a factor w that depends on the sectioning plane. Namely, if and , for a given , then the section is
where
In particular, if is 1, the horizontal cross-sections are circles, and the horizontal stretching of the vertical sections is 1 for all planes. In that case, the superellipsoid is a solid of revolution, obtained by rotating the Lamé curve with exponent around the vertical axis.
Superellipsoid
The basic shape above extends from −1 to +1 along each coordinate axis. The general superellipsoid is obtained by scaling the basic shape along each axis by factors , , , the |
https://en.wikipedia.org/wiki/Journal%20of%20Educational%20and%20Behavioral%20Statistics | The Journal of Educational and Behavioral Statistics is a peer-reviewed academic journal published by SAGE Publications on behalf of the American Educational Research Association and American Statistical Association. It covers statistical methods and applied statistics in the educational and behavioral sciences. The journal was established in 1976 as the Journal of Educational Statistics and obtained its current name in 1994. The journal's editors are Steven Andrew Culpepper (University of Illinois at Urbana-Champaign) and Gongjun Xu (University of Michigan).
Mission statement
The Journal of Educational and Behavioral Statistics (JEBS) provides an outlet for papers that are original and useful to those applying statistical approaches to problems and issues in educational or behavioral research. Typical papers will present new methods of analysis. In addition, critical reviews of current practice, tutorial presentations of less well known methods, and novel applications of already-known methods will be published. Papers discussing statistical techniques without specific educational or behavioral interest will have lower priority.
Abstracting and indexing
Journal of Educational and Behavioral Statistics is abstracted and indexed in, among other databases, SCOPUS and the Social Sciences Citation Index. According to the Journal Citation Reports, the journal has a 2022 impact factor of 2.4.
Editors-in-chief
The following is a list of the people who have recently been the editor-in-chief of Journal of Educational and Behavioral Statistics:
Steven Culpepper
Li Cai
Daniel McCaffrey
Sandip Sinharay
Matthew Johnson
David Rindskopf
David Thissen
Howard Wainer
References
External links
Education journals
Statistics journals
American Statistical Association academic journals
SAGE Publishing academic journals
Bimonthly journals
English-language journals
Academic journals established in 1976
Mathematical and statistical psychology journals
Psychology journals associated with learned and professional societies |
https://en.wikipedia.org/wiki/Walter%20Gautschi | Walter Gautschi (born December 11, 1927) is a Swiss-born American mathematician, writer and professor emeritus of Computer science and Mathematics at Purdue University in West Lafayette, Indiana. He is primarily known for his contributions to numerical analysis and has authored over 200 papers in his area and published four books.
Early life and education
Gautschi was born December 11, 1927 in Basel, Switzerland. His paternal family originally hailed from Reinach. He had one twin brother Werner (1927-1959). He completed a Ph.D. in mathematics from the University of Basel on the thesis Analyse graphischer Integrationsmethoden advised by Alexander Ostrowski and Andreas Speiser (1953).
Career
Since then, he did postdoctoral work as a Janggen-Pöhn Research, Fellow at the Istituto Nazionale per le Applicazioni del Calcolo in Rome (1954) and at the Harvard Computation Laboratory (1955). He had positions at the National Bureau of Standards (1956–59), the American University in Washington D.C., the Oak Ridge National Laboratory (1959–63) before joining Purdue University where he has worked from 1963 to 2000 and now being professor emeritus. He has been a Fulbright Scholar at the Technical University of Munich (1970) and held visiting appointments at the University of Wisconsin–Madison (1976), Argonne National Laboratory, the Wright-Patterson Air Force Base, ETH Zurich (1996-2001), the University of Padova (1997), and the University of Basel (2000).
As well-known (e.g. Gerhard Wanner, Geneva ca. 2011 and the well-known first-hand sources and subsequent reports such as Math. Intelligencer, etc), one of Gautschi's most important contributions on numerical simulation of special functions offered evidence and confidence to de Branges's tour-de-force attack on the elusive Bieberbach conjecture on the magnitude of coefficients of schlicht functions, which hitherto received only slow, difficult and partial progress by work of Bieberbach, Loewner, Gabaredian and Schiffer.
Personal life
In 1960, Gautschi married Erika, who was previously married to his twin brother Werner (1927-1959). Werner was also an academic professor and lecturer and emigrated to the United States with his wife in 1956. After his sudden death, Erika returned to Switzerland, while being pregnant with her child to Basel were she met Walter and married him in 1960. They had three daughters;
Theresa (1961-2018), married Ainsworth, two children; Emily Ainsworth (b. 1994) and Keith (b. 1997), formerly of Camas, Washington.
Doris (b. 1965)
Caroline Cari (b. 1969)
Through his predeceased twin brother, he has stepson/nephew, Thomas (b. 1960). Gautschi still resides in West Lafayette, Indiana.
Books
Colloquium approximatietheorie, MC Syllabus 14, Mathematisch Centrum Amsterdam, 1971. With H. Bavinck and G. M. Willems
Numerical analysis: an introduction, Birkhäuser, Boston, 1997; 2nd edition, 2012.
Orthogonal polynomials: computation and approximation, Oxford University Press, Oxford, 200 |
https://en.wikipedia.org/wiki/Arsen%20Goshokov | Arsen Aslanovich Goshokov (; born 5 June 1991) is a Russian former footballer.
Career statistics
Statistics accurate as of matches played on 22 August 2014
External links
References
1991 births
Living people
Russian men's footballers
Russia men's youth international footballers
Russia men's under-21 international footballers
PFC Spartak Nalchik players
Russian Premier League players
FC KAMAZ Naberezhnye Chelny players
Men's association football forwards
FC Ural Yekaterinburg players
Circassian people of Russia |
https://en.wikipedia.org/wiki/Seo%20Deok-kyu | Seo Deok-Kyu (Hangul: 서덕규, October 22, 1978) is a South Korean football player.
Seo was a part of South Korea who of the 2001 Confederations Cup.
Club career statistics
External links
1978 births
Living people
Men's association football defenders
South Korean men's footballers
South Korea men's international footballers
Ulsan Hyundai FC players
Gimcheon Sangmu FC players
K League 1 players
2001 FIFA Confederations Cup players
Footballers from Seoul |
https://en.wikipedia.org/wiki/British%20Society%20for%20Research%20into%20Learning%20Mathematics | The British Society for Research into Learning Mathematics is a United Kingdom association for people interested in research in mathematics education.
Purpose
BSRLM organises the Special Interest Group (SIG) on mathematics education for the British Educational Research Association (BERA). It is a participating society of the Joint Mathematical Council (JMC), and has close connections with teacher associations through British Congress for Mathematics Education (BCME).
Events
BSRLM organises a day conference each academic term (three times each year) where members present reports of recently completed studies, work in progress or more speculative thinking. There are also on-going workshops run on a collaborative basis to investigate issues in different areas of interest. The summer meeting is usually preceded by a new researchers’ day and the autumn meeting includes an AGM. Many members of BSRLM are also involved in the Congress of the European Society for Research in Mathematics Education (CERME), the International Group for the Psychology of Mathematics Education (IGPME) and other international research organisations.
Publications
Proceedings of meetings of the Society are available online several weeks after each day conference.
A peer reviewed journal, Research in Mathematics Education (RME) published by Taylor and Francis three times a year. Each year the Janet Duffin Award is made to the author or authors judged to making the most outstanding contribution to the journal.
Structure
The membership of BSRLM is made up mostly of researchers, students, teachers and education advisors. BSRLM is run by an elected executive committee of nine members. The current chair is Professor Jeremy Hodgen (University College London).
Origins of BSRLM
In the late 1960s and early 1970s there was a number of teacher trainers and university lecturers in the UK, principally teaching mathematics and psychology, who became involved in research in the learning and teaching of mathematics, and in the movement to establish mathematics education as an academic discipline. Two groups who were meeting during this period were the Psychology of Mathematics Education Workshop, principally in London, and the British Society for the Psychology of Learning Mathematics, in locations in the Midlands. Since many individuals attended both of these groups, they eventually came together as BSRLM in January 1985.
References
External links
Society website
1985 establishments in the United Kingdom
Learned societies of the United Kingdom
Mathematics education in the United Kingdom
Mathematical societies
Organizations established in 1985
Professional associations based in the United Kingdom |
https://en.wikipedia.org/wiki/Polyhedral%20combinatorics | Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes.
Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for instance, they seek inequalities that describe the relations between the numbers of vertices, edges, and faces of higher dimensions in arbitrary polytopes or in certain important subclasses of polytopes, and study other combinatorial properties of polytopes such as their connectivity and diameter (number of steps needed to reach any vertex from any other vertex). Additionally, many computer scientists use the phrase “polyhedral combinatorics” to describe research into precise descriptions of the faces of certain specific polytopes (especially 0-1 polytopes, whose vertices are subsets of a hypercube) arising from integer programming problems.
Faces and face-counting vectors
A face of a convex polytope P may be defined as the intersection of P and a closed halfspace H such that the boundary of H contains no interior point of P. The dimension of a face is the dimension of this hull. The 0-dimensional faces are the vertices themselves, and the 1-dimensional faces (called edges) are line segments connecting pairs of vertices. Note that this definition also includes as faces the empty set and the whole polytope P. If P itself has dimension d, the faces of P with dimension d − 1 are called facets of P and the faces with dimension d − 2 are called ridges. The faces of P may be partially ordered by inclusion, forming a face lattice that has as its top element P itself and as its bottom element the empty set.
A key tool in polyhedral combinatorics is the ƒ-vector of a polytope, the vector (f0, f1, ..., fd − 1) where fi is the number of i-dimensional features of the polytope. For instance, a cube has eight vertices, twelve edges, and six facets, so its ƒ-vector is (8,12,6). The dual polytope has a ƒ-vector with the same numbers in the reverse order; thus, for instance, the regular octahedron, the dual to a cube, has the ƒ-vector (6,12,8). Configuration matrices include the f-vectors of regular polytopes as diagonal elements.
The extended ƒ-vector is formed by concatenating the number one at each end of the ƒ-vector, counting the number of objects at all levels of the face lattice; on the left side of the vector, f−1 = 1 counts the empty set as a face, while on the right side, fd = 1 counts P itself.
For the cube the extended ƒ-vector is (1,8,12,6,1) and for the octahedron it is (1,6,12,8,1). Although the vectors for these example polyhedra are unimodal (the coefficients, taken in left to right order, increase to a maximum and then decrease), there are higher-dimensional polytopes for which this is not true.
For simplicial polytopes (polytopes in which every facet is a simplex), it is often convenien |
https://en.wikipedia.org/wiki/Axis-aligned%20object | In geometry, an axis-aligned object (axis-parallel, axis-oriented) is an object in n-dimensional space whose shape is aligned with the coordinate axes of the space.
Examples are axis-aligned rectangles (or hyperrectangles), the ones with edges parallel to the coordinate axes. Minimum bounding boxes are often implicitly assumed to be axis-aligned. A more general case is rectilinear polygons, the ones with all sides parallel to coordinate axes or rectilinear polyhedra.
Many problems in computational geometry allow for faster algorithms when restricted to (collections of) axis-oriented objects, such as axis-aligned rectangles or axis-aligned line segments.
A different kind of example are axis-aligned ellipsoids, i.e., the ellipsoids with principal axes parallel to the coordinate axes.
References
Geometry |
https://en.wikipedia.org/wiki/Associahedron | In mathematics, an associahedron is an -dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of letters, and the edges correspond to single application of the associativity rule. Equivalently, the vertices of an associahedron correspond to the triangulations of a regular polygon with sides and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal. Associahedra are also called Stasheff polytopes after the work of Jim Stasheff, who rediscovered them in the early 1960s after earlier work on them by Dov Tamari.
Examples
The one-dimensional associahedron K3 represents the two parenthesizations ((xy)z) and (x(yz)) of three symbols, or the two triangulations of a square. It is itself a line segment.
The two-dimensional associahedron K4 represents the five parenthesizations of four symbols, or the five triangulations of a regular pentagon. It is itself a pentagon and is related to the pentagon diagram of a monoidal category.
The three-dimensional associahedron K5 is an enneahedron with nine faces (three disjoint quadrilaterals and six pentagons) and fourteen vertices, and its dual is the triaugmented triangular prism.
Realization
Initially Jim Stasheff considered these objects as curvilinear polytopes. Subsequently, they were given coordinates as convex polytopes in several different ways; see the introduction of for a survey.
One method of realizing the associahedron is as the secondary polytope of a regular polygon. In this construction, each triangulation of a regular polygon with n + 1 sides corresponds to a point in (n + 1)-dimensional Euclidean space, whose ith coordinate is the total area of the triangles incident to the ith vertex of the polygon. For instance, the two triangulations of the unit square give rise in this way to two four-dimensional points with coordinates (1, 1/2, 1, 1/2) and (1/2, 1, 1/2, 1). The convex hull of these two points is the realization of the associahedron K3. Although it lives in a 4-dimensional space, it forms a line segment (a 1-dimensional polytope) within that space. Similarly, the associahedron K4 can be realized in this way as a regular pentagon in five-dimensional Euclidean space, whose vertex coordinates are the cyclic permutations of the vector (1, 2 + φ, 1, 1 + φ, 1 + φ) where φ denotes the golden ratio. Because the possible triangles within a regular hexagon have areas that are integer multiples of each other, this construction can be used to give integer coordinates (in six dimensions) to the three-dimensional associahedron K5; however (as the example of K4 already shows) this construction in general leads to irrational numbers as coordinates.
Another realization, due to Jean-Louis Loday, is based on the correspondence of the vertices of the associahedron with n-leaf rooted binary trees, and directly produces integer coordinates in (n − 2 |
https://en.wikipedia.org/wiki/Tree%20network | A tree topology, or star-bus topology, is a hybrid network topology in which star networks are interconnected via bus networks. Tree networks are hierarchical, and each node can have an arbitrary number of child nodes.
Regular tree networks
A regular tree network's topology is characterized by two parameters: the branching, , and the
number of generations, . The total number of the nodes, , and the number of peripheral nodes , are given by
Random tree networks
Three parameters are crucial in determining the statistics of random tree networks, first, the branching probability, second the maximum number of allowed progenies at each branching point, and third the maximum number of generations, that a tree can attain. There are a lot of studies that address the large tree networks, however small tree networks are seldom studied.
Tools to deal with networks
A group at MIT has developed a set of functions for Matlab that can help in analyzing the networks. These tools could be used to study the tree networks as well.
References
Network topology
Trees (data structures) |
https://en.wikipedia.org/wiki/Hypertree%20network | A hypertree network is a network topology that shares some traits with the binary tree network. It is a variation of the fat tree architecture.
A hypertree of degree k depth d may be visualized as a 3-dimensional object whose front view is the top-down complete k-ary tree of depth d and the side view is the bottom-up complete binary tree of depth d.
Hypertrees were proposed in 1981 by James R. Goodman and Carlo Sequin.
Hypertrees are a choice for parallel computer architecture, used, e.g., in the connection machine CM-5.
References
Network topology |
https://en.wikipedia.org/wiki/Gil%20Kalai | Gil Kalai (born 1955) is an Israeli mathematician and computer scientist. He is the Henry and Manya Noskwith Professor Emeritus of Mathematics at the Hebrew University of Jerusalem, Israel, Professor of Computer Science at the Interdisciplinary Center, Herzliya, and adjunct Professor of mathematics and of computer science at Yale University, United States.
Biography
Kalai received his PhD from Hebrew University in 1983, under the supervision of Micha Perles, and joined the Hebrew University faculty in 1985 after a postdoctoral fellowship at the Massachusetts Institute of Technology. He was the recipient of the Pólya Prize in 1992, the Erdős Prize of the Israel Mathematical Society in 1993, and the Fulkerson Prize in 1994. He is known for finding variants of the simplex algorithm in linear programming that can be proven to run in subexponential time, for showing that every monotone property of graphs has a sharp phase transition, for solving Borsuk's problem (known as Borsuk's conjecture) on the number of pieces needed to partition convex sets into subsets of smaller diameter, and for his work on the Hirsch conjecture on the diameter of convex polytopes and in polyhedral combinatorics more generally.
From 1995 to 2001, he was the Editor-in-Chief of the Israel Journal of Mathematics. In 2016, he was elected honorary member of the Hungarian Academy of Sciences. In 2018 he was a plenary speaker with talk Noise Stability, Noise Sensitivity and the Quantum Computer Puzzle at the International Congress of Mathematicians in Rio de Janeiro.
Kalai's conjectures on quantum computing
Kalai is a quantum computing skeptic who argues that true (classically unattainable) quantum computing will not be achieved because the necessary quality of quantum error correction cannot be reached.
Conjecture 1 (No quantum error correction). The process for creating a quantum error-correcting code will necessarily lead to a mixture of the desired codewords with undesired codewords. The probability of the undesired codewords is uniformly bounded away from zero. (In every implementation of quantum error-correcting codes with one encoded qubit, the probability of not getting the intended qubit is at least some δ > 0, independently of the number of qubits used for encoding.)
Conjecture 2. A noisy quantum computer is subject to noise in which information leaks for two substantially entangled qubits have a substantial positive correlation.
Conjecture 3. In any quantum computer at a highly entangled state there will be a strong effect of error-synchronization.
Conjecture 4. Noisy quantum processes are subject to detrimental noise.
Recognition
Kalai was the winner of the 2012 Rothschild Prize in mathematics. He was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to combinatorics, convexity, and their applications, as well as to the exposition and communication of mathematics".
See also
Kalai's 3d conjecture
Entropy influence conj |
https://en.wikipedia.org/wiki/Hans%20Heinrich%20B%C3%BCrmann | Hans Heinrich Bürmann (died 21 June 1817, in Mannheim) was a German mathematician and teacher. He ran an "academy of commerce" in Mannheim since 1795 where he used to teach mathematics. He also served as a censor in Mannheim. He was nominated Headmaster of the Commerce Academy of the Grand Duchy of Baden in 1811. He did scientific research in the area of combinatorics and he contributed to the development of the symbolic language of mathematics. He discovered the generalized form of the Lagrange inversion theorem. He corresponded and published with Joseph Louis Lagrange and Carl Hindenburg.
Iterate function composition notation
The compositional notation for the iterate of function was originally introduced by Bürmann and later independently suggested by John Frederick William Herschel in 1813.
See also
Bürmann series
Lagrange–Bürmann formula
References
1817 deaths
Year of birth missing
18th-century German mathematicians
19th-century German mathematicians
People from the Grand Duchy of Baden |
https://en.wikipedia.org/wiki/Oleg%20Vlasov | Oleg Sergeyevich Vlasov (; born 10 December 1984) is a Russian former football player.
Career statistics
Notes
Achievements
Russian Premier League runner-up: 2003
External links
Player page on the official FC Terek Grozny website
1984 births
People from Kirovsky District, Leningrad Oblast
Living people
Russian men's footballers
Russia men's under-21 international footballers
Men's association football midfielders
FC Zenit Saint Petersburg players
FC Leon Saturn Ramenskoye players
FC Akhmat Grozny players
Russian Premier League players
FC Torpedo Moscow players
FC Mordovia Saransk players
FC Arsenal Tula players
FC Dynamo Saint Petersburg players
FC Leningradets Leningrad Oblast players
Sportspeople from Leningrad Oblast |
https://en.wikipedia.org/wiki/Michel%20Mandjes | Michael Robertus Hendrikus "Michel" Mandjes (born 14 February 1970 in Zaandam) is a Dutch mathematician, known for several contributions to queueing theory and applied probability theory. His research interests include queueing models for telecommunications,traffic management and analysis, and network economics.
He holds a full-professorship (Applied Probability and Queueing Theory) at the University of Amsterdam (Korteweg-de Vries Institute for Mathematics). From September 2004 he is advisor of the "Queueing and Performance Analysis" theme at EURANDOM, Eindhoven.
He is author of the book "Large deviations for Gaussian queues", and is associate editor of the journals Stochastic Models and Queuing Systems.
He contributed to the book Queues and Lévy fluctuation theory, published in 2015.
Books
"Large deviations for Gaussian queues" (2007)
References
External links
Homepage
1970 births
Living people
Dutch mathematicians
Queueing theorists
Academic staff of the University of Amsterdam
People from Zaanstad |
https://en.wikipedia.org/wiki/External | External may refer to:
External (mathematics), a concept in abstract algebra
Externality, in economics, the cost or benefit that affects a party who did not choose to incur that cost or benefit
Externals, a fictional group of X-Men antagonists
See also
Internal (disambiguation) |
https://en.wikipedia.org/wiki/Directed%20graph | In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.
Definition
In formal terms, a directed graph is an ordered pair where
V is a set whose elements are called vertices, nodes, or points;
A is a set of ordered pairs of vertices, called arcs, directed edges (sometimes simply edges with the corresponding set named E instead of A), arrows, or directed lines.
It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called edges, links or lines.
The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a multiset). Sometimes these entities are called directed multigraphs (or multidigraphs).
On the other hand, the aforementioned definition allows a directed graph to have loops (that is, arcs that directly connect nodes with themselves), but some authors consider a narrower definition that does not allow directed graphs to have loops.
Directed graphs without loops may be called simple directed graphs, while directed graphs with loops may be called loop-digraphs (see section Types of directed graph).
Types of directed graphs
Subclasses
Symmetric directed graphs are directed graphs where all edges appear twice, one in each direction (that is, for every arrow that belongs to the digraph, the corresponding inverse arrow also belongs to it). (Such an edge is sometimes called "bidirected" and such graphs are sometimes called "bidirected", but this conflicts with the meaning for bidirected graphs.)
Simple directed graphs are directed graphs that have no loops (arrows that directly connect vertices to themselves) and no multiple arrows with same source and target nodes. As already introduced, in case of multiple arrows the entity is usually addressed as directed multigraph. Some authors describe digraphs with loops as loop-digraphs.
Complete directed graphs are simple directed graphs where each pair of vertices is joined by a symmetric pair of directed arcs (it is equivalent to an undirected complete graph with the edges replaced by pairs of inverse arcs). It follows that a complete digraph is symmetric.
Semicomplete multipartite digraphs are simple digraphs in which the vertex set is partitioned into sets such that for every pair of vertices x and y in different sets, there is an arc between x and y. There can be one arc between x and y or two arcs in opposite directions.
Semicomplete digraphs are simple digraphs where there is an arc between each pair of vertices. Every semicomplete digraph is a semicomplete multipartite digraph in a trivial way, with each vertex constituting a set of the partition.
Quasi-transitive digraphs are simple |
https://en.wikipedia.org/wiki/Vasili%20Yanotovsky | Vasili Grigoryevich Yanotovsky (; born 2 January 1976) is a Russian former footballer.
Career statistics
Club
References
External links
1976 births
People from Zabaykalsky Krai
Living people
Russian men's footballers
FC Tom Tomsk players
Russian Premier League players
Simurq PIK players
Russian expatriate men's footballers
Expatriate men's footballers in Azerbaijan
FC SKA-Khabarovsk players
Men's association football midfielders
Sportspeople from Zabaykalsky Krai
FC Zarya Leninsk-Kuznetsky players
FC Kuzbass Kemerovo players |
https://en.wikipedia.org/wiki/Viktor%20Stroyev | Viktor Viktorovich Stroyev (, born 16 January 1987) is a Russian former footballer.
Career statistics
External links
Player page on the official FC Tom Tomsk website
1987 births
Footballers from Voronezh
Living people
Russian men's footballers
Russia men's youth international footballers
Russia men's under-21 international footballers
Men's association football defenders
FC Tom Tomsk players
Russian Premier League players
FC Fakel Voronezh players
FC Zenit Saint Petersburg players |
https://en.wikipedia.org/wiki/Rational%20number | In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g., The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold
A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ).
A real number that is not rational is called irrational. Irrational numbers include the square root of 2 , , and the golden ratio (). Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.
Rational numbers can be formally defined as equivalence classes of pairs of integers with , using the equivalence relation defined as follows:
The fraction then denotes the equivalence class of .
Rational numbers together with addition and multiplication form a field which contains the integers, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and a field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite extensions of are called algebraic number fields, and the algebraic closure of is the field of algebraic numbers.
In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals (see Construction of the real numbers).
Terminology
The term rational in reference to the set refers to the fact that a rational number represents a ratio of two integers. In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational coefficients, although the term "polynomial over the rationals" is generally preferred, to avoid confusion between "rational expression" and "rational function" (a polynomial is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a rational curve is not a curve defined over the rationals, but a curve which can be parameterized by rational functions.
Etymology
Although nowadays rational numbers are defined in terms of ratios, the te |
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